(This blog post is a slightly expanded copy of a thread I wrote on Twitter in 2021.)
I have met people before who say that statistics and applied maths are easy for students because it’s just subbing numbers into the right formulas. Sometimes it’s not quite so overt, and I just see people express frustration because the sudents just aren’t succeeding, even though all they need to do is sub things into formulas. Either way, it makes me sad and angry, because “just” subbing numbers into formulas is not easy, in multiple different ways.
I will now describe a number of those ways, numbered idiosyncratically in the order I thought of them.
1. For many students studying especially intro stats, they have very little experience with maths and what they have has been traumatic. The very act or substituting things into formulas is actually hard for them, if not triggering. Just looking at the formula will create a rush of feelings that make it hard to think.
2. For the students with little or traumatic experience with maths, one thing that really holds them back from succeeding is not knowing the order of operations. They don’t know that addition is after multiplication, or that their calculator will in general know the correct order. This means even if they put the numbers in the right places, they won’t get the right answer every time.
3. The formulas often use letters that the students have never seen before and don’t know what they’re called. I can’t count the number of students I’ve met who don’t know the name of the letter μ. There is nothing like not being able to read the formula to make you feel like an idiot.
4. Even if you can read the formula, finding which numbers in the problem go into it is often difficult. It’s like one of those reading comprehension tasks that ask about technical details you probably missed. You have to go searching again to find the numbers you need hidden in the text.
5. The search for the right numbers also requires you to know what the words mean. If your formula has a standard deviation, then it’s useful to know what standard deviation means, if only to know how people talk about it so you can find it in the problem text. This is further complicated by the fact that the context can change the meaning. If it’s implied in the text that the standard deviation came from the data not somehow from the population, that changes everything.
6. Some might argue that you don’t need to know what the words mean. Can’t they just find the numbers next to those words in the text and put those numbers into the formula? But imagine having six meaningless words to look for every time. That’s exhausting. And do you really want students using formulas without meaning? That doesn’t seem like a good idea to me even in the short term.
6b. Many applied maths courses introduce new applications in every assignment, with a whole raft of brand-new terminologies that have letters and/or numbers to go with them, and it’s a big load to deal with even if you decide not to try to understand the context. It’s one of those difficult technical reading comprehension tasks again.
7. If there are even a handful of formulas to choose from, and they’re spread out through the course materials, it’s actually a large task to search for the right one. Flicking through all the materials, you might easily miss one, and even if you have a handy cheat sheet with them all there, you still have to find the right one among the list.
8. In order to choose which formula is the right one, you need to be able to distinguish between different situations that formulas might apply to. What features do the situations have that make them different enough to need different formulas? What features are irrelevant? This is actually hard to notice and to a novice is not obvious at all. It’s also hard to notice because it’s rare for teachers to explicitly point out those features.
8b. Indeed, to notice how the situations are similar and different, you need to have enough examples so that you can compare them to each other, and many courses only have one or two examples of each situation/formula. That’s just not enough to glean the unspoken rules of deciding between them.
9. When they’ve chosen the right formula and put the right numbers in it and gotten an answer, they still have to know what to do with the answer. This is a separate skill, often requiring more understanding of the context and also skills with written language.
10. Language is a whole set of problems of its own. In the genre of maths assignment questions, information is presented in terse sentences that assume you can tell the difference between technical words and ordinary words and keywords that indicate expectations. And you can’t always do that, especially on the fly without specific instruction.
11. Many many students actually do want to understand, despite you telling them they don’t need to. They want to make connections between ideas and have reasons why the choices were made the way they were. If this is unrequited, they come to resent the whole thing.
11b. They don’t necessarily need the whole mathematical theory, but they do need reasons and connections. And actually you yourself decide between methods based on these reasons and connections anyway.
I’ll stop there. But I hope I’ve made it clear that choosing and using formulas is a long long way from easy, so we need to support our students and give them the credit they deserve when they struggle with it. Because it is actually hard and they’re trying.
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