What Is Conceptual Understanding Good For?
Trying to be less fuzzy and more specific when I aim for conceptual understanding
What is the point of conceptual understanding? I often find myself teaching something in the name of conceptual understanding but without a very clear goal of what I want students to learn. There can be a lot of “students might learn” or “I want students to think about” and not much “here is how this lesson will help students solve problems in the future.” I also constantly struggle to figure out whether procedures or concepts should come first. I wrote this post to try and be specific and clear about what conceptual understanding is good for.
I started by trying to name some specific ways I think conceptual understanding can benefit students, with examples for where they come up. I spent a bunch of time reading about conceptual understanding and I didn’t find any single resource I’d recommend. Too many didn’t offer many examples or were full of vague statements about making connections or something. I don’t think this is exhaustive, but this does capture all the ideas I found while I tried to understand this idea better.
Course Correction
Humans forget things. If students only have a procedural understanding of something, when the procedure gets hazy they are prone to make silly mistakes. Conceptual understanding can clarify what to do and why in these situations.
When finding -8 + -5, students might be unsure whether two negatives make a positive. A conceptual understanding of adding integers using the number line or another representation can help them clarify what to do.
When working with circles, students often need to convert between radius and diameter. Without understanding what those things mean they might start randomly guessing at whether they multiply by two or divide by two. A conceptual understanding of the circle helps to avoid errors and remind students of the relationship.
When using integration by parts, students might forget the exact integration formula. A conceptual understanding of integration by parts as the opposite of the product rule can help them either reconstruct the formula, or realize that they are looking for two terms multiplied together, where one term of the derivative is the integrand.
Flexibility
Standard algorithms are great in lots of situations, but they are also inefficient in others. Conceptual understanding can offer students an alternate path in certain situations.
When subtracting 103 - 96, students might realize the numbers are 7 apart rather than stacking and borrowing to subtract.
When solving a system of equation like the one above, students might realize that both equations represent lines with a y-intercept of 5 and avoid a messier strategy.
When finding 17% of 200, students might realize that since 200 is twice as big as 100, the answer must be 34.
Why It Works
Students want to feel like math makes sense and to know why a rule works or why a procedure is what it is. Conceptual understanding help students understand the why in these situations.
The derivation of the circle area formula can help students understand that the formula isn't just made up out of nowhere.
Long division can help students understand why certain decimals repeat indefinitely.
Drawing diagrams that break fractions up into unit fraction chunks can help students understand why the "flip and multiply" strategy works for fraction division.
Which Strategy
Some problems are straightforward — solve this system of equations, add these fractions, find the volume. Others are less straightforward, and conceptual understanding can help students figure out what knowledge to draw on to solve a problem.
Understanding a bunch of different contexts for percents and the language we use to talk about percents help students understand if they are finding a part, a whole, a percent, an increase, a decrease, or some combination.
Understanding what inequalities correspond to in the world helps students realize when to use an inequality to solve a problem.
Understanding the ways area, average value, and accumulation are related helps students realize when to use an integral to solve a problem.
Connect to Prior Knowledge
New knowledge sticks best when it's connected to prior learning. Conceptual understanding is the stuff that connects concepts and creates those links.
Percents are really just fractions with a denominator of 100, and a conceptual understanding of percents can help students bring their fraction knowledge to the table.
Students are often much better at "fill in the missing number" problems like 10 + ___ = 15 than 10 + x = 15. A conceptual understanding of equations can help students connect equation solving to what they already know about arithmetic.
Linear functions are just proportional relationships that might begin at a value other than 0. Understanding this relationship helps students draw on prior knowledge of proportions, equivalent fractions, and linearity.
Why is this important?
One reason I find this framework helpful is that I often find myself engaging in fuzzy thinking around conceptual understanding, doing an activity without clear goals that doesn't actually lead anywhere. These are clear, concrete goals that I can work toward, or use to decide whether an activity is worthwhile. The other reason I think this is helpful is to clarify whether conceptual understanding should come first, or if it's ok to teach a procedure first.
Concepts First
If I'm teaching a concept for course correction or to connect with prior knowledge the conceptual understanding should usually come first. I want the connections to prior knowledge up front to help the learning stick and explicitly draw on what students already know. If concepts help to course correct, I want students to get in the habit of thinking about them from the beginning.
Procedures First
If I'm teaching a concept to help students understand why or to know which strategy to apply, those should usually come later. Every student should see the derivation of the circle formula, but tossing it in too early can just be confusing when there's a lot to remember. Discerning which strategy applies is best once the strategy itself is secure — if I start going on about when elimination vs substitution is easier, students are likely to lose the thread of the strategy itself.
It Depends
Finally, if I'm teaching a concept for flexibility it really depends and the two probably go together. Sometimes it's helpful to start with the conceptual approach, other times it's helpful to bring it in after a more efficient approach.
These aren’t hard and fast rules. All this stuff depends on the topic and the context. The big idea is: I want to have clear goals when I try to teach conceptual understanding to ensure students are learning something useful and not foundering with fuzzy knowledge. Thinking about those goals can also help me to figure out whether the conceptual understanding needs to come first.
In my opinion, this is an smart and very useful reflection. Thank you.