Mathematical modeling, MP.4, is a powerful practice that engages students in problems worth solving and enriches their learning experiences. School leaders have said that a pain point for their teachers is getting students to engage with this math practice. Teachers generally do not have a lot of experience creating math models, developing them takes a long time to do, and finding good examples is difficult. However, while it does take longer than simply teaching students to implement an algorithm, the rewards are greater. Students who engage in modeling develop deeper conceptual understanding and stronger procedural fluency.

Here are some things teachers can do to incorporate mathematical modeling in the classroom.

GETTING STARTED

Start small. The first modeling task of the year should be completed in one class. This will minimize the risk of overwhelming your students. Then increase the scope to projects that last two or three days, and by the end of the year, your students can tackle a much larger project.

For example, you don’t have to start with a large-scale, open-ended project like,“What’s the relationship between speed and gas mileage?” Instead, start with a smaller activity, such as this one, adapted from Discovery Education's Math TechbookTM, Grade 7:

When converting a book to a movie, a page of description lasts 10 seconds, a page of dialogue lasts 1 minute, and a page of action lasts 90 seconds. Analyze a few pages of your favorite book, and predict how long its movie version would run. What parts of the book would you skip to make the movie shorter?

Use Problems Worth Solving. Modeling tasks should be authentic problems that engage students as problem-solvers. Alone or in teams, students should work through the tasks, create models, interpret, and revise results.

Vary the tasks. Some modeling problems have an open beginning, middle, and end. Others are closed in various parts. Use a variety of task types. Decide which task type is best for each scenario based on whether the goal is to prepare students for a new concept or to practice and reinforce skills.

Help students be successful. Your first activity should not be so challenging that students shut down. Encourage success with these techniques:

• Scaffold the problem by suggesting the types oftools to use

• Restrict the size of the problem by providing alimited set of information

• Ask “What are questions that someone might have?”• Prepare students for variables by asking “What would

change? What would stay the same?”• Identify alternative entry points for students who

have trouble getting started

Manage the conversation. The effectiveness of a modeling activity is directly proportional to the quality of the classroom discussion.

USING MATH MODELING TO DEVELOP PROCEDURAL FLUENCY

Model often. Modeling is a skill that must be practiced, just like solving an equation or graphing a line. Modeling doesn’t need to happen every day, but students only get better by tackling authentic problems often. Make modeling a regular part of your instruction.

Mathematical Modeling to Develop Procedural Fluency How to implement this powerful practice in your classroomBy Patrick Vennebush

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Address multiple standards. Good modeling tasks address multiple standards. For instance, students completing the task, “What Does 2,000 Calories Look Like?” might engage in five standards (Common Core Standards 6.EE.3, 6.EE.6, 6.RP.2, 7.EE.1, and 7.EE.4) simultaneously, which is more efficient, integrated, and impactful than teaching one standard each day for five days.

Align activities with goals. Determine class activities based on your instructional goals. Choose fun and engaging activities that apply and reinforce concepts.

Share alternate strategies. For problems with computation, demonstrate multiple solutions. For instance, show how the equation 7/12 = x/84 can be handled with cross-multiplying or equivalent fractions. Although cross-multiplying might be considered more advanced, it involves finding the product of 7 × 84. On the other hand, using equivalent ratios only requires calculating 7 × 7.

Nix the tricks. FOIL is limiting — it only teaches students how to multiply binomials. Instead, teach students to use the distributive property no matter how many terms are in each factor. Likewise, teaching students to “invert and multiply” when dividing fractions is not nearly as effective as having them use common denominators and discover shortcuts on their own.

Give it time. Procedural fluency follows a developmental progression, and the expectations for students at various ages and grades will differ. For instance, when playing a dice game, a kindergartener who rolls three 5’s might count the dots, but a fluent third-grader would recognize three groups of 5 as 3 × 5 = 15. Likewise, a middle school student might use an area model for multiplying a monomial by a binomial, but a high school student should apply the distributive property using symbols.

ASSESSING MATH MODELING ACTIVITIES

Encourage self and peer reviews. Students can assess each others’ work, and often they’re more critical of their work than you’d ever be. Use your time effectively by allowing students to review with a partner. Students improve their modeling and communication skills as they engage in MP.3, critiquing the reasoning of others.

Keep it simple. Problem- and project-specific rubrics are precise, but they take time to create. Instead, concentrate your efforts on finding and reviewing tasks, and then use a general modeling rubric that can work

for many different activities. You can use a general scoring rubric such as these:

• 2 points: correct conclusion + sufficient reasoning

• 1 point: correct conclusion + insufficient reasoningOR incorrect conclusion + valid reasoning

Alternatively, you can use a rubric that evaluates each portion of the modeling process.

Use both formative and summative assessments. Not every task needs a grade. Math modeling is a developmental process. Students should continually show progress, but that growth takes time. It may be unfair to assess their work from the beginning. Instead, formatively assess students by asking lots of questions (“What did you do?” and “Why did you do that?”), and use that information to guide selection of future tasks. The collective data that you gather can be compiled to yield a summative grade or score when the time comes.

Teachers need support and time to implement mathematical modeling effectively. Professional development should give teachers opportunities to practice mathematical modeling on their own and hone skills to teach it with confidence. When trying to get students engaged in modeling, one of the hard things for teachers to do is ask the right questions when students say something that is incorrect or state a misconception. In the traditional mode, teachers tell them the answer and hope students internalize it. The better thing to do is to ask students effective questions so they can think about it and form accurate conclusions. Asking the right questions is a skill that requires development and practice and school leaders need to ensure that teachers have opportunities to develop mastery.

PATRICK VENNEBUSHDirector of Mathematics Discovery Education

Patrick began his career as a middle and high school mathematics teacher in Pennsylvania. He served as associate director for PBS TeacherLine, Online Projects Manager for NCTM Illuminations, and Chair of MathCounts Question Writing Committee.