There’s a good reason why educators often talk about the need to move beyond explanations. People who don’t know much about teaching think all the action in teaching is about the clarity of the explanation. (That, and getting kids to listen to your ultra-clear explanation.)

There’s much more to the job than that, of course. Michael Fenton puts this nicely in a recent post:

In my first few years in the classroom, I held the notion that the best way to improve as a teacher was to

hone my explaining skills. I figured that if I could explain things more clearly, then my students would learn more. […]

The best way to grow as a teacher is to develop my capacity to listen, to hear, to understand.[…] This doesn’t mean that I’ll stop working on those other skills. But it does mean I have a new passion for learning about listening—really listening—to students and their thinking.

I think this focus on listening is wonderful, and Michael did say that he’s going to keep working on his other skills, which is a nuanced take. But what about the title of the post, “Beyond Explaining, Beyond Engaging”? Philosopher Eric Schwitzgebel has a helpful distinction between a piece’s *headline view* and its *nuanced view:*

Here’s what I think the typical reader — including the typical academic reader — recalls from their reading, two weeks later: one sentence. […] As an author, you are responsible for both the headline view and the nuanced view. Likewise, as a critic, I believe it’s fair to target the headline view as long as one also acknowledges the nuance beneath.

So let’s take on that headline: should we go “beyond explaining”? If we’re trying to improve our teaching, it could be that getting better at listening has a higher payoff than getting better at explaining. But my experience has been that there isn’t any strict hierarchy of payoffs in teaching. Teaching evolves in funny ways. Last year I taught an 8th Grade class that pushed on my classroom management. This year I’ve spent a good deal of time learning how to tutor students with learning disabilities. I’d hate to say that explaining is some sort of basic teaching skill, the sort of thing novices focus on but more experienced teachers don’t need. Teaching is weirder, more cyclical, more web-like than that.

Maybe, though, we should move beyond explaining because it’s *easy — *or because pretty much everybody knows how to do it well after their first two years on the job.

That might be true, for all I know. If I doubt it, it’s only because it’s only over the past year that I’ve really started to understand some of the things that make a good explanation hum and lead to great student thinking, instead of slack-eyed drooling from the back rows of the classroom.

Besides, a lot of what I’ve learned about explaining comes from outside math education writers and speakers. Which started me thinking that maybe this knowledge (if it’s even true) isn’t as well known to math teachers as it could be.

Here’s what I think I know about giving good explanations to kids:

## 1. Study Complete Mathematical Thoughts; Don’t “Roll Them Out”

The first students I taught were subject to an especially painful type of instruction. I don’t know exactly how to describe it. Maybe an analogy would help. Imagine a magician (*Ze Great Pershan-o*) who is waaaay too detail-oriented: OK AND NOW CHECK IT OUT: I PUT MY HAND IN THE HAT! AND NEXT, I CLENCH MY FIST! HERE COMES THAT HAND SLOWLY COMING OUT OF THE HAT! ETC!

This is basically also how my explanations worked. HERE IS AN EQUATION! (WRITES AN EQUATION.) WE WANT TO SOLVE FOR X! (WRITES ‘X = ?’) HOW SHOULD WE DO THIS, CLASS? WELL, WE *COULD *DIVIDE BY 2. BUT THAT WOULDN’T HELP US VERY MUCH. LET’S INSTEAD SUBTRACT 2, ETC!

The phony enthusiasm was a problem. Another problem was that I was feeding the math one mini-idea at a time rather than presenting them the complete mathematical thought. I’ve come to think that when we do this — when we *roll out *the explanation, line by line — we lose a lot of kids.

Do you know that thing you do when you’re trying to understand something hard in a math textbook? How you put one finger at the top of the explanation, and then go line-by-line to make sure you understand each piece? But then you go back and try to make sense of the whole? That sort of self-explanation is where the learning can come from in an explanation, I think, and if we roll the explanation out, we’re making it harder for kids to look at how the pieces fit together.

So, when we’re ready to explain an idea to students, we ought to be offering them a complete mathematical thought. No need to dice it down to the atomic level, like Dumb Houdini or whatever.

Practically, this means that if I intend to “show the steps” in an explanation, I make a real effort to reveal them all at once. I project or photocopy artifacts like the ones that Algebra by Example create. If I’m working one-on-one, I’ll scribble a full example down on the page, rather than coaching a kid through that example step-by-step.

