We’ve come to the last two Common Core Standards for Mathematical practice: CCSMP7 / Look for and make use of structure and CCSMP8 / Look for and express regularity in repeated reasoning. Structure and pattern form much of the foundations of mathematics. What is the quadratic formula but a shortcut for the repeated task of completing the square? What are exponents but repeated multiplication? Math sets in place a system for efficient, streamlined calculations that work in all situations.
CCSMP7 involves analyzing and identifying the underlying structures of algebraic and geometric constructions. Students should be able to step back from a complicated expression like (y2 – 25)2 + 9(y2 – 25) + 14 and see that it’s very similar to x2 + 9x + 14. Lots of what goes wrong with algebra involves difficulty seeing symbolic structure. The language of math can seem opaque. I thought (x + 3)/(x + 5) = 3/5 because you can cancel the x‘s, right? What do you mean x + 3/7 isn’t the same as (x + 3)/7? Why isn’t (x + 5)2 equivalent to (x2 + 25)? Many of these common student errors can be attributed in part to a lack of understanding of algebraic structures.
CCSMP8 is about finding general methods (read: shortcuts, formulas, and algorithms) to accomplish tasks efficiently. Note that learning an algorithm that someone else has designed, like the standard multiplication algorithm, does not meet this standard. CCSMP8 asks us to let students notice and document the patterns created when they repeat a process many times. Of course, we don’t have the time for students to develop every single formula and algorithm they’ll use, but we need to recognize the difference in learning that comes from internally generated versus externally imposed structural reasoning.
Ways to teach CCSMP7 and CCSMP8
- Color! Color is a great way to emphasize structure in many settings. For example, students often struggle with when terms can be cancelled in rational expressions. Have students highlight or use colored pencils to circle factors and terms to help them see the structure of a fraction.
- Emphasize algebraic structure by replacing complex expressions with silly symbols. In the example above, I might have rewritten (y2 – 25)2 + 9(y2 – 25) + 14 as J2 + 9J + 14 to emphasize that the first term has “something” squared, and that same “something” is multiplied by 9 in the second term.
- Require students solve problems “the hard way” before moving into the easy way. This is where “drill and kill” actually comes in handy. Having students drill through several similar problems of, say, expanding and simplifying exponent problems allows them to see for themselves the patterns and regularities that develop. They will generate their own internal rules for working with exponents. Class discussions can codify and fact-check students’ rules. When I’ve given students the opportunity, I’ve seen them develop inventive, accurate, non-traditional formulas and algorithms.
- Give concrete examples whenever possible. For example, many students picture a triangle as having a straight line at the bottom (the base), and a peak at the top. But when triangles appear in different orientations, some students struggle to see the structure. To help students visualize, say, altitudes of triangles from different vertices, they can cut out triangles and set them upright on their desks using different sides as the base. Other examples of concrete demonstrations of structure include algebra tiles, a piece of string the length of one radian, and cutouts of reference triangles for the unit circle.
- Ask students to deconstruct formulas. A great example is the compound interest formula, A = P(1 + r/k)kt. This formula seems mysterious and handed down from on high, but can be broken down. What does the kt mean? Well, that is the total number of compounding periods. What does an exponent do? Oh, we are just multiplying as many times as there are compounding periods!
Seeing the structure and regularity in math empowers students to view themselves as mathematicians. Student-generated algorithms resonate deeply with kids and create pride of ownership. Together, these standards help ensure that kids have a good grasp of the foundations of mathematics.