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Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

$how does math improve problem solving skills$

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub .

Try Our Free Confidence Boosters

How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients.

Learning Math with the Help of Artificial Intelligence (AI)

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions.

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

Get Math Help with Khanmigo Right Now

You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support.

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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Open Access

Peer-reviewed

Research Article

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Roles Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Resources, Writing – original draft, Writing – review & editing

Affiliation School of Mathematics and Statistics, The University of Sydney, Sydney, Australia

Roles Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliation School of Arts and Humanities, Edith Cowan University, Joondalup, Australia

Clio Cresswell,
Craig P. Speelman

Published: July 29, 2020
https://doi.org/10.1371/journal.pone.0236153
Peer Review
Reader Comments

Mathematics is often promoted as endowing those who study it with transferable skills such as an ability to think logically and critically or to have improved investigative skills, resourcefulness and creativity in problem solving. However, there is scant evidence to back up such claims. This project tested participants with increasing levels of mathematics training on 11 well-studied rational and logical reasoning tasks aggregated from various psychological studies. These tasks, that included the Cognitive Reflection Test and the Wason Selection Task, are of particular interest as they have typically and reliably eluded participants in all studies, and results have been uncorrelated with general intelligence, education levels and other demographic information. The results in this study revealed that in general the greater the mathematics training of the participant, the more tasks were completed correctly, and that performance on some tasks was also associated with performance on others not traditionally associated. A ceiling effect also emerged. The work is deconstructed from the viewpoint of adding to the platform from which to approach the greater, and more scientifically elusive, question: are any skills associated with mathematics training innate or do they arise from skills transfer?

Citation: Cresswell C, Speelman CP (2020) Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors. PLoS ONE 15(7): e0236153. https://doi.org/10.1371/journal.pone.0236153

Editor: Jérôme Prado, French National Center for Scientific Research (CNRS) & University of Lyon, FRANCE

Received: January 13, 2020; Accepted: June 30, 2020; Published: July 29, 2020

Copyright: © 2020 Cresswell, Speelman. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Mathematics is often promoted as endowing those who study it with a number of broad thinking skills such as: an ability to think logically, analytically, critically and abstractly; having capacity to weigh evidence with impartiality. This is a view of mathematics as providing transferable skills which can be found across educational institutions, governments and corporations worldwide. A view material to the place of mathematics in curricula.

Consider the UK government’s commissioned inquiry into mathematics education “Making Mathematics Count” ascertaining the justification that “mathematical training disciplines the mind, develops logical and critical reasoning, and develops analytical and problem-solving skills to a high degree” [ 1 p11]. The Australian Mathematical Sciences Institute very broadly states in its policy document “Vision for a Maths Nation” that “Not only is mathematics the enabling discipline, it has a vital productive role planning and protecting our well-being” (emphasis in original) [ 2 ]. In Canada, British Columbia’s New 2016 curriculum K-9 expressly mentions as part of its “Goals and Rationale”: “The Mathematics program of study is designed to develop deep mathematical understanding and fluency, logical reasoning, analytical thought, and creative thinking.” [ 3 ]. Universities, too, often make such specific claims with respect to their teaching programs. “Mathematics and statistics will help you to think logically and clearly, and apply a range of problem-solving strategies” is claimed by The School of Mathematical Sciences at Monash University, Australia [ 4 ]. The School of Mathematics and Statistics at The University of Sydney, Australia, directly attributes as part of particular course objectives and outcomes skills that include “enhance your problem-solving skills” as part of studies in first year [ 5 ], “develop logical thinking” as part of studies in second year, which was a statement drafted by the lead author in fact [ 6 ], and “be fluent in analysing and constructing logical arguments” as part of studies in third year [ 7 ]. The University of Cambridge’s Faculty of Mathematics, UK, provides a dedicated document “Transferable Skills in the Mathematical Tripos” as part of its undergraduate mathematics course information, which again lists “analytic ability; creativity; initiative; logical and methodical reasoning; persistence” [ 8 ].

In contrast, psychological research, which has been empirically investigating the concept of transferability of skills since the early 1900s, points quite oppositely to reasoning skills as being highly domain specific [ 9 ]. Therefore, support for claims that studying mathematics engenders more than specific mathematics knowledge is highly pertinent. And yet it is largely absent. The 2014 Centre for Curriculum Redesign (CCR) four part paper “Mathematics for the 21st Century: What Should Students Learn?” concludes in its fourth paper titled “Does mathematics education enhance higher-order thinking skills?” with a call to action “… there is not sufficient evidence to conclude that mathematics enhances higher order cognitive functions. The CCR calls for a much stronger cognitive psychology and neuroscience research base to be developed on the effects of studying mathematics” [ 10 ].

Inglis and Simpson [ 11 ], bringing up this very issue, examined the ability of first-year undergraduate students from a high-ranking UK university mathematics department, on the “Four Cards Problem” thinking task, also known as the Wason Selection Task. It is stated as follows.

Each of the following cards have a letter on one side and a number on the other.

how does math improve problem solving skills

Here is a rule: “if a card has a D on one side, then it has a 3 on the other”. Your task is to select all those cards, but only those cards, which you would have to turn over in order to find out whether the rule is true or false. Which cards would you select?

This task involves understanding conditional inference, namely understanding the rule “If P then Q” and with this, deducing the answer as “P and not Q” or “D and 7”. Such logical deduction indeed presents as a good candidate to test for a potential ability of the mathematically trained. This task has also been substantially investigated in the domain of the psychology of reasoning [ 12 p8] revealing across a wide range of publications that only around 10% of the general population reach the correct result. The predominant mistake being to pick “D and 3”; where in the original study by Wason [ 13 ] it is suggested that this was picked by 65% of people. This poor success rate along with a standard mistake has fuelled interest in the task as well as attempts to understand why it occurs. A prevailing theory being the so named matching bias effect; the effect of disproportionately concentrating on items specifically mentioned in the situation, as opposed to reasoning according to logical rules.

Inglis and Simpson’s results isolated mathematically trained individuals with respect to this task. The participants were under time constraint and 13% of the first-year undergraduate mathematics students sampled reached the correct response, compared to 4% of the non-mathematics (arts) students that was included. Of note also was the 24% of mathematics students as opposed to 45% of the non-mathematics students who chose the standard mistake. The study indeed unveiled that mathematically trained individuals were significantly less affected by the matching bias effect with this problem than the individuals without mathematics training. However, the achievement of the mathematically trained group was still far from masterful and the preponderance for a non-standard mistake compared with non-mathematically trained people is suggestive. Mathematical training appears to engender a different thinking style, but it remains unclear what the difference is.

Inglis, Simpson and colleagues proceeded to follow up their results with a number of studies concentrated on conditional inference in general [ 14 , 15 ]. A justification for this single investigatory pathway being that if transfer of knowledge is present, something subtle to test for in the first place, a key consideration should be the generalisation of learning rather than the application of skills learned in one context to another (where experimenter bias in the choice of contexts is more likely to be an issue). For this they typically used sixteen “if P then Q” comprehension tasks, where their samples across a number of studies have included 16-year-old pre-university mathematics students (from England and Cyprus), mathematics honours students in their first year of undergraduate university study, third year university mathematics students, and associated control groups. The studies have encompassed controls for general intelligence and thinking disposition prior to training, as well as follows ups of up to two years to address the issue of causation. The conclusive thinking pattern that has emerged is a tendency of the mathematical groups towards a greater likelihood of rejecting the invalid denial of the antecedent and affirmation of the consequent inferences. But with this, and this was validated by a second separate study, the English mathematics group actually became less likely to endorse the valid modus tollens inference. So again, mathematical training appears to engender a different thinking style, but there are subtleties and it remains unclear what the exact difference is.

This project was designed to broaden the search on the notion that mathematics training leads to increased reasoning skills. We focused on a range of reasoning problems considered in psychological research to be particularly insightful into decision making, critical thinking and logical deduction, with their distinction in that the general population generally struggles with answering them correctly. An Australian sample adds diversity to the current enquiries that have been European focussed. Furthermore, in an effort to identify the impact of mathematics training through a possible gradation effect, different levels of mathematically trained individuals were tested for performance.

Well-studied thinking tasks from a variety of psychological studies were chosen. Their descriptions, associated success rates and other pertinent details follows. They were all chosen as the correct answer is typically eluded for a standard mistake.

The three-item Cognitive Reflection Test (CRT) was used as introduced by Frederick [ 16 ]. This test was devised in line with the theory that there are two general types of cognitive activity: one that operates quickly and without reflection, and another that requires not only conscious thought and effort, but also an ability to reflect on one’s own cognition by including a step of suppression of the first to reach it. The three items in the test involve an incorrect “gut” response and further cognitive skill is deemed required to reach the correct answer (although see [ 17 ] for evidence that correct responses can result from “intuition”, which could be related to intelligence [ 18 ]).

In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

Bat and ball

A bat and a ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?

The solutions are: 47 days for the Lily Pads problem, 5 minutes for the Widgets problem and 5 cents for the Bat and Ball problem. The considered intuitive, but wrong, answers are 24 days, 100 minutes and 10 cents, respectively. These wrong answers are attributed to participants becoming over focused on the numbers so as to ignore the exponential growth pattern in the Lily Pads problem, merely complete a pattern in numbers in the Widgets problem, and neglect the relationship “more than” in the Bat and Ball problem [ 19 ]. The original study by Frederick [ 16 ] provides a composite measure of the performance on these three items, with only 17% of those studied (n = 3428) reaching the perfect score. The CRT has since been studied extensively [ 19 – 21 ]. Research using the CRT tends not to report performance on the individual items of the test, but rather a composite measure of performance. Attridge and Inglis [ 22 ] used the CRT as a test for thinking disposition of mathematics students as one way to attempt to disentangle the issue of filtering according to prior thinking styles rather than transference of knowledge in successful problem solving. They repeat tested 16-year old pre-university mathematics students and English literature students without mathematics subjects at a one-year interval and found no difference between groups.

Three problems were included that test the ability to reason about probability. All three problems were originally discussed by Kahneman and Tversky [ 23 ], with the typically poor performance on these problems explained by participants relying not on probability knowledge, but a short-cut method of thinking known as the representativeness heuristic. In the late 1980s, Richard Nisbett and colleagues showed that graduate level training in statistics, while not revealing any improvement in logical reasoning, did correlate with higher-quality statistical answers [ 24 ]. Their studies lead in particular to the conclusion that comprehension of, what is known as the law of large numbers, did show improvement with training. The first of our next three problems targeted this law directly.

(a). the larger hospital
(b). the smaller hospital
(c). about the same (that is, within 5 percent of each other)

Kahneman and Tversky [ 23 ] reported that, of 50 participants, 12 chose (a), 10 chose (b), and 28 chose (c). The correct answer is (b), for the reason that small samples are more likely to exhibit extreme events than large samples from the same population. The larger the sample, the more likely it will exhibit characteristics of the parent population, such as the proportion of boys to girls. However, people tend to discount or be unaware of this feature of sampling statistics, which Kahneman and Tversky refer to as the law of large numbers. Instead, according to Kahneman and Tversky, people tend to adhere to a fallacious law of small numbers, where even small samples are expected to exhibit properties of the parent population, as illustrated by the proportion of participants choosing the answer (c) in their 1972 study. Such thinking reflects use of the representativeness heuristic, whereby someone will judge the likelihood of an uncertain event based on how similar it is to characteristics of the parent population of events.

Birth order

(a). What is your estimate of the number of families surveyed in which the exact order of births was BGBBBB?
(b). In the same survey set, which, if any, of the following two sequences would be more likely: BBBGGG or GBBGBG?

All of the events listed in the problem have an equal probability, so the correct answer to (a) is 72, and to (b) is “neither is more likely”. Kahneman and Tversky [ 23 ] reported that 75 of 92 participants judged the sequence in (a) as less likely than the given sequence. A similar number (unspecified by Kahneman and Tversky, but the statistical effect was reported to be of the same order as in (a)) reported that GBBGBG was the more likely sequence. Again, Kahneman and Tversky suggested that these results reflected use of the representativeness heuristic. In the context of this problem, the heuristic would have taken the following form: some birth orders appear less patterned than others, and less patterned is to be associated with the randomness of birth order, making them more likely.

Coin tosses

(a). H T H T H T H T
(b). H H H H T T T T
(c). T T H H T T H H
(d). H T T H T H H T
(e). all of the above are equally likely

The correct answer in this problem is (e). Kahneman and Tversky [ 23 ] reported that participants tend to choose less patterned looking sequences (e.g., H T T H T H H T) as more likely than more systematic looking sequences (e.g., H T H T H T H T). This reasoning again reflects the representativeness heuristic.

Three further questions from the literature were included to test problem solving skill.

Two drivers

(a). Driver A would win the race
(b). Driver B would win the race
(c). the two drivers would arrive at the same time (within a few seconds of one another)

This problem was developed by Pelham and Neter [ 25 ]. The correct answer is (a), which can be determined by calculations of driving times for each Driver, using time = distance/velocity. Pelham and Neter argue, however, that (c) is intuitively appealing, on the basis that both drivers appear to have the same overall average speed. Pelham and Neter reported that 67% of their sample gave this incorrect response to the problem, and a further 13% selected (b).

Petrol station

Imagine that you are driving along the road and you notice that your car is running low on petrol. You see two petrol stations next to each other, both advertising their petrol prices. Station A’s price is 65c/litre; Station B’s price is 60c/litre. Station A’s sign also announces: “5c/litre discount for cash!” Station B’s sign announces “5c/litre surcharge for credit cards.” All other factors being equal (for example, cleanliness of the stations, number of cars waiting at each etc), to which station would you choose to go, and why?

This problem was adapted from one described by Galotti [ 26 ], and is inspired by research reported by Thaler [ 27 ]. According to Thaler’s research, most people prefer Station A, even though both stations are offering the same deal: 60c/litre for cash, and 65c/litre for credit. Tversky and Kahneman [ 28 ] explain this preference by invoking the concept of framing effects. In the context of this problem, such an effect would involve viewing the outcomes as changes from some initial point. The initial point frames the problem, and provides a context for viewing the outcome. Thus, depending on the starting point, outcomes in this problem can be viewed as either a gain (in Station A, you gain a discount if you use cash) or a loss (in Station B, you are charged more (a loss) for using credit). Given that people are apparently more concerned about a loss than a gain [ 29 ], the loss associated with Station B makes it the less attractive option, and hence the preference for Station A. The correct answer, though, is that the stations are offering the same deal and so no station should be preferred.

And finally, a question described by Stanovich [ 30 , 31 ] as testing our predisposition for cognitive operations that require the least computational effort.

Jack looking at Anne

(c). Cannot be determined

Stanovich reported that over 80% of people choose the “lazy” answer (c). The correct answer is (a).

The above questions survey, in a clear problem solving setting, an ability to engage advanced cognitive processing in order to critically evaluate and possibly override initial gut reasoning, an ability to reason about probability within the framework of the law of large numbers and the relationship between randomness and patterning, an ability to isolate salient features of a problem and, with the last question in particular, an ability to map logical relations. It might be hypothesised that according to degrees of mathematical training, in line with the knowledge base provided and the claims of associated broad and enhanced problem-solving abilities in general, that participants with greater degrees of such training would outperform others on these questions. This hypothesis was investigated in this study. In addition, given that no previous study on this issue has examined the variety of problems used in this study, we also undertook an exploratory analysis to investigate whether there exist any associations between the problems in terms of their likelihood of correct solution. Similarities between problems might indicate which problem solving domains could be susceptible to the effects of mathematics training.

Introductory—First year, second semester, university students with weak high school mathematical results, only enrolled in the current unit as a compulsory component for their chosen degree, a unit not enabling any future mathematical pathway, a typical student may be enrolled in a Biology or Geography major;
Standard—First year, second semester, university students with fair to good high school mathematical results, enrolled in the current mathematics unit as a compulsory component for their chosen degree with the possibility of including some further mathematical units in their degree pathway, a typical student may be enrolled in an IT or Computer Science major;
Advanced1—First year, second semester, university mathematics students with very strong interest as well as background in mathematics, all higher year mathematical units are included as possible future pathway, a typical student may be enrolled in a Mathematics or Physics major;
Advanced2—Second year, second semester, university mathematics students with strong interest as well as background in mathematics, typically a direct follow on from the previously mentioned Advanced1 cohort;
Academic—Research academics in the mathematical sciences.

Participants

123 first year university students volunteered during “help on demand” tutorial times containing up to 30 students. These are course allocated times that are supervised yet self-directed by students. This minimised disruption and discouraged coercion. 44 second year university students completed the questionnaire during a weekly one-hour time slot dedicated to putting the latest mathematical concepts to practice with the lecturer (whereby contrast to what occurs in tutorial times the lecturer does most of the work and all students enrolled are invited). All these university students completed the questionnaire in normal classroom conditions; they were not placed under strict examination conditions. The lead author walked around to prevent discussion and coercion and there was minimum disruption. 30 research academics responded to local advertising and answered the questionnaire in their workplace while supervised.

The questionnaires were voluntary, anonymous and confidential. Participants were free to withdraw from the study at any time and without any penalty. No participant took this option however. The questionnaires gathered demographic information which included age, level of education attained and current qualification pursued, name of last qualification and years since obtaining it, and an option to note current speciality for research academics. Each problem task was placed on a separate page. Participants were not placed under time constraint, but while supervised, were asked to write their start and finish times on the front page of the survey to note approximate completion times. Speed of completion was not incentivised. Participants were not allowed to use calculators. A final “Comments Page” gave the option for feedback including specifically if the participants had previously seen any of the questions. Questionnaires were administered in person and supervised to avoid collusion or consulting of external sources.

The responses were coded four ways: A) correct; B) standard error (the errors discussed above in The Study); C) other error; D) left blank.

The ethical aspects of the study were approved by the Human Research Ethics Committee of the University of Sydney, protocol number [2016/647].

The first analysis examined the total number of correct responses provided by the participants as a function of group. Scores ranged from 1 to 11 out of a total possible of 11 (Problem 6 had 2 parts) ( Fig 1 ). An ANOVA of this data indicated a significant effect of group (F(4, 192) = 20.426, p < .001, partial η 2 = .299). Pairwise comparisons using Tukey’s HSD test indicated that the Introductory group performed significantly worse than the Advanced1, Advanced2 and Academic groups. There were no significant differences between the Advanced1, Advanced2 and Academic groups.

