Previously, I’ve written about some of the Common Core Standards for Mathematical Practice. Today I’ll focus on tightly-linked Standards 2 and 4: Reason abstractly and quantitatively and model with mathematics. Both involve the processes of de- and re-contextualization, which are a foundation for abstract thought in math.
An example: say a caterer charges a $200 base fee plus $12 for each person at a party. We can create a mathematical model of the caterer’s income: the equation y = 200 + 12x. This equation is an abstract, decontextualized entity; it can be manipulated without any reference to the context. We can graph the line, calculate the x– and y-intercepts, and find where it intersects other lines, all without once considering the caterer. In fact, the equation could represent something totally different, like the relationship between your starting bank balance and how many $12 pizzas you can buy if your account has a $200 minimum. After manipulating the model (say, substituting 500 for y and solving to get x = 25), the situation can be recontextualized: the caterer needs 25 people to attend the party in order to make $500.
Our ability to think abstractly doesn’t fully develop until our early to mid-teens (or even later). Students in middle and high school classes will vary wildly in both their developmental stage and their naturally wired propensity for one type of thinking versus another. Many kids, especially those who tend to struggle in math, learn math best when it’s contextual. It gives them a framework on which to hang abstract mathematical concepts. As they grow in their ability to transfer back and forth between concrete and abstract, they will need the concrete less and less.
Here are some things students can do in math class to build their modeling and abstract thinking skills.
- Represent quantitative relationships in several ways (graphs, tables, equations, words) and explain how they all express the same thing
- Match equations to word problems, explaining their reasoning
- Describe how the numbers and variables in an expression relate to the real life scenario they are modeling
- Translate word expressions like “four more than double the square of a number” into math symbols
- Given problems with extraneous or insufficient information, identify information that is either unnecessary or lacking
- Make up different stories for a decontextualized math expression or equation; compare with other students’
- Build physical models of algebraic expressions (using algebra tiles for completing the square, for example)
- Build physical models of geometric objects, abstracting them in various ways. For example, use marshmallows and toothpicks to represent vertices and edges of a cube, while other students mold clay into a solid cube. How are the representations the same and different? How do they relate to the expression y = x3? How about the graph of the function f(x) = x3?
These skills will clearly translate to other fields, as students wrestle with metaphor in English class and abstract concepts like “liberty” in Civics. An entry into the world of abstract thinking is one of the great gifts that mathematics gives us.
How have you taught abstract reasoning? Leave your thoughts in the comments.