Whoever designed the world of math wasn’t an architect. The nature of math curriculum is that it is fundamentally unstable because it is top-heavy. Nearly every math concept we learn is part of the foundation of many more concepts. Every door we open in math reveals three more. Math is an upside-down pyramid.
As students move up the inverted pyramid, frustration becomes increasingly common. Normally, this is attributed to the difficulty of the advanced content, and there is some truth to that. But it is much more common for the frustration to arise from holes in the base of the pyramid. The upper levels of the pyramid are very hard to build if you’ve forgotten the ideas upon which they are built. High level concepts are nearly impossible to learn without the proper foundation.
Frustration of this sort is especially common in the spring. The end of the schoolyear is when knowledge gaps get revealed most dramatically. For example, in the latter third of the year in high school geometry, most of the new concepts are combinations of several old ideas and a couple of new ones. If you don’t have the old ideas down, you’re going to be lost. Comprehensive finals can be particularly punishing for students who have holes in the lower levels of their math pyramid.
Math isn’t the only subject that is built this way. Science classes, especially chemistry, are upside-down pyramids, too. Learning a foreign language works this way as well. If you only did the bare minimum to keep up in Spanish I, you can expect Spanish II to be brutally hard. Do yourself a favor and get caught up.
You don’t want to wind up with a knowledge pyramid that looks like Swiss cheese:
The solution to this issue is to walk the mastery path. As soon as new content arises, pursue understanding aggressively. Once understanding is achieved, keep practicing. Regularly revisit old concepts from earlier in the year and from previous years. If you don’t, you’ll forget.
The inverted pyramid of mathematics is constantly being eroded, so it needs constant upkeep. Patch holes in the base as soon as you notice them.
Returning to the fundamentals of math is never a waste of time. Content at the bottom of the pyramid is too important to ignore. We can prop up a weak foundation with a calculator, which is often an appropriate accommodation. This comes back to bite you, however, when you run into no-calculator tests. The new SAT, by the way, includes a no-calculator section.
When you know the fundamentals very well, moving up the pyramid takes much less effort. A good target to aim for is automaticity, which means knowing something so deeply that you can do it without thought, as a reflex. For example, you know the answer to 2+2 instinctively; that sort of arithmetic takes no effort at all. For every foundational math problem you can solve in this automatic way, you’ll have extra brain power leftover to figure out new math concepts.
In other words, learning and practicing the fundamentals of mathematics dramatically reduces cognitive load, the amount of mental effort required for the task at hand. Climbing up the pyramid is hard, so taking the time to lighten your load is smart. You’ll work harder now, but you’ll save energy later. Solving harder math problems takes a profound effort if you don’t have the basics down. For many students, that is the primary challenge they face when moving forward in math.
It’s never too late to get caught up. Summer is an excellent time to patch holes in your foundation. You can’t afford to have a weak base. Too much rests upon it.
Title Image: Loper, Chris. 2016.
Swiss Cheese: Loper, Chris. 2016.
Math Pyramid: Loper, Chris. “Math is an Upside-Down Pyramid.” 2016.