Whether or not you teach in a state that has adopted the Common Core State Standards for Mathematics (or a modified version of them), the Common Core 8 Standards for Mathematical Practice should be something every math teacher fosters in their instructional practice. These practices are based on NCTM’s processing standards and NRC’s standards for mathematical proficiency in the Add It Up report. They are about helping students become problem-solvers, creative thinkers, communicators, users of multiple resources, and most importantly, able to apply what they know in multiple ways. That’s what teaching math should be about – helping students use what they learn in the world around them, now and in the future.
I am NOT going to get into a debate with anyone on the pros or cons of the CC Math Standards themselves. That is a politically charged hot mess. Whether you are for or against these standards is irrelevant. How you teach and support student understanding – i.e. your TEACHING PRACTICES, is what makes the difference, NOT the standards you follow in your curriculum. No matter the content standards, they way you help students learn, understand and apply those standards is important and vital, and is what the practices are all about. Lack of how to incorporate effective practices is what I have found, from years of working with teachers, is one of the biggest deterrents in student learning. And, as evidenced in many articles and classrooms I have observed, there is a great deal of misunderstanding of the Standards for Mathematical Practice and how to incorporate them effectively.
In a previous post, I highlighted Mathematical Practice #5, Use Appropriate Tools Strategically, where I tried to explain what the practice meant and provide some examples. Today I’d like to do the same thing with Mathematical Practice #4, Model with Mathematics as I think this is one of the most misunderstood, or ‘misused’ practices. I myself, before doing an in-depth study of the practices, interpreted this practice wrong. In my mind, I thought it meant that I should be using manipulatives and ‘models’ (i.e. technology simulations, physical models, etc.) while teaching and I would therefore be modeling with mathematics. That is part of the standard, but NOT the true purpose. Remember, these standards for practice are what we, as teachers, are trying to foster in our students – meaning, we are trying to help our students model problem situations with mathematics to help them better understand it and/or solve it.
Let’s look at the actual standard. I have highlighted key phrases that help clarify this standard:
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.
Notice the different ways “modeling” is represented: apply the mathematics; write an addition problem; apply proportional reasoning; use geometry; make assumptions and approximations; map relationships; use tools. This standard is not talking about physical models, but rather helping students look at a real situation (so context is important) and use mathematics to help understand it, explain it, make it simpler, relate it to something else, etc. Can they use physical manipulatives? Sure – geometry, tables, graphs, manipulatives – but all of those tools are to help them make sense and model the real world situation. As a teacher, what does this mean YOU should be doing to help foster students ability to model with mathematics? Well, for one thing, give them relevant and real problems to solve that are not “naked math” (see my previous post on this!) but contextual problems that force students to think, analyze and decide what mathematics will help them solve the problem. And then, provide them opportunities to collaborate, use a variety of tools, ask questions, and approach these problems in a variety of ways. Modeling with math means students apply mathematics and tools in ways that make sense to them so they can apply their mathematical understanding. That’s how YOU know they really understand the math, and that’s how THEY know math is relevant.
Obviously, the key here is making sure you provide relevant problems that “arise in every day situations” (as the standard emphasizes). Students learn and are more engaged if what they are doing is relevant. And there are real problems every where.
So – here’s an every-day situation that arose just last night. There were 3 winners in the 1.6 billion dollar Power Ball lottery. Depending on the grade of your students, you could ask them a simple question – should these 3 winners take an annuity or take a lump sum? (Naturally – expect explanations and support for their answers)! In order to answer this question, a lot of decisions need to be made concerning how to model the situation with mathematics. What, if any, equations would be helpful? Do they need a table to organize the data? What are the taxes and how will that impact the amounts? These are big numbers – so could they make it simpler by using smaller numbers or proportions? Think of all the great mathematics that will happen and all the modeling with mathematics students will do within the context of this real-life problem. Think of the engagement. Think of the conversations. Hard to get that with a worksheet!