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Modeling mathematical ideas include real-world problems in mathematics and have different interpretations at multiple levels of learning. For example, modeling mathematics in elementary grades can simply refer to writing number sentences or multistep equations using the four basic operations to describe mathematical situations involving addition, subtraction, multiplication, and division. In addition, modeling mathematics can be used to mean employing mathematics to analyze a real-world problem in their school and community.

At the secondary-school level, modeling mathematics can refer to using principles from geometry to create an architectural design or using calculus to obtain marginal cost and revenue in economics. At the college or university level, modeling typically helps students to research an important problem in their field that can impact society. To develop students to become mathematically and conceptually proficient in modeling, it is important to teach them how to: apply their prior knowledge to observe and theorize from a situation; make suitable assumptions and formulate a problem; identify essential inputs and outputs to describe the problem; make educated approximations to simplify complex problems and perform related analysis; implement the approximated problem via simulations and validate against benchmark solutions; and validate the model by comparing with true experimental data that can help predict the evolution of a more efficient model.

The process described above for modeling can be implemented using the National Council of Teachers of Mathematics (NCTM) process standards, which include problem solving, reasoning and proof, communication, representations, and connections. Related to these process standards is the framework called the Five Strands of Mathematical Proficiency described by the National Research Council’s (2001) report called Adding It Up. The five strands of mathematical proficiency include conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001). These strands are interwoven to represent the interconnectedness of these proficiency strands (see Figure 1.1).

In particular, the strands of conceptual understanding and procedural fluency are essential for students in all grade levels in order to comprehend the mathematical concepts, perform operations, and identify appropriate relationships. Strategic competence and adaptive reasoning are especially important when students make sense of mathematical ideas as they problem solve, problem pose, and justify one’s reasoning. One of the last but most essential strands of mathematical proficiency is having a productive disposition toward mathematics. This is something that we refer to as developing students’ mathematical hearts and minds. That means, mathematics should be a sense-making experience that is positive where students feel exhilaration from discovering and making mathematical connections marked by the “Aha” and Eureka moments. Many of our former pre-service teachers, who reflect on their elementary and middle-school mathematics, recount how they started disliking mathematics later in middle and high school because mathematics became a series of senseless steps of operations and rules that they memorized without constructing meaning.

Each of the strands of mathematical proficiency discussed above connects directly to the Standards for Mathematical Practice. Standard 4 of Common Core State Standards in Mathematics (CCSS-M) (NGACBP & CCSSO, 2010) describes mathematically proficient students who can apply what they know to simplify a complicated

Five strands of math proficiency. National Research Council, 2001

Figure 1.1 Five strands of math proficiency. National Research Council, 2001.

Table 1.1 Common Core Mathematical Practices (NGA Center and CCSSO, 2010, p. 6) with question prompts for encouraging mathematical practices (Suh & Seshaiyer, 2014)

Mathematical Practices

Questioning Prompts

(MP1) Make sense of problems and persevere in solving them.

Does the problem make sense? What do you need to find out? What

information do you have? What strategies are you going to use? Does this problem require you to use your numeric ability, spatial reasoning, and/or logical reasoning? What can you do when you are stuck?

(MP2) Reason abstractly and quantitatively.

What do the numbers in the problem mean? What is the relationship among the numbers in the problem? How can you use number sense to help you check for reasonableness of your solution? What operations or algorithms are involved? Can you generalize the problem using symbols?

  • (MP3) Construct viable arguments and critique the reasoning of others.
  • (MP4) Model with mathematics.

How can you justify or prove your thinking? Do you agree with your classmate's solution? Why or why not? Does anyone have the same answer but a different way to explain it? How are some of your classmates strategies related and are some strategies more efficient than others?

How is this math concept used in a real-world context? Where have you seen similar problems happening in everyday life? Can you take a real- world problem and model it using mathematics? What data or information is necessary to solve the problem? How can you formulating a model by selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables?

(MP5) Use appropriate tools strategically.

What tools or technology can you use to solve the problem? Are certain manipulatives or representations more precise, efficient, and clear than others? How could you model this problem situation with pictures, diagrams, numbers, words, graphs, and/or equations? What representations might help you visualize the problem?

(MP6) Attend to precision.

What specific math vocabulary, definitions, and representations can you use in your explanation to be more accurate and precise? What are important math concepts that you need to include in your justification and proof to communicate your ideas clearly?

(MP7) Look for and make use of structure.

What patterns and structures do you notice in the problem? Are there logical steps that you need to take to solve the problem? Is this problem related to a class of problems (i.e., multi-step, work backwards, algebraic, etc.)? Can you use a particular algorithmic process to solve this problem?

(MP8) Look for and express regularity in repeated reasoning.

Do you see a repeating pattern? Can you explain the pattern? Is there a pattern that can be generalized to a rule? Can you predict the next one? What about the last one?

situation and who can “apply the mathematics ... to solve problems in everyday life.” In addition to modeling with mathematics as described in the standards, the process of selecting and using tools to visualize and explore the task connects with CCSS-M standard 5: use appropriate tools strategically. Furthermore, as students struggle to make sense of their task by translating within and among multiple representations, they develop an important aspect of strategic competence in mathematics. To help us discuss the important ways that modeling mathematical ideas support teachers’ and students’ strategic competence, we created the visual in Figure 1.3 to show how all these different modeling activities support the “ability to formulate, represent, and solve mathematical problems” (NRC 2001, p. 116).

In one of our chapters entitled, Developing Strategic Competence by Teaching through Mathematical Practices (Suh & Seshaiyer, 2014), we discussed what strategic competence means for teachers. We defined strategic competence for teachers to include the ability to (a) formulate, represent, and solve problems; (b) model mathematical ideas using engaging real-life problems; and (c) demonstrate representational fluency by translating and connecting within and among students’ multiple representations and strategies with accuracy, efficiency, and flexibility. One of the ways that teachers can help students advance their mathematical reasoning is through explicit use of questioning and mathematical discourse. To bring out the Common Core Mathematical Practices, we developed some questions that teachers could use in their classroom to prompt their students (see Table 1.1).

These question prompts can help students make sense of problems and persevere in solving them (MP1) and sustain students’ interest in the problem-solving process by engaging them in mathematics conversation where they can discuss the reasoning of their classmates (MP3). Teachers can focus on having students’ reason abstractly and quantitatively (MP2) so that they can better connect the conceptual understanding with the procedural fluency. Students model mathematics by visualizing the problem situation and relating the problems to real-life scenarios (MP4) while using appropriate tools to represent their thinking (MP5).

Students use mathematical structures (MP7) in computational work, data analysis, algebraic, and geometric structures to look for patterns and methods in the repeated reasoning (MP8). To develop these important mathematical practices, teachers need to provide time and space for students to reason, critique classmates’ reasoning, and value different perspectives to problem solving.

In addition, teachers need to develop their own strategic competence as they evaluate strategies and judiciously select the strategies as pedagogical content tools (Rasmussen & Marrongelle, 2006) to help advance students’ mathematical thinking. Developing proficiency in teaching mathematics with a focus on eliciting students’ strategic competence requires more than the analysis of students’ diverse strategies; more time and space is needed for students to reason through by sharing, arguing, and justifying their strategic thinking as well.

Think about it!

What are some classroom strategies you use to promote mathematics proficiency as shown in the Five Strands for Math Proficiency?

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