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!