Math Conversations at HomeApril 1, 2020
School is canceled, you barely leave the house, and you have a child you’re trying to help learn math. You’re going to be having more math conversations than you used to. Let’s talk about how to make them as joyful and productive as possible.
Here are three guidelines for math conversations.
- Start from where things make sense
- Be curious
- Keep it light and nonjudgmental
Start from where things make sense
Let’s say you’re trying to solve 6 x 25 with your third or fourth grader. You want to help, so you show your child what you learned how to do. So maybe you set it up vertically.
This gets you the right answer. But when you try to explain it to your child, you realize you don’t actually understand it. That 5 x 6 = 30 part makes sense to you, but the 2 x 6 = 12 part makes less sense. Why do you move that 12 over, and why do you write in (or not, depending on how you learned it) a ghost zero? The answer is right, so it works, but WHY does it work?
In his wonderful book Arithmetic, Paul Lockhart wrote “adults tend to confuse familiarity with understanding.” Arithmetic, like many ideas in math, is powerful in part because it hides so much. So we need to steer toward understanding rather than saying “do it like this because that’s how I learned to do it.”
“Adults tend to confuse familiarity with understanding.”
— Paul Lockhart
Some questions that move us toward understanding:
- I don’t understand why that works. Can you explain it to me?
- Why did you put the twelve there (or any specific point that doesn’t make sense)?
- What would a picture of that look like?
Drawing a picture is often the critical move. The ability to draw pictures, models, diagrams, etc. is a sign that you have a robust understanding built on things actually making sense. (Could you draw a picture of 6 x 25 yourself? What would you draw?)
The other critical move is: be willing to back up until things DO make sense.
Can’t draw a picture of 6 x 25? What about:
- 6 x 20?
- 6 x 10?
- 6 x 5?
- 3 x 5?
- 3 x 2?
- 3 x 1?
Maybe you feel silly for going back to such an easy problem. But mathematicians do this ALL the time. Go to a “baby case” and try to get a feel for what’s going on. Maybe you realize you don’t really understand what multiplication even means. But if you do, you should be able to draw a picture of 3 x 2 that illuminate it. For example:
In my drawing, I’ve got 3 x 2 representing 3 GROUPS OF 2. Could I make a drawing of 3 x 5 then? Of 6 x 5? Of 6 x 10? Suddenly it doesn’t seem as hard. By making things simpler, you’ve actually created a ladder to climb down, and then to climb back up again!
I’ll talk more in future posts about some of the most useful ways to understand and draw diagrams of operations. But for now, let’s stick with the overview.
Start from where things make sense: two takeaways
- Draw a picture and see if you can explain what’s going on.
- If you can’t draw a picture, make the problem simpler until you can.
There are really two things to be curious about when you’re learning and teaching math: the math, and the learning process. So try to make curiosity a habit.
I wonder why that works?
To begin with, don’t be satisfied with answers, right or wrong, if you don’t understand them. You deserve to actually know why things work! And so does your child. That means you need to start noticing if you are just memorizing steps in an obscure-feeling process, or if what you’re doing actually makes sense to you. It also means that you might get to an answer, and decide that you’d like to do things again a totally different way.
Let’s return to the 6 x 25 example again. Maybe you already used the algorithm to find the answer (150). But because it didn’t make sense, you went down the ladder of easier problems I discussed above. As you went back up, you noticed from your pictures that 6 groups of 5 was really the same as 3 groups of 10.
That’s cool! And you can use the same idea for the big question: 6 x 25 must be the same as 3 x 50.
That’s a way easier problem for me to solve. I can see it’s 150, since counting by 50s is easier for me. And now it makes way more sense too. In fact, my understanding is becoming more robust, since I’m not just memorizing one method for moving forward; I’m connecting multiple approaches to create a network of strategies and insights. Kids who have multiple approaches that they can switch between end up being able to do this kind of work faster, more accurately, and with less raw effort, because they see how they can replace tricky questions with easier versions that get them where they want to go.
(Notice that my diagram gets more abstract, and I don’t bother to draw 25 dots. That’s natural – we tend to drop scaffolding once we feel more solid. But if your child wanted to draw 6 groups of 25 dots, that would be okay too. Though I’d encourage drawing them in an organized way—clusters of 10, say, and using different colors, rather than a mishmash of dots—so it’s easier to see what’s happening. And maybe my diagram was too simple to start. Whatever. Just make it work for you and your child!)
What are you thinking?
The other thing to be curious about is what’s actually happening in your child’s mind. If we look at math as nothing more than a series of facts that we need to make sure you know, there’s nothing to be curious about. But math is about thinking and making observations and having ideas. So what ideas is your child having? What does your child notice? They are going to see things we might miss, and it’s exciting to be able to have a conversation that leads us to new understandings of what’s going on in each other’s mind. This is what’s so wonderful about real math conversations!
To give that a chance to happen, though, there’s one more thing to remember.
Be Curious: two takeaways
- Solve problems in multiple ways, and see how the strategies relate
- You and your child DESERVE to understand what’s going on!
Keep it playful and light
Curiosity and sense-making are the hallmarks of play. And if you want math conversations to happen, don’t make them high-stakes. It’s hard to have a conversation with someone who is constantly judging you. No one needs to have their intelligence called into question because of how they understand (or don’t understand) a new idea.
- Judge intelligence based on how long it takes to solve a math problem
- Get overly intense
- Keep it playful, keep it light, keep it fun
- Take a step back or pause if it feels like things are getting too intense
- I’m not good at math/I’m not smart
- Ask your father/ask your mother
- I’m not a math person
- Why aren’t you getting this? We just talked about this yesterday.
- It’s obvious!
- You should know this by now.
- I don’t know – let’s find out!
- Interesting idea. Why do you think that’s true?
- I don’t get it yet. Can you draw a picture that would help me get it?
- These numbers are confusing me. Let’s do an easier problem first.
- I wonder if this (tool, model, drawing, manipulative) would help.
I’ll be following up soon with specifics on mathematical representations that are particularly helpful for understanding arithmetic and other central elementary math ideas.
Until then, I encourage you to make a habit of saying some variation of the six statements below. Here’s to having productive and joyful math conversations at home!
Start from where things make sense
- “How could we draw a picture so this made sense?”
- “This feels confusing. Let’s start with an easier problem.”
- “That’s the right answer, but I don’t see how you got it. How did you do that?”
- “Let’s try to do it a totally different way. How many ways could we come up with?”
Keep it light and nonjudgmental
- “It doesn’t matter if we get it wrong. Let’s just mess with it.”
- “I have no idea what to do. Let’s figure it out together!”