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Error bars are one standard error of the mean.

https://doi.org/10.1371/journal.pone.0236153.g001

Overall solution time, while recorded manually and approximately, was positively correlated with group, such that the more training someone had received, the longer were these solution times (r(180) = 0.247, p = .001). However, as can be seen in Fig 2 , this relationship is not strong.

https://doi.org/10.1371/journal.pone.0236153.g002

A series of chi-squared analyses, and their Bayesian equivalents, were performed on each problem, to determine whether the distribution of response types differed as a function of group. To minimise the number of cells in which expected values in some of these analyses were less than 5, the Standard Error, Other Error and Blank response categories were collapsed into one category (Incorrect Response). For three of the questions, the expected values of some cells did fall below 5, and this was due to most people getting the problem wrong (Four Cards), or most people correctly responding to the problem (Bat and Ball, Coin Tosses). In these cases, the pattern of results was so clear that a statistical analysis was barely required. Significant chi-squared results were examined further with pairwise posthoc comparisons (see Table 1 ).

https://doi.org/10.1371/journal.pone.0236153.t001

The three groups with the least amount of training in mathematics were far less likely than the other groups to give the correct solution (χ 2 (4) = 31.06, p < .001; BF 10 = 45,045) ( Table 1 ). People in the two most advanced groups (Advanced2 and Academic) were more likely to solve the card problem correctly, although it was still less than half of the people in these groups who did so. Further, these people were less likely to give the standard incorrect solution, so that most who were incorrect suggested some more cognitively elaborate answer, such as turning over all cards. The proportion of people in the Advanced2 and Academic groups (39 and 37%) who solved the problem correctly far exceeded the typical proportion observed with this problem (10%). Of note, also, is the relatively high proportion of those in the higher training groups who, when they made an error, did not make the standard error, a similar result to the one reported by Inglis and Simpson [ 11 ].

The cognitive reflection test

In the Lily Pads problem, although most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, it was also the case that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 27.28, p < .001; BF 10 = 15,554), with the standard incorrect answer being the next most prevalent response for the two lower ability mathematics groups ( Table 1 ).

Performance on the Widgets problem was similar to performance on the Lily Pads problem in that most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, but that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 23.76, p< .001; BF 10 = 516) ( Table 1 ). As with the Lily Pads and Widget problems, people in the Standard, Advanced1, Advanced2 and Academic groups were highly likely to solve the Bat and Ball problem (χ 2 (4) = 35.37, p < .001; BF 10 = 208,667). Errors were more likely from the least mathematically trained people (Introductory, Standard) than the other groups ( Table 1 ).

To compare performance on the CRT with previously published results, performance on the three problems (Lily Pads, Widgets, Bat and Ball) were combined. The number of people in each condition that solved 0, 1, 2, or 3 problems correctly is presented in Table 2 . The Introductory group were evenly distributed amongst the four categories, with 26% solving all three problems correctly. Around 70% of the rest of the groups solved all 3 problems correctly, which is vastly superior to the 17% reported by Frederick [ 16 ].

https://doi.org/10.1371/journal.pone.0236153.t002

Responses to the Hospitals problem were almost universally split between correct and standard errors in the Standard, Advanced1, Advanced2 and Academic groups. Although this pattern of responses was also evident in the Introductory group, this group also exhibited more non-standard errors and non-responses than the other groups. However, the differences between the groups were not significant (χ 2 (4) = 4.93, p = .295; BF 10 = .068) ( Table 1 ). Nonetheless, the performance of all groups exceeds the 20% correct response rate reported by Kahneman and Tversky [ 23 ].

The two versions of the Birth Order problem showed similar results, with correct responses being more likely in the groups with more training (i.e., Advanced1, Advanced2 and Academic), and responses being shared amongst the various categories in the Introductory and Standard groups (χ a 2 (4) = 24.54, p < .001; BF 10 = 1,303; χ b 2 (4) = 25.77, p < .001; BF 10 = 2,970) ( Table 1 ). Nonetheless, performance on both versions of the problem in this study was significantly better than the 82% error rate reported by Kahneman and Tversky [ 23 ].

The Coin Tosses problem was performed well by all groups, with very few people in any condition committing errors. There were no obvious differences between the groups (χ 2 (4) = 3.70, p = .448; BF 10 = .160) ( Table 1 ). Kahneman and Tversky [ 23 ] reported that people tend to make errors on this type of problem by choosing less patterned looking sequences, but they did not report relative proportions of people making errors versus giving correct responses. Clearly the sample in this study did not perform like those in Kahneman and Tversky’s study.

Responses on the Two Drivers problem were clearly distinguished by a high chance of error in the Introductory and Standard groups (over 80%), and a fairly good chance of being correct in the Advanced1, Advanced2 and Academic groups (χ 2 (4) = 46.16, p < .001; BF 10 = 1.32 x 10 8 ) ( Table 1 ). Academics were the standout performers on this problem, although over a quarter of this group produced an incorrect response. Thus, the first two groups performed similarly to the participants in the Pelham and Neter [ 25 ] study, 80% of whom gave an incorrect response.

Responses on the Petrol Station problem were marked by good performance by the Academic group (73% providing a correct response), and just over half of each of the other groups correctly solving the problem. This difference was not significant (χ 2 (4) = 4.68, p = .322: BF 10 = .059) ( Table 1 ). Errors were fairly evenly balanced between standard and other, except for the Academic group, who were more likely to provide a creative answer if they made an error. Thaler [ 27 ] reported that most people get this problem wrong. In this study, however, on average, most people got this problem correct, although this average was boosted by the Academic group.

Responses on the Jack looking at Anne problem generally were standard errors, except for the Advanced2 and Academic groups, which were evenly split between standard errors and correct responses (χ 2 (4) = 18.03, p = .001; BF 10 = 46) ( Table 1 ). Thus, apart from these two groups, the error rate in this study was similar to that reported by Stanovich [ 30 ], where 80% of participants were incorrect.

A series of logistic regression analyses were performed in order to examine whether the likelihood of solving a particular problem correctly could be predicted on the basis of whether other problems were solved correctly. Each analysis involved selecting performance (correct or error) on one problem as the outcome variable, and performance on the other problems as predictor variables. Training (amount of training) was also included as a predictor variable in each analysis. A further logistic regression was performed with training as the outcome variable, and performance on all of the problems as predictor variables. The results of these analyses are summarised in Table 3 . There were three multi-variable relationships observed in these analyses, which can be interpreted as the likelihood of solving one problem in each group being associated with solving the others in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. Training also featured in each of these sets, moderating the relationships as per the results presented above for each problem.

https://doi.org/10.1371/journal.pone.0236153.t003

The final “Comments Page” revealed the participants as overwhelmingly enjoying the questions. Any analysis of previous exposure to the tasks proved impossible as there was little to no alignment on participant’s degree of recall, if any, and even perceptions of what exposure entailed. For example, some participants confused being exposed to the particular tasks with being habitually exposed to puzzles, or even mathematics problems, more broadly.

In general, the amount of mathematics training a group had received predicted their performance on the overall set of problems. The greater the training, the more problems were answered correctly, and the slower the recorded response times. There was not an obvious difference between the Advanced1, Advanced2 and Academic groups on either of these measures, however there were clear differences between this group and the Introductory and Standard groups, with the former exhibiting clearly superior accuracy. While time records were taken approximately, so as to avoid adding time pressure as a variable, that the Advanced1, Advanced2 and Academic groups recorded more time in their consideration of the problems, may suggest a “pause and consider” approach to such problems is a characteristic of the advanced groups. This is in line with what was suggested by an eye-movement tracking study of mathematically trained students attempting the Four Cards Problem; where participants that had not chosen the standard error had spent longer considering the card linked to the matching bias effect [ 14 ]. It is important to note, however, that longer response times may reflect other cognitive processes than deliberation [ 32 ].

Performance on some problems was associated with performance on other problems. That is, if someone correctly answered a problem in one of these sets, they were also highly likely to correctly answer the other problems in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. This is different with how these problems have been typically clustered a priori in the research literature: (I) Lily Pads, Widgets and Bat and Ball (CRT); (II) Hospitals and Two Drivers (explained below); (III) Hospitals, Birth Order and Coin Tosses (representativeness heuristic); (IV) Birth Order and Coin Tosses (probability theory). Consideration of these problem groupings follows.

Correctly answering all three problems in (I) entailed not being distracted by particular pieces of information in the problems so as to stay focused on uncovering the real underlying relationships. The Lily Pads and Widget problems can mislead if attention is over focused on the numbers, and conversely, the Petrol Station problem can mislead if there is too much focus on the idea of a discount. While the Lily Pads and Widget problems are traditionally paired with the Bat and Ball problem in the CRT, it may be that performance on the Bat and Ball problem did not appear as part of this set due to an added level of difficulty. With the problems in (I), avoiding being distracted by certain parts of the questions at the expense of others almost leads directly to the correct answer. However, with the Bat and Ball problem, further steps in mathematical reasoning still need to occur in answering which two numbers add together to give a result while also subtracting one from the other for another.

With the problems in (II) it is of interest that the Two Drivers problem was created specifically to be paired with the Hospitals problem to test for motivation in problem solving [ 23 ]. Within this framework further transparent versions of these problems were successfully devised to manipulate for difficulty. The Two Drivers problem was amended to have Driver B travelling at exactly 5 mph during the first half of the race and at exactly 95 mph during the last half of the race. The Hospitals problem was amended so the smaller hospital would have “only 2” babies born each day and where for a period of one year the hospitals recorded the number of days on which all of the babies born were boys. Could the association in (II) be pointing to how participants overcome initial fictitious mathematical rules? Maybe they reframe the question in simpler terms to see the pattern. The Four Cards Problem also elicited a high number of incorrect answers where, associated with mathematical training, the standard incorrect solution was avoided for more cognitively elaborate ones. Indeed, a gradation effect appeared across the groups where the standard error of the “D and 3” cards becomes “D only” ( Table 4 ). Adrian Simpson and Derrick Watson found a comparable result across their two groups [14 p61]. This could again be pointing to having avoided an initial fictitious rule of simply concentrating on items directly found in the question, participants then seek to reframe the question to unearth the logical rule to be deduced. An added level of difficulty with this question may be why participants become trapped in a false answer. The eye-movement tracking study mentioned above supports this theory.

https://doi.org/10.1371/journal.pone.0236153.t004

The problems in (III) fit naturally together as part of basic probability theory, a topic participants would have assimilated, or not, as part of various education curricula. While the equal likelihood of all possible outcomes with respect to a coin toss may be culturally assimilated, the same may not be as straightforward for birth gender outcomes where such assumptions could be swayed by biological hypothesis or folk wisdom [ 33 ]. The gradation of the results in terms of mathematical training does not support this possibility.

The effect of training on performance accuracy was more obvious in some problems compared to others, and to some extent, this was related to the type of problem. For instance, most of the problems in which performance was related to training (Four Cards, CRT [Lily Pads, Widgets, Bat and Ball], Two Drivers, Jack looking at Anne) could be classed as relying on logical and/or critical thinking. The one exception was the Birth Order problems, which are probability related.

In contrast, two of the three problems in which training did not appear to have much impact on performance (Hospitals and Coin Tosses) require domain-specific knowledge. The Hospitals problem requires a degree of knowledge about sampling statistics. This is a topic of quite distinct flavour that not all mathematically trained individuals gain familiarity with. On the other hand, all groups having performed well on the Coin Tosses problem is in line with a level of familiarity with basic probability having been originally presented at high school. While the questioning of patterning as negatively correlated with randomness is similar to that appearing in the Birth Order question, in the Birth Order question this aspect is arguably more concealed. These results and problem grouping (III) could be pointing to an area for improvement in teaching where the small gap in knowledge required to go from answering the Coin Tosses problem correctly to achieving similarly with the Birth Order problem could be easily addressed. A more formal introduction to sampling statistics in mathematical training could potentially bridge this gap as well as further be extended towards improvement on the Hospitals problem.

The other problem where performance was unrelated to training, the Petrol Station problem, cannot be characterised similarly. It is more of a logical/critical thinking type problem, where there remains some suggestion that training may have impacted performance, as the Academic group seemed to perform better than the rest of the sample. An alternate interpretation of this result is therefore that this problem should not be isolated but grouped with the other problems where performance is affected by training.

The Introductory group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Introduction to Differentiation, Applications of the Derivative, Antiderivatives, Areas and the Definite Integral), Financial Mathematics, Statistical Analysis. The Introductory group then explored concepts in mathematical modelling with emphasis on the importance of calculus in their first semester of mathematical studies.
The Standard group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Rates of Change, Integration including the method of substitution, trigonometric identities and inverse trigonometric functions, Areas and Volumes of solids of revolution, some differential equations), Combinatorics, Proof (with particular focus on Proof by Mathematical Induction), Vectors (with application to projectile motion), Statistical Analysis. In first semester their mathematical studies then covered a number of topics the Advanced1 group studied prior to gaining entrance at university; further details on this are given below.
The Advanced1 group’s mathematics high school syllabus studied prior to first semester course entry covered: the same course content the Standard group covered at high school plus extra topics on Proof (develop rigorous mathematical arguments and proofs, specifically in the context of number and algebra and further develop Proof by Mathematical Induction), Vectors (3 dimensional vectors, vector equations of lines), Complex Numbers, Calculus (Further Integration techniques with partial fractions and integration by parts), Mechanics (Application of Calculus to Mechanics with simple harmonic motion, modelling motion without and with resistance, projectiles and resisted motion). The Standard group cover these topics in their first semester university studies in mathematics with the exclusion of further concepts of Proof or Mechanics. In first semester the Advanced1 group have built on their knowledge with an emphasis on both theoretical and foundational aspects, as well as developing the skill of applying mathematical theory to solve practical problems. Theoretical topics include a host of theorems relevant to the study of Calculus.

In summary, at the point of our study, the Advanced1 group had more knowledge and practice on rigorous mathematical arguments and proofs in the context of number and algebra, and more in-depth experience with Proofs by Induction, but the bulk of extra knowledge rests with a much deeper knowledge of Calculus. They have had longer experience with a variety of integration techniques, and have worked with a variety of applications of calculus to solve practical problems, including a large section on mechanics at high school. In first semester at university there has been a greater focus on theoretical topics including a host of theorems and associated proofs relevant to the topics studied. As compared to the Introductory and Standard groups, the Advanced1 group have only widened the mathematics knowledge gap since their choice of post-compulsory mathematics at high school. The Advanced2 group come directly from an Advanced1 cohort. And the Academics group would have reached the Advanced1 group’s proficiency as part of their employment. So, are specific reasoning skills resulting from this level of abstract reasoning? Our findings suggest this should certainly be an area of investigation and links in interestingly with other research work. In studying one of the thinking tasks in particular (the Four Cards Problem) and its context of conditional inference more specifically, Inglis and Simpson [ 15 ] found a clear difference between undergraduates in mathematics and undergraduates in other university disciplines, yet also showed a lack of development over first-year university studies on conditional inference measures. A follow up study by Attridge and Inglis [ 22 ] then zeroed in on post-compulsory high school mathematical training and found that students with such training did develop their conditional reasoning to a greater extent than their control group over the course of a year, despite them having received no explicit tuition in conditional logic. The development though, whilst demonstrated as not being the result of a domain-general change in cognitive capacity or thinking disposition, and most likely associated with the domain-specific study of mathematics, revealed a complex pattern of endorsing more of some inferences and less of others. The study here focused on a much broader problem set associated with logical and critical thinking and it too is suggestive of a more complex picture in how mathematics training may be contributing to problem solving styles. A more intricate pattern to do with the impact of mathematical training on problem solving techniques is appearing as required for consideration.

There is also a final interpretation to consider: that people in the Advanced 1, Advanced2 and Academic groups did not gain anything from their mathematics training in terms of their ability to solve these problems. Instead, with studies denying any correlation of many of these problems with what is currently measured as intelligence [ 30 ], they might still be people of a particular intelligence or thinking disposition to start with, who have been able to use that intelligence to not only solve these problems, but also survive the challenges of their mathematics training.

That the CRT has been traditionally used as a measure of baseline thinking disposition and that performance has been found to be immutable across groups tested is of particular interest since our results show a clear possible training effect on these questions. CRT is tied with a willingness to engage in effortful thinking which presents as a suitable ability for training. It is beyond the scope of this study, but a thorough review of CRT testing is suggestive of a broader appreciation and better framework to understand thinking disposition, ability and potential ability.

Mathematical training appears associated with certain thinking skills, but there are clearly some subtleties that need to be extricated. The thinking tasks here add to the foundational results where the aim is for a firmer platform on which to eventually base more targeted and illustrative inquiry. If thinking skills can be fostered, could first year university mathematics teaching be improved so that all samples from that group reach the Advanced1 group level of reasoning? Do university mathematics courses become purely about domain-specific knowledge from this point on? Intensive training has been shown to impact the brain and cognition across a number of domains from music [ 34 ], to video gaming [ 35 ], to Braille reading [ 36 ]. The hypothesis that mathematics, with its highly specific practice, fits within this list remains legitimate, but simply unchartered. With our current level of understanding it is worth appreciating the careful wording of the NYU Courant Institute on ‘Why Study Math?’ where there is no assumption of causation: “Mathematicians need to have good reasoning ability in order to identify, analyze, and apply basic logical principles to technical problems.” [ 37 ].

Limitations

One possible limitation of the current study is that the problems may have been too easy for the more advanced people, and so we observed a ceiling effect (i.e., some people obtained 100% correct on all problems). This was most obvious in the Advanced1, Advanced2 and Academic groups. It is possible that participants in these groups had developed logical and critical thinking skills throughout their mathematical training that were sufficient to cope with most of the problems used in this study, and so this would support the contention that training in mathematics leads to the development of logical and critical thinking skills useful in a range of domains. Another interpretation is that participants in these groups already possessed the necessary thinking skills for solving the problems in this study, which is why they are able to cope with the material in the advanced units they were enrolled in, or complete a PhD in mathematics and hold down an academic position in a mathematics department. This would then suggest that training in mathematics had no effect on abstract thinking skills—people in this study possessed them to varying extents prior to their studies. This issue might be settled in a future study that used a greater number of problems of varying difficulties to maximise the chances of finding a difference between the three groups with the most amount of training. Alternatively, a longitudinal study that followed people through their mathematics training could determine whether their logical and critical thinking abilities changed throughout their course.

A further limitation of the study may be that several of the reasoning biases examined in this study were measured by only one problem each (i.e., Four Cards Problem, Two Drivers, Petrol Station, Jack looking at Anne). A more reliable measure of these biases could be achieved by including more problems that tap into these biases. This would, however, increase the time required of participants during data collection, and in the context of this study, would mean a different mode of testing would likely be required.

