There is a great need to promote the awareness of modeling mathematical ideas across the curriculum in order for students to gain conceptual understanding of the methods employed to solve real-world problems. What does modeling mathematical ideas mean to elementary and middle-grade teachers and students? The term “model” is used in many different ways in our everyday language and also has many interpretations within our academic vocabulary.

The release of the CCSS-M has also drawn a lot of attention to the word model and modeling. Some of the ways in which models and modeling are used are from the specific grade-level standards, for example, from the third-grade CCSS-M standard, 3.OA. develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models and another, 3.NF.3 recognize and generate simple equivalent fractions, for example, 1 / 2 = 2/4, 4 / 6 = 2 / 3. Explain why the fractions are equivalent, for example, using a visual fraction model.

The term “model” is used differently in the fourth Standard for Mathematical Practice, “Model with Mathematics” and also in the high-school content standard, “Mathematical Modeling.” To understand the CCSS-M, let us take a look at the description for the mathematical practices standard 4: model with mathematics:

Text Box 1.2 Model with Mathematics. From Standard 4 of CCSS-M

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. (NGACBP & CCSSO, 2010)

“Model with mathematics” calls for an important mathematical practice to connect between real-world scenarios and mathematical representations to solve problems where students “can apply the mathematics that they know to solve problems arising in everyday life, society, and the workplace.” In the description, it also states, “They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.”

While we worked on defining what strategic competence was for our teachers, we noticed that there were many parallels with the recently released Mathematical Practices from the CCSS (NGACBP & CCSSO, 2010). In particular, we saw a direct connection to standard 4: model with mathematics and standard 5: use appropriate tools strategically. As students develop their algebraic habits of mind, they need to be encouraged to effectively communicate mathematical ideas using language efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.

As we consider all the important and different ways that we model mathematical ideas in elementary and middle grades, we offer Developing Strategic Competence in Modeling Mathematical Ideas as a balanced approach to teaching and learning mathematics. Developing Strategic Competence in Modeling Mathematical Ideas (see Figure 1.2), as we define it in our book, includes (1) the important representational models, tools, technology, and manipulatives that bring mathematical thinking visibly

Figure 1.2 Developing strategic competence in modeling mathematical ideas. Source:Authors.

as student communicate their mathematical understanding; (2) the interpretative models of mathematical ideas critical to understanding a specific mathematics concept; (3) the application of mathematics for problem solving; (4) problem posing; and (5) mathematical modeling (more about it in chapter 3).

In the figure above, we use a metaphor of gears that must work together, because the different models and modeling must all work together to provide students’ mathematical understanding, fluency, and mathematical power.

One of the ways that we discuss developing strategic competence in modeling mathematics ideas is through problem solving. According to the NCTM, problem solving refers to mathematical tasks that have the potential to provide intellectual challenges that can enhance students’ mathematical development. Such tasks can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity (Van de Walle, Karp, & Bay-Williams, 2014).

Throughout our book, we engage our readers in a math happening, which is a simple yet powerful mathematical routine to include in a mathematical classroom to bring mathematics closer to each individual’s daily lives. Math happenings present a realistic problem that can happen to an individual in their immediate world (see Appendix MMI Toolkit 4—Sharing Math Happenings).

For students, they may encounter a math happening when deciding on which cell phone plan works best for their family’s usage, or when calculating their statistics for their favorite athlete or game. Polya’s (1945) famous book How to Solve It introduced the dynamic, cyclic nature of problem solving, where a student may begin with a problem and engage in thought and activity to understand it. The student attempts to make a plan and in the process may discover a need to understand the problem better. Or when a plan has been formed, the student may attempt to carry it out and be unable to do so.

The next activity may be attempting to make a new plan, or going back to develop a new understanding of the problem, or posing a new (possibly related) problem to work on. Schoenfeld’s (1985) work on problem solving described and demonstrated the importance of metacognition, an executive, or monitor component to his problemsolving theory. His problem-solving courses included explicit attention to a set of guidelines for reflecting about the problem-solving activities in which the students were engaged. Effective problem-solving instruction must provide the students with an opportunity to reflect during problem-solving activities in a systematic and constructive way.

The second way, we discuss developing strategic competence in modeling mathematics ideas, is through the use of important representational models, tools, technology, and manipulatives that bring mathematical thinking visibly as students’ problem solving and communicate their mathematical understanding. We use the five star representations (see Figure 1.3) to reinforce the notion that having representational fluency, being able to translate a mathematical idea among these representations, strengthens one’s strategic competence and conceptual understanding.

