Hidden Skills

The small things that hold students back

I'm a big proponent of math fact practice. Fact fluency is important, both because fluency helps students to do math more quickly and efficiently and because fact fluency frees up space in working memory and helps students learn new math. There's more to foundational number skills than being fast and accurate with math facts, though, and I worry that a focus on fact fluency alone misses lots of other stuff that's just as important.

Here's an example of a skill that some students have, others don't, and that holds back the students who don't have it.

If I tell you that 8 * 16 = 128, tell me what 128 / 8 is. What about 128 / 16? 16 * ___ = 128? Mathematically proficient humans can answer these questions pretty easily. Some students can’t.

In the US Common Core standards this idea of inverse operations is 3.OA.6. It's a 3rd grade standard. And the thing is, if a student doesn't understand this in 3rd grade, they aren't going to see it much in the few years after that. It comes up incidentally, and some kids will figure it out without being explicitly taught it. But others don’t. Then in middle school math the idea of inverse operations suddenly plays a huge role. It comes up a lot in proportional reasoning. If apples cost $2.11 each, I multiply 2.11 by the number of apples to find the cost, and I divide by 2.11 to find the number of apples I can buy for a given price. That skill, and the related skill for addition and subtraction, play a big role in understanding inverse operations and solving equations. I can solve the equation 4x = 12 by knowing that 4 * 3 = 12, but I can also solve it by knowing that 12 / 4 gives me the same answer. That second strategy is crucial for more complicated equations. It comes up when working with percentages, as well as a few places in geometry and probability. I see a huge difference between the students who understand this idea about inverse operations and those who don't. If you don't, all that learning about proportions and equations and more isn't very likely to stick because it's built on a shaky foundation.

I'm lucky to have a bit of intervention time with most of my students. That inverse operations skill has been my focus recently, figuring out which kids don't understand it and working with them one-on-one whenever I have a few minutes. We're making some progress. It's really awesome to see. It's a slow road, though.

I'm just now understanding some of the hidden skills that some students miss, that hold them back into the future. I wish teachers had a better bank of skills like this to better understand where the common stumbling blocks are for students. Does fact fluency matter? Yes, absolutely. It's hard to understand this idea of inverse operations without solid fact fluency. Shaky fact fluency holds back lots of students. But there's more to foundational skills in math than fact fluency.