Broad sweeping intuitive claims of the transferable skills endowed by a study of mathematics require evidence. Our study uniquely covers a wide range of participants, from limited mathematics training through to research academics in the mathematical sciences. It furthermore considered performance on 11 well-studied thinking tasks that typically elude participants in psychological studies and on which results have been uncorrelated with general intelligence, education levels and other demographic information [ 15 , 16 , 30 ]. We identified different performances on these tasks with respect to different groups, based on level of mathematical training. This included the CRT which has developed into a method of measuring baseline thinking disposition. We identified different distributions of types of errors for the mathematically trained. We furthermore identified a performance threshold that exists in first year university for those with high level mathematics training. This study then provides insight into possible changes and adjustments to mathematics courses in order for them to fulfil their advertised goal of reaching improved rational and logical reasoning for a higher number of students.

It is central to any education program to have a clear grasp of the nature of what it delivers and how, but arguably especially so for the core discipline that is mathematics. In 2014 the Office of The Chief Scientist of Australia released a report “Australia’s STEM workforce: a survey of employers” where transferable skills attributed to mathematics were also ones that employers deemed as part of the most valuable [ 38 ]. A better understanding of what mathematics delivers in this space is an opportunity to truly capitalise on this historical culture-crossing subject.

Supporting information

https://doi.org/10.1371/journal.pone.0236153.s001

Acknowledgments

The authors would like to thank Jacqui Ramagge for her proof reading and input, as well as support towards data collection.

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Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Clio cresswell.

1 School of Mathematics and Statistics, The University of Sydney, Sydney, Australia

Craig P. Speelman

2 School of Arts and Humanities, Edith Cowan University, Joondalup, Australia

Associated Data

All relevant data are within the paper and its Supporting Information files.

Introduction

Each of the following cards have a letter on one side and a number on the other.

Here is a rule: “if a card has a D on one side, then it has a 3 on the other”. Your task is to select all those cards, but only those cards, which you would have to turn over in order to find out whether the rule is true or false. Which cards would you select?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

Bat and ball

A bat and a ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?

A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day, and in the smaller hospital, about 15 babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower. For a period of one year, each hospital recorded the number of days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days? (Circle one letter.)

(a) the larger hospital
(b) the smaller hospital
(c) about the same (that is, within 5 percent of each other)

Birth order

All families of six children in a city were surveyed. In 72 families the exact order of births of boys and girls was GBGBBG.

(a) What is your estimate of the number of families surveyed in which the exact order of births was BGBBBB?
(b) In the same survey set, which, if any, of the following two sequences would be more likely: BBBGGG or GBBGBG?

Coin tosses

In a sequence of coin tosses (the coin is fair) which of the following outcomes would be most likely (circle one letter):

(a) H T H T H T H T
(b) H H H H T T T T
(c) T T H H T T H H
(d) H T T H T H H T
(e) all of the above are equally likely

Three further questions from the literature were included to test problem solving skill.

Two drivers

Two drivers set out on a 100-mile race that is marked off into two 50-mile sections. Driver A travels at exactly 50 miles per hour during the entire race. Driver B travels at exactly 45 mph during the first half of the race (up to the 50-mile marker) and travels at exactly 55 mph during the last half of the race (up to the finish line). Which of the two drivers would win the race? (Circle one letter.)

(a) Driver A would win the race
(b) Driver B would win the race
(c) the two drivers would arrive at the same time (within a few seconds of one another)

Petrol station

And finally, a question described by Stanovich [ 30 , 31 ] as testing our predisposition for cognitive operations that require the least computational effort.

Jack looking at Anne

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person? (Circle one letter.)

Stanovich reported that over 80% of people choose the “lazy” answer (c). The correct answer is (a).

A questionnaire was constructed containing the problems described in the previous sections plus the Four Cards Problem as tested by Inglis and Simpson [ 11 ] for comparison. The order of the problems was as follows: 1) Lily Pads; 2) Hospitals; 3) Widgets; 4) Four Cards; 5) Bat and Ball; 6) Birth Order; 7) Petrol Station; 8) Coin Tosses; 9) Two Drivers; 10) Jack looking at Anne. It was administered to five groups distinctive in mathematics training levels chosen from a high-ranking Australian university, where the teaching year is separated into two teaching semesters and where being a successful university applicant requires having been highly ranked against peers in terms of intellectual achievement:

Introductory—First year, second semester, university students with weak high school mathematical results, only enrolled in the current unit as a compulsory component for their chosen degree, a unit not enabling any future mathematical pathway, a typical student may be enrolled in a Biology or Geography major;
Standard—First year, second semester, university students with fair to good high school mathematical results, enrolled in the current mathematics unit as a compulsory component for their chosen degree with the possibility of including some further mathematical units in their degree pathway, a typical student may be enrolled in an IT or Computer Science major;
Advanced1—First year, second semester, university mathematics students with very strong interest as well as background in mathematics, all higher year mathematical units are included as possible future pathway, a typical student may be enrolled in a Mathematics or Physics major;
Advanced2—Second year, second semester, university mathematics students with strong interest as well as background in mathematics, typically a direct follow on from the previously mentioned Advanced1 cohort;
Academic—Research academics in the mathematical sciences.

Participants

The responses were coded four ways: A) correct; B) standard error (the errors discussed above in The Study); C) other error; D) left blank.

The ethical aspects of the study were approved by the Human Research Ethics Committee of the University of Sydney, protocol number [2016/647].

An external file that holds a picture, illustration, etc.
Object name is pone.0236153.g001.jpg

Error bars are one standard error of the mean.

An external file that holds a picture, illustration, etc.
Object name is pone.0236153.g002.jpg

Superscripts label the groups (e.g., Introductory = a). Within the table, these letters refer to which other group a particular group was significantly different to according to a series of pairwise post hoc chi squared analyses (Bonferroni corrected α = .005) (e.g., ‘d’ in the Introductory column indicates the Introductory and the Advanced2 (d) group were significantly different for a particular problem).

The cognitive reflection test

P = Problem (1 = Four Cards; 2 = Lily Pads; 3 = Widgets; 4 = Bat & Ball; 5 = Hospitals; 6a = Birth Order (a); 6b = Birth Order (b); 7 = Coin Tosses; 8 = Two Drivers; 9 = Petrol Station; 10 = Jack looking at Anne).

training = Amount of training condition.

p = significance level of logistic regression model.

% = percentage of cases correctly classified by the logistic regression model.

✓ = significant predictor, α < .05.

* = logistic regression for the training outcome variable is multinomial, whereas all other logistic regressions are binomial.

Although several aspects of the data suggest mathematics training improves the chances that someone will solve problems of the sort examined here, differences in the performance of participants in the Advanced1, Advanced2 and Academic groups were not obvious. This is despite the fact that large differences exist in the amount of training in these three groups. The first two groups were undergraduate students and the Academic group all had PhDs and many were experienced academic staff. One interpretation of this result is current mathematics training can only take someone so far in terms of improving their abilities with these problems. There is a point of demarcation to consider in terms of mathematical knowledge between the Advanced1, Advanced2 and Academic groups as compared to the Introductory and Standard groups. In Australia students are able to drop mathematical study at ages 15–16 years, or choose between a number of increasingly involved levels of mathematics. For the university in this study, students are filtered upon entry into mathematics courses according to their current knowledge status. All our groups involved students who had opted for post-compulsory mathematics at high school. And since our testing occurred in second semester, some of the mathematical knowledge shortfalls that were there upon arrival were bridged in first semester. Students must pass a first semester course to be allowed entry into the second semester course. A breakdown of the mathematics background of each group is as follows:

The Introductory group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Introduction to Differentiation, Applications of the Derivative, Antiderivatives, Areas and the Definite Integral), Financial Mathematics, Statistical Analysis. The Introductory group then explored concepts in mathematical modelling with emphasis on the importance of calculus in their first semester of mathematical studies.
The Standard group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Rates of Change, Integration including the method of substitution, trigonometric identities and inverse trigonometric functions, Areas and Volumes of solids of revolution, some differential equations), Combinatorics, Proof (with particular focus on Proof by Mathematical Induction), Vectors (with application to projectile motion), Statistical Analysis. In first semester their mathematical studies then covered a number of topics the Advanced1 group studied prior to gaining entrance at university; further details on this are given below.
The Advanced1 group’s mathematics high school syllabus studied prior to first semester course entry covered: the same course content the Standard group covered at high school plus extra topics on Proof (develop rigorous mathematical arguments and proofs, specifically in the context of number and algebra and further develop Proof by Mathematical Induction), Vectors (3 dimensional vectors, vector equations of lines), Complex Numbers, Calculus (Further Integration techniques with partial fractions and integration by parts), Mechanics (Application of Calculus to Mechanics with simple harmonic motion, modelling motion without and with resistance, projectiles and resisted motion). The Standard group cover these topics in their first semester university studies in mathematics with the exclusion of further concepts of Proof or Mechanics. In first semester the Advanced1 group have built on their knowledge with an emphasis on both theoretical and foundational aspects, as well as developing the skill of applying mathematical theory to solve practical problems. Theoretical topics include a host of theorems relevant to the study of Calculus.

Limitations

Supporting information

Acknowledgments.

The authors would like to thank Jacqui Ramagge for her proof reading and input, as well as support towards data collection.

Funding Statement

The authors received no specific funding for this work.

Data Availability

PLoS One. 2020; 15(7): e0236153.

Decision Letter 0

17 Mar 2020

PONE-D-20-01159

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Dear Professor Speelman,

Thank you for submitting your manuscript to PLOS ONE. I have sent it to two expert reviewers and have received their comments back. As you can see at the bottom of this email, both reviewers are positive about your manuscript but raise some issues that you would need to address before the manuscript can be considered for publication. Notably, reviewer #1 points out that the manuscript should include a discussion on the reasons why individuals with math training may have improved reasoning skills (e.g., logical intuitions versus deliberate thinking). The reviewer also rightly mentions that your sample sizes are limited, notably for the most advanced groups. This should be discussed and acknowledged. Reviewer #2 has a number of conceptual and methodological points that you will also have to address. The reviewer provides very thorough comments and I will not reiterate the points here. However, note that both reviewers suggest that you need to improve the figures and I agree with them.

We would appreciate receiving your revised manuscript by May 01 2020 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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Academic Editor

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Reviewers' comments:

Reviewer #1: I think this is a very good and interesting manuscript trying to answer an important research question. I propose some changes that I believe should be applied before publication.

1. Each reasoning bias is measured with only one problem. In reasoning research, it is rather common to measure each type of reasoning problem with a series of structurally equivalent reasoning problems, so the results will be independent of contexts effects and will be generalizable to that type of problem. Here, the authors only measured each reasoning bias with one single problem and this might be problematic (see, for example: Fiedler & Hertel, 1994). I think this can be addressed by simply discussing it in the limitation section.

2. This is rather a minor issue, but the discussion on the CRT problems is not up-to-date (page 7). Most recent experiments on dual process theory suggest that people who are able to correctly solve these reasoning problems (including the CRT) do so intuitively, and not because they engaged in careful deliberation (Bago & De Neys, 2019). Intelligence made people have better intuitive responses (Thompson, Pennycook, Trippas & Evans, 2018). Similarly, this problems persists in the discussion about reaction times (page 25). Longer reaction times does not necessarily mean that people engaged in deliberation (see: Evans, Kyle, Dillon & Rand, 2015). Response time might be driven by decision conflict or response rationalization. These issues could be clarified with some changes in the wording or some footnotes on page 7 and 25. Furthermore, it would be interesting to have a discussion on how mathematical education helps people overcome their biases. Is it because it creates better intuition, or helps people engage in deliberation? An interesting question this manuscript does not discuss. It’s on the authors whether or not they discuss this latter point now, but the changes on page 7 and 25 should be made.

3. A more serious problem is the rather small sample size (especially in the more advanced groups). This small sample size makes the appearance of both false negatives and false positives more likely. Perhaps, the authors could compute the Bayes Factors for the chi-square or logistic regression test, so we can actually see how strong the evidence is for or against the null. This is especially important as the authors run a great number of explorative analysis (Table 3), and some of those results might need to be interpreted with great caution (depending on the Bayes Factor).

The graphs are not looking good, they should comply with APA formatting. At the very least, the axis titles should be meaningful and measure units should be written there.

The presentation order of the problems is quite unusual; why isn’t it random? Why did the authors decide on this order?

Reviewer #2: The study reported in this paper compared five groups of participants with varying levels of mathematical expertise on a set of reasoning tasks. The study is interesting and informative. It extends the current literature on this topic (which is reviewed very nicely in the introduction). However, there are some issues with the current analysis and interpretation that should be resolved prior to publication. I have therefore recommended major revisions. My comments are organised in the order in which they came up in the paper and they explain my responses to the questions above.

1. Line 114 – “general population” a bit misleading – they were also students but from other disciplines.

2. Line 124 onwards reads:

“The ultimate question to consider here is: are any skills associated with mathematics training innate or do they arise from skills transfer? Though to investigate how mathematical training affects reasoning skills, randomised sampling and randomised intervention to reveal causal relationships are clearly not viable. With so many possible confounding variables and logistical issues, it is even questionable what conclusions such studies might provide. Furthermore, a firm baseline from which to propose more substantive investigations is still missing.”

I find this paragraph slightly problematic because the current study doesn’t inform us on this ultimate question, so it makes the outline of the current study in the following paragraph feel unsatisfactory. I think the current study is important but prefacing it with this paragraph underplays that importance. And I think a randomised controlled study, although not viable, would give the answers we need because the random allocation to groups would allow us to rule out any confounding variables. Finally, the last sentence in this paragraph is unclear to me.

3. In the descriptions of the five participants groups the authors refer to the group’s level of interest in mathematics, but this seems like an overgeneralisation to me. Surely the introductory group could contain a biology student who also happens to be good at mathematics and very much enjoy it? I would be more comfortable with the descriptions if the parts about interest level were removed.

4. How many of the 123 first year students were in each of the three first year groups?

5. Line 313 – the standard group is referred to as “university mathematics students”, but they are not taking mathematics degreed.

6. Line 331 - what is a practice class?

7. Were the data collection settings quiet? From the description it sounds like groups of participants were completing the study at the same time in the same room, but the authors should make this explicit for the sake of the method being reproducible. E.g. how many students were in the room at the time?

8. Line 355-356 – the authors should not use the term “marginally worse” because this is statistically inappropriate – in a frequentist approach results are either significant or non-significant.

9. Line 340 – “approximate completion times were noted.”

This doesn’t sound rigorous enough to justify analysing them. Their analysis is interesting, but the authors should remind readers clearly whenever the response times are analysed or discussed that their recording was only manual and approximate.

10. I suggest replacing Figure 1 with a bar chart showing standard error of the mean on the error bars. A table with mean score out of 11 and the standard deviation for each group may also be useful. Figure 2 should be a scatterplot rather than a box and whisker plot.

11. Was the 0-11 total correct score approximately normally distributed across the full sample?

12. Chi square analysis requires at least 5 cases in each cell, was this met? It seems not since Table 1 shows lots of cells in the “no response” row having 0% of cases.

13. The chi-square analyses should be followed up with post hoc tests to see exactly where the differences between groups are. The descriptions as they stand aren’t that informative (as readers can just look at Table 1) without being backed up by post hoc tests.

14. For each chi square analysis in the text, I would find it easier to read if the test statistics came at the top of the paragraph, before the description.

15. Line 381-383 – “Of note, also, is the relatively low proportion of those in the higher training groups who, when they made an error, did not make the standard error, a similar result to the one reported by Inglis and Simpson [11]."

I think this is supposed to say that a low proportion did make the standard error or that a high proportion did not make the standard error.

16. Line 403 - p values this small should be reported as p < .001 rather than p = .000 since they aren’t actually 0.

17. Line 476 – “…if a particular outcome variable was predicted significantly by a particular predictor variable, the converse relationship was also observed”

Isn’t that necessarily the case with regression analyses, like with correlations?

18. I don’t think the logistic regression analyses add much to the paper and at the moment they come across as potential p-hacking since they don’t clearly relate to the research question. To me they make the paper feel less focused. Having said that, there is some interesting discussion of them in the Discussion section. I’d recommend adding some justification to the introduction for why it is interesting to look at the relationships among tasks (without pretending to have made any specific hypotheses about the relationships, of course).

19. Line 509 would be clearer if it read “between these groups and the introductory and standard groups”

20. Lines 597 – 620 - This is an interesting discussion, especially the suggestion that advanced calculus may be responsible for the development. No development in reasoning skills from the beginning of a mathematics degree onwards was also found by Inglis and Simpson (2009), who suggested that the initial difference between mathematics and non-mathematics undergraduates could have been due to pre-university study of mathematics. Attridge & Inglis (2013) found evidence that this was the case (they found no difference between mathematics and non-mathematics students at age 16 but a significant difference at the end of the academic year, where the mathematics students had improved and the non-mathematics students had not).

Could the authors add some discussion of whether something similar may have been the case with their Australian sample? E.g. do students in Australia choose whether, or to what extent, to study mathematics towards the end of high school? If not, the description of the groups suggests that there were at least differences in high school mathematics attainment between groups 1-3, even if they studied the same mathematics curriculum. Do the authors think that this difference in attainment could have led to the differences between groups in the current study?

21. Line 617 – “Intensive training has been shown to impact the brain and cognition across a number of domains from music, to video gaming, to Braille reading [31].”

Reference 31 appears to only relate to music. Please add references for video gaming and Braille reading.

22. I recommend editing the figures from SPSS’s default style or re-making them in Excel or DataGraph to look more attractive.

23. I cannot find the associated datafile anywhere in the submission. Apologies if this is my mistake.

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/ . PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at gro.solp@serugif . Please note that Supporting Information files do not need this step.

Author response to Decision Letter 0

20 Apr 2020

All responses are detailed against the specific reviewers' comments in the Response to Reviewers document

Submitted filename: Response to Reviewers.docx

Decision Letter 1

11 Jun 2020

PONE-D-20-01159R1

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors.

Dear Dr. Speelman,

Thank you for submitting your revised manuscript to PLOS ONE. I have sent it to reviewer #2 and have now received the reviewer's comment. As you can see, the reviewer thinks that the manuscript is improved but has some outstanding issues that you would need to address in another round of revision. I notably agree with the reviewer that you should provide the raw data, allowing readers to replicate your analyses. Therefore, I invite you submit a revised version of your manuscript.

Please submit your revised manuscript by Jul 26 2020 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at gro.solp@enosolp . When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Reviewer #2: The manuscript has improved but there are still a few issues that should be resolved prior to publication.

1. On lines 96, 97, 100 and 102, the references to “general population” should be changed to reflect the fact that these participants were non-mathematics (arts) students.

2. Line 306 – change “mathematics students” to “university students”.

3. The method section doesn’t specify the gender split and mean age of the sample.

4. Table 3 - values the p values listed as .000 should be changed to <.001.

5. Table 3 - I suggest repeating the list of problem numbers and names in the legend. It may make for a long legend but would make it much easier for the reader to interpret the table.

