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Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.  

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.” 

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry 

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says. 

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on. 

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry: 

• How much does the pencil sharpener remove? 
• What is the length of a brand new, unsharpened pencil? 
• Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch. 

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.” 

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.” 

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.” 

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them. 

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it. 

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs. 

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.” 

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result. 

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry. 

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day. 

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives. 

Mathematics for Teaching

This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.

What is mathematical investigation?

Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.

I first heard about math investigations in 1990 when I attended a postgraduate course in Australia.  I love it right away and it has since become one of my favorite mathematical activity for my students who were so proud of themselves when they finished their first investigation.

Problem solving is a convergent activity. It has definite goal – the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems.

Teaching through mathematical investigation allows  for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind .

Although  students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point.  The “open-endedness” of many investigation also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:

• systematic exploration of the given situation
• formulating problems and conjectures
• attempting to provide mathematical justifications for the conjectures.

In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions. To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation .

Some parents and even teachers complain that students are not learning mathematics in this kind of activity. Indeed they won’t if the teacher will not discuss the results of the investigation, highlight and correct the misconceptions, synthesize students’ findings and help students make connection among the math concepts covered in the investigation. This goes without saying that teachers should try the investigation first before giving it to the students.

I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions .

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23 thoughts on “ What is mathematical investigation? ”

Can anyone give me an example of it?

I think there’s a need for students to conduct investigations, especially in math, in order for them to really learn the language of mathematics. We also have to note that those young ones have their own way of doing things; which in some ways may simplify processes. Naci John

Is it possible to teach all math topics using investigation method?

No. Only those topics in math that involves generalizing and conjecturing using the mathematics students at that grade level know may be taught by math investigation. However, you can always start any topic by math investigation but it doesn’t have to be an investigation all the way. The students may see patterns and be able to state a conjecture but they will not always have the capability to explain or justify it mathematically. Doing math investigation is more for developing disposition to think mathematically. Don’t overdo it.

I find this article very informative. I hope UP NISMED can organize a workshop for teachers and students for this essential undertaking in Math. I think reconciling LS and MI will be a great boost in the teaching strats of teachers and mathematical abilities of the students.

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Thank you for posting it.

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In elementary school I had to take investigations. In all 956 lessons standard addition is mentionedONCE and not truly taught. same sor addition and multiplication. long division is never properly taught. they never even mentioned how to do it. they teach a guess and check method leaving you with a remainder- no decimals. This lleaves students with lower grades on timed tests because they are rushed due to the innefficient method they first learned. The board of education deemed the program “sub standard”. Many students who used the program in elementary school needed help with math in middle school. So answer this, why do we continually use investigations?

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please, pki investigate naman to: How many other rectangles can be made with twelve tiles?

To me, Math investigation is an ally of constructivism. Students are engaged in tasks that allow them to put in more on what they know as they tinker on a math problem. They do exploration and research to be able to shed more light to the problem. Creative and critical thinking come to the fore. The investigation may take time. But, at the end of the process, students defend what they got, defend it, and share it.

Math investigation takes time. It is quality time. The teacher stands by for consultation and extends assistance, as needed by the students.

My book “High School Geometry with Practical Work and Portfolio Making” is rich on math investigations. My other book “Science Research and Statistics” has much more on investigations particularly on Science Investigatory projects.

hello! i am trying to teach my students to appreciate the process of mathematical investigation and to develop the skills needed. however, i am having a hard time organizing my lessons. i want to them to develop the thinking skills necessary for them to create their own math IPs or math research papers later on. 1. do you have any suggestions or activities or lesson plans that i can refer to so that the skills can be developed gradually? 2.can you recommend any textbook that can be used for this purpose alone including different mathematical model?

i hope that you could answer my queries. thank you very much.

It’s best to expose students to problems with many solutions/answers first before giving them mathematical investigation tasks. You can also convert some of your exercises to problems and mathematical investigation. Check out my post on exercises, problems and math investigations .

I think our use of the Connected Math series really falls in line with this post. Curriclum planning needs to become as strignent as lessons planning that we as teachers work extremely hard on.

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Mathematics investigation

At its core, mathematics is about problem-solving and modelling the world around us. By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding. Using a guided investigation model ensures that students stay focused on the mathematics being used and make connections to other areas of learning.

