Mathematics and Reading are Different

On the Science of Mathematics

A bunch of different structures with a similar overall style—a useful metaphor?

Here’s a question that I wrote myself, but is extremely similar to a bunch of things I’ve seen people actually ask:

I’m a 3^rd Grade teacher (or: parent, or: journalist) and I’m enthusiastic about the Science of Reading. It’s totally changing how I think about reading, especially for struggling learners. But what about math? Isn’t it time for a Science of Mathematics movement?

And I always want to answer with a very respectful, no, I don’t think so.

There certainly is a body of research that I think math teachers should know about. And I’m not a both-sider about the so-called Math Wars. I basically think that explicit instruction is an important part of teaching, and that there is a lot of evidence to support this view. You could certainly call that research “science,” as this group does. I have no issue with that.

But implicit in the question is a comparison to reading. And I think there are important differences that make it much, much harder to talk about a Science of Mathematics. There are two related issues:

1. There are big research gaps in math.

2. Those gaps exist in part because math is not one big skill.

To be clear, there are some research results that definitely are relevant to all of mathematics teaching. I fully admit that one might call this a “Science of Mathematics.” I don’t really want to argue over terms.

But what has allowed research on learning to read to capture the imagination of journalists, parents, teachers, etc., is a single, unified story that we can tell about how children learn to read. That story has educational implications that everyone can understand.

The situation in math is just very different. There isn’t that kind of unity, which means we have to rebuild a “Science of _____” for each mathematical topic. And we haven’t gotten to many of them yet.

We Don’t Have a Science of Fractions

Fractions are hard. We have good reason to think that fractions are a gatekeeper for algebra, which is a gatekeeper for everything else. So there is a lot of energy to help students learn fractions in schools.

So, what does the story look like? What is the Science of Fractions?

To get a sense of the state of the field, you can check out a 2015 review titled Why is learning fraction and decimal arithmetic so difficult? An explicit aim of the review is to “stimulate greater amounts of research in the area,” a sign that this isn’t a fully developed picture.

A lot of the work that I’ve encountered feels preliminary. For example, in 2013, a large team including Lynn Fuchs and Robert Siegler (who seems to have given fractions a big research boost in the 2010s) published a paper reporting on their intervention for at-risk students. They report positive results, but they also emphasize the lack of prior work:

“Only a handful of studies have assessed the efficacy of fraction instruction or intervention…Even fewer studies have examined the processes by which intervention effects occur.”

And then there’s the actual chaos. Between interventions and curricula published in either psychology or math education, I’m aware of like five completely different ideas about what good early fraction instruction should look like. Some say it should start with circular wholes. Some say they should be rectangular. Or maybe you should emphasize part-whole models at first (like “2 out of 4” for 2/4). Or maybe you shouldn’t. Maybe it should all start with the number line.

This is a mess as far as I can tell, and even if you’re totally on board with explicit instruction, there is very little guidance that research can give you about how children learn fractions and how to teach them. Your hardest questions are unanswered by research.

Even The Science of Math Facts is Complicated

I think there really is a body of knowledge about math facts that you could call a science. There’s a story you can tell about how students go from counting, to using certain more complex counting strategies, to solving addition problems, to learning their addition facts, to beginning to learn about multiplication.

For a good time you can go back and read my posts summarizing addition and multiplication fact research. I want to highlight an interesting little nugget buried in those posts.

We have very good reason to think that all students should be taught an addition strategy called “counting on.” That’s when, to solve 7 + 3, you start at 7 and then count up from there: 8, 9, 10. It has been documented that the majority of students learn this strategy before commiting any addition facts to memory.

And in fact, in an intervention where this “counting on” was taught to struggling students, they also ended up memorizing more facts, with results comparable to a group that focused on drill and practice:

We examined the effects of drill and practice to encourage automatic retrieval, conceptual lessons to promote decomposition strategies, and the teaching of efficient counting strategies. Regardless of intervention approach, effect sizes were of similar magnitude, suggesting the potential efficacy of all three approaches.