This idea, of course, isn’t my own. I came to understand this from reading about cognitive science, and especially cognitive load theory. There’s more to math than explaining, but there are ways of explaining things that preserve the math and others that make it harder for students to make meaning. Fully worked-out examples can help kids make meaning from our explanations, I think.

## 2. Use Arrows to Emphasize Process, Change and Action

This was an aha moment for me. I came to think people across education were saying similar things about how it’s easier for people to think about *actions*, rather than *properties*. You can hear this idea bouncing around research on how kids solve word problems, how our minds especially remember narratives, and what constitutes good writing.

Nobody has told me that these ideas are related. I’m a bit worried that I’m connecting totally unrelated people and ideas. Still, here’s the idea:

An interesting result of Cognitively Guided Instruction is that numerically equivalent word problems are often handled very differently by children. Consider these two problems:

**Problem 1: **Jill has 8 jellies, but a raccoon eats 5 of them. How many does she have left?

**Problem 2: **Jill has 8 jellies. A raccoon has 5 jellies. How many more jellies does Jill have?

Numerically equivalent problems — 8 minus 5 — but the first problem contains an *action* that is easy to represent. It would occur to a lot of kids that they could solve the first problem by dealing out 8 counters (or whatever) and then removing 5 off them from their pile. That’s directly modeling the action of Problem 1.

Problem 2 doesn’t contain an easy-to-notice action, so direct modeling would be less likely to occur to children. Carpenter and pals found that, in fact, kids didn’t use direct modeling strategies for Problem 2, and as a result Problem 2 was a bit trickier for kids to handle.

(This is 100% true, in my experience.)

Word problems with actions, essentially, contain stories that are easy for us to represent and understand. And stories are the sort of thing that our minds most easily grasp and symbolically represent. As Dan Willingham notes:

Research from the last 30 years shows that stories are indeed special. Stories are easy to comprehend and easy to remember, and that’s true not just because people pay close attention to stories; there is something inherent in the story format that makes them easy to understand and remember. Teachers can consider using the basic elements of story structure to organize lessons and introduce complicated material, even if they don’t plan to tell a story in class.

Incidentally, Tom Newkirk makes a similar observation about what constitutes good writing in his book *Minds Made for Stories*:

Later, Newkirk argues that part of what makes good non-fiction writers good is they find subjects and actions to metaphorically represent abstract structures. They turn “evolution is a process whereby genes are randomly mutated” into “mutagens are constantly attacking our genetic material, altering it in ways that have the power to change the direction of an entire species” or whatever. Action, action, action.

(I’m also pretty sure this connects to Anna Sfard’s work on the way we tend to turn mathematical processes into mathematical objects but I’m not sure I have all the pieces put together yet.)

Back to Planet Classroom: What does this mean for my teaching? Practically, a lot of **annotated arrows**.

Instead of an example of simplifying expressions that looks like this…

…I try to recreate it with annotated arrows, to emphasize the actions involved:

Our minds privilege stories, which means that they privilege change over inaction. If our explanations can include more doing *things *to *things*, this can help kids see what we mean a bit more easily.

For a lot of fun use of arrows to represent the actions in otherwise static representations, check out David Wees’ new tumblr for mathematical annotations. A prime example:

Arrows — especially annotated arrows — can help transform examples (i.e. what correct work might look like) into explanations that help someone understand the examples.

## 3. Describe Mathematical Pictures, Though This is Harder Than You Might Expect

The What Works Clearinghouse (WWC) is this big federal initiative to try to sort through the evidence for various educational claims and give clear recommendations. The thing is, there is a ton of dissatisfaction with their standards for recommendations. Some people think their standards of evidence are weirdly strict. Others say they privilege large experimental or quasi-experimental studies over other forms of evidence.

Anyway, they have this report on helping students who struggle with math, and I like it. Their fifth recommendation is all about visuals:

Why are visuals important? At least partly because words don’t distract you from pictures — you can pay attention to both at once. (Unlike reading a slide and hearing it explained to you, where the words interfere with each other. You might find yourself doubly distracted in that situation.)

This relates to dual-coding theory, a theory from cognitive science that deserves to be better know in math education. Like worked-out examples or the privileged role of narrative, it’s a legitimately useful bit of cognitive science to know.

If you’re looking to teach a strategy, describing a (complete!) mathematical picture (with arrows!) can help.

