How to Explain Stuff Well
The components of an effective explanation
What does an effective explanation look like?
Explanations are a neglected part of math teaching. Every teacher explains stuff, yet I don't see much advice about how to explain well. Here’s my best attempt.
One note. These principles apply to explaining concepts to a full class, but they also apply to explanations I give to students one-on-one. That second case is sometimes more important to get right. My time is limited. If I'm going to pull up a chair and try to help a student understand something they're confused about, I want to make sure that explanation lands.
Before the explanation
If I want students to understand something, I don't want to launch right into the explanation. There are a few things that come first, to make sure students are ready.
One thing at a time
Explanations work best when they are explaining exactly one idea. Trying to explain thing A, then thing B which depends on thing A, and so on, is a recipe for confusion. It’s fine to explain multiple ideas in a class, but they should be separate explanations, with a chance to check for understanding and apply the learning in between. Before an explanation, I need to ask myself: am I trying to explain one idea, or several ideas?
Check for prior knowledge (ideally on a previous day)
In every explanation, there are some things I assume students know. If I’m explaining something about solving systems of equations, I probably assume students know how to add and subtract like terms. If I’m explaining something about cross-sections, I probably assume students know the names of different shapes. Check if students know these things beforehand. If they don't, reteach with a bit of practice. This works best a day or several days before. I don’t want too many tangents checking for prior knowledge right before an explanation. That prior knowledge should be firmly in long-term memory by the time students need it, which means they should have learned it longer than five minutes ago.
Activate prior knowledge
If your explanation assumes students know some things, even if they know those things pretty well, it's helpful to give students a few problems so those ideas are fresh going into the explanation. This reminds students of stuff they need to know, and it can also provide some positive momentum that helps students feel confident. Activating prior knowledge is especially helpful if I’ve done the knowledge check on a previous day so I don’t have to worry too much about going back and reteaching some skill before the explanation. I can ask a few quick questions, remind students of some things they’ll need in a moment, and dive in.
During the explanation
Attention
Make sure students are paying attention. This sounds obvious but is harder than it sounds. There are lots of things that can be distracting. There’s the obvious, like announcements on the intercom, phones, and side conversations. But other things that seem like regular parts of learning can be a distraction, too. Having students copy notes while they should be listening can be a distraction, even though it might look like they’re learning on the surface. Asking too many questions during an explanation can lead to tangents that take away from the main idea. The structure of the learning environment also matters. Make sure there isn’t too much for students to take in at once, like having lots of text on a screen while the teacher is talking, or showing a diagram that has unnecessary details. There’s a whole branch of research studying these ideas if you want to learn more.
Connect to what students know
We learn best by building on what we already know. What previous ideas does this concept build on? How is it similar or different? What do students know that can help them understand the new idea?
Multiple examples/non-examples
It’s easy to underestimate how many examples students need to understand a new idea. Multiple examples, and non-examples where a concept doesn’t apply, are important. One way to make this efficient is to use examples like, “what would happen if this positive changed to a negative?” or “what if this 10 was a 20?” These can give students the benefits of multiple examples while being efficient with time.
Check for understanding
Pick a few key moments and check to see whether students understand. A whole-class response system like finger voting or mini-whiteboards is helpful here. It’s often best to pick a specific moment and have every student respond, rather than asking questions to individual students throughout. Those individual questions can lead to false positives, rather than a systematic check for understanding at the most important moment.
Keep it short
Keep the explanation as short as possible while doing all that other stuff. One key here is making sure you’re teaching one thing at a time. Long explanations are often long because they’re trying to teach too many things at once.
After the explanation
Apply it
Do something with the math! Solve problems. See whether students are applying what they know. Give feedback. Solve more problems. If you don’t use it you lose it.
An example
Here’s an example of an explanation put together with these principles.
Imagine I’m teaching students to simplify fractions. First, simplifying fractions isn’t one skill, it’s actually lots of different little skills wrapped up together. Maybe I break it down and start with finding an equivalent fraction by using divisibility rules for 2, 3, 5, and 10.
First, I make sure students know those divisibility rules. I also want them to know that we can find equivalent fractions by multiplying the numerator and denominator by the same number, for instance turning 3/4 into 6/8. I’ll put some time toward those skills before we start to simplify fractions.
Immediately before I explain the concept, I’ll give students some warmup questions. What’s a fraction that’s equivalent to 1/2? 5/6? What is the divisibility rule for 10s? 2s? What number are both 6 and 8 divisible by?
I make sure students are paying attention — nothing on the board, nothing on their desks, Chromebooks away. Then I launch:
One way to find equivalent fractions that you’ve seen before is to multiply numerator and denominator by the same number. In the same way, we can divide the numerator and denominator by the same number to make an equivalent fraction. For instance, here’s a fraction:
You all told me a minute ago that 6 and 8 are both divisible by 2. So I can do this:
And that’s equal to
If, instead, the fraction is 6/9, then I divide by 3, since both numbers are divisible by 3, to give me 2/3. If the fraction is 60/90, then I divide by 10, since both numbers are divisible by 10. If the fraction is 5/9, I can’t find an equivalent fraction because 5 and 9 don’t have any factors in common besides 1.
Here’s one for you to try: find a fraction equivalent to 10/12 by dividing numerator and denominator by a common factor.
Some final thoughts
I’m not saying that’s the perfect way to teach simplifying fractions. That skill involves lots and lots of pieces that fit together in the right ways; my explanation is only one piece. There’s also guided practice, checking for understanding, responding to common misconceptions, and more. There are other ways to teach this skill besides an explanation. I’m sure lots of teachers reading this are thinking, “ooh, I think it would work better if you changed…” You’re probably right! There’s no one perfect explanation. My goal is to illustrate the principles of an effective explanation. In that example, I broke simplifying fractions down into a small, manageable chunk, planned the prior knowledge I needed to check and activate before the explanation, connected the new idea to something they already learned, used multiple examples and non-examples, kept it short, and checked for understanding. It’s easy, especially when teachers get busy in the thick of the school year, to forget those pieces. That’s the point of this post.
I’m also not saying that explanations are the only ingredient in effective teaching. They aren’t even the only ingredient in effective explicit instruction. But every teacher explains things, and it’s worth trying to explain things well.
Finally, explicit instruction is easy to caricature. Lots of people in education like to say, “don’t lecture at students, instead do my personal favorite thing that you can read about if you buy my book!” Those people are correct that extended one-directional lectures are not a very effective way to teach. Part of my point in this post is that explaining stuff well is actually hard. Obviously if you do it badly lots of kids don’t learn. Obviously there’s more to teaching than good explanations. But it’s also easy to be bad at explanations, watch students not learn stuff from them, and conclude that explanations are worthless. Don’t do that.