Now if you’re a big fan of the Math Wars, you may find this very exciting! One of the biggest debates about multiplication facts is whether strategies should be taught prior to trying to commit facts to memory. A particularly contentious question is whether students should just do a lot of “Number Talks” to develop their mental strategies. There are progressive curricula that really emphasize the importance of these strategies—using a known fact (5 x 5 = 25) to derive a nearby one (5 x 6 = 30).

Doesn’t this settle it, then? Can’t we just rely on this addition result to show the importance and efficacy of giving strategy instruction to struggling students?

Absolutely not. Because multiplication is nothing like addition. For adding, numbers are all pretty much the same. 9 + 2 and 8 + 3 and 12 + 5, the same strategies work the same way. So when you’re learning addition, you’re learning about numbers and the entire number system. But learning to multiply by 5s and by 4s are really different from each other—skip-counting works so well and easily for 5s, less so for 4s.

I love how Bruce Sherin and Karen Fuson put it:

Stated simply, students acquire a great deal of knowledge about specific numbers-such as 4, 12, and 32-and this knowledge allows of new strategies or the use of old strategies in new contexts. For this reason the central issues associated with the learning of single-digit multiplication are very different from those associated with addition.

Because multiplication differs from addition, we can’t just port over all the results from addition. And that’s too bad, because there really is a Science of Addition Facts. But here are some things we simply don’t have research on for multiplication:

• We have very little evidence as to whether any strategy instruction is helpful.

• We have very little research on better/worse orders for learning multiplication facts. (We do know which ones students tend to find easiest…I guess teach the easiest first?)

• We don’t know if multiplication should be taught alongside division from a relatively early stage or not.

These are signficant gaps, but my point isn’t so much that the gaps exist—it’s that they exist because multiplication is different from addition.

Multiplication is not like addition, because it requires a lot of number-specific knowledge. But mathematics as a whole is not like reading, because to teach it you need a ton of topic-specific knowledge. That’s the difference.

The Many Sciences of Math

Given all the above, there isn’t really an easy or satisfying answer for 3rd Grade teachers looking to stick to the science. I mean, I’m a 3rd Grade math teacher, and over the next 8 months I’ve got to figure out how to help kids struggling with addition facts, get everybody moving on multiplication, lay the groundwork for fractions, teach a handful of geometric concepts, etc.

There is guidance from research that’s relevant for teaching all these things. I know that story. But what I’d really need is a Science of Fractions, Science of Addition, Science of Multiplication, Science of Geometry, and so on.

All that considered, I’d be pretty shocked if Science of Math really captures the public imagination, makes it to the front page, animates a movement. Though I might be wrong.

An interesting way for me to be wrong, though, would be if Science of Reading looks a lot more like the gap-filled research I described above. Here is reading expert Tim Shanahan in Vox:

“Often phonics advocates promote the use of research since research supports phonics instruction … but when it comes to specific prescriptions about how phonics should be taught, they make all kinds of claims that come down to: do it my way, I know best, don’t worry about what the science has to say,” said Shanahan. “There are features of effective phonics instruction suggested by the research and there are aspects on which there is either no research or the research rejects the advocate’s prescription.”

And here is Mark Seidenberg, who wrote the book on this stuff:

In an interview with Vox, Seidenberg expounded on his criticism: “It’s a difficult situation because people want to adopt better practices, they understand the idea that what was done before was not really based on solid ideas … but now you have a huge demand for science-based practices pursued by advocacy groups and people who don’t have a great understanding of the science.”

And then there’s Matt Barnum on how a major component of reading success—the importance of background knowledge—hasn’t been part of this Science of Reading movement/moment:

But there aren’t clear research-based answers here. Although there is solid evidence that knowledge is an important part of reading, there is less research on how schools should go about building knowledge in a way that translates into improved reading skills. 

I still think that math has more of these problems—gaps in research, unclear solutions, a discipline of many skills rather than a big one—but it could be that math and reading are more similar in that way than it might seem. I’d be wrong, but in a way that might be troubling for Science of X advocates.