This particular meaning for modeling mathematics ideas relates to the Common Core Mathematical Practice Standard 5: use tools appropriately, as outlined in the description, “When making mathematical models, they [students] know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.” An example would be using the virtual coin toss simulator to generate outcomes to discuss the difference between experimental and theoretic probability, taking advantage of technology’s ability to generate large sets of trials in an efficient manner to reveal the “law of large numbers.” Developing representational fluency is important for a mathematics teacher and learner. Representations is an important mathematical process in the NCTM (2000) standard:

Mathematical ideas can be represented in a variety of ways: pictures, concrete materials, tables, graphs, number and letter symbols, spreadsheet displays, and so on. The ways in which mathematical ideas are represented is fundamental to how people understand and use those ideas. Many of the representations we now take for granted are the result of a process of cultural refinement that took place over many years. When students gain access to mathematical representations and the ideas they express and when they can create representations to capture mathematical concepts or relationships, they acquire a set of tools that significantly expand their capacity to model and interpret physical, social, and mathematical phenomena. (NCTM, 2000, p. 4)

A way to think about representations is that they allow for construction of knowledge from “models of thinking to models for thinking” (Gravemeijer, 1999). The Association of Mathematics Teacher Educator’s standards for Pedagogical Knowledge for Teaching Mathematics includes the ability to “construct and evaluate multiple representations of mathematical ideas or processes, establish correspondences between representations, understand the purpose and value of doing so, and use various instructional tools, models, technology, in ways that are mathematically and pedagogically grounded” (AMTE, 2010, p. 4).

Throughout our book, we will focus on strengthening representational fluency through visible thinking strategies in the classroom that will offer opportunities for students to use representations, tools, technology, and manipulatives to show their mathematical thinking.

Figure 1.3 Five star representations and math tools. Source: Authors.

The third way, we discuss developing strategic competence in modeling mathematics ideas, is understanding and interpreting important conceptual models within and between mathematics topics. For example, we discuss the different interpretations within division when we give examples of partitive and measurement division.

Partitive division would be a story problem about fair share to see how many items each person gets, whereas the measurement division model would be a story problem about repeated subtraction or measuring off a number of items to find out the number of groups. Another example would be used when teaching fractions and exposing students to the region model, set model, measurement model, and the area model for fractions. In addition, in rational-number conceptual models, Lesh, Laudau, and Hamilton (1983) cited an example of how a youngster’s rational-number conceptual model can have a within-concept network associated with each rational-number subconstruct, for example, ratio, number line, part whole, operator, rate, and quotient model. An example of a between-concept systems links rational-number ideas with other concepts such as measurement, whole-number division, and intuitive geometry concepts related to areas and number lines. In our book, we will visit these important conceptual models and interpretations within and between concepts by delving deeply in understanding the learning progressions of mathematical ideas.

Finally, but one of the most important ways that we focus on developing strategic competence in modeling mathematics ideas is by introducing elementary and middle- grade students to the nature of mathematical modeling that involves problem posing through real-world problem scenarios (more in chapter 3). These authentic mathematical modeling tasks involve students engaging in unique stages of the mathematical modeling process. The following six steps were modified from the steps as defined by the Society for Industrial and Applied Mathematics (SIAM)—Moody’s Mega Math Challenge—which is a national mathematical modeling contest for high-school students sponsored by The Moody’s Foundation (see http://m3challenge.siam.org/ resources/modeling-handbook).

1. Posing the Problem Statement: Is it real-world and does it require math modeling? What mathematical questions come to mind?

2. Making Assumptions to Define, and Simplify the Real-world Problem: What assumptions do you make? What are the constraints that help you define and simplify the problem?

3. Considering the Variables: What variables will you consider? What data/infor- mation is necessary to answer your question?

4. Building Solutions: Generate solutions.

5. Analyzing and Validating their Conclusions: Does your solutions make sense? Now, take your solution and apply it to the real-world scenario. How does it fit? What do you want to revise?

6. Presenting and Justifying the Reasoning for Your Solution: (From Bliss, K.M., Fowler, K.R., & Galluzzo, B.J. (2014). Math Modeling. Getting Started & Getting Solutions. SIAM: Society for Industrial and Applied Mathematics.)

In the following chapters, we share how these important ways models and modeling are used in teaching and learning mathematics through our research lessons, and we consider how teachers, students, math educators, researchers, and mathematicians use models and modeling to understand mathematics teaching and learning. Using this holistic approach allows one to appreciate all the different facets of modeling math ideas.

Think about it!

A close look at our national and state standards and you will see the word “model” appears throughout these documents. Take the time to search for the word model within your grade level standard. What are some examples from your current grade/standards that address these different and important ways to model mathematical ideas? Cite the specific standards and give an example.