6. I am not sure what the new post hoc tests are comparing. What I expected was to see group 1 compared to groups 2, 3, 4 and 5, and so on. This would tell us which groups are statistically different from each other. At the moment we only know from the overall chi square tests whether there are any differences among the groups or not, we don’t know specifically which groups are statistically different from each other and which ones are not. We only have the authors’ interpretations based on the observed counts.

7. Line 584 - change “performance was correlated with training” to “performance was related to training” to avoid any confusion since a correlation analysis was not performed.

8. Data file – I had expected the data file to give the raw data rather than summary data, i.e. with each participant in a separate row, and a column indicating their group membership, a column giving their age, a column for sex etc (including all the demographics mentioned in the method), and a column for each reasoning question. This would allow other researchers to replicate the regression analyses and look at other relationships within the dataset. Without being able to replicate all analyses in the paper, the data file does not meet the minimal data set definition for publication in PLOS journals: https://journals.plos.org/plosone/s/data-availability .

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/ . PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at gro.solp@serugif . Please note that Supporting Information files do not need this step.

Author response to Decision Letter 1

16 Jun 2020

Please see "Response to Reviewers" document

Decision Letter 2

PONE-D-20-01159R2

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Additional Editor Comments (optional):

Acceptance letter

Dear Dr. Speelman:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

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9 Ways to Improve Math Skills Quickly & Effectively

$Overhead view of a child using a piece of paper, a pen, and a calculator to do math homework and improve their math skills$

Written by Ashley Crowe

Help your child improve their math skills with the game that makes learning an adventure!

Parent Resources

The importance of understanding basic math skills

9 Ways to improve math skills
How to use technology to improve math skills

Math class can move pretty fast. There’s so much to cover in the course of a school year. And if your child doesn’t get a new math idea right away, they can quickly get left behind.

If your child is struggling with basic math problems every day, it doesn’t mean they’re destined to be bad at math. Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it’s best to take each year step-by-step, lesson by lesson.

This article has tips and tricks to improve your child’s math skills while minimizing frustrations and struggles. If your child is growing to hate math, read on for ways to improve their skills and confidence, and maybe even make math fun!

But first, the basics.

Math is a subject that builds on itself. It takes a solid understanding of past concepts to prepare for the next lesson.

That’s why math can become frustrating when you’re forced to move on before you’re ready. You’re either stuck trying to catch up or you end up falling further behind.

But with a strong understanding of basic math skills, your child can be set up for school success. If you’re unfamiliar with the idea of sets or whole numbers , this is a great place to start.

What are considered basic math skills?

The basic math skills required to move on to higher levels of math learning are:

Addition — Adding to a set.
Subtraction — Taking away from a set.
Multiplication — Adding equal sets together in groups (2 sets of 3 is the same as 2x3, or 6).
Division — How many equal sets can be found in a number (12 has how many sets of two in it? 6 sets of 2).
Percentages — A specific amount in relation to 100.
Fractions & Decimals — Fractions are equal parts of a whole set. Decimals represent a number of parts of a whole in relation to 10. These both contrast with whole numbers.
Spatial Reasoning — How numbers and shapes fit together.

How to improve math skills

People aren’t bad at math — many just need more time and practice to gain a thorough understanding.

How can you help your child improve their math abilities? Use our top 9 tips for quickly and effectively improving math skills .

1. Wrap your head around the concepts

Repetition and practice are great, but if you don’t understand the concept , it will be difficult to move forward.

Luckily, there are many great ways to break down math concepts . The trick is finding the one that works best for your child.

Math manipulatives can be a game-changer for children who are struggling with big math ideas. Taking math off the page and putting it into their hands can bring ideas to life. Numbers become less abstract and more concrete when you’re counting toy cars or playing with blocks. Creating these “sets” of objects can bring clarity to basic math learning.

2. Try game-based learning

During math practice, repetition is important — but it can get old in a hurry. No one enjoys copying their times tables over and over and over again. If learning math has become a chore, it’s time to bring back the fun!

Game-based learning is a great way to practice new concepts and solidify past lessons. It can even make repetition fun and engaging.

Game-based learning can look like a family board game on Friday night or an educational app , like Prodigy Math .

$A glimpse of the Prodigy Math Game world and a sample math question a kid could receive to help improve their math skills while playing.$

Take math from frustrating to fun with the right game, then watch the learning happen easily!

3. Bring math into daily life

You use basic math every day.

As you go about your day, help your child see the math that’s all around them:

Tell them how fast you’re driving on the way to school
Calculate the discount you’ll receive on your next Target trip
Count out the number of apples you need to buy at the grocery store
While baking, explain how 6 quarter cups is the same amount of flour as a cup and a half — then enjoy some cookies!

Relate math back to what your child loves and show them how it’s used every day. Math doesn’t have to be mysterious or abstract. Instead, use math to race monster trucks or arrange tea parties. Break it down, take away the fear, and watch their interest in math grow.

4. Implement daily practice

Math practice is important. Once you understand the concept, you have to nail down the mechanics. And often, it’s the practice that finally helps the concept click. Either way, math requires more than just reading formulas on a page.

Daily practice can be tough to implement, especially with a math-averse child. This is a great time to bring out the game-based learning mentioned above. Or find an activity that lines up with their current lesson. Are they learning about squares? Break out the math link cubes and create them. Whenever possible, step away from the worksheets and flashcards and find practice elsewhere.

5. Sketch word problems

Nothing causes a panic quite like an unexpected word problem. Something about the combination of numbers and words can cause the brain of a struggling math learner to shut down. But it doesn’t have to be that way.

Many word problems just need to be broken down, step by step . One great way to do this is to sketch it out. If Doug has five apples and four oranges, then eats two of each, how many does he have left? Draw it, talk it out, cross them off, then count.

If you’ve been talking your child through the various math challenges you encounter every day, many word problems will start to feel familiar.

6. Set realistic goals

If your child has fallen behind in math, then more study time is the answer. But forcing them to cram an extra hour of math in their day is not likely to produce better results. To see a positive change, first identify their biggest struggles . Then set realistic goals addressing these issues .

Two more hours of practicing a concept they don’t understand is only going to cause more frustration. Even if they can work through the mechanics of a problem, the next lesson will leave them feeling just as lost.

Instead, try mini practice sessions and enlist some extra help. Approach the problem in a new way, reach out to their teacher or try an online math lesson . Make sure the extra time is troubleshooting the actual problem, not just reinforcing the idea that math is hard and no fun.

Set Goals and Rewards in Prodigy Math

Did you know that parents can set learning goals for their child in Prodigy Math? And once they achieve them, they'll unlock in-game rewards of your choice!

7. Engage with a math tutor

If your child is struggling with big picture concepts, look into finding a math tutor . Everyone learns differently, and you and your child’s teacher may be missing that “aha” moment that a little extra time and the right tutor can provide.

It’s amazing when a piece of the math puzzle finally clicks for your child. If you’re ready to get that extra help, try a free 1:1 online session from Prodigy Math Tutoring. Prodigy’s tutors are real teachers who know how to connect kids to math. With the right approach, your child can become confident in math — and who knows, they may even begin to enjoy it.

8. Focus on one concept at a time

Math builds on itself. If your child is struggling through their current lesson, they can’t skip it and come back to it later. This is the time to practice and repeat — re-examining and reinforcing the current concept until it makes sense.

Look for other ways to approach new math ideas. Use math manipulatives to bring numbers off the page. Or try a learning app with exciting rewards and positive reinforcement to encourage extra practice.

Take a step back when frustrations get high — but resist the temptation to just let it go. Once the concept clicks, they’ll be excited to forge ahead.

9. Teach others math you already know

Even if your child is struggling in math, they’ve still learned so much since last year. Focus on the improvements they’ve made and let them showcase their knowledge. If they have younger siblings, your older child can demonstrate addition or show them how to use a number line. This is a great way to build their confidence and encourage them to keep going.

Or let them teach you how they solve new problems. Have your child talk you through the process while you solve a long division problem . You’re likely to find yourself a little rusty on the details. Play it up and get a little silly. They’ll love teaching you the ropes of this “new math.”

Child using movable numbers and math symbols on a table to show a 5x5 formula and help someone else improve their math skills

Embracing technology to improve math skills

Though much of your math learning was done with pencil to paper, there are many more ways to build number skills in today’s tech world.

Your child can take live, online math courses to work through tough concepts. Or play a variety of online games, solving math puzzles and getting consistent practice while having fun.

These technical advances can help every child learn math, no matter their preferred learning or study style. If your child is a visual learner, there’s an app for that. Do they process best while working in groups? Jump online and find one. Don’t keep repeating the same lessons from their math class over and over. Branch out, try something new and watch the learning click.

Look online for more math help

There are so many online resources, it can be hard to know where to start.

At Prodigy, we’re happy to help you get the ball rolling on your child’s math learning, from kindergarten through 8th grade. It’s free to sign up, fun to play and exciting to watch as your child’s math understanding grows.

Sign up for a free parent account and get instant data on your child’s progress as they build more math skills with Prodigy Math Game . It’s time to take the math struggle out of your home and enjoy learning together!

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Math Activities

How to Improve Problem-Solving Skills in Math

$how does math improve problem solving skills$

Importance of Problem-Solving Skills in Math

Problem-solving skills are crucial in math education , enabling students to apply mathematical concepts and principles to real-world situations. Here’s why problem-solving skills are essential in math education:

1. Application of knowledge: Problem-solving in math requires encouraging students to apply the knowledge they acquire in the classroom to tackle real-life problems. It helps them understand the relevance of math in everyday life and enhances their critical thinking skills.

2. Developing critical thinking: Problem-solving requires students to analyze, evaluate, and think critically about different approaches and strategies to solve a problem. It strengthens their mathematical abilities and improves their overall critical thinking skills.

3. Enhancing problem-solving skills: Math problems often have multiple solutions, encouraging students to think creatively and explore different problem-solving strategies. It helps develop their problem-solving skills, which are valuable in various aspects of life beyond math.

4. Fostering perseverance: Problem-solving in math often requires persistence and resilience. Students must be willing to try different approaches, learn from their mistakes, and keep trying until they find a solution. It fosters a growth mindset and teaches them the value of perseverance.

Benefits of strong problem-solving skills

Having strong problem-solving skills in math offers numerous benefits for students:

1. Improved academic performance: Students with strong problem-solving skills are likelier to excel in math and other subjects that rely on logical reasoning and critical thinking.

2. Enhanced problem-solving abilities: Strong problem-solving skills extend beyond math and can be applied to various real-life situations. It includes decision-making, analytical thinking, and solving complex problems creatively.

3. Increased confidence: Successfully solving math problems boosts students’ self-confidence and encourages them to tackle more challenging tasks. This confidence spills over into other areas of their academic and personal lives.

4. Preparation for future careers: Problem-solving skills are highly sought after by employers in various fields. Developing strong problem-solving skills in math sets students up for successful careers in engineering, technology, finance, and more.

Problem-solving skills are essential for math education and have numerous benefits for students. By fostering these skills, educators can empower students to become confident, critical thinkers who can apply their mathematical knowledge to solve real-world problems.

Understand the Problem

Breaking down the problem and identifying the key components.

To improve problem-solving math skills, it’s essential to first understand the problem at hand. Here are some tips to help break down the problem and identify its key components:

1. Read the problem carefully: Take your time to read it attentively and ensure you understand what it asks. Pay attention to keywords or phrases that indicate what mathematical operation or concept to use.

2. Identify the known and unknown variables: Determine what information is already given in the problem (known variables) and what you need to find (unknown variables). This step will help you analyze the problem more effectively.

3. Define the problem in your own words: Restate the problem using your own words to ensure you clearly understand what needs to be solved. It can help you focus on the main objective and eliminate any distractions.

4. Break the problem into smaller parts: Complex math problems can sometimes be overwhelming. Breaking them down into smaller, manageable parts can make them more approachable. Identify any sub-problems or intermediate steps that must be solved before reaching the final solution.

Reading and interpreting math word problems effectively

Many math problems are presented as word problems requiring reading and interpreting skills. Here are some strategies to help you effectively understand and solve math word problems:

1. Highlight key information: As you read the word problem, underline or highlight any important details, such as numbers, units of measurement, or specific keywords related to mathematical operations.

2. Visualize the problem: Create visual representations, such as diagrams or graphs, to help you understand the problem better. Visualizing the problem can make determining what steps to take and how to approach the solution easier.

3. Translate words into equations: Convert the information in the word problem into mathematical equations or expressions. This translation step helps you transform the problem into a solvable math equation.

4. Solve step by step: Break down the problem into smaller steps and solve each step individually. This approach helps you avoid confusion and progress toward the correct solution.

Improving problem-solving skills in math requires practice and patience. By understanding the problem thoroughly, breaking it into manageable parts, and effectively interpreting word problems, you can confidently enhance your ability to solve math problems.

Use Visual Representations

Using diagrams, charts, and graphs to visualize the problem.

One effective way to improve problem-solving skills in math is to utilize visual representations. Visual representations , such as diagrams, charts, and graphs, can help make complex problems more tangible and easily understood. Here are some ways to use visual representations in problem-solving:

1. Draw Diagrams: When faced with a word problem or a complex mathematical concept, drawing a diagram can help break down the problem into more manageable parts. For example, suppose you are dealing with a geometry problem. In that case, sketching the shapes involved can provide valuable insights and help you visualize the problem better.

2. Create Charts or Tables: For problems that involve data or quantitative information, creating charts or tables can help organize the data and identify patterns or trends. It can be particularly useful in analyzing data from surveys, experiments, or real-life scenarios.

3. Graphical Representations: Graphs can be powerful tools in problem-solving, especially when dealing with functions, equations, or mathematical relationships. Graphically representing data or equations makes it easier to identify key features that may be hard to spot from a numerical representation alone, such as intercepts or trends.

Benefits of visual representation in problem-solving

Using visual representations in problem-solving offers several benefits:

1. Enhances Comprehension: Visual representations provide a visual context for abstract mathematical concepts, making them easier to understand and grasp.

2. Encourages Critical Thinking: Visual representations require active engagement and critical thinking skills. Students can enhance their problem-solving and critical thinking abilities by analyzing and interpreting visual data.

3. Promotes Pattern Recognition: Visual representations simplify identifying patterns, trends, and relationships within data or mathematical concepts. It can lead to more efficient problem-solving and a deeper understanding of mathematical principles.

4. Facilitates Communication: Visual representations can be shared and discussed, helping students communicate their thoughts and ideas effectively. It can be particularly useful in collaborative problem-solving environments.

Incorporating visual representations into math problem-solving can significantly enhance understanding, critical thinking, pattern recognition, and communication skills. Students can approach math problems with a fresh perspective and improve their problem-solving abilities using visual tools.

Work Backwards

Understanding the concept of working backward in math problem-solving.

Working backward is a problem-solving strategy that starts with the solution and returns to the given problem. This approach can be particularly useful in math, as it helps students break down complex problems into smaller, more manageable steps. Here’s how to apply the concept of working backward in math problem-solving:

1. Identify the desired outcome : Start by clearly defining the goal or solution you are trying to reach. It could be finding the value of an unknown variable, determining a specific measurement, or solving for a particular quantity.

2. Visualize the result : Imagine the final step or solution. It will help you create a mental image of the steps needed to reach that outcome.

3. Trace the steps backward : Break down the problem into smaller steps, working backward from the desired outcome. Think about what needs to happen immediately before reaching the final solution and continue tracing the steps back to the beginning of the problem.

4. Check your work : Once you have worked backward to the beginning of the problem, double-check your calculations and steps to ensure accuracy.

Real-life examples and applications of working backward

Working backward is a valuable problem-solving technique in math and has real-life applications. Here are a few examples:

1. Financial planning : When creating a budget, you can work backward by determining your desired savings or spending amount and then calculating how much income or expenses are needed to reach that goal.

2. Project management : When planning a project, you can work backward by setting a fixed deadline and then determining the necessary steps and timelines to complete the project on time.

3. Game strategy : In games like chess or poker, working backward can help you anticipate your opponent’s moves and plan your strategy accordingly.

4. Recipe adjustments : When modifying a recipe, you can work backward by envisioning the final taste or texture you want to achieve and adjusting the ingredients or cooking methods accordingly.

By practicing working backward in math and applying it to real-life situations, you can enhance your problem-solving abilities and find creative solutions to various challenges.

Try Different Strategies

When solving math problems, it’s essential to have a repertoire of problem-solving strategies. You can improve your problem-solving skills and tackle various mathematical challenges by trying different approaches. Here are some strategies to consider:

Exploring Various Problem-Solving Strategies

1. Guess and Check: This strategy involves making an educated guess and checking if it leads to the correct solution. It can be useful when dealing with trial-and-error problems.

2. Drawing a Diagram: Visually representing the problem through diagrams or graphs can help you understand and solve it more effectively. This strategy is particularly useful in geometry and algebraic reasoning.

3. Using Logic: Using logical reasoning is useful for breaking down complicated problems into smaller, more manageable components. This strategy is especially useful in mathematical proofs and logical puzzles.

4. Working Backwards: Start with the desired outcome and return to the given information. When dealing with equations or word problems, this approach can assist.

5. Using Patterns: Look for patterns and relationships within the problem to determine a solution. This approach can be used for different mathematical problems, such as sequences and numerical patterns.

When and How to Apply Different Strategies in Math Problem-Solving

Knowing when and how to apply different problem-solving strategies is crucial for success in math. Here are some tips:

Understand the problem: Read the problem carefully and identify the key information and requirements.
Select an appropriate strategy: Choose the most appropriate problem-solving strategy for the problem.
Apply the chosen strategy: Implement the selected strategy, following the necessary steps.
Check your solution: Verify your answer by double-checking the calculations or applying alternative methods.
Reflect on the process: After solving the problem, take a moment to reflect and evaluate your problem-solving approach. Identify areas for improvement and consider alternative strategies that could have been used.

By exploring different problem-solving strategies and applying them to various math problems, you can enhance your problem-solving skills and develop a versatile toolkit for tackling mathematical challenges. Practice and persistence are key to honing your problem-solving abilities in math.

Key takeaways and tips for improving problem-solving skills in math

In conclusion, developing strong problem-solving skills in math is crucial for success in this subject. Here are some key takeaways and tips to help you improve your problem-solving abilities:

Practice regularly: The more you practice solving math problems, the better you will become at identifying patterns, applying strategies, and finding solutions.
Break down the problem: When faced with a complex math problem, break it into smaller, more manageable parts. It will make it easier to understand and solve.
Understand the problem: Before diving into a solution, fully understand the problem. Identify what information is given and what you are asked to find.
Draw diagrams or visualize: Use visual aids, such as diagrams or sketches, to help you better understand the problem and visualize the solution.
Use logical reasoning: Apply logical reasoning skills to analyze the problem and determine the most appropriate approach or strategy.
Try different strategies: If one approach doesn’t work, don’t be afraid to try different strategies or methods. There are often multiple ways to solve a math problem.
Seek help and collaborate: Don’t hesitate to seek help from your teacher, classmates, or online resources. Collaborating with others can provide different perspectives and insights.
Learn from mistakes: Mistakes are a valuable learning opportunity. Analyze your mistakes, understand where you went wrong, and learn from them to avoid making the same errors in the future.
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Differentiated Teaching

5 Ways to Build Math Problem Solving Skills (based on brain research)

Whether talking about state tests or meeting with your team to plan the next math unit, the conversation inevitably turns to word problems. But knowing how to build math problem-solving skills without resorting to pages of boring story problem practice can be hard.

