President, Colorado Council of Teachers of Mathematics
This issue of the Colorado Mathematics Teacher continues the focus on the Standards for Mathematical Practice and highlights Math Practice 4: Model with mathematics.
In searching the Common Core State Standards for Mathematics from kindergarten to high school, I find over 150 references to modeling, in both the practice standards and content standards. Examples include: concrete models, model with sets of objects, area models, visual models, models for decimals, probability model, linear model, exponential model, and model relationships. Further, the word model is used as both a verb and a noun. In all instances, the word model refers to what students do rather than what teachers do.
The word, as you see from above, has various and different meanings. When we ask students to model with sets of object, we are asking them to demonstrate a concept or to act out a situation (e.g., with manipulatives). Further, model can also refer to using visual diagrams to present a mathematical idea. When asking students to use a linear model, we are focusing their attention on translating a situation into mathematics. Further, we might be referring to the modeling process as described in the Standards. The basic modeling process (see pages 72-73 in CCSS) begins with a problem and requires that students move fluidly between mathematical representations and the context of the real-world situation represented in the problem as they formulate, compute, validate, and interpret their results.
I encourage you to take a moment now to read an excerpt of Math Practice 4:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. . . . . 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. (CCSS, p. 7)
Note some key phrases in the Practice:
- Apply mathematics to solve problems arising in everyday life,
- Make assumptions and approximations,
- Identify important quantities,
- Map relationships using tools,
- Analyze relationships,
- Interpret mathematical results,
- Reflect on whether results make sense, and
- Improve the model.
These phrases, then, capture the essence of Math Practice 4.
If these are the behaviors we want students to engage in, with respect to the modeling practice, how do we support our students in developing their expertise in modeling? I suggest we begin by providing students with a variety of real-world problems or situations, support them as they try to make sense of the situation, and encourage students to represent these problems mathematically. As students mathematize the problems, provide opportunities for students to share solutions. Encourage students to justify why their model makes sense and construct a viable argument to support their reasoning.
To prepare for this support for students, it is important to work with colleagues to anticipate what students might do in order to facilitate classroom discussions in which students share multiple ways of modeling problem situations. Through conversation with colleagues, you will want to focus on what would constitute a viable argument for a solution method and explore questions that would support and further student thinking. While teaching modeling may be a challenge, it directly connects the mathematics our students learn in school with authentic, real-world problems they may encounter outside of school.
Be sure to explore this issue of the Colorado Mathematics Teacher that further develops the idea of mathematical modeling. Karen Fuson addresses Math Practice 4 through the eyes of elementary students and teachers. Ellen Whitesides, keynote speaker at our 2013 CCTM conference, shares more on the practice of modeling. And, mark your calendars now for the 2014 CCTM Conference: September 25–26, 2014.