The Power of Process: Using Strategic Tools for Modeling and Data Analysis - Compass PD, LLC

By Dr. Sherri Lorton

Welcome back to our blog series on Mathematical Practices (MPs) in elementary classrooms! In our first post, we dove into Communicating Reasoning (MP3 and MP6), highlighting the need for students to build clear, precise arguments. This second installment shifts our focus from Communication to the question of relevance and application with the Modeling and Data Analysis standards: MP2, MP4, and MP5.

This trio of practices is essential for transforming mathematics from an abstract exercise in the classroom into a powerful tool for analyzing the real world. As we explore how these practices evolve in grades K-5, remember that this conversation pairs with a parallel series focusing on secondary level instruction, underscoring the vital, continuous nature of these mathematical habits.

Understanding the Trio of Modeling and Data Analysis Practices

Modeling and Data Analysis ensures students analyze complex, real-world scenarios and use mathematical models to solve problems and interpret results. Each component plays a distinct role in this larger process.

MP4: Model with mathematics.

This is the action of taking a messy problem—one arising in everyday life or the workplace—and representing it in a physical form. Modeling involves identifying important quantities in a practical situation and mapping their relationships using diagrams, tables, graphs, or formulas. For students to be proficient, they must interpret their answers within the context of the situation and reflect on whether the results make sense, making improvements to the model if necessary. Modeling occurs in a contextual situation; using tools in a real-world scenario is necessary to use MP4.

MP2: Reason abstractly and quantitatively.

This is the cognitive process behind modeling. It requires students to bring two complementary abilities to the problem: decontextualizing and contextualizing. When students decontextualize, they “abstract a given situation and represent it symbolically” (e.g., turning a word problem into an equation like 2 + 5 or 3 x 4). When they contextualize, they pause to relate the symbols back to the real-world meaning of the numbers, units, and operations. This practice ensures students don’t just manipulate symbols blindly; they maintain sense-making throughout the problem-solving process.

MP5: Use appropriate tools strategically.

This practice is the necessary skill set for executing modeling and reasoning effectively. Mathematically proficient students consider the range of tools available—which could include concrete objects, drawings, rulers, calculators, virtual manipulatives, or graph paper—and select the tool(s) relevant and useful for the specific problem. As students advance, they must learn to recognize the power and limitations of each tool, making decisions about when and why a particular tool is appropriate. For instance, once efficient methods (like standard algorithms) are available, using diagrams or concrete objects just for calculation might not be strategic, although they remain valuable for explaining thinking (MP3).

Similarities and Key Differences

• MP4 (Modeling) relies on MP2 (Reasoning): A student cannot effectively model a situation (MP4) without first decontextualizing the problem to identify the quantities and relationships (MP2) and then contextualizing the solution back into the real world.
• Both MP4 and MP2 rely on MP5 (Tools): The formulation, computation, and interpretation phases of modeling (MP4) all rely on the strategic use of tables, diagrams, symbols, and physical objects (MP5).
• The clearest difference is their focus: MP4 focuses on the contextual problem application (real-world story), MP2 focuses on the meaning of the quantities (the abstract numbers and units), and MP5 focuses on selection efficiency (choosing the right representation).

The Progression of Modeling and Data Analysis (K-5)

The ability to model with mathematics matures substantially across the elementary grades. Let’s look at how this development unfolds through a few grade-band examples.

In Kindergarten and First Grade, students use concrete objects to connect mathematical concepts to simple, real-world situations.

• Task: A kindergarten student models by building shapes like houses or animals from component shapes (e.g., sticks and clay balls), modeling the world around them. A first-grader student uses concrete models to represent adding within 100 and explains the reasoning, showing early place value modeling.
• MP4 in action: Modeling is centered on physical objects or drawings to model actions like adding or sharing.
• MP2 in action: Reasoning involves making sense of small, whole quantities.
• MP5 in action: Tools are primarily manipulatives or simple drawings.

In Second and Third Grades, students use concrete objects and drawings to represent procedures and data.

• Task: A third-grade student solves a two-step word problem that requires using a scaled bar graph or equation. They must decontextualize the problem steps and select the most appropriate method.
• MP4 in action: Modeling expands to encompass data analysis. Students gather measurement data and create line plots or bar graphs to represent data sets.
• MP2 in action: Reasoning advances as students reason with larger numbers (up to 1,000) and properties of operations.
• MP5 in action: Tools include rulers (now using standard units) and graphical tools like line plots.

In Fourth and Fifth Grades, students show deeper sophistication with numbers and quantity.

• Task: A fourth-grade student solves a word problem involving fractions, like determining the total amount of flour needed if 15 students each need 3/8 of a cup. They use symbolic representation (3/8 x 15) and visual models, such as number line diagrams, to show the relationship between 15 groups of 3/8. The ability to interpret the result, such as converting cups to cups and recognizing this amount is less than half of 15, demonstrates strong contextual reasoning.
• MP4 in action: Modeling becomes highly conceptual, applied to complex systems like fractions, decimals, and volume.
• MP2 in action: Reasoning focuses on reasoning quantitatively about relationships, such as multiplicative comparisons (Grade 4) or the effect of scaling (Grade 5 fractions).
• MP5 in action: The use of tools shifts toward abstract tools like number lines for fractions/decimals, formulas, and possibly geometry software.

Building Modeling Together

Prioritizing conceptual understanding, procedural fluency, and application (solving real-world problems) requires intentional planning. MP2, MP4, and MP5 can be the cornerstone to this focus on all three important areas. Here are a few reflective questions for your role:

Teachers

How often do you assign tasks that truly require students to build a model (MP4) rather than just execute a procedure? When students encounter difficulty, do you prompt them to step back and contextualize the problem (MP2) or to select a different tool (MP5) to gain insight?

Coaches

Look for evidence of students moving flexibly between the real-world context and the mathematical abstract world (MP2). Are students choosing tools strategically, or are they only using the tool provided in the instructions (MP5)? Advocate for rich, complex tasks that truly require modeling.

Administrators

Are your instructional materials designed to support rigorous application, ensuring students have access to problems that require complex reasoning and modeling techniques? High-quality instructional materials are pivotal in nurturing critical thinking, modeling, and mathematical reasoning.

By intentionally cultivating these practices, we empower students to not only master the math content but to apply their knowledge confidently and flexibly to solve problems arising in their world, paving the way for success in middle school and beyond. Additionally, secondary math teachers build on your work in elementary to prepare their students for college-level work, professional environments, and analytical thinking in all aspects of life. Our final blog in this series will explore Problem Solving (MP1, MP7, and MP8).