These days word problems aren't the basic one-step wonders that many of us dealt with as students. Instead, multi-step story problems that require students to apply multiple concepts and skills are incorporated into instruction and state assessments.

Understanding brain research can help simply the process of teaching this challenging format of math problem-solving to students, including those who struggle.

step-by-step math problem-solving for word problems

What research says about building master problem solvers in math

Have you seen how many math skills we must teach these days? No teacher has enough time to build critical math skills AND effectively teach problem-solving…or do they?

Research would argue we are going about these tasks all wrong. They say there are many reasons students struggle with math word problems , but one big one is that we aren't doing what's best for the brain. Instead, here's what the brain research says about the must-have elements for building step-by-step math problem-solving mastery.

Finding #1: Becoming a master problem solver requires repetition.

Duh, right? Any good teacher knows this…but what's the best recipe for repetition if you want students to master math word problems? How much practice? How often?

Let's start with the concept of mastery.

How do you develop math problem solving skills?

In the 1990's, Anders Ericsson studied experts to explore what made some people excel. Findings showed a positive correlation between the amount of deliberate practice (activities that require a high level of concentration and aren't necessarily inherently fun) and skill level.

In other words, the more practice someone gets, the more they improve. This became the basis of Malcolm Gladwell's 10,000-hour rule, which stated that it takes 10,000 hours to make you an expert in a field.

But what should that practice look like for students who struggle with word problems? Is it better to have a deep dive into story problems, or do short bursts of practice do more for long-term understanding?

Designing Better Word Problem Activities: Building Step-by-step Math Problem-Solving Practice

We can look at Ebbinghaus' work on memory & retention to answer that. He found spacing practice over time decreased the number of exposures needed. In other words, small amounts of practice over several days, weeks, or even months actually means you need LESS practice than if you try to cram it all in at once.

For over 80 years, this finding has stood the test of time. While research has shown that students who engage in mass practice (lots of practice all at once) might do better on an assessment that takes place tomorrow, students who engage in repeated practice over a period of time retain more skills long-term (Bloom & Shuell, 1981; Rea & Modigliani, 1985).

And how long does the research say you should spend reviewing?

How long should should students be practicing with story problems to build math problem solving skills ?

How long should problem-solving practice really be?

Shorter is better. As discussed earlier, peak attention required for deliberate practice can only be maintained for so long. And the majority of research supports 8-10 minutes as the ideal lesson length (Robertson, 2010).

This means practice needs to be focused so that during those minutes of discussion, you can dive deep – breaking down the word problem and discussing methods to solve it.

Teacher Tip: Applying this finding to your classroom

Less is actually more as long as you plan to practice regularly. While students who struggle with word problems may need a great deal of practice to master word problems, ideally, this practice should be provided in short, regular intervals with no more than 8-10 minutes spent in whole group discussion.

Here are a few simple steps to apply these findings to your math classroom:

Find 8-12 minutes in your daily schedule to focus on problem-solving – consider this time sacred & only for problem-solving.
Select only 1-2 word problems per day. Target step-by-step math problem-solving to build math problem-solving skills through a less-is-more approach using Problem of the Day .

Finding #2: Students who are challenged & supported have better outcomes.

Productive struggle, as it is called in the research, focuses on the effortful practice that builds long-term understanding.

Important to this process are opportunities for choice, collaboration, and the use of materials or topics of interest (which will be discussed later).

This productive struggle also helps students build flexible thinking so that they can apply previously learned skills to new or unfamiliar tasks (Bransford, Brown, & Cocking, 2000).

“Meaningful learning tasks need to challenge ever student in some way. It is crucial that no student be able to coast to success time after time; this experience can create the belief that you are smart only if you can succeed without effort.” -Carol Dweck

It is also critical to provide support and feedback during the challenging task (Cimpian, Arce, Markman, & Dweck, 2007). This prevents frustration and fear of failure when the goal seems out of reach or when a particularly challenging task arises.

Simple ways to build productive struggle into your math classroom

Giving students who struggle with word problems a chance to struggle with challenging word problems is critical to building confidence and skills. However, this challenge must be reasonable, or the learner's self-esteem will falter, and students need support and regular feedback to achieve their potential.

Here are a few simple things to try:

Select problems that are just at the edge of students' Zone of Proximal Development.
Scaffold or model with more challenging problems to support risk-taking.
Give regular feedback & support – go over the work and discuss daily.

Finding #3: Novelty & variation are keys to engagement.

When it comes to standardized testing (and life in general), problems that arise aren't labeled with the skills and strategies required to solve them.

This makes it important to provide mixed practice opportunities so students are focused on asking themselves questions about what the problem is asking and what they are trying to find.

This type of variation not only supports a deeper level of engagement, it also supports the metacognitive strategies needed to analyze and develop a strategy to solve (Rohrer & Taylor, 2014).

The benefits of novelty in learning

A 2013 study also supports the importance of novelty in supporting reinforcement learning (aka review). The findings suggested that when task variation was provided for an already familiar skill, it offered the following benefits:

reduced errors due to lack of focus
helped learners maintain attention to task
motivated and engaged student

Using variety to build connections & deepen understanding

In addition, by providing variations in practice, we can also help learners understand the skills and strategies they are using on a deeper level.

When students who struggle with word problems are forced to apply their toolbox of strategies to novel problem formats, they begin to analyze and observe patterns in how problems are structured and the meaning they bring.

This requires much more engagement than being handed a sheet full of multiplication story problems, where students can pull the numbers and compute with little focus on understanding.

Designing word problems that incorporate variety & novelty

Don't be afraid to shake things up!

Giving students practice opportunities with different skills or problem formats mixed in is a great way to boost engagement and develop meta-cognitive skills.

Here are a few tips for trying it out in the classroom:

Change it up! Word problem practice doesn't have to match the day's math lesson.
Give opportunities to practice the same skill or strategy in via different formats.
Adjust the wording and/or topic in word problems to help students generalize skills.

Finding #4: Interest and emotion increase retention and skill development.

Attention and emotion are huge for learning. We've all seen it in our classroom.

Those magical lessons that hook learners are the ones that stick with them for years to come, but what does the research say?

build problem solving for students who struggle with word problems

The Science Behind Emotion & Learning

Neuroscientists have shown that emotions create connections among different sections of the brain (Immordino-Yang, 2016) . This supports long-term retrieval of the skills taught and a deeper connection to the learning.

This means if you can connect problem-solving with a scenario or a feeling, your students will be more likely to internalize the skills being practiced. Whether this is by “wowing” them with a little-known fact or solving real-world problems, the emotional trigger can be huge for learning.

What about incorporating student interests?

As for student interests, a long line of research supports the benefits of using these to increase educational outcomes and student motivation, including for students who struggle with word problems (Chen, 2001; Chen & Ennis, 2004; Solomon, 1996).

Connecting classwork with student interests has increased students' intentions to participate in future learning endeavors (Chen, 2001).

And interests don't just mean that love of Pokemon!

It means allowing social butterflies to work collaboratively. Providing students with opportunities to manipulate real objects or create models. Allowing kids to be authentic while digging in and developing the skills they need to master their learning objectives.

What this looks like in a math class

Evoke emotion and use student interests to engage the brain in deep, long-lasting learning whenever possible.

This will help with today's learning and promote long-term engagement, even when later practice might not be as interesting for students who struggle with word problems.

Here's how to start applying this research today:

Find word problems that match student interests.
Connect real-life situations and emotions to story problem practice.
Consider a weekly theme to connect practice throughout the week.

Finding #5: Student autonomy builds confidence & independence.

Autonomy is a student's ability to be in control of their learning. In other words, it is their ability to take ownership over the learning process and how they demonstrate mastery.

Why students need to control their learning

Research shows that providing students a sense of control and supporting their choices is way to help engage learners and build independent thinking. It also increased intrinsic motivation (Reeve, Nix, & Hamm, 2003).

However, this doesn't mean we just let kids learn independently. Clearly, some things require repeated guidance and modeling. Finding small ways that students can take control of the learning process is much better in these instances.

We know that giving at least partial autonomy has been linked to numerous positive student learning outcomes (Wielenga-Meijer, Taris, Widboldus, & Kompier, 2011).

But how can we foster this independence and autonomy, especially with those students who struggle to self-regulate behavior?

Fostering independence in students who struggle to stay on task

Well, the research says several conditions support building toward independence.

The first (and often neglected) is to explain unappealing choices and why they are one of the options.

When it comes to word problems, this might include explaining the rationale behind one of the strategies that appears to be a lot more work than the others.

It is also important to acknowledge students' negative feelings about a task or their ability to complete it. While we want them to be able to build independence, we don't want them to drown in overwhelm.

By providing emotional support, we can help determine whether a student is stuck with the learning or with the emotions from the cognitive challenge.

Finally, giving choices is recommended. Identifying choices you and your students who struggle with word problems can live with is an important step.

Whether this is working in partners, trying an alternative method, or skipping a problem and coming back, students need to feel like they have some ownership over the challenge they are working through.

By building in opportunities for autonomy, and choice, teachers help students build a sense of self-efficacy and confidence in their ability to be successful learners across various contexts (McCombs, 2002,2006).

We know this leads to numerous positive outcomes and has even been linked to drop-out prevention (Christenson & Thurlow, 2004).

Fostering autonomy in your classroom

You're not going to be able to hold their hands forever.

Giving opportunities to work through challenges independently and to feel ownership for their choices will help build both confidence and skills.

Here's how to get started letting go:

Give students time to tackle the problem independently (or in partners).
Don't get hyper-focused on a single method to solve – give opportunities to share & learn together.
Provide appropriate support (where needed) to build autonomy for all learners – like reading the problem orally.

Finding #6: Students need to be taught how to fail & recover from it.

Despite Ericsson's findings discussed early on in this post, talent does matter, and it is important to teach students to recover from failure because those are the moments when they learn the most.

A 2014 study by Brooke Macnamara analyzed 88 studies to determine how talent factored into deliberate practice.

Her findings show what we (as teachers) already know, students may require different amounts of practice to reach the same skill level…but how do we keep those struggling students from keeping up?

Failure Quote 1 build math problem solving skills

Growth mindset research gives us insight into ways to support students who struggle with word problems, encourage all students in math problem-solving, and harness the power of failure through “yet.”

You might not be able to do something yet, but if you keep trying, you will. This opens the door for multiple practice opportunities where students learn from each other.

And what about the advanced students?

Many of these students have not experienced failure, but they may have met their match when it comes to complex word problems.

To support these students, who may be experiencing their first true challenge, we need to have high standards and provide constructive, supportive feedback on how to grow.

Then we need to give them space to try again.

There is great power in allowing students to revise and try again, but our grading system often discourages being comfortable with failure.

Building the confidence to fail in your classroom

Many students feel the pressure always to have the right answer. Allowing students to fail safely means you can help them learn from these failures so they don't make the same mistake twice.

Here's how you can safely foster growth and build math problem solving skills through failure in your classroom:

Build in time to analyze errors & reflect.
Reward effort & growth as much as, if not more than, accuracy.
At least initially, skip the grading so students aren't afraid to be wrong.

Getting started with brain-based problem solving

The brain research is clear.

Spending 45 minutes focused on a sheet of word problems following the same format isn't the answer.

By implementing this research, you can save yourself time and the frustration from a disengaged class.

Based on this research, I've created Daily Problem Solving bundles to save you time and build math problem-solving skills. You can get each month separately or buy the full-year bundle at a major discount.

Currently, I offer these bundles for several grade levels, including:

Try Daily Problem Solving with your Learners

Of course, you do! Start working to build step-by-step math problem-solving skills today by clicking the button below to sign up for a free set of Daily Problem Solving.

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Engaging Maths

Dr catherine attard, promoting creative and critical thinking in mathematics and numeracy.

by cattard2017
Posted on June 25, 2017

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300 (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

higher-level thinking within authentic and meaningful contexts;
complex problem solving;
open-ended responses; and
substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

Number and algebra

The Number System and Place Value
Calculations and Numerical Methods
Fractions, Decimals, Percentages, Ratio and Proportion
Properties of Numbers
Patterns, Sequences and Structure
Algebraic expressions, equations and formulae
Coordinates, Functions and Graphs

Geometry and measure

Angles, Polygons, and Geometrical Proof
3D Geometry, Shape and Space
Measuring and calculating with units
Transformations and constructions
Pythagoras and Trigonometry
Vectors and Matrices

Probability and statistics

Handling, Processing and Representing Data
Probability

Working mathematically

Thinking mathematically
Mathematical mindsets
Cross-curricular contexts
Physical and digital manipulatives

For younger learners

Early Years Foundation Stage

Advanced mathematics

Decision Mathematics and Combinatorics
Advanced Probability and Statistics

Published 2018

The Problem-solving Classroom

Visualising
Working backwards
Reasoning logically
Conjecturing
Working systematically
Looking for patterns
Trial and improvement.

stage of the lesson
level of thinking
mathematical skill.
The length of student response increases (300-700%)
More responses are supported by logical argument.
An increased number of speculative responses.
The number of questions asked by students increases.
Student - student exchanges increase (volleyball).
Failures to respond decrease.
'Disciplinary moves' decrease.
The variety of students participating increases. As does the number of unsolicited, but appropriate contributions.
Student confidence increases.
conceptual understanding
procedural fluency
strategic competence
adaptive reasoning
productive disposition

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Mathematics

How to Improve Math Skills

Last Updated: March 29, 2024 Approved

This article was co-authored by Daron Cam and by wikiHow staff writer, Hannah Madden . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. wikiHow marks an article as reader-approved once it receives enough positive feedback. This article received 28 testimonials and 85% of readers who voted found it helpful, earning it our reader-approved status. This article has been viewed 410,334 times.

There’s no doubt about it: math is tough. As a result, a lot of kids (and adults!) struggle with math at some point in their lives. By building up your skills and practicing every day, you can make math a little less frustrating and have a higher chance of success. Use these tips and tricks during school, while you’re studying, and when you’re out and about to break down and complete math problems easily.

Play math games.

$Build up your skills while having some fun!$

DragonBox 5+ which lets you gradually build your algebra skills until you’re able to master more and more advanced equations.
Prodigy, a game targeted at elementary-school students, that integrates math practice into a role-playing game that allows players to use math to make their way through an appealing fantasy world.
Polyup, a calculator-based math game for more advanced high school and college students.

Practice math in everyday scenarios.

$Make math part of your daily life to practice it without even thinking.$

Or, if you plan to hike a new trail that’s 7 miles long and it takes you 20 minutes to walk a mile, how long should you plan for your hike to take? (2 hours and 36 minutes).

Use mental math if you can.

$Doing math in your head can help you remember key concepts.$

If you’re worried about your mental math skills, you can always double check your answer on your phone or computer.

Joseph Meyer

Develop your mental math skills. Mental math is when you perform mathematical calculations without using calculators, paper, or counting aids. Use your mind, memory, lessons, and discussions with your classmates to refine your math skills and build strong problem-solving strategies.

Review math concepts every day.

$Practice makes perfect, and math is no exception.$

Make note cards. Write out important concepts and formulas on note cards so that you can easily refer to them while doing problems and use them for study guides before exams.
Study in a quiet place. Distractions, whether aural or visual, will detract both from your ability to pay attention and to retain information.
Study when you’re alert and rested. Don’t try to force yourself to study late at night or when you’re sleep-deprived.

Show your work, not just your answers.

$Writing it all out can help you spot mistakes.$

Showing your work can also help you check your answers on homework and test problems.
Don’t solve math problems with a pen! Use a pencil so you can erase and correct mistakes if they happen.

Sketch out word problems to give yourself a visual.

$Word problems are usually tougher than straight math problems.$

For example, a problem might say, “If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?” You could draw 2 squares to represent the bags, then fill in 4 circles split between them to represent the candy.

Practice with example problems.

$You can find tons of example problems online.$

Your teacher might also be able to give you some extra example problems if you ask for them.
Using example problems is a great way to practice for a test.

Look up lessons online.

$Sometimes you need a little more explanation to really understand.$

PatrickJMT on YouTube, a college math professor
Khan Academy, a website with video lessons and interactive study guides
Breaking Math, a podcast for math concepts

Master one concept before moving onto the next.

$In math, each topic builds upon the last one.$

Teach math problem or concept to someone else.

$Learning by teaching someone is a great way to solidify concepts.$

Have your friend or family member ask you questions, too. Try to answer them as best you can to really practice.

Expert Q&A

Reader Videos

Try not to fall behind in your homework or schoolwork. The more you keep up in class, the easier it will be. Thanks Helpful 4 Not Helpful 0

$Improve Your Math Grade$

↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
↑ http://www.schoolfamily.com/school-family-articles/article/10785-mastering-math
↑ https://www.edutopia.org/article/5-tips-improving-students-success-math
↑ https://math.osu.edu/undergrad/non-majors/resources/study-math-college
↑ https://www.youtube.com/watch?t=96&v=aIRh_15O2S0&feature=youtu.be
↑ https://www.mathgoodies.com/articles/improve_your_grades

About This Article

To improve your math skills, start by taking good notes in class and asking lots of questions to understand the material. Then, schedule time each day to study from your notes and do your homework. When you study, do practice problems to cement your comprehension of the math. In addition to studying, try playing math games online, such as DragonBox 5+ or Prodigy, which will help hone your math skills in a fun way. For ways to incorporate math into your everyday life, read on! Did this summary help you? Yes No

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Please note you do not have access to teaching notes, developing problem-solving skills in mathematics: a lesson study.

International Journal for Lesson and Learning Studies

ISSN : 2046-8253

Article publication date: 3 January 2017

Problem solving is a skill in mathematics which although always relevant has heightened priority due to the changes in the new mathematics GCSE (Department for Education, 2013). It has previously been a skill which is deemed underdeveloped within mathematics and therefore is a theme which teachers are seeking to improve and nurture in order to align with the new changes. The GCSE is the formal qualification that students take at the end of Key Stage 4 (KS4) in the UK. The paper aims to discuss these issues.

