In this final post about curricular coherence, I’m pinch-hitting for Bill, who is busy reorganizing his wine cellar. This time, we talk about coherence of mathematical practice.
We value coherence of content because we believe that a coherently arranged curriculum makes it possible for a student to see the subject as a whole, to understand the logical connections and deep structures, and to use that understanding for more efficient problem-solving and better retention of knowledge and procedures. But making it possible does not make it probable. The way students do mathematics, their mathematical practice, may have an effect on their ability to take advantage of a coherent curriculum. The CCSSM describes eight aspects of the complex construct of mathematical practice. Here we focus on two aspects, using structure (MP7) and abstraction (MP8).
Structure in arithmetic and algebraic expressions reveals what might be called “hidden meaning.” For example, writing $x^2-6x-7$ as $(x-3)^2-16$ reveals that, for real values of $x$, the expression assumes values greater than or equal to $-16$ (and it assumes that value only when $x=3$). Writing it as $(x-7)(x+1)$ highlights the values of $x$ that make the expression 0.
Treating pieces of expressions as a single “chunk” can simplify calculations; seeing that $4x^2-8x+3$ can be written as $(2x)^2-4(2x)+3$ helps one obtain the factorization from the (easier) factorization of $z^2-4z+3.$ This example can be generalized to encompass all polynomial expressions, providing students with a general purpose tool that can be used to transform a general polynomial into one with leading coefficient~1. It amounts to a change of variable in order to hide complexity, a practice that is useful all over mathematics.
Hidden meaning in geometric figures often involves the creation of auxiliary lines not originally part of a given figure. Two classic examples are the construction of a line through a vertex of a triangle parallel to the opposite side as a way to see that the angle measures of a triangle add to $180^\circ$ and the introduction of a symmetry line in an isosceles triangle to see that the base angles are congruent. Another kind of hidden structure makes use of the invariance of area when it is calculated in more than one way—finding the length of the altitude to the hypotenuse of a right triangle, given the lengths of its legs, for example.
A final example of using structure is in the view that students form of the base ten notational system. The compactness and regularity of this system make it useful for efficient computation and estimation. But in that compactness there is also the danger of superficial, and therefore fragile, grasp of procedures. The Number and Operations in Base Ten domain in CCSSM lays out a progression designed to help students learn to see the decimal expansion of a rational number as, in advanced language, a linear combination of powers of 10 with coefficients taken from integers between 0 and 9 helps. Similarly, viewing a polynomial in $x$ as a linear combination of powers of $x$ can lead to an understanding of polynomial algebra as a system in its own right. Writing $3x^2-7x + 5$ “in base $(x-2)$” as
$$
3(x-2)^2+5(x-2) + 3
$$
reveals another kind of hidden meaning in the expression.
Another theme that runs throughout a coherent curriculum is a cross-grade emphasis that helps students develop and use the many faces of abstraction. One of the most important uses of abstraction is captured in the CCSSM Standard for Mathematical Practice no.~8 (MP8), “Look for and express regularity in repeated reasoning.” It asks students to abstract a process from several instances of that process in a way that doesn’t refer to the inputs to any particular instance. Describing that process in precise algebraic language allows one to create general algorithms, equations, expressions, and functions. This practice can bring coherence to many seemingly different areas of the curriculum that often cause students difficulty.
The description of MP8 in CCSSM gives the following example:
By paying attention to the calculation of slope as they repeatedly check whether points are on the line through $(1, 2)$ with slope 3, middle school students might abstract the equation $\frac{y – 2}{x – 1} = 3$.
Helping students develop the habit of testing several numerical points to see if they are on the line and then looking for and expressing the “rhythm” in their calculations gives them a way to find the equation of a line between two points without leaning on formulas (“point slope form,” for example), and, more importantly, it gives them a general purpose tool for finding Cartesian equations of geometric objects, given some defining geometric conditions.
As another example, consider the task of building an equation. Teachers know that building is much harder for students than checking. The same practice of abstracting from numerical examples is useful here, too. For example, consider the stylized story problem:
Emilio drives from Tucson to Phoenix at an an average speed of 60MPH and returns at an average speed of 50MPH. If the total time on the road is 4.4 hours, how far is Tucson from Phoenix?
The practice of abstracting regularity from repeated actions can be used to build an equation whose solution is the answer to the problem: One takes several guesses (for the distance) and checks them, focusing on the steps that are common to each of the checks. The goal isn’t to stumble on (or approximate) an answer by “guess and check;” the goal is to come up with a general “guess checker” expressed as an algebraic equation:
$$
\frac {\text{guess}}{60} + \frac {\text{guess}}{50}= 4.4
$$
These two examples seem quite different, but coherence comes from the fact that exactly the same mathematical practice is used to find an algebraic equation whose solution solves the problem.