How to Use Clear Mental Models

In my last post, I discussed the pitfalls of fuzzy mental models﹣mental models that do not provide a clear, unambiguous picture of what operations and symbols mean. But getting a clear mental model is only the beginning of the story.

Let’s come back to our example from last time, multiplication. Say you decided to give a simple, unambiguous definition: 3 ⨉ 5 is equal to 3 5s, 6 ⨉ 7 is equal to 6 7s, and so on, so forth. What happens next? How do you move from a clear mental model to actually being able to multiply with fluidity?

The first thing you’ll notice when you start being mindful of mental models is that they take surprisingly long to absorb. It may take a child a few weeks to become comfortable with a new definition. Even after the definition can be fluidly recited, you’ll see that some “obvious” things may not actually seem obvious to your child. For example, anyone who’s used to multiplying knows that 1 ⨉ 5 is 5, and that in general, multiplying a number by 1 results in that same number. However, this may very well not be immediately obvious to your child! That’s because it can take a while to get comfortable with the idea of “putting together copies of a number.” At this stage, any kind of gentle practice will help the mental model become fully integrated and absorbed.

After the model is at your child’s fingertips, it’s time to start making observations using the model. Some observations turn out to be a lot closer to the surface than others, and they should be worked on first. However, it’s imperative to keep the mental model in mind when deciding what’s obvious and what’s not. For instance, the fact that

3 ⨉ 5 + 6 ⨉ 5 = 9 ⨉ 5

turns out to be much more obvious than the fact that 3 ⨉ 5 = 5 ⨉ 3! That’s because 3 copies of 5 and another 6 copies of 5 clearly make 9 copies of 5, whereas the fact that 3 copies of 5 is the same as 5 copies of 3 requires something else: the realization that an array shows a multiplication. The benefit of having an explicit mental model is that it becomes clearer what things should and what things should not be apparent to your child.

As you and your child make observations, it’s important to constantly refer back to the mental model. People already fluent with a model have a tendency to think that once a student has seen a definition a few times, the student understands it and is comfortable with it. Nothing could be further from the truth. Mental models take a long time to be internalized, and it takes even longer even for all the “obvious” observations to be made. And only after the “obvious” observations are made are you ready for the more sophisticated observations that make calculations easier.

You may well wonder whether this means that progress using mental models is very slow. Won’t a student take too long to learn to learn elementary mathematics if every single observation needs to be built on top of other observations, and if you constantly need to refer back to the original mental models?

The short answer is “no.” The longer answer is that the sequence of attained skills using these methods is very different. At the beginning, you sacrifice facility with calculation for conceptual fluency. However, in the long term, the student gains robust knowledge that is retained well and easily applied. While this starts out slowly, your student will eventually speed up and surpass learners who can calculate quickly immediately but whose mental models are fuzzy.

As a small case study, I have applied these methods with my 8-year-old since she was about 4.5. In the first few years, she was neither ahead nor behind: she had started on all four operations young and she had early exposure to algebraic concepts, but she had not spent time on the standard algorithms and she was slow on many of the standard 1st grade skills. We spent our math lessons making sure her mental models were well-integrated and making observations about how the various concepts we were working with related to each other.

The further we moved into mathematics, the more rewards we’ve reaped from her clear mental models. When we finally covered the standard algorithms, they each took about a day to learn, due to the clear models of the operations and place value. When we covered fractions, the time we had spent developing models of division made things much simpler. Despite our “slow” start, she’s currently working on high school topics such as algebra and proof-based geometry. Furthermore, she is very confident in her mathematics and she is able to write down clear explanations for everything she uses.

Of course, not every child will wind up accelerated through these approaches ﹣ acceleration in mathematics is a mix of both nature and nurture. However, every child will reap the benefits of the confidence and logical thinking that clear mental models bring. Clear mental models make math make sense.

The next question, of course, is a practical one. How does one use this idea to actually teach one’s child? In the coming weeks, I’ll provide lots of examples of the (deceptively simple) lessons I give to my children and other children I teach, and I will explain how they fit into this framework. This should make the ideas clearer. Stay tuned!

This is the third and last post in a series of posts about mental models. Here are the first and second entries in the series.

Is “Counting On” Obvious?

I started teaching my older girl when she was 4.5. At this point, she could count, subitize, and show numbers on her fingers and that was about all. We started slowly and gently by introducing addition. And as soon as we got started, I got my first inkling of the fact that obvious things were not in fact so obvious. Since my goal was to make sure that my kids didn’t use any shortcuts that they didn’t thoroughly understand, this was a dilemma.