Design/methodology/approach

The focus of the enquiry was to explore, using lesson studies, the differences in students’ approaches to problem solving. Consequently, key themes relating to the mediation of gender, ability, and academic motivation surfaced. Considering these themes, the paper subsequently reflects upon pedagogical practices which might effectively develop students’ ability to problem solve. The study took part in a mixed gender comprehensive secondary school with students taking part in the observation lesson ranging in age from 11 to 12 years old. The authors are the teachers who took part in the lesson study. The teachers implemented observation techniques in the form of video and peer observation with the accompanying teacher. In addition, students provided feedback on how they approached the problem-solving tasks through a form of semi-structured interviews, conducted via the use of video diaries where no teachers were present to prevent power bias. Following this, a thematic analysis of both the observations and student video diaries generated conclusions regarding how said key themes shaped the students’ approaches to problem solving.

Students’ frustration and competitive need to find a specific answer inhibited their ability to thoroughly explore the problem posed thus overseeing vital aspects needed to solve the problem set. Many students expressed a passion for problem solving due to its freedom and un-rigid nature, which is something teachers should nurture.

Originality/value

Generally, teachers are led by a culture in which attainment is the key. However, an atmosphere should be developed where the answer is not the key and students can explore the vibrant diversity mathematics and problem solving can offer.

Lesson study
Mathematics
Problem-solving skills
Video diaries

Bradshaw, Z. and Hazell, A. (2017), "Developing problem-solving skills in mathematics: a lesson study", International Journal for Lesson and Learning Studies , Vol. 6 No. 1, pp. 32-44. https://doi.org/10.1108/IJLLS-09-2016-0032

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How to improve your problem solving skills and build effective problem solving strategies

Design your next session with SessionLab

Join the 150,000+ facilitators  using SessionLab.

What is a problem solving process?

What are the problem solving steps I need to follow?

Problem solving strategies

What skills do i need to be an effective problem solver, how can i improve my problem solving skills.

Solving problems is like baking a cake. You can go straight into the kitchen without a recipe or the right ingredients and do your best, but the end result is unlikely to be very tasty!

Using a process to bake a cake allows you to use the best ingredients without waste, collect the right tools, account for allergies, decide whether it is a birthday or wedding cake, and then bake efficiently and on time. The result is a better cake that is fit for purpose, tastes better and has created less mess in the kitchen. Also, it should have chocolate sprinkles. Having a step by step process to solve organizational problems allows you to go through each stage methodically and ensure you are trying to solve the right problems and select the most appropriate, effective solutions.

What are the problem solving steps I need to follow?

All problem solving processes go through a number of steps in order to move from identifying a problem to resolving it.

Depending on your problem solving model and who you ask, there can be anything between four and nine problem solving steps you should follow in order to find the right solution. Whatever framework you and your group use, there are some key items that should be addressed in order to have an effective process.

We’ve looked at problem solving processes from sources such as the American Society for Quality and their four step approach , and Mediate ‘s six step process. By reflecting on those and our own problem solving processes, we’ve come up with a sequence of seven problem solving steps we feel best covers everything you need in order to effectively solve problems.

1. Problem identification

The first stage of any problem solving process is to identify the problem or problems you might want to solve. Effective problem solving strategies always begin by allowing a group scope to articulate what they believe the problem to be and then coming to some consensus over which problem they approach first. Problem solving activities used at this stage often have a focus on creating frank, open discussion so that potential problems can be brought to the surface.

2. Problem analysis

Though this step is not a million miles from problem identification, problem analysis deserves to be considered separately. It can often be an overlooked part of the process and is instrumental when it comes to developing effective solutions.

The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem . As part of this stage, you may look deeper and try to find the root cause of a specific problem at a team or organizational level.

Remember that problem solving strategies should not only be focused on putting out fires in the short term but developing long term solutions that deal with the root cause of organizational challenges.

Whatever your approach, analyzing a problem is crucial in being able to select an appropriate solution and the problem solving skills deployed in this stage are beneficial for the rest of the process and ensuring the solutions you create are fit for purpose.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or problem solving activities designed to produce working prototypes of possible solutions.

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold.

4. Solution development

No solution is likely to be perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically.

During this stage, you will often ask your team to iterate and improve upon your frontrunning solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose.

5. Decision making

Nearly there! Once your group has reached consensus and selected a solution that applies to the problem at hand you have some decisions to make. You will want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

The decision making stage is a part of the problem solving process that can get missed or taken as for granted. Fail to properly allocate roles and plan out how a solution will actually be implemented and it less likely to be successful in solving the problem.

Have clear accountabilities, actions, timeframes, and follow-ups. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group.

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved.

6. Solution implementation

This is what we were waiting for! All problem solving strategies have the end goal of implementing a solution and solving a problem in mind.

Remember that in order for any solution to be successful, you need to help your group through all of the previous problem solving steps thoughtfully. Only then can you ensure that you are solving the right problem but also that you have developed the correct solution and can then successfully implement and measure the impact of that solution.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way.

7. Solution evaluation

So you and your team developed a great solution to a problem and have a gut feeling its been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback. You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives.

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization.

It’s worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time. Data and insight is invaluable at every stage of the problem solving process and this one is no different.

Problem solving workshops made easy

Problem solving strategies are methods of approaching and facilitating the process of problem-solving with a set of techniques , actions, and processes. Different strategies are more effective if you are trying to solve broad problems such as achieving higher growth versus more focused problems like, how do we improve our customer onboarding process?

Broadly, the problem solving steps outlined above should be included in any problem solving strategy though choosing where to focus your time and what approaches should be taken is where they begin to differ. You might find that some strategies ask for the problem identification to be done prior to the session or that everything happens in the course of a one day workshop.

The key similarity is that all good problem solving strategies are structured and designed. Four hours of open discussion is never going to be as productive as a four-hour workshop designed to lead a group through a problem solving process.

Good problem solving strategies are tailored to the team, organization and problem you will be attempting to solve. Here are some example problem solving strategies you can learn from or use to get started.

Use a workshop to lead a team through a group process

Often, the first step to solving problems or organizational challenges is bringing a group together effectively. Most teams have the tools, knowledge, and expertise necessary to solve their challenges – they just need some guidance in how to use leverage those skills and a structure and format that allows people to focus their energies.

Facilitated workshops are one of the most effective ways of solving problems of any scale. By designing and planning your workshop carefully, you can tailor the approach and scope to best fit the needs of your team and organization.

Problem solving workshop

Creating a bespoke, tailored process
Tackling problems of any size
Building in-house workshop ability and encouraging their use

Workshops are an effective strategy for solving problems. By using tried and test facilitation techniques and methods, you can design and deliver a workshop that is perfectly suited to the unique variables of your organization. You may only have the capacity for a half-day workshop and so need a problem solving process to match.

By using our session planner tool and importing methods from our library of 700+ facilitation techniques, you can create the right problem solving workshop for your team. It might be that you want to encourage creative thinking or look at things from a new angle to unblock your groups approach to problem solving. By tailoring your workshop design to the purpose, you can help ensure great results.

One of the main benefits of a workshop is the structured approach to problem solving. Not only does this mean that the workshop itself will be successful, but many of the methods and techniques will help your team improve their working processes outside of the workshop.

We believe that workshops are one of the best tools you can use to improve the way your team works together. Start with a problem solving workshop and then see what team building, culture or design workshops can do for your organization!

Run a design sprint

Great for:

aligning large, multi-discipline teams
quickly designing and testing solutions
tackling large, complex organizational challenges and breaking them down into smaller tasks

By using design thinking principles and methods, a design sprint is a great way of identifying, prioritizing and prototyping solutions to long term challenges that can help solve major organizational problems with quick action and measurable results.

Some familiarity with design thinking is useful, though not integral, and this strategy can really help a team align if there is some discussion around which problems should be approached first.

The stage-based structure of the design sprint is also very useful for teams new to design thinking. The inspiration phase, where you look to competitors that have solved your problem, and the rapid prototyping and testing phases are great for introducing new concepts that will benefit a team in all their future work.

It can be common for teams to look inward for solutions and so looking to the market for solutions you can iterate on can be very productive. Instilling an agile prototyping and testing mindset can also be great when helping teams move forwards – generating and testing solutions quickly can help save time in the long run and is also pretty exciting!

Break problems down into smaller issues

Organizational challenges and problems are often complicated and large scale in nature. Sometimes, trying to resolve such an issue in one swoop is simply unachievable or overwhelming. Try breaking down such problems into smaller issues that you can work on step by step. You may not be able to solve the problem of churning customers off the bat, but you can work with your team to identify smaller effort but high impact elements and work on those first.

This problem solving strategy can help a team generate momentum, prioritize and get some easy wins. It’s also a great strategy to employ with teams who are just beginning to learn how to approach the problem solving process. If you want some insight into a way to employ this strategy, we recommend looking at our design sprint template below!

Use guiding frameworks or try new methodologies

Some problems are best solved by introducing a major shift in perspective or by using new methodologies that encourage your team to think differently.

Props and tools such as Methodkit , which uses a card-based toolkit for facilitation, or Lego Serious Play can be great ways to engage your team and find an inclusive, democratic problem solving strategy. Remember that play and creativity are great tools for achieving change and whatever the challenge, engaging your participants can be very effective where other strategies may have failed.

LEGO Serious Play

Improving core problem solving skills
Thinking outside of the box
Encouraging creative solutions

LEGO Serious Play is a problem solving methodology designed to get participants thinking differently by using 3D models and kinesthetic learning styles. By physically building LEGO models based on questions and exercises, participants are encouraged to think outside of the box and create their own responses.

Collaborate LEGO Serious Play exercises are also used to encourage communication and build problem solving skills in a group. By using this problem solving process, you can often help different kinds of learners and personality types contribute and unblock organizational problems with creative thinking.

Problem solving strategies like LEGO Serious Play are super effective at helping a team solve more skills-based problems such as communication between teams or a lack of creative thinking. Some problems are not suited to LEGO Serious Play and require a different problem solving strategy.

Card Decks and Method Kits

New facilitators or non-facilitators
Approaching difficult subjects with a simple, creative framework
Engaging those with varied learning styles

Card decks and method kids are great tools for those new to facilitation or for whom facilitation is not the primary role. Card decks such as the emotional culture deck can be used for complete workshops and in many cases, can be used right out of the box. Methodkit has a variety of kits designed for scenarios ranging from personal development through to personas and global challenges so you can find the right deck for your particular needs.

Having an easy to use framework that encourages creativity or a new approach can take some of the friction or planning difficulties out of the workshop process and energize a team in any setting. Simplicity is the key with these methods. By ensuring everyone on your team can get involved and engage with the process as quickly as possible can really contribute to the success of your problem solving strategy.

Source external advice

Looking to peers, experts and external facilitators can be a great way of approaching the problem solving process. Your team may not have the necessary expertise, insights of experience to tackle some issues, or you might simply benefit from a fresh perspective. Some problems may require bringing together an entire team, and coaching managers or team members individually might be the right approach. Remember that not all problems are best resolved in the same manner.

If you’re a solo entrepreneur, peer groups, coaches and mentors can also be invaluable at not only solving specific business problems, but in providing a support network for resolving future challenges. One great approach is to join a Mastermind Group and link up with like-minded individuals and all grow together. Remember that however you approach the sourcing of external advice, do so thoughtfully, respectfully and honestly. Reciprocate where you can and prepare to be surprised by just how kind and helpful your peers can be!

Mastermind Group

Solo entrepreneurs or small teams with low capacity
Peer learning and gaining outside expertise
Getting multiple external points of view quickly

Problem solving in large organizations with lots of skilled team members is one thing, but how about if you work for yourself or in a very small team without the capacity to get the most from a design sprint or LEGO Serious Play session?

A mastermind group – sometimes known as a peer advisory board – is where a group of people come together to support one another in their own goals, challenges, and businesses. Each participant comes to the group with their own purpose and the other members of the group will help them create solutions, brainstorm ideas, and support one another.

Mastermind groups are very effective in creating an energized, supportive atmosphere that can deliver meaningful results. Learning from peers from outside of your organization or industry can really help unlock new ways of thinking and drive growth. Access to the experience and skills of your peers can be invaluable in helping fill the gaps in your own ability, particularly in young companies.

A mastermind group is a great solution for solo entrepreneurs, small teams, or for organizations that feel that external expertise or fresh perspectives will be beneficial for them. It is worth noting that Mastermind groups are often only as good as the participants and what they can bring to the group. Participants need to be committed, engaged and understand how to work in this context.

Coaching and mentoring

Focused learning and development
Filling skills gaps
Working on a range of challenges over time

Receiving advice from a business coach or building a mentor/mentee relationship can be an effective way of resolving certain challenges. The one-to-one format of most coaching and mentor relationships can really help solve the challenges those individuals are having and benefit the organization as a result.

A great mentor can be invaluable when it comes to spotting potential problems before they arise and coming to understand a mentee very well has a host of other business benefits. You might run an internal mentorship program to help develop your team’s problem solving skills and strategies or as part of a large learning and development program. External coaches can also be an important part of your problem solving strategy, filling skills gaps for your management team or helping with specific business issues.

Now we’ve explored the problem solving process and the steps you will want to go through in order to have an effective session, let’s look at the skills you and your team need to be more effective problem solvers.

Problem solving skills are highly sought after, whatever industry or team you work in. Organizations are keen to employ people who are able to approach problems thoughtfully and find strong, realistic solutions. Whether you are a facilitator , a team leader or a developer, being an effective problem solver is a skill you’ll want to develop.

Problem solving skills form a whole suite of techniques and approaches that an individual uses to not only identify problems but to discuss them productively before then developing appropriate solutions.

Here are some of the most important problem solving skills everyone from executives to junior staff members should learn. We’ve also included an activity or exercise from the SessionLab library that can help you and your team develop that skill.

If you’re running a workshop or training session to try and improve problem solving skills in your team, try using these methods to supercharge your process!

Active listening

Active listening is one of the most important skills anyone who works with people can possess. In short, active listening is a technique used to not only better understand what is being said by an individual, but also to be more aware of the underlying message the speaker is trying to convey. When it comes to problem solving, active listening is integral for understanding the position of every participant and to clarify the challenges, ideas and solutions they bring to the table.

Some active listening skills include:

Paying complete attention to the speaker.
Removing distractions.
Avoid interruption.
Taking the time to fully understand before preparing a rebuttal.
Responding respectfully and appropriately.
Demonstrate attentiveness and positivity with an open posture, making eye contact with the speaker, smiling and nodding if appropriate. Show that you are listening and encourage them to continue.
Be aware of and respectful of feelings. Judge the situation and respond appropriately. You can disagree without being disrespectful.
Observe body language.
Paraphrase what was said in your own words, either mentally or verbally.
Remain neutral.
Reflect and take a moment before responding.
Ask deeper questions based on what is said and clarify points where necessary.

Active Listening #hyperisland #skills #active listening #remote-friendly This activity supports participants to reflect on a question and generate their own solutions using simple principles of active listening and peer coaching. It’s an excellent introduction to active listening but can also be used with groups that are already familiar with it. Participants work in groups of three and take turns being: “the subject”, the listener, and the observer.

Analytical skills

All problem solving models require strong analytical skills, particularly during the beginning of the process and when it comes to analyzing how solutions have performed.

Analytical skills are primarily focused on performing an effective analysis by collecting, studying and parsing data related to a problem or opportunity.

It often involves spotting patterns, being able to see things from different perspectives and using observable facts and data to make suggestions or produce insight.

Analytical skills are also important at every stage of the problem solving process and by having these skills, you can ensure that any ideas or solutions you create or backed up analytically and have been sufficiently thought out.

Nine Whys #innovation #issue analysis #liberating structures With breathtaking simplicity, you can rapidly clarify for individuals and a group what is essentially important in their work. You can quickly reveal when a compelling purpose is missing in a gathering and avoid moving forward without clarity. When a group discovers an unambiguous shared purpose, more freedom and more responsibility are unleashed. You have laid the foundation for spreading and scaling innovations with fidelity.

Collaboration

Trying to solve problems on your own is difficult. Being able to collaborate effectively, with a free exchange of ideas, to delegate and be a productive member of a team is hugely important to all problem solving strategies.

Remember that whatever your role, collaboration is integral, and in a problem solving process, you are all working together to find the best solution for everyone.

Marshmallow challenge with debriefing #teamwork #team #leadership #collaboration In eighteen minutes, teams must build the tallest free-standing structure out of 20 sticks of spaghetti, one yard of tape, one yard of string, and one marshmallow. The marshmallow needs to be on top. The Marshmallow Challenge was developed by Tom Wujec, who has done the activity with hundreds of groups around the world. Visit the Marshmallow Challenge website for more information. This version has an extra debriefing question added with sample questions focusing on roles within the team.

Communication

Being an effective communicator means being empathetic, clear and succinct, asking the right questions, and demonstrating active listening skills throughout any discussion or meeting.

In a problem solving setting, you need to communicate well in order to progress through each stage of the process effectively. As a team leader, it may also fall to you to facilitate communication between parties who may not see eye to eye. Effective communication also means helping others to express themselves and be heard in a group.

Bus Trip #feedback #communication #appreciation #closing #thiagi #team This is one of my favourite feedback games. I use Bus Trip at the end of a training session or a meeting, and I use it all the time. The game creates a massive amount of energy with lots of smiles, laughs, and sometimes even a teardrop or two.

Creative problem solving skills can be some of the best tools in your arsenal. Thinking creatively, being able to generate lots of ideas and come up with out of the box solutions is useful at every step of the process.

The kinds of problems you will likely discuss in a problem solving workshop are often difficult to solve, and by approaching things in a fresh, creative manner, you can often create more innovative solutions.

Having practical creative skills is also a boon when it comes to problem solving. If you can help create quality design sketches and prototypes in record time, it can help bring a team to alignment more quickly or provide a base for further iteration.

The paper clip method #sharing #creativity #warm up #idea generation #brainstorming The power of brainstorming. A training for project leaders, creativity training, and to catalyse getting new solutions.

Critical thinking

Critical thinking is one of the fundamental problem solving skills you’ll want to develop when working on developing solutions. Critical thinking is the ability to analyze, rationalize and evaluate while being aware of personal bias, outlying factors and remaining open-minded.

Defining and analyzing problems without deploying critical thinking skills can mean you and your team go down the wrong path. Developing solutions to complex issues requires critical thinking too – ensuring your team considers all possibilities and rationally evaluating them.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Data analysis

Though it shares lots of space with general analytical skills, data analysis skills are something you want to cultivate in their own right in order to be an effective problem solver.

Being good at data analysis doesn’t just mean being able to find insights from data, but also selecting the appropriate data for a given issue, interpreting it effectively and knowing how to model and present that data. Depending on the problem at hand, it might also include a working knowledge of specific data analysis tools and procedures.

