Conceptual Understanding Doesn't Happen All At Once

It happens slowly, in bits and pieces

One idea that has become popular in math education in the last ten years is the idea that conceptual understanding should come before procedural fluency. Many of the curricula that are popular right now take this approach to a fault.¹ One way this goes badly is these programs assume conceptual understanding can happen all at once: if the tasks are rich enough, if they connect to real-world contexts that students relate to, if the teacher leads an effective discussion, then students will suddenly have that conceptual understanding and the procedures will be easy.

In the real world, conceptual understanding develops gradually. It happens in fits and starts over time. Here's an example of what I mean, from my current unit on integer addition.

One piece of conceptual understanding that's important for adding integers is to visualize addition as moving around on a number line.² If students have a strong conceptual understanding of adding integers on a number line, they can use it to solve more or less any problem. For instance, if I want to add -2 + 11, I start at -2, and then I move 11 to the right. If I want to add -4 + -8, I start at -4, then move 8 to the left. The idea doesn't seem too hard, and if students understand that idea, the logic goes, they can solve any problem.

The way many (not all, but many) popular curricula do this is they pick some context, often money or temperature, and have students do some sort of task that leads to using number lines. The teacher leads a discussion, the students understand number lines, and then students do some problems or tasks designed to practice the concept.

That approach hasn't worked very well for me. Number lines are a great tool, but when I try to introduce them and start solving all sorts of different problems students get tripped up. They try to count 11 to the right from -2, but that's a lot of counting and it's easy to make a mistake, often an "off-by-one" mistake. The negative part of the number line can feel scary, some students do great with problems on the positive side but get confused on the negative side. Crossing 0 is tricky. That's not every student — some students pick up the number line stuff right away. But that can honestly make it worse. Some students are confused and frustrated that they're getting problems wrong, and seeing students who just get it can make them feel like they will never figure out this negative numbers thing.

My approach is to emphasize number lines, but to build up that conceptual understanding gradually, over lots of days, by breaking integer addition down into lots of small pieces and slowly building up. Here are the pieces I break integer addition into:

1. Get students really comfortable placing positive and negative numbers on number lines. Don't take this for granted. I start practicing this the last week of the previous unit, and mix in quick chunks of practice putting numbers on number lines and ordering negative numbers. We also use a few contexts: thermometers and moving up and down in elevation are good ones. All this helps students get comfortable with the negative part of the number line.

2. Do "small addend" problems starting on the positive side. These are problems like 8 + -1 or 12 + -3 where the second number is close to 0.³ This helps students get used to moving around on the number line, and understand that negative numbers mean moving left, without getting too complicated. Mix in some positives like 7 + 3 so it isn't always moving to the left.

3. Do "small addend" problems starting on the negative side. At first I use a positive addend, like -6 + 2 or -10 + 3. This helps students get used to moving around on the negative side of the number line. Do some mixed practice with the last skill.

4. Now mix in "small addend" problems with two negatives. Practice these for a bit. Students will like the fact that it's just like adding with positive numbers, but the answer is negative. Use money as a context to help understand why this is true (this context will be important to revisit later, as students often feel good about this subskill until we get to multiplication). If I owe one person $3 and another person $5, I owe $8 in total. Do some mixed practice with the last few skills.

5. Solve additive inverse problems like -5 + 5 and 8 + -8. Again, money is a good context here. These are surprisingly tricky at first!

6. Now look at how addition is commutative, or order doesn't matter. 8 + 3 = 3 + 8. -1 + 2 = 2 + -1. Therefore, -2 + 11 = 11 + -2. That's a lot easier! Number lines are a great way to understand this idea.

7. Finally, get to "large addend" problems on the number line. These are really tricky, and I teach students another representation and strategy for these that I'll put in a footnote.⁴ But the key to doing something like -8 + 10 on a number line is to decompose. I start at -8. 10 will take me past 0, so I start by jumping 8 to 0. 2 are left over from the 10, so the answer is 2. That decomposition looks a bit different depending on the specifics of the problem, and for these it's good to have another strategy handy, but it's also a great way to deepen students' understanding of the number line.

The key to making this all work is lots of number lines early on that gradually go away, lots of mixed practice, and lots of chances for conversation and feedback. There’s no one grand reveal when everyone arrives at the same understanding, that’s not realistic.

Final Thoughts

One of the hardest things about teaching math is that, for most teachers, we know the content we are teaching so well that all of the little pieces become invisible. It's like reading: it's a small miracle that the human mind can turn scratches into words and ideas so fluently, with so little conscious processing. Early elementary teachers are the most skillful educators, taking an incredibly complex skill and breaking it down into small pieces and then building students up to become fluent readers. In math we have lots and lots of skills that, just like reading, are composed of tiny pieces that become invisible to us. That's the blessing and the curse of the human brain: it's a blessing that those pieces become invisible because it frees up our minds to do more and more complex math, but it's a curse in that it makes teaching much harder.

It's easy to fool yourself into thinking the conceptual-understanding-all-at-once approach works. It works for lots of students. Those are the students who arrive with a decent foundation thinking about number lines, working with negative numbers, decomposing numbers, and more. For those students, number lines just fit with everything else they know and it clicks quickly. But for other students who have a longer way to go, they need this skill broken down into pieces, and lots of time to build up their conceptual understanding. There's a way of teaching this that takes the conceptual-first approach, watches a bunch of students struggle, and blames them. They weren't paying attention during the discussion, or they're answer-getting, or whatever. That doesn't help students. Breaking skills into pieces is a much better way to help every student learn. Not every student needs it, but some students can’t learn without it.

1. It’s not every curriculum, but this idea has taken a pretty large market share compared with 15 years ago.

2. There’s a lot more to integer addition than number lines, this is just one important strand from the unit.

3. I realize that technically both numbers in an addition problem are addends so this isn’t mathematically precise but there’s no name for the second number in an addition problem so “small addend” it is.

4. For a problem like -8 + 10, the non-number-line strategy is to ask two questions. First, are there more positives or negatives? And second, how many more? In that problem there are more positives, and there are 2 more positives than negatives so the answer is 2. Money is a good context for this one, and so is this Desmos lesson “Floats and Anchors.” I prefer this strategy for “large addend” problems, some students who aren’t very good at decomposing really struggle with problems like -8 + 10 or 7 + -11 if their only strategy is the number line.