Let me illustrate. Do you know what 9+1 is? If you’re reading a math blog, you almost certainly know that it’s 10. Furthermore, you know that to add 1 to any number, you can simply take the next number in the counting sequence: 65+1 is 66, 118+1 is 119, and so on. Similarly, to add 3 to a number, we can count 1, 2, 3 starting from the number following our number. In fact, this works in general: to add something to a number, we can start at the number following our original number and count up to that something. This method of addition is called “counting on.”

Why does this work?

This is the kind of question that tends to stump people. For most of us, that fact is either completely intuitively obvious or memorized by rote. There is no “why.” It just is.

Except that if you’ve tried to teach this to young kids, you’ve probably noticed that they do not feel this way. If you ask little kids to put together 9 and 1, kids tend to start counting at 1. Assuming they don’t lose track, they’ll say the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 before they arrive at the conclusion that the answer is 10. They will not use the shortcut. It is in no way obvious to them.

That brings us to the next question. Since it’s not obvious to a little kid that 9 and 1 together make 10, what do we do? Most answers to this question fall into the following two categories:

(a) Teach a kid directly that this works. Instruct a kid that a number plus 1 is the next number and then practice, practice, practice.


(b) Let a kid discover this fact through experience. A typically developing child will eventually figure out that if you put a number and 1 together, you get the next number. After all, kids often have to figure out what happens if you take one more of a thing…

Out of those two options, I prefer (b). That’s the option that doesn’t rob kids of their innate numerical sense and instead replace it with “it’s true because the teacher told me so.” Unfortunately, option (b) isn’t very efficient, and it also tends to break down when a child is faced with concepts that aren’t often encountered in the wild. When that happens, even teachers who started out preferring option (b) tend to default to option (a). First, we’ll teach it, then we’ll cross our fingers that it’ll eventually make sense!

My preferred option is neither (a) or (b): it involves working with a child’s mental model of the objects and operations to derive each new step. Like option (a), it is directed and efficient. Like option (b), it makes deep sense to the child and leaves them with their mathematical confidence intact. It is, in fact, the best of both worlds.

This is the first in a series of posts about counting on. Here are the second, third, and fourth entries in the series. These lay out precisely how I teach counting on.

How It All Began

I’ve been teaching my kids math since before I became a homeschooler. I had never planned to pull my kids out of school (ah, the best laid plans of mice and men…), but I always figured I’d need to supplement their educations, and accordingly, I started my 8-year-old’s math lessons when she was about 4.5 and attending preschool.

As I had taught many products of our educational system over the years, I had deep knowledge of what could go wrong with their mathematical education. The main issue I’d seen over the years is something I like to call “symbol shuffling.” When they were doing math, they weren’t thinking logically: instead, they had a poorly memorized set of rules, and they tried to apply them. This would lead to equations like (﹣4)+(﹣5) = 9, because, after all, “two negatives make a positive.”

The problem, as I saw it, was that a lot of the mathematical symbols weren’t producing mental images of anything in particular. They were simply symbols on a page, and they were to be manipulated accordingly. The rules were rules not because they were logical inevitabilities: they were rules because the teacher had said so. In the context of that mindset, (﹣4)+(﹣5) = 9 makes just as much sense as (﹣4)+(﹣5) =﹣9. After all, how can something make sense when the symbols don’t actually refer to anything?

I was resolved to do better with my kids. So when my 8-year-old was 4.5, we started some very gentle math lessons, starting with simple addition and subtraction. My one overriding concern was that she needed to understand everything she used﹣I’d seen the outcome of the opposite approach far too many times. So I was determined to teach her to true conceptual mastery, and to let everything take as long as it needed to…

As you’ll see if you keep reading my blog, this worked far better than even I expected. Stay tuned!

The Blog Begins!

I’m starting this blog when my kids are 8 and (just turned) 5. In my next few posts, I’ll summarize what I’ve done with them both so far. They are both mathematically precocious, so they are developmentally ready for certain concepts earlier than standard. However, I’ve found that the tools that I’ve developed work very well with a wide range of kids.

After I post a summary of our current mathematical whereabouts, I plan to update this blog a few times a week with my 5-year-old’s work, and occasionally with my 8-year-old’s work. I’m hoping this explains our method better than a simply theoretical explanation!