Having a solid grasp of data analysis techniques is useful if you’re leading a problem solving workshop but if you’re not an expert, don’t worry. Bring people into the group who has this skill set and help your team be more effective as a result.

Decision making

All problems need a solution and all solutions require that someone make the decision to implement them. Without strong decision making skills, teams can become bogged down in discussion and less effective as a result.

Making decisions is a key part of the problem solving process. It’s important to remember that decision making is not restricted to the leadership team. Every staff member makes decisions every day and developing these skills ensures that your team is able to solve problems at any scale. Remember that making decisions does not mean leaping to the first solution but weighing up the options and coming to an informed, well thought out solution to any given problem that works for the whole team.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

Dependability

Most complex organizational problems require multiple people to be involved in delivering the solution. Ensuring that the team and organization can depend on you to take the necessary actions and communicate where necessary is key to ensuring problems are solved effectively.

Being dependable also means working to deadlines and to brief. It is often a matter of creating trust in a team so that everyone can depend on one another to complete the agreed actions in the agreed time frame so that the team can move forward together. Being undependable can create problems of friction and can limit the effectiveness of your solutions so be sure to bear this in mind throughout a project.

Team Purpose & Culture #team #hyperisland #culture #remote-friendly This is an essential process designed to help teams define their purpose (why they exist) and their culture (how they work together to achieve that purpose). Defining these two things will help any team to be more focused and aligned. With support of tangible examples from other companies, the team members work as individuals and a group to codify the way they work together. The goal is a visual manifestation of both the purpose and culture that can be put up in the team’s work space.

Emotional intelligence

Emotional intelligence is an important skill for any successful team member, whether communicating internally or with clients or users. In the problem solving process, emotional intelligence means being attuned to how people are feeling and thinking, communicating effectively and being self-aware of what you bring to a room.

There are often differences of opinion when working through problem solving processes, and it can be easy to let things become impassioned or combative. Developing your emotional intelligence means being empathetic to your colleagues and managing your own emotions throughout the problem and solution process. Be kind, be thoughtful and put your points across care and attention.

Being emotionally intelligent is a skill for life and by deploying it at work, you can not only work efficiently but empathetically. Check out the emotional culture workshop template for more!

Facilitation

As we’ve clarified in our facilitation skills post, facilitation is the art of leading people through processes towards agreed-upon objectives in a manner that encourages participation, ownership, and creativity by all those involved. While facilitation is a set of interrelated skills in itself, the broad definition of facilitation can be invaluable when it comes to problem solving. Leading a team through a problem solving process is made more effective if you improve and utilize facilitation skills – whether you’re a manager, team leader or external stakeholder.

The Six Thinking Hats #creative thinking #meeting facilitation #problem solving #issue resolution #idea generation #conflict resolution The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Flexibility

Being flexible is a vital skill when it comes to problem solving. This does not mean immediately bowing to pressure or changing your opinion quickly: instead, being flexible is all about seeing things from new perspectives, receiving new information and factoring it into your thought process.

Flexibility is also important when it comes to rolling out solutions. It might be that other organizational projects have greater priority or require the same resources as your chosen solution. Being flexible means understanding needs and challenges across the team and being open to shifting or arranging your own schedule as necessary. Again, this does not mean immediately making way for other projects. It’s about articulating your own needs, understanding the needs of others and being able to come to a meaningful compromise.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Working in any group can lead to unconscious elements of groupthink or situations in which you may not wish to be entirely honest. Disagreeing with the opinions of the executive team or wishing to save the feelings of a coworker can be tricky to navigate, but being honest is absolutely vital when to comes to developing effective solutions and ensuring your voice is heard.

Remember that being honest does not mean being brutally candid. You can deliver your honest feedback and opinions thoughtfully and without creating friction by using other skills such as emotional intelligence.

Explore your Values #hyperisland #skills #values #remote-friendly Your Values is an exercise for participants to explore what their most important values are. It’s done in an intuitive and rapid way to encourage participants to follow their intuitive feeling rather than over-thinking and finding the “correct” values. It is a good exercise to use to initiate reflection and dialogue around personal values.

Initiative

The problem solving process is multi-faceted and requires different approaches at certain points of the process. Taking initiative to bring problems to the attention of the team, collect data or lead the solution creating process is always valuable. You might even roadtest your own small scale solutions or brainstorm before a session. Taking initiative is particularly effective if you have good deal of knowledge in that area or have ownership of a particular project and want to get things kickstarted.

That said, be sure to remember to honor the process and work in service of the team. If you are asked to own one part of the problem solving process and you don’t complete that task because your initiative leads you to work on something else, that’s not an effective method of solving business challenges.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Impartiality

A particularly useful problem solving skill for product owners or managers is the ability to remain impartial throughout much of the process. In practice, this means treating all points of view and ideas brought forward in a meeting equally and ensuring that your own areas of interest or ownership are not favored over others.

There may be a stage in the process where a decision maker has to weigh the cost and ROI of possible solutions against the company roadmap though even then, ensuring that the decision made is based on merit and not personal opinion.

Empathy map #frame insights #create #design #issue analysis An empathy map is a tool to help a design team to empathize with the people they are designing for. You can make an empathy map for a group of people or for a persona. To be used after doing personas when more insights are needed.

Being a good leader means getting a team aligned, energized and focused around a common goal. In the problem solving process, strong leadership helps ensure that the process is efficient, that any conflicts are resolved and that a team is managed in the direction of success.

It’s common for managers or executives to assume this role in a problem solving workshop, though it’s important that the leader maintains impartiality and does not bulldoze the group in a particular direction. Remember that good leadership means working in service of the purpose and team and ensuring the workshop is a safe space for employees of any level to contribute. Take a look at our leadership games and activities post for more exercises and methods to help improve leadership in your organization.

Leadership Pizza #leadership #team #remote-friendly This leadership development activity offers a self-assessment framework for people to first identify what skills, attributes and attitudes they find important for effective leadership, and then assess their own development and initiate goal setting.

In the context of problem solving, mediation is important in keeping a team engaged, happy and free of conflict. When leading or facilitating a problem solving workshop, you are likely to run into differences of opinion. Depending on the nature of the problem, certain issues may be brought up that are emotive in nature.

Being an effective mediator means helping those people on either side of such a divide are heard, listen to one another and encouraged to find common ground and a resolution. Mediating skills are useful for leaders and managers in many situations and the problem solving process is no different.

Conflict Responses #hyperisland #team #issue resolution A workshop for a team to reflect on past conflicts, and use them to generate guidelines for effective conflict handling. The workshop uses the Thomas-Killman model of conflict responses to frame a reflective discussion. Use it to open up a discussion around conflict with a team.

Planning

Solving organizational problems is much more effective when following a process or problem solving model. Planning skills are vital in order to structure, deliver and follow-through on a problem solving workshop and ensure your solutions are intelligently deployed.

Planning skills include the ability to organize tasks and a team, plan and design the process and take into account any potential challenges. Taking the time to plan carefully can save time and frustration later in the process and is valuable for ensuring a team is positioned for success.

3 Action Steps #hyperisland #action #remote-friendly This is a small-scale strategic planning session that helps groups and individuals to take action toward a desired change. It is often used at the end of a workshop or programme. The group discusses and agrees on a vision, then creates some action steps that will lead them towards that vision. The scope of the challenge is also defined, through discussion of the helpful and harmful factors influencing the group.

Prioritization

As organisations grow, the scale and variation of problems they face multiplies. Your team or is likely to face numerous challenges in different areas and so having the skills to analyze and prioritize becomes very important, particularly for those in leadership roles.

A thorough problem solving process is likely to deliver multiple solutions and you may have several different problems you wish to solve simultaneously. Prioritization is the ability to measure the importance, value, and effectiveness of those possible solutions and choose which to enact and in what order. The process of prioritization is integral in ensuring the biggest challenges are addressed with the most impactful solutions.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

Project management

Some problem solving skills are utilized in a workshop or ideation phases, while others come in useful when it comes to decision making. Overseeing an entire problem solving process and ensuring its success requires strong project management skills.

While project management incorporates many of the other skills listed here, it is important to note the distinction of considering all of the factors of a project and managing them successfully. Being able to negotiate with stakeholders, manage tasks, time and people, consider costs and ROI, and tie everything together is massively helpful when going through the problem solving process.

Record keeping

Working out meaningful solutions to organizational challenges is only one part of the process. Thoughtfully documenting and keeping records of each problem solving step for future consultation is important in ensuring efficiency and meaningful change.

For example, some problems may be lower priority than others but can be revisited in the future. If the team has ideated on solutions and found some are not up to the task, record those so you can rule them out and avoiding repeating work. Keeping records of the process also helps you improve and refine your problem solving model next time around!

Personal Kanban #gamestorming #action #agile #project planning Personal Kanban is a tool for organizing your work to be more efficient and productive. It is based on agile methods and principles.

Research skills

Conducting research to support both the identification of problems and the development of appropriate solutions is important for an effective process. Knowing where to go to collect research, how to conduct research efficiently, and identifying pieces of research are relevant are all things a good researcher can do well.

In larger groups, not everyone has to demonstrate this ability in order for a problem solving workshop to be effective. That said, having people with research skills involved in the process, particularly if they have existing area knowledge, can help ensure the solutions that are developed with data that supports their intention. Remember that being able to deliver the results of research efficiently and in a way the team can easily understand is also important. The best data in the world is only as effective as how it is delivered and interpreted.

Customer experience map #ideation #concepts #research #design #issue analysis #remote-friendly Customer experience mapping is a method of documenting and visualizing the experience a customer has as they use the product or service. It also maps out their responses to their experiences. To be used when there is a solution (even in a conceptual stage) that can be analyzed.

Risk management

Managing risk is an often overlooked part of the problem solving process. Solutions are often developed with the intention of reducing exposure to risk or solving issues that create risk but sometimes, great solutions are more experimental in nature and as such, deploying them needs to be carefully considered.

Managing risk means acknowledging that there may be risks associated with more out of the box solutions or trying new things, but that this must be measured against the possible benefits and other organizational factors.

Be informed, get the right data and stakeholders in the room and you can appropriately factor risk into your decision making process.

Decisions, Decisions… #communication #decision making #thiagi #action #issue analysis When it comes to decision-making, why are some of us more prone to take risks while others are risk-averse? One explanation might be the way the decision and options were presented. This exercise, based on Kahneman and Tversky’s classic study , illustrates how the framing effect influences our judgement and our ability to make decisions . The participants are divided into two groups. Both groups are presented with the same problem and two alternative programs for solving them. The two programs both have the same consequences but are presented differently. The debriefing discussion examines how the framing of the program impacted the participant’s decision.

Team-building

No single person is as good at problem solving as a team. Building an effective team and helping them come together around a common purpose is one of the most important problem solving skills, doubly so for leaders. By bringing a team together and helping them work efficiently, you pave the way for team ownership of a problem and the development of effective solutions.

In a problem solving workshop, it can be tempting to jump right into the deep end, though taking the time to break the ice, energize the team and align them with a game or exercise will pay off over the course of the day.

Remember that you will likely go through the problem solving process multiple times over an organization’s lifespan and building a strong team culture will make future problem solving more effective. It’s also great to work with people you know, trust and have fun with. Working on team building in and out of the problem solving process is a hallmark of successful teams that can work together to solve business problems.

9 Dimensions Team Building Activity #ice breaker #teambuilding #team #remote-friendly 9 Dimensions is a powerful activity designed to build relationships and trust among team members. There are 2 variations of this icebreaker. The first version is for teams who want to get to know each other better. The second version is for teams who want to explore how they are working together as a team.

Time management

The problem solving process is designed to lead a team from identifying a problem through to delivering a solution and evaluating its effectiveness. Without effective time management skills or timeboxing of tasks, it can be easy for a team to get bogged down or be inefficient.

By using a problem solving model and carefully designing your workshop, you can allocate time efficiently and trust that the process will deliver the results you need in a good timeframe.

Time management also comes into play when it comes to rolling out solutions, particularly those that are experimental in nature. Having a clear timeframe for implementing and evaluating solutions is vital for ensuring their success and being able to pivot if necessary.

Improving your skills at problem solving is often a career-long pursuit though there are methods you can use to make the learning process more efficient and to supercharge your problem solving skillset.

Remember that the skills you need to be a great problem solver have a large overlap with those skills you need to be effective in any role. Investing time and effort to develop your active listening or critical thinking skills is valuable in any context. Here are 7 ways to improve your problem solving skills.

Share best practices

Remember that your team is an excellent source of skills, wisdom, and techniques and that you should all take advantage of one another where possible. Best practices that one team has for solving problems, conducting research or making decisions should be shared across the organization. If you have in-house staff that have done active listening training or are data analysis pros, have them lead a training session.

Your team is one of your best resources. Create space and internal processes for the sharing of skills so that you can all grow together.

Ask for help and attend training

Once you’ve figured out you have a skills gap, the next step is to take action to fill that skills gap. That might be by asking your superior for training or coaching, or liaising with team members with that skill set. You might even attend specialized training for certain skills – active listening or critical thinking, for example, are business-critical skills that are regularly offered as part of a training scheme.

Whatever method you choose, remember that taking action of some description is necessary for growth. Whether that means practicing, getting help, attending training or doing some background reading, taking active steps to improve your skills is the way to go.

Learn a process

Problem solving can be complicated, particularly when attempting to solve large problems for the first time. Using a problem solving process helps give structure to your problem solving efforts and focus on creating outcomes, rather than worrying about the format.

Tools such as the seven-step problem solving process above are effective because not only do they feature steps that will help a team solve problems, they also develop skills along the way. Each step asks for people to engage with the process using different skills and in doing so, helps the team learn and grow together. Group processes of varying complexity and purpose can also be found in the SessionLab library of facilitation techniques . Using a tried and tested process and really help ease the learning curve for both those leading such a process, as well as those undergoing the purpose.

Effective teams make decisions about where they should and shouldn’t expend additional effort. By using a problem solving process, you can focus on the things that matter, rather than stumbling towards a solution haphazardly.

Create a feedback loop

Some skills gaps are more obvious than others. It’s possible that your perception of your active listening skills differs from those of your colleagues.

It’s valuable to create a system where team members can provide feedback in an ordered and friendly manner so they can all learn from one another. Only by identifying areas of improvement can you then work to improve them.

Remember that feedback systems require oversight and consideration so that they don’t turn into a place to complain about colleagues. Design the system intelligently so that you encourage the creation of learning opportunities, rather than encouraging people to list their pet peeves.

While practice might not make perfect, it does make the problem solving process easier. If you are having trouble with critical thinking, don’t shy away from doing it. Get involved where you can and stretch those muscles as regularly as possible.

Problem solving skills come more naturally to some than to others and that’s okay. Take opportunities to get involved and see where you can practice your skills in situations outside of a workshop context. Try collaborating in other circumstances at work or conduct data analysis on your own projects. You can often develop those skills you need for problem solving simply by doing them. Get involved!

Use expert exercises and methods

Learn from the best. Our library of 700+ facilitation techniques is full of activities and methods that help develop the skills you need to be an effective problem solver. Check out our templates to see how to approach problem solving and other organizational challenges in a structured and intelligent manner.

There is no single approach to improving problem solving skills, but by using the techniques employed by others you can learn from their example and develop processes that have seen proven results.

Try new ways of thinking and change your mindset

Using tried and tested exercises that you know well can help deliver results, but you do run the risk of missing out on the learning opportunities offered by new approaches. As with the problem solving process, changing your mindset can remove blockages and be used to develop your problem solving skills.

Most teams have members with mixed skill sets and specialties. Mix people from different teams and share skills and different points of view. Teach your customer support team how to use design thinking methods or help your developers with conflict resolution techniques. Try switching perspectives with facilitation techniques like Flip It! or by using new problem solving methodologies or models. Give design thinking, liberating structures or lego serious play a try if you want to try a new approach. You will find that framing problems in new ways and using existing skills in new contexts can be hugely useful for personal development and improving your skillset. It’s also a lot of fun to try new things. Give it a go!

Encountering business challenges and needing to find appropriate solutions is not unique to your organization. Lots of very smart people have developed methods, theories and approaches to help develop problem solving skills and create effective solutions. Learn from them!

Books like The Art of Thinking Clearly , Think Smarter, or Thinking Fast, Thinking Slow are great places to start, though it’s also worth looking at blogs related to organizations facing similar problems to yours, or browsing for success stories. Seeing how Dropbox massively increased growth and working backward can help you see the skills or approach you might be lacking to solve that same problem. Learning from others by reading their stories or approaches can be time-consuming but ultimately rewarding.

A tired, distracted mind is not in the best position to learn new skills. It can be tempted to burn the candle at both ends and develop problem solving skills outside of work. Absolutely use your time effectively and take opportunities for self-improvement, though remember that rest is hugely important and that without letting your brain rest, you cannot be at your most effective.

Creating distance between yourself and the problem you might be facing can also be useful. By letting an idea sit, you can find that a better one presents itself or you can develop it further. Take regular breaks when working and create a space for downtime. Remember that working smarter is preferable to working harder and that self-care is important for any effective learning or improvement process.

Want to design better group processes?

Over to you

Now we’ve explored some of the key problem solving skills and the problem solving steps necessary for an effective process, you’re ready to begin developing more effective solutions and leading problem solving workshops.

Need more inspiration? Check out our post on problem solving activities you can use when guiding a group towards a great solution in your next workshop or meeting. Have questions? Did you have a great problem solving technique you use with your team? Get in touch in the comments below. We’d love to chat!

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$Building Problem-solving skills for math$

Building Problem-solving Skills for 7th-Grade Math

Mathematics is a subject that requires problem-solving skills to excel. In 7th grade, students begin to encounter more complex math concepts, and the ability to analyze and solve problems becomes increasingly important. Building problem-solving skills for math not only helps students to master math concepts but also prepares them for success in higher-level math courses and in life beyond academics.

In this article, we will several key skills that are needed for success in 7th-grade math, and also explore how they can benefit students both academically and personally. We will also provide tips and strategies to help students develop and improve their problem-solving skills. Let’s dive in!

Building Analytical Skills

The first of seven important skills to build is that of analytical skills. These allow students to analyze a problem and break it down into smaller parts. From there, they’re able to identify the key components that need to be addressed. Analytical skills also hone students’ abilities to identify patterns. Students should be able to identify patterns in mathematical data, such as in number sequences, geometric shapes, and graphs. Importantly, students should not just be able to recognize the pattens, but they should be able to describe them (more on that in communication) and use them to make predictions and solve problems.

We alluded to this earlier, but breaking down problems is an essential component of analytical skills. Students with strong analytical skills can break problems down into smaller and more manageable parts. They are then able to identify key components of a problem and use this information to develop a strategy for solving it.

