Building Understanding

You would think that most curriculum would be designed to build student understanding, but that isn't the case. Many people, including teachers and curriculum designers, continue to advocate procedural learning over concept development. This is partly due to a natural bias to teach the way we were taught, but also partly due to the build up of anecdotal evidence suggesting that concept development is either less effective or less efficient than procedural learning.

How is that possible? How could memorization be a more effective way to learn something than understanding it? Understanding definitely pays off in the long run. Students who rely on memorization eventually get stuck while students who understand what they learn are able to move on to more advanced studies. The problem isn't that building understanding is ineffective, it's that curricula designed to build understanding are ineffective. I would argue that if we were actually building understanding with our curricula, then it would quickly become our new normal because it would be so much more effective than what we are doing now.

Let's consider the laws of exponents:

In a procedural approach, students are given the laws to memorize, and then asked to apply those laws in dozens of practice problems. To avoid interference, each law is typically introduced on a separate day. While students will still have to pick out which law to apply for a given problem on a cumulative test, few problems will require the student to apply more than one law at a time. At most, a student may be asked to simplify an expression like this:

When a procedural approach is taken, the expectation is that students will be introduced to these laws in pre-algebra, but that they won't remember them beyond the test. The laws will then be re-taught in Algebra and Algebra II, and only mastered in Algebra II after students are applying them on a daily basis. This means that we are relying on brute force and repetition to get students to mastery, and we aren't expecting the low-performing or learning disabled students who will never get through Algebra II to master these laws at all.

In a conceptual approach, the laws of exponents are first introduced by deriving them, and demonstrating how and why they work:

The theory is that if you can derive a rule and explain it, then you understand it. However, once a student can do that, the curriculum then returns to the traditional path: each law is practiced in isolation to minimize interference and mastery is not expected until two years later in Algebra II after students are applying the laws on a daily basis. So even though we are teaching for understanding, we are not expecting a different outcome. This is why many people see concept development as a waste of time.

I don't believe that curriculum designers are simply going through the motions or paying lip service to building understanding by slapping a different introduction onto the traditional procedural approach; I think that they don't know what else to do. We want the students to apply the laws of exponents. They understand the laws of exponents (at least well enough to explain and justify them). So what else is there?

The problem is that curriculum designers are conflating the laws of exponents with the task that students need to perform. Because of this, they view the derivation of the laws simply as a bridge to the laws themselves, and once the students understand the laws, they think their job is done. The goal isn't to understand the laws of exponents, but to understand how to simplify exponential expressions well enough to debug things if something goes wrong or to reason through things if something new and unexpected pops up. To get there, the steps in the task needed to be explicit and grounded enough so that students can try things out, and problems need to be complex enough so that students need to try things out. We don't want automaticity and abstraction at this point.

To launch into a complex problem, students need to understand their goal and a set of basic tools and strategies for reaching that goal.

So what does a student need to know and be able to do to perform this task?

The student needs to know that, in order to write an expression in simplest terms, he or she must be able to identify, re-group, and count all of the factors in the expression.

To re-group and count the factors, the student needs to be able to interpret exponents, which means that the student needs to know what an exponent is and apply that definition to write out the factors in an exponential expression using multiplication:

To do that, a student needs to be able to identify the base for an exponent. Our expression has six exponents. What is the base for the second exponent (6)? Is it 3a or a? What is the base for the fifth exponent (3)? If you are fluent with exponential expressions and the laws of exponents, you may be able to pick out bases intuitively, but how would you figure out a base if you couldn't see it right away?

To identify the base of an exponent, you need to apply the rules for order of operations. To make this easier, let's make each operation in the expression explicit (another sub-task that students need to be able to perform that we don't teach):

Now I can tell that the base of the second exponent (6) is a and not 3a because I wouldn't multiply by 3 until after I've evaluated the exponent.

Once all of the exponents have been written out as multiplication, then the factors can be re-arranged (commutative and associative properties of multiplication) and re-grouped using exponents.

While this process may seem more complicated than memorizing and applying the laws of exponents, all we are doing is making every step explicit. If you were to attempt to solve the same problem using the laws of exponents, you would find yourself going through the same process. You may not be aware of some of the steps because they are so automatic and you are doing them in your head subconsciously, but they are there. Making them explicit actually makes it easier for students who don't yet have that level of automaticity. It also makes it easier to catch any errors you may be making.

A common error is treating coefficients like exponents. When students are adding exponents, they instinctively want to add the coefficients:

And when they are multiplying exponents, they want to multiply the coefficients:

Their pattern recognition causes them to hone in on the numbers and to start operating on them instead of thinking in terms of factors. Thinking in terms of the laws of exponents encourages students to go straight to computation. You identify which law applies, and then you add or multiply.

Stepping back and explicitly thinking in terms of factors instead should help students catch and correct these errors:

1. What factors are there in this expression? A 6 and two a's, and a 3 and five a's.
2. Can I re-arrange and group any of the factors? I can re-arrange the factors to group the a's, so now I have a 6, a 3, and seven a's.

The laws of exponents don't enter the student's thinking at all except as a short cut for combining two a's and five a's.

1. What factors are there in this expression? Four 3's, four groups of five a's, and four groups of two b's.
2. How many a's and b's do I have? Four groups of five a's is twenty a's and four groups of two b's is eight b's.
3. Can I re-arrange and group any of the factors? Nope.

Again, the laws of exponents only serve as an efficient way to count factors.

When building understanding, we need to be clear on the tasks that we want students to perform. We don't want them to derive or apply the laws of exponents. We want them to simplify expressions containing exponents. When we do that, we uncover critical sub-tasks that students need to be able to do, such as identify the bases of exponents. If a student cannot identify the base of an exponent, the laws of exponents are not going to help, and the student will be unable to complete the task.

The mistake that many curriculum designers make is thinking that breaking down a task represents a form of scaffolding. Scaffolding are supports that we put into place when building a structure, but that are removed when the structure is complete. That is not at all what we are attempting to do. The sub-tasks that we identify through a task analysis represent what students need to know and be able to do; they aren't a bridge to a higher level abstraction that will eventually replace the sub-tasks, but a foundation for reasoning and problem solving that enables abstraction.