Brady Fukumoto

It starts as early as arithmetic. Students who are slow on their basic math facts begin falling behind their faster peers as soon as multi-digit arithmetic. That's what researchers found in the study "Lack of Automaticity in the Basic Addition Facts as a Characteristic of Arithmetic Learning Problems and Instructional Needs." "The slowest cluster reported use of counting strategies on many bundles; the fastest cluster reported use of retrieval or efficient-thinking strategies. Cluster group was the best predictor of performance on multidigit tasks. ... Analysis by error type showed most errors on the multidigit sums were due to fact inaccuracy, not algorithmic errors. The implication is that the cognitive demands caused by inefficient solutions of basic facts made the multidigit sums inaccessible."

When should students memorize math facts? I wouldn't ask a student to memorize times tables until: 1. They understand that multiplication is repeated addition. 2. They’re able to calculate times tables using that definition. But students don’t always have to derive things before memorizing them. For instance, in university math it’s efficient to: 1. Memorize derivative rules first. 2. Later derive them from scratch. In general, a student should loosely understand what something means before memorizing it. This does not always have to be a derivation / full computation. It’s just that in arithmetic, full computation turns out to be the most straightforward way to understand things. In calculus, if you want to loosely explain what the derivative means, then derivations are not as conceptually straightforward. Instead, it’s more straightforward to just explain that the derivative represents the instantaneous rate of change, or the slope of the tangent line to the curve. With that information, it makes sense that, e.g.: - The derivative of x^2 should get larger and larger as x increases. - The derivative of sin(x) oscillates up and down. That’s the main thing I’m trying to get at: The reason why it’s helpful to “loosely understand what something means” before memorizing it is that it makes the memorization process a lot easier. And the goal should always be to approach that baseline understanding in the most quick / easy / straightforward / simple way, whatever it may be.

For the vast majority of kids, memorizing basic math facts is not a big ask. Math education is a weird world where people fight wars against the mathematical equivalent of learning to tie one's shoes. Like, it's obviously super useful and it's not that freaking hard! Just do it!

The way you make hard things easy is by mastering subskills. When you're automatic on basic arithmetic, solving equations feels like swiftly moving puzzle pieces around, and you just need to identify how they fit together. But without automaticity on basic arithmetic, each puzzle piece is a heavy weight. You struggle to move them at all, much less figure out where they're supposed to go. It's the same with all skill trees.

Right answers do matter in mathematics. And furthermore, answers that are not just right, but also reasonably fast. Why? Because you gotta develop automaticity on low-level skills in order to free up mental resources for higher-level thinking! If you don't, then low-level actions will constantly be interrupting your flow of thought and you won't be able to see the forest for the trees. Think about all the skills that a basketball player has to execute in parallel: they have to run around, dribble the basketball, and think about strategic plays, all at the same time. If they had to consciously think about the mechanics of running and dribbling, they would not be able to do both at the same time, and they would not have enough brainspace to think about strategy. The same thing happens in math. For instance, solving equations feels smooth when basic arithmetic is automatic -- it's like moving puzzle pieces around, and you just need to identify how they fit together. But without automaticity on basic arithmetic, each puzzle piece is a heavy weight. You struggle to move them at all, much less figure out where they're supposed to go. It’s the same all the way up the ladder of math: from arithmetic, to algebra, to calculus, to all sorts of university-level math and beyond. If you try to advance to the next rung on the ladder without standing stable on the previous rung, then you’re going to fall off the ladder. And even if you get saved by grade inflation and manage to make it up a step without completely falling off, it’s the next step that will get you.