The thing, though, is that it’s very easy to mess this up. A “mathematical picture” is not “a bunch of written numbers or words.” That’s not describing a picture with words — that’s just using *spoken* words to describe *written *words. I don’t think that helps as much, according to dual-coding. Words can distract you from words.

I’m not saying that board work in the above tweet is bad at all. My point is just that these equations are going to compete for attention with any spoken explanation in a way that (dual-coding says) a picture wouldn’t. (Though check out those annotated arrows!)

A problem: when I think about it, there are a lot of mathematical topics that I can’t think of a good picture for. And even for ones where I can (e.g. the connection between area and multiplication) those visual representations aren’t *obviously *connected to their numerical ones to kids. Those connections need to be carefully taught. Ideally, they’re built into a curriculum.

I mean, it’s obvious to me that you can carve up a rectangle into 4 quarters and this represents multiplying (x + 3)(x +7), but anyone who’s tried knows that this isn’t obvious to high school math students.

So while it’s great to aim for “describing a mathematical picture” as an ideal for explanation, we’re limited by the mathematical pictures that kids understand and that we know.

I love learning new pictures for mathematical ideas. I feel like this year I really realized the power of the visual representation of the Pythagorean Theorem to help my kids understand its meaning and use:

But there are a lot of topics where I don’t know good visuals to go with the numbers, equations or words. I’d love if we could find more of them.

## What Beyond Explanations Shouldn’t Mean

I could be wrong, but I don’t see a lot of people writing or thinking about how to give good mathematical explanations. This is despite the fact that the vast majority of teachers I talk to say they give explanations often, even if they are a bit embarrassed by this. (They shouldn’t be, I think.)

And the vast majority of thinkers and writing about teaching would tell me that they aren’t anti-explanation, just against a mono-focus on explaining as the core of teaching.

So why doesn’t anyone write about giving good mathematical explanations? Three possibilities I can think of.

- Unlike me, pretty much already knows how to give good mathematical explanations.
- People don’t think that improving our explanations is worth the ink. It’s a low pay-off instructional improvement.
- Anti-explanation ideals make it trickier to talk about improving our explanations.

I’m pretty sure it’s not Possibility 1. I think Possibility 2 sounds good, and Possibility 3 is a solid maybe and is anyway related to 2.

Either way, now you know what I’ve recently figured out about explaining stuff to kids:

- Represent complete mathematical thoughts
- Use annotated arrows to emphasize action and change in those representations
- If possible, describe mathematical pictures

So, what’s next? Just last week I tried out a new representation of solving trig problems with my geometry students. It flopped:

But then I made a little tweak, and it went better. Which got me thinking: we’ve got this whole internet thing. Why aren’t people sharing more of these images? Is it less fun for us to share pictures of our own work? Does it seem self-promoting in a way that sharing other things (e.g. activities) doesn’t?

I’m not sure. But I think that this work is valuable, and is worth sharing. Explanations are nitty gritty, but it’s important nitty gritty.

Rearranging the terms of the two parts at the same time as doing -(x^2 – 4) is too clever! No wonder they got it wrong.

I like the visual approach, and also the “Get the whole story”. In addition I find that the lack of connecting dialogue odd, and the development of meaning in algebra is often lacking.

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I don’t talk about *how* to explain something, but I think I do a lot of writing about my explanations. And I usually include the pictures.

So I’m modeling how I explain things, if not spelling it out.

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I’ve finally done a diagrammatic account of a (simple) trig example. Is this what you are after?

It’s in my latest post:

https://howardat58.wordpress.com/2017/04/17/sin2x-less-than-or-equal-to-2sinx-for-smallish-x/

Here is a verse on the matter:

https://howardat58.wordpress.com/2014/07/08/why-cant-i-just-get-the-answer/

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I tend to think the floor and ceiling on explanations are pretty close together. I can only optimize explanations so much before I ask myself if there isn’t a better medium for student learning, or better preconditions for that explanation, or or or …

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Yeah. To me, that sounds like my second option (“It’s a low pay-off instructional improvement.”)

I want to say “I’m not sure how to respond to this” but what I mean is not that there’s some unbridgeable difference of opinion between us here, just that I really don’t know how we can have a conversation about

how muchsomething helps classroom learning.Like, I could say “yeah it’s a small difference but small differences compound over a school year.” Or that hitting the ceiling on explanations is one easy way to help kids feel comfortable in your room.

But I just don’t know.

