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Modeling With Mathematics: What it Is and How it Aligns With the Standards-Driven Classroom
I recently facilitated a teacher training at a school district with Instructional Empowerment. The training included College and Career Ready (CCR) Standards for Mathematics, and we discussed the eight standards for mathematical practice. For more information on all eight practices you can check out CoreStandards.org.
Unlike the Next Generation Science Standards, the mathematics practices are not embedded in the content standards and are listed separately within the CCR documents. However, the mathematical practices are designed to be integrated into math lessons as student behaviors to promote critical thinking and reasoning.
Needless to say, the importance of ensuring these practices are embedded into daily lessons and engaged in by students is crucial but often neglected because they are a separate document and their purpose may be misunderstood, especially the practice of modeling with mathematics.
Modeling is not modeling
For years, the common practice of teachers has been to model through the “I do, we do, you do” process by clearly describing the concept, modeling by showing the desired outcome using different instructional techniques while thinking aloud, and providing examples and non-examples to show students the expectations. This is not modeling with mathematics.
The Common Core Initiative for Model with Mathematics states that mathematically proficient students can apply the math they know to solve problems that arise in everyday life, society, and the workplace. Seems simple enough, right? But what does this look like in the classroom?Modeling w/ mathematics: Common Core requires students to apply #math to solve problems that arise in everyday life, society & the workplace. What does this look like in the #k12 classroom? Click To Tweet
What does modeling with mathematics mean?
When I posed this question to the teachers, I heard the following answers:
- “Using base ten blocks!”
- “Using an array to model multiplication!”
- “I do, now you do.”
- “Using manipulatives to represent mathematical concepts!”
- “Showing students how to graph an equation!”
- “Solving a real-world math problem!”
If you are an inquiry and 5E purist, this is the moment you wait for – exposing the misconception! I expected most of these answers, since I knew most of these examples were in the next slide as non-examples of modeling with mathematics… and I was essentially telling them they were all wrong. We discussed some examples for elementary and secondary mathematics and I revealed the definition:
Modeling mathematics is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.
Seems straight-forward, right?
I was not expecting the vehemence with which the teachers defended their answers after the examples and the definition of modeling with mathematics was revealed. From their perspective, modeling a number with manipulatives, representing a function with table of values, or solving a system of equations with graphs would totally fit into this definition!
But these examples were not modeling with mathematics. I had touched a nerve and, the worst part, it was right before lunch so everyone was hungry! I decided it was time for a much-needed lunch break, more for my benefit than the theirs because I needed time to regroup!
I would never have allowed my students to leave my classroom at that level of frustration and misunderstanding. Although I knew the teachers would come back – I hoped they would all come back – I also knew that I needed to make sure everyone was constructing an understanding of this important concept so they could implement modeling with mathematics into their daily lessons with students. When they returned, I was ready. I had decided to approach this from a science perspective – go figure!
The basic modeling cycle is summarized in the following steps:
- Identify the question or problem dealing with a situation in the real world that deals with everyday life. Ask the question!
- Identify variables in the situation and select those that represent essential features of the model. Identify the constraints!
- Formulate a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. Research the problem!
- Analyze and perform operations on these relationships to draw conclusions. Develop possible solutions!
- Interpret the results of the mathematics in terms of the original situation. Select a solution!
- Validate the conclusions by comparing them with the situation, and then either improving the model. Build a prototype and improve and redesign as needed!
- Report on the conclusions using evidence and the reasoning behind them.
For us STEM geeks, this is the engineering design process – a series of steps that guides teams as they solve problems. My “go to” engineering website is TeachEngineering.org. This process also incorporates the science and engineering practices identified in the Framework for K-12 Science Education, (National Academies Press, 2012) and can be downloaded for free here.
So we were able to grapple with our misunderstanding about modeling with mathematics, and the teachers were able to discuss how to integrate modeling with mathematics into their daily instruction. Modeling with mathematics is simply the application of mathematics (and engineering) using the practices and habits of mind to solve real-world problems. This problem-solving process is integral to learning mathematics conceptually. In the real world, it’s what drives the development of innovative new technology that solves the human problem. We know it as STEM, but that is another blog.