After students have been taught a skill or concept it is important that they apply it in a meaningful context as this will reveal the depth of their understanding. A guided inquiry is one way to do this. By scaffolding the problem, the teacher can lead students through a problem-solving exercise, highlighting the mathematics they need to use. Starting with a prompt – either a picture or a question – students can explore different ways of approaching the problem and communicating their thinking. Mathematics investigation links directly to the Critical and Creative Capability in the Australian Curriculum (Version 9.0) .

Investigations can be differentiated by adjusting the complexity of the original problem or the expected outcomes. Investigations can be assessed and guided through the use of rubrics. A rubric for assessment is usually in the form of a matrix or grid. It is used to interpret and assess a student's work against identified criteria. A student can use the rubric to self-assess their own work. 

By using a particular scaffold or approach regularly, students learn what is expected of them and are able to improve and develop their problem-solving skills. Teachers should explicitly teach how to approach a problem using such a scaffold. Popular scaffolds or protocols for guided investigations include:

Inquiry Maths

• Three-Act Tasks

To teach a Three-Act Task on the topic of ‘proportion and ratio’ include the following components, noting a Three-Act Task is a task consisting of three distinct sections.

• An engaging and perplexing Act One, often a video. Students discuss what they saw, pose questions and decide what information is needed for them to be able to answer the questions they have posed.
• An information and solution-seeking Act Two. Students are given the information they need and work, either in small groups or individually, to find a solution.
• A solution-discussion and revealing Act Three. Students discuss their findings and the solution is revealed. Students may then discuss and pose further questions.

The teacher:

• explicitly teaches the mathematics needed for the task
• explicitly teaches how to use a scaffold or protocol
• makes the learning intention and success criteria clear
• invites questions and wonderings
• focuses the students on the mathematical ideas being developed
• makes connections to students’ previous learning
• makes connections to other subject areas
• encourages students to work collaboratively
• monitors students, giving specific feedback to guide the students further.

The students:

• know how to use a particular scaffold or approach when working on a maths investigation to solve a problem
• know how to work collaboratively and respectfully
• communicate their ideas and thinking clearly
• ask questions and respectfully respond to other’s questions
• are prepared to struggle and make mistakes in order to make progress in their learning.

Examples of the strategy in action

Thinking mathematically

In this video, a teacher discusses how they used an investigation to reinforce the students’ understanding of estimation and prediction.

Nana's chocolate milk

A Three-Act Task on proportion and ratio. 

The Inquiry Maths website showcases a model of guided inquiry that encourages students to apply their knowledge based on a given prompt.

Open Middle problems

In an Open Middle problem an expression is presented and students are challenged to find the numbers that will make the expression true. They are challenging, suitable for all year levels, cover most maths topics, and promote problem-solving and reasoning skills. On this site you will find a comprehensive range of problems.

Transforming tasks strategy: from procedure to problem solving

Tasks are presented as following a ‘before and after’ approach. It demonstrates how to move from a procedural approach to posing questions that incorporate more of a problem-solving approach and mathematical investigation.

Problem Solving and Mathematical Modeling

• First Online: 11 January 2023

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• Parikshit Narendra Mahalle 7 ,
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• Sachin R. Sakhare 9 &
• Atul P. Kulkarni 10  

Part of the book series: Studies in Autonomic, Data-driven and Industrial Computing ((SADIC))

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A problem is a puzzle or task that requires a logical thought process or fundamental mathematical steps to solve it. The puzzle generally represents the set of questions on the underlined use case which also consist of complete description of the use case along with the set of constraints about the use case. Logic is very importantly used to solve the puzzle or problem. Logic is defined as a method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved. Logic is a subjective matter and varies from person to person. Logic is directly linked to the natural intelligence of human being and is a language of reasoning. Logic also represents the set of rules we use when we do reasoning. It is observed that majority of the employment of fresh engineering graduates across the globe is in software and information technology sector.

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Mahalle, P.N., Ambritta P., N., Sakhare, S.R., Kulkarni, A.P. (2023). Problem Solving and Mathematical Modeling. In: Foundations of Mathematical Modelling for Engineering Problem Solving. Studies in Autonomic, Data-driven and Industrial Computing. Springer, Singapore. https://doi.org/10.1007/978-981-19-8828-8_2

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Problem Solving, Mathematical Investigation, and Modeling (6081)

This course intends to enhance the students’ knowledge and skills in dealing with real-life and/or non-routine applications of mathematics. Students will have the opportunity to explore the use of problem-solving strategies or heuristics as they engage in mathematical investigations, formulate and justify conjectures, make generalizations, and communicate mathematical ideas.

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