Speaking personally, it’s not that giving better explanations has been some huge thing. It’s a small thing (though I don’t know how small). But I’m starting to think that we need to spend more time talking about the small things in math education. If we only talk about the big things we end up with this weird disconnect between practice and talk.

And another thing too: I’m not the only one that finds the small things the most interesting, or am I?

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Is this based on finding the Instructional Explanations literature (which is often nearby the Worked Examples literature)? A former colleague at the Forum, Ellen Clay, did a fair amount of work on Instructional Explanations and had both a grant and some grad courses where the main work of the course was to generate and critique video of instructional explanations for big ideas like adding and subtracting integers (she used chips and making zeroes), reading algebraic expressions with meaning, and solving equations using chains of reasoning.

I’m not great at finding canonical articles on Google, but it seems like an A Renkl is oft cited and writes about the effectiveness of instructional explanations — both what makes for a good explanation and what the role of explanation is in relation to other activities. Not working at a University anymore has made it harder for me to find literature, too…

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I’ve never heard of this before, but it sounds up my alley. Thanks!

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I agree with your Dumb Houdini idea. Stringing out your first modeling of a process is bad practice that loses kids. I also agree with your Study Complete Mathematical Thoughts idea. However, if what the students are trying to learn is a process, like simplifying an algebraic expression (Algebra By Example), then the “complete mathematical thought” is not the final version of the work presented in its entirety. More accurately, the complete mathematical thought is a smooth, uninterrupted video or live performance of someone simplifying the algebraic expression correctly while speaking out loud about what they are doing (so the viewer can perceive the inner monologue that is present in the simplifyer and therefore part of the complete mathematical idea).

I have tried many an arrow diagram in my day and have gotten similar, flopped results. The reality is, all processes are perceived as stories. In your first example simplifying 5x^5/3x^3, the story is simple, satisfying and filled with “doing things to things”. First the x^5 and the x^3 are EXPANDED (like a slinky or something. It’s oddly satisfying.) Then, terms are SLASHED OUT (which is, again oddly, very satisfying). Finally, terms are REGROUPED. As I said, it is a very clean, satisfying story filled with “doing things to things”. In this story, a person is doing actions to symbols.

The arrow version is more complicated and confusing for two reasons:

1) It presents more written information than the previous method. Instead of just the four lines of math, there are three arrow symbols with words attached in addition. This almost doubles the information. You conjecture that this is made up for by the fact that it taps into the powerful story framework that humans remember better. But the fact is the first method is perceived as a story just as much as this one, so there is no advantage to the arrows from that perspective.

2) There is confusion regarding who is doing the actions in the story. In the first example, it is very clear that the person is doing the EXPANDING, SLASHING, and REGROUPING. In the second version, it is unclear who is doing the UNMULTIPLYING, for example. It should be the person, but the arrow with the words UNMULTIPLY seem to allude to the fact that the process is being “carried out” in a sterile, anonymous way. So the human is carrying out the UNMULTIPLY action but also referring to that action as if its being carried out anonymously. It’s subtle, but certainly muddy’s the clarity of the story.

This brings me to one final realization I’ve had while reading your post and writing this comment. One problem with mathematics is that the little micro-stories that the symbols seem to play out (expanding, being cancelled, regrouping, flipping over, etc) are fun to engage and interact with. Plus, they’re happening right before your eyes as you perform them. So ignoring them in favor of the ACTUAL stories they refer to, the stories played out by the quantities and concepts those symbols represent, is very difficult.

Thanks for the post.

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Thanks for the comment! Lots that I’m still thinking about but…

This, of course, is entirely possible. It actually went pretty well for my kids in class, so I’m not sure if I’m ready to buy the “more confusing” theory. As you note, my explanation packs a lot more info into the diagram, but I think that’s important. The issue with canceling is that it’s a shaky principle on which to conduct your exponential simplifications — do bases mix? how do numbers work? — so I like helping kids see how unmultiplying can help sort of chunk up the expression.

But I’m not strongly committed to any of this. There’s no one, true, best explanation, I’m pretty sure.

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I agree with the shakiness of the cancelling, for sure, as a sound technique for understanding what’s happening when you simplify. All I’m saying is that, as a story element, things being slashed out is an easy action to understand. That’s actually the danger of it. Like a drug, its easy to do and it feels good when you do it. So learners prefer it. Once you show them a method that refers to the meaning of what is happening (which requires more information be considered), they often have an initial distaste for it.

Keep churning away!

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