Part 3 of 8, Strategies for Integrating the Mathematical Practices into Instruction

By Dr. Michele Douglass

Our series on the mathematical practices continues by looking at a practice that is often grouped with the one we will discuss in our next blog. Both require an understanding of the content in a way that allows you to represent it. If you are like me, when you read this blog post’s title, “Models with Mathematics,” you think manipulatives. However, the practice we will discuss here is not about manipulatives; it is about using mathematical symbols to represent a situation.

Students proficient in building models of mathematics can apply math concepts to real-world problems. They are able to write an equation to model a situation or create a graph of the data. They use a variety of representations (formulas, graphs, tables, problems, number lines) and can move easily between them to figure out solutions. They are able to see when different representation doesn’t seem to align or lead to a solution, and change their course of action. As they are solving a problem, they reflect on whether the results are making sense. Often, a student is asking herself, “How can I represent this mathematically?”

As you think about students who might be doing this already, your list of names is probably short. Most students don’t just begin solving problems this way. Therefore, tasks must be designed and scaffolded for students to move from forms of dependency in problem solving to being able to come up with models for the problems. As I work with teachers planning lessons and tasks to specifically promote math modeling, I look for the following in their instruction:

• Provides real-world problems for students to solve daily (story problems, school, or community problems)
• Explicitly connects an equation to match the real-world problem
• Provides opportunities for students to go back and forth between different math tools—function tables, flow charts, Venn diagrams, number lines, 100 charts, etc. should be readily available, and students should be gaining familiarity in how and when to use each model
• Asks students to take complex math and make it simpler by creating models
• Requires students to identify the unknowns or variables, compute results, report findings, and justify results and procedures used

Designing tasks of this nature is not easy. However, that is only part of the method in building this mathematical practice. It is equally critical to think about what I, the teacher, am doing while students are working through the task. Here are a few things I consider during the instructional time:

• After reading, and before solving a problem, have students predict what the answer should be about. What would be a logical answer?
• Facilitate discussions about what the problem is asking and what might be logical answers.
• Facilitate discussions about what the symbols or variables mean in the equation. For example, "Why are we writing the equation 57 + ___ = 97?”
• Monitor student work as they solve problems, asking them if this tool is going to help solve the problem and how it works to help solve the problem.

Many times I find that the types of questions asked during instruction impact students’ abilities to develop modeling with mathematics. I have found several helpful documents that list questions that can support a student’s progression from a naïve modeler to a more sophisticated modeler. Here are a few such questions:

• What number sentence can be used to describe this situation?
• What are ways to represent the situation?
• What is an equation or expression that matches the diagram (number line, chart, table)?
• What do you already know about solving this problem?
• What tools have we used that might help you organize the information from the problem?
• How might a picture or math tool help you solve the problem? What are some ways to visually represent the situation that show the mathematics?
• How might you represent what the problem is asking?
• What formulas might make sense to use?
• Is this working, or do you need to change your model?

I will end with several examples of problems that will support students in learning to be modelers of mathematics. These examples represent tasks across various grade levels. For the primary grades, modeling is writing an equation for a story problem or even identifying the correct number sentence to describe a situation. As students progress in their learning, modeling becomes far more sophisticated. As you read each example, think about how to use the prompts above to facilitate learning and pose productive questions.

Example 1: If your allowance is going to increase by the same amount each month, do you want it to increase by the same number or the same percent? Is there a difference? How would you model this to convince others of your solution?

By the time students begin working with ratios and proportions, they are beginning their thinking of additive versus multiplicative comparisons. Models are wonderful ways to support students as they are building these comparisons. Think about how students might derive a solution using tables or graphs. What might they do for equations? How would they even begin to make a prediction?

Example 2: Two siblings—a brother and a sister—attend the same school. Walking at constant rates, the brother takes 40 minutes to walk home from school, while the sister takes only 30 minutes on the same route. If she leaves school 6 minutes after her brother, how many minutes has he traveled before she catches up to him?

Students might begin by drawing a number line and marking the school on one end of the line and home on the other end, showing the brother above the line and the sister below the line. If students start this way, ask how they are showing or accounting for the 6-minute delay for the sister. Another way for students to model each situation would be graphing these as lines if they recognize the math of walking at a constant rate.

Example 3: Selling Soup – Martha wants to set up a soup stand at the Farmer’s Market. Her profits will go to a charity. She is selling a cup of soup with a roll for $1.25. Her costs include the following: (1) she buys the soup for $5 per 2.5-liter bottle, which serves 10 people; (2) the rolls cost $2 for a pack of 10. She is only going to offer one type of soup, so she uses a survey. If she wants to sell 500 cups of soup, what should she buy to make the most profit and have as little left over as possible? (More of this task can be found at http://map.mathshell.org/materials/download.php?fileid=1542)

This example included multiple models. You can usually use real situations with students that reach this level of modeling. How often does your site/district work with fundraisers? These are wonderful opportunities to give to your students to solve as problems. Ask them which one will bring in the most profit for the school. You can extend this further by having students work with the profits to plan field trips using only the profits from the fundraisers. In both of these types of tasks, students will be using multiple models and various math concepts to arrive at a solution.

Example 4: Ways to Make Ten – This is played as a game in various ways. Students are shown a manipulative that represents part of a ten. It could be using linking cubes or cubes in a bag. It could be drawing a ten-frame that has some of the squares filled with circles. Students write the model for making ten by writing the number sentence that would use the part they see with the unknown part to make ten.

References and Additional Resources:

Common Core State Standards Initiative website: http://www.corestandards.org/Math/Practice/

Implementing the Mathematical Practice Standards website: http://mathpractices.edc.org/

Inside Mathematics website: http://www.insidemathematics.org/index.php/mathematical-practice-standards

Mathematics Assessment Resource Service website: http://map.mathshell.org/materials/index.php

Michele Douglass, Ph.D., is the president of MD School Solutions, Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as technology programs and professional development, including NUMBERS® with John Woodward, Ph.D., and Mary Stroh.