Along with identifying patterns comes identifying relationships. Students with good mathematical analytical skills can identify relationships between different mathematical concepts, such as the relationship between addition and subtraction, or the relationship between angles and shapes. Through strengthening this skill, students will be able to describe these relationships and use them to solve problems.

An important part of analytical skills is the ability to analyze data. Students should be able to analyzeand interpret data presented in a variety of formats, such as graphs, charts, and tables. They should be able to use this data to make predictions, draw conclusions, and solve problems.

Speaking of conclusions, reaching sound conclusions based on mathematical data is a fundamental skill needed for making predictions based on trends in a graph, or drawing inferences from a set of data.

Another skill students should master is the ability to compare and contrast mathematical concepts, such as the properties of different shapes or the strategies for solving different types of problems. Through this, they’ll be able to use the information they gather to solve problems.

With all these skills at play comes arguably the most important: Critical thinking. This is an indicator that a student really grasps the concepts and it’s just repeating them back to you on command. Critical thinking is the ability to evaluate information and arguments, and make judgements and decisions based on evidence, and apply logic and reasoning to solve problems.

Building Creative Thinking

This is the ability for students (or anyone, really) to think outside the box and come up with innovative solutions to problems. This involves being able to approach problems from different angles and to consider multiple perspectives. For a 7th-grader, this skill can be exercised through the following:

Thinking Outside the Box: Students should be encouraged to think creatively and come up with innovative solutions to problems. This involves thinking outside the box and considering multiple perspectives.
Finding Multiple Solutions: Students should be able to come up with multiple solutions to a problem and evaluate each one to determine which is the most effective.
Developing Original Ideas: Students should be able to develop original ideas and approaches to solving problems. This involves being able to come up with unique and innovative solutions that may not have been tried before.
Making Connections: Students should be able to make connections between different mathematical concepts and apply these connections to solve problems. This involves looking for similarities and differences between concepts and using this information to make new connections.
Visualizing Solutions : Students should be able to visualize solutions to problems and use diagrams, charts, and other visual aids to help them solve problems.
Using Metaphors and Analogies: Students should be able to use metaphors and analogies to help them understand complex mathematical concepts. This involves using familiar concepts to explain unfamiliar ones and making connections between different ideas.

Building Problem-Solving Strategies

It may sound like the same thing, but building problem-solving strategies is not the same as building problem-solving skills. Building strategies for problem-solving lends itself to actual problem-solving. Let’s expand on this: Say your student is presented a problem that they’re struggling with, these are some of the problem-solving strategies they may use in order to solve the puzzle.

Identify the problem: The first step in problem-solving is to identify the problem and understand what is being asked. Students should carefully read the problem and make sure they understand the question before attempting to solve it.
Draw a diagram: Students can draw a diagram to help visualize the problem and better understand the relationships between different parts of the problem.
Use logic: Students can use logic to identify patterns and relationships in the problem. They can use this information to develop a plan to solve the problem.
Break the problem down: Students can break a complex problem down into smaller, more manageable parts. They can then solve each part of the problem individually before combining the solutions to get the final answer.
Guess and check: Students can guess and check different solutions to the problem until they find the correct answer. This method involves trying different solutions and evaluating the results until the correct answer is found.
Use algebra: Algebraic equations can be used to solve a variety of mathematical problems. Students can use algebraic equations to represent the problem and solve for the unknown variable.
Work backward: Students can work backward from the final answer to determine the steps required to solve the problem. This method involves starting with the end goal and working backward to determine the steps needed to get there.

Building Persistence and Perseverance

In an increasingly instant-gratification world with apps, searches and AI chatbots just a click away, this is an important skill not just in the math classroom, but for life in general. Problem-solving, whether that’s a math problem or a life challenge, often requires persistence and perseverance. Student need to learn to be able to stick with a problem even when it seems challenging, difficult, or seemingly impossible. Here are ways you can encourage your students to stick it out when working on problems:

Trying multiple approaches: When faced with a challenging problem, students can demonstrate persistence by trying multiple approaches until they find one that works. They don’t give up after one attempt but keep trying until they find a solution.
Reframing the problem: If a problem seems particularly difficult, students can demonstrate perseverance by reframing the problem in a different way. This can help them see the problem from a new perspective and come up with a different approach to solve it.
Asking for help: Sometimes, even with persistence, a problem may still be difficult to solve. In these cases, students can demonstrate perseverance by asking for help from their teacher or classmates. This shows that they are willing to put in the effort to find a solution, even if it means seeking assistance.
Learning from mistakes: Making mistakes is a natural part of the problem-solving process, but students can demonstrate persistence by learning from their mistakes and using them to improve their problem-solving skills. They don’t get discouraged by their mistakes, but instead, they use them as an opportunity to learn and grow.
Staying focused: In order to solve complex math problems, it’s important for students to stay focused and avoid distractions. Students can demonstrate perseverance by staying focused on the problem at hand and not getting distracted by other things.

Building Communication Skills

Communication is essential for building problem-solving skills for math

We alluded to this earlier, but a central part of building problem-solving skills is building the ability to articulate a problem or a solution. This isn’t just for the sake of personal understanding, but critical for collaboration. Students need to be able to explain their thinking, ask questions, and work with others to solve problems. Here are some examples of communication skills that can be used to build problem-solving skills:

Clarifying understanding: Students can ask questions to clarify their understanding of the problem. They can seek clarification from their teacher or classmates to ensure they are interpreting the problem correctly.
Explaining their reasoning: When solving a math problem, students can explain their reasoning to show how they arrived at a particular solution. This can help others understand their thought process and can also help students identify errors in their own work.
Collaborating with peers: Problem-solving can be a collaborative effort. Students can work together in groups to solve problems and communicate their ideas and solutions with each other. This can lead to a better understanding of the problem and can also help students learn from each other.
Writing clear explanations: When presenting their solutions to a math problem, students can write clear and concise explanations that are easy to understand. This can help others follow their thought process and can also help them communicate their ideas more effectively.
Using math vocabulary: Math has its own language and using math vocabulary correctly is essential for effective communication. Students can demonstrate their understanding of math concepts by using correct mathematical terms and symbols when explaining their solutions.

Building Mathematical Knowledge

Building problem-solving skills for math involves solid understanding of mathematical concepts

This would seem like a no-brainer, since you’re a math educator clicking on an article about building math problem-solving skills. However, it’s worth being explicit that problem-solving in math requires a solid understanding of mathematical concepts, including arithmetic, algebra, geometry, and data analysis. Students need to be able to apply these concepts to solve problems in real-world contexts.

7th-grade math covers a wide range of mathematical concepts and skills. Here are some examples of mathematical knowledge that 7th-grade math students should have:

Algebraic expressions and equations: Students should be able to write and simplify algebraic expressions and solve one-step and two-step equations.
Proportional relationships: Students should be able to understand and apply proportional relationships, including identifying proportional relationships in tables, graphs, and equations.
Geometry: Students should have a solid understanding of geometry concepts such as angles, triangles, quadrilaterals, circles, and transformations.
Statistics and probability: Students should be able to analyze and interpret data using measures of central tendency and variability, and understand basic probability concepts.
Rational numbers: Students should have a solid understanding of rational numbers, including ordering, adding, subtracting, multiplying, and dividing fractions and decimals.
Integers: Students should be able to perform operations with integers, including adding, subtracting, multiplying, and dividing.
Ratios and proportions: Students should be able to understand and use ratios and proportions in a variety of contexts, including scale drawings and maps.

In conclusion, problem-solving skills are essential for success in 7th grade math. Analytical skills, critical and creative thinking, problem-solving strategies, persistence, communication skills, and mathematical knowledge are all important components of effective problem-solving. By developing these skills, students can approach math problems with confidence and achieve their full potential.

If you enjoyed this read, be sure to browse more of our articles . More importantly, if you want to save yourself hours of preparation time by having full math curriculums, review guides and tests available at the click of a button, be sure to sign up to our 7th Grade Newsletter . You’ll receive loads of free lesson resources, tips and advice and exclusive subscription offers!

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3 Simple Strategies to Improve Students’ Problem-Solving Skills

These strategies are designed to make sure students have a good understanding of problems before attempting to solve them.

Research provides a striking revelation about problem solvers. The best problem solvers approach problems much differently than novices. For instance, one meta-study showed that when experts evaluate graphs , they tend to spend less time on tasks and answer choices and more time on evaluating the axes’ labels and the relationships of variables within the graphs. In other words, they spend more time up front making sense of the data before moving to addressing the task.

While slower in solving problems, experts use this additional up-front time to more efficiently and effectively solve the problem. In one study, researchers found that experts were much better at “information extraction” or pulling the information they needed to solve the problem later in the problem than novices. This was due to the fact that they started a problem-solving process by evaluating specific assumptions within problems, asking predictive questions, and then comparing and contrasting their predictions with results. For example, expert problem solvers look at the problem context and ask a number of questions:

What do we know about the context of the problem?
What assumptions are underlying the problem? What’s the story here?
What qualitative and quantitative information is pertinent?
What might the problem context be telling us? What questions arise from the information we are reading or reviewing?
What are important trends and patterns?

As such, expert problem solvers don’t jump to the presented problem or rush to solutions. They invest the time necessary to make sense of the problem.

Now, think about your own students: Do they immediately jump to the question, or do they take time to understand the problem context? Do they identify the relevant variables, look for patterns, and then focus on the specific tasks?

If your students are struggling to develop the habit of sense-making in a problem- solving context, this is a perfect time to incorporate a few short and sharp strategies to support them.

3 Ways to Improve Student Problem-Solving

1. Slow reveal graphs: The brilliant strategy crafted by K–8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students’ questioning, sense-making, and evaluating predictions.

For instance, in one third-grade class, students are given a bar graph without any labels or identifying information except for bars emerging from a horizontal line on the bottom of the slide. Over time, students learn about the categories on the x -axis (types of animals) and the quantities specified on the y -axis (number of baby teeth).

The graphs and the topics range in complexity from studying the standard deviation of temperatures in Antarctica to the use of scatterplots to compare working hours across OECD (Organization for Economic Cooperation and Development) countries. The website offers a number of graphs on Google Slides and suggests questions that teachers may ask students. Furthermore, this site allows teachers to search by type of graph (e.g., scatterplot) or topic (e.g., social justice).

2. Three reads: The three-reads strategy tasks students with evaluating a word problem in three different ways . First, students encounter a problem without having access to the question—for instance, “There are 20 kangaroos on the grassland. Three hop away.” Students are expected to discuss the context of the problem without emphasizing the quantities. For instance, a student may say, “We know that there are a total amount of kangaroos, and the total shrinks because some kangaroos hop away.”

Next, students discuss the important quantities and what questions may be generated. Finally, students receive and address the actual problem. Here they can both evaluate how close their predicted questions were from the actual questions and solve the actual problem.

To get started, consider using the numberless word problems on educator Brian Bushart’s site . For those teaching high school, consider using your own textbook word problems for this activity. Simply create three slides to present to students that include context (e.g., on the first slide state, “A salesman sold twice as much pears in the afternoon as in the morning”). The second slide would include quantities (e.g., “He sold 360 kilograms of pears”), and the third slide would include the actual question (e.g., “How many kilograms did he sell in the morning and how many in the afternoon?”). One additional suggestion for teams to consider is to have students solve the questions they generated before revealing the actual question.

3. Three-Act Tasks: Originally created by Dan Meyer, three-act tasks follow the three acts of a story . The first act is typically called the “setup,” followed by the “confrontation” and then the “resolution.”

This storyline process can be used in mathematics in which students encounter a contextual problem (e.g., a pool is being filled with soda). Here students work to identify the important aspects of the problem. During the second act, students build knowledge and skill to solve the problem (e.g., they learn how to calculate the volume of particular spaces). Finally, students solve the problem and evaluate their answers (e.g., how close were their calculations to the actual specifications of the pool and the amount of liquid that filled it).

Often, teachers add a fourth act (i.e., “the sequel”), in which students encounter a similar problem but in a different context (e.g., they have to estimate the volume of a lava lamp). There are also a number of elementary examples that have been developed by math teachers including GFletchy , which offers pre-kindergarten to middle school activities including counting squares , peas in a pod , and shark bait .

Students need to learn how to slow down and think through a problem context. The aforementioned strategies are quick ways teachers can begin to support students in developing the habits needed to effectively and efficiently tackle complex problem-solving.

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Mathematics Improves Your Critical Thinking and Problem-Solving
Mathematics provides a systematic and logical framework for problem-solving and critical thinking. The study of math helps to develop analytical skills, logical reasoning, and problem-solving abilities that can be applied to many areas of life.By using critical thinking skills to solve math problems, we can develop a deeper understanding of concepts, enhance our problem-solving skills, and ...
Unlocking the Power of Math Learning: Strategies and Tools for Success
A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills. Moreover, the importance of math learning goes beyond solving equations and formulas.
Why It's So Important to Learn a Problem-Solving Approach to Mathematics
The skills the problem solvers developed in math transferred, and these students flourished. We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument.
Does mathematics training lead to better logical thinking and ...
Mathematics is often promoted as endowing those who study it with transferable skills such as an ability to think logically and critically or to have improved investigative skills, resourcefulness and creativity in problem solving. However, there is scant evidence to back up such claims. This project tested participants with increasing levels of mathematics training on 11 well-studied rational ...
How to Improve Problem-Solving Skills: Mathematics and Critical
Decision Making: Choose the most suitable method to address the problem. Implementation: Put the chosen solution into action. Evaluation: Reflect on the solution's effectiveness and learn from the outcome. By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to ...
Does mathematics training lead to better logical thinking and reasoning
The School of Mathematics and Statistics at The University of Sydney, Australia, directly attributes as part of particular course objectives and outcomes skills that include "enhance your problem-solving skills" as part of studies in first year , "develop logical thinking" as part of studies in second year, which was a statement drafted ...
9 Ways to Improve Math Skills Quickly & Effectively
Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it's best to take each year step-by-step, lesson by lesson. This article has tips and tricks to improve your child's math skills ... solving math puzzles and ...
6 Tips for Teaching Math Problem-Solving Skills
1. Link problem-solving to reading. When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools ...
How to Improve Problem-Solving Skills in Math
To improve problem-solving math skills, it's essential to first understand the problem at hand. Here are some tips to help break down the problem and identify its key components: 1. Read the problem carefully: Take your time to read it attentively and ensure you understand what it asks. Pay attention to keywords or phrases that indicate what ...
Build math problem solving: a brain-based routine
Here's how you can safely foster growth and build math problem solving skills through failure in your classroom: Build in time to analyze errors & reflect. Reward effort & growth as much as, if not more than, accuracy. At least initially, skip the grading so students aren't afraid to be wrong. Getting started with brain-based problem solving
How does learning more about mathematics improve one's problem-solving
The neurological process of mathematics is logical and so is problem-solving. There exist two portions of logical use in mathematics: in practice and in rigor. Realizing logic constantly provides for better awareness when problem-solving. Intuition is more about fuzzy-logical interrelations.
12 Ways to Improve Problem Solving Skills
On the other hand, you might try to save by cutting your spending or by lowering other costs. Use some strategies to help you come up with solutions: Divide and conquer. Break the problem into smaller problems and brainstorm solutions for them separately, one by one. Use analogies and similarities.
Problem Solving Maths: Strengthening Mathematical Skills
Maths problem solving is when a mathematical task challenges pupils to apply their knowledge, logic and reasoning in unfamiliar contexts. Problem solving questions often combine several elements of maths. We know from talking to the hundreds of school leaders and maths teachers that we work with as one to one online maths tutoring providers ...
Promoting Creative and Critical thinking in Mathematics and Numeracy
The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it's mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies: Problem Solving, Reasoning, Fluency ...
PDF Problem solving in mathematics
Therefore, high-quality assessment of problem solving in public tests and assessments1 is essential in order to ensure the effective learning and teaching of problem solving throughout primary and secondary education. Although the focus here is on the assessment of problem solving in mathematics, many of the ideas will be directly transferable ...
The Problem-solving Classroom
The Problem-solving Classroom. This article forms part of our Problem-solving Classroom Feature, exploring how to create a space in which mathematical problem solving can flourish. At NRICH, we believe that there are four main aspects to consider: • Highlighting key problem-solving skills. • Examining the teacher's role.
Developing Mathematical Reasoning: How To Teach Effectively
These are as follows, and are a great starting point to developing math reasoning at the elementary school level. 1. Start math lessons with a question. 2. Students need to wonder and struggle. 3. You are not the answer key. 4. Say yes to your students original ideas (but not yes to methodical answers)
Mathematics as a Complex Problem-Solving Activity
Problem-solving in mathematics supports the development of: The ability to think creatively, critically, and logically. The ability to structure and organize. The ability to process information. Enjoyment of an intellectual challenge. The skills to solve problems that help them to investigate and understand the world.
10 Ways to Improve Math Skills
Math Teacher. Develop your mental math skills. Mental math is when you perform mathematical calculations without using calculators, paper, or counting aids. Use your mind, memory, lessons, and discussions with your classmates to refine your math skills and build strong problem-solving strategies. 4.
Developing problem-solving skills in mathematics: a lesson study
Purpose. Problem solving is a skill in mathematics which although always relevant has heightened priority due to the changes in the new mathematics GCSE (Department for Education, 2013). It has previously been a skill which is deemed underdeveloped within mathematics and therefore is a theme which teachers are seeking to improve and nurture in ...
PDF Developing mathematical problem-solving skills in primary school by
mathematical problem-solving by providing visual tools called Problem-solving Keys, would improve students' performance in tasks and skills in justifying their reasoning. To map students' problem -solving skills and strategies, data from 25 fifth graders' pre-tests and post-tests with non-routine mathematical tasks were analysed.
How to improve your problem solving skills and strategies
6. Solution implementation. This is what we were waiting for! All problem solving strategies have the end goal of implementing a solution and solving a problem in mind. Remember that in order for any solution to be successful, you need to help your group through all of the previous problem solving steps thoughtfully.
"New Math" Explained: Tips for Parents and Students ...
According to US News, " Today's math is more applied, and how students work the problem matters ." (Armijo, 2024). It's not just the solution that's important but how you got there that matters. In addition, "new math" focuses on problem-solving strategies and encourages students to approach mathematical problems creatively.
Building Problem-solving Skills for 7th-Grade Math
Ratios and proportions: Students should be able to understand and use ratios and proportions in a variety of contexts, including scale drawings and maps. In conclusion, problem-solving skills are essential for success in 7th grade math. Analytical skills, critical and creative thinking, problem-solving strategies, persistence, communication ...
3 Ways to Improve Student Problem-Solving
3 Ways to Improve Student Problem-Solving. 1. Slow reveal graphs: The brilliant strategy crafted by K-8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students' questioning, sense-making, and evaluating predictions.