**Me**: What’s 91 divided by 7?

**Her**: [Draws hands on board.]

**Me**: What are these for?

**Her**: For counting.

My move was to nail the question down on a context and ask her the question again.

**Me: **Hold on. Let’s make up a division story for this question. Let’s say that 7 people are equally sharing 91 crackers.

**Her: **Can we change it to mushrooms?

**Me: **Sure.

And she starts counting on the hands. She hadn’t done this for smaller numbers, like 30 divided by 3. There she articulated that 30 divided by 3 is 10, because 3 times 10 is 30. That doesn’t seem to be on her mind right now, so I try to ask a suggestive question.

**Me: **[Draws 7 stick figures.] Here are the 7 people. They don’t have any arms though.

**Her: **Can you make one super tall and one super short?

**Me: **Not this time. They’re all the same armless height. Anyway, how many mushrooms can we *definitely* give to each person?

**Her: **10.

**Me: **Cool, and that would take care of a bunch of the mushrooms. That would take care of 70 of the mushrooms. And how many left would there be for us to take care of?

**Her: **21.

**Me: **Nice. So, how many more mushrooms can we give to each person?

And then she goes back to her hands and does a bunch of counting. I interrupt her and ask her whether we could give them each 4. She says no, after some thought. She says that it would have to be more than 2. It takes a little bit of thinking before she tries and confirms that 3 works.

—

I think that this picture, and this dialogue, captures an important step in learning multiplication and division, and how awkward it all is.

I’m very new to all of this, so I’d appreciate some comments. As is our custom on this site, here are a few prompts:

- Umm…how did that dialogue go? What worked? What could’ve gone better, in your view?
- I feel like there’s some wisdom here about how people learn division and multiplication that I’m not able to articulate particularly well. Maybe you can?
- How do you ween kids off of relatively slow and sloppy methods like counting?

Looking forward to your thoughts.

## 9 replies on “91 mushrooms, 7 people”

1. I’m curious about how she did 21/7 on her hands. Can you explain that a bit?

2. I thought breaking the number 91 into two parts and dividing each part and finally adding the ‘mushrooms’ was an interesting step.

3. Because she waited and said that 2 mushrooms for each person won’t work (in response to your question about if 4 mushrooms can be given to each of the 7 persons if there were 21 mushrooms left) suggests that she is seeing division as an inverse operation of multiplication.

1. She knew that 7 times 2 was 14, and then she counted on the fingers on the whiteboard to see if adding another 7 would get to 21.

2. I think it helps. A lot of my kids think about division successfully in that way.

3. I don’t think there’s an On/Off switch for kids understanding of division as the inverse of multiplication. It’s a relationship that kids can see and use with varying degrees of success in different contexts. This student can use the inverse relationship successfully with smaller numbers, but isn’t able to use it with larger division problems yet.

Wow, thank you for sharing that. Drawing her hands on the board was a rather strange step, I still can’t figure out why she would feel the need to do that, so it is very interesting. By drawing her hands on the board, it is like she is trying to “show all her working”.

I also think the dialogue went well. I was wondering if it might help to have an explicit discussion about “breaking numbers apart” to make multiplications and divisions easier. Maybe something along the lines of one of the number talks mentioned in Boaler’s “Elephant in the Classroom” (or the US equivalent). There was one mentioned in there about having the class demonstrate as many ways as they can to solve a multiplication like 18*7.

After something like that, I was thinking it might be helpful to have a challenge of “ok, this time no counting on fingers or toes”.

I wish I knew how the “leap to abstraction” is made. I noticed in the Jo Boaler course that students who continued to count on their fingers were unsuccessful in Algebra, but students who could partition numbers so as to find the answer without counting each one would be successful. I liked that this was an example of a transition phase.

I am not sure that teachers can do anything. I liked that you supported the separation of parts of the division, and that you didn’t make her feel bad abut the fingers, mushrooms, or counting.

I don’t know why kids draw what they draw. Drawing kids seem to find all sorts of sketches calming to them. I am all about supporting them for as long as it takes until the math panic subsides.

Very curious. I assume this is your 4th graders? If it’s your trig class, you have problems 😉

So what’s bothering me here is that she’s adding to divide, but she must already know how to multiply.

If you asked her what’s 7*3 and she added on her fingers, I would say the problem is she doesn’t have her multiplication facts down. But if she does, then what she needs to do (with smaller numbers) is satisfy herself that multiplication really is the inverse of division. Then she can stop adding and at least try trial multiplication.

21/7 = ? means 7 * ? = 21

I’ll bet she knows that, you taught her that. She just doesn’t believe it yet. Maybe if she sees it working with 2,3,6 she would generalize? Or maybe just wiggle her fingers under the desk?

A lot of intriguing things here, as usual. One thought: you ask her, “How many mushrooms can we definitely give to each person?” which is like asking, what’s the biggest number you’re confident/sure we can give each person [and have at least enough to go around]?”

That’s not quite the same as one of Dan Meyer’s typical questions to get kids to guess a lower and upper bound, if memory serves: he usually asks, “What’s a number you know is too small?” and “What’s a number you know is too big?” Those questions are also good, of course, and you might have gotten her to pick an upper bound, too, and maybe she’d have said 20.

So when she was clear that there were 21 left and you asked her how many more mushrooms each person gets, she ‘retreated’ to her finger approach. Seems like she didn’t have instant access to 21 divided by 7 is 3 or that 7 * 3 = 21. I think that here, rather than ask if we could give 4 more, you might have stayed closer to the first iteration, asking again, “How many more we can definitely give each person?”

Maybe that isn’t any better, but my thinking is that it keeps the thinking similar to the previous thinking, and develops potentially the idea of partial quotients: “Keep taking multiples of the divisor away from the dividend, using the biggest power of 10 you can. Reduce the power of ten (possibly more than once) and repeat the process until you have zero remaining or less than the divisor.” Of course, I wouldn’t use that language with elementary students, but that really is how “long division” works. Of course, your estimate might be low, but that’s okay, because you can take more away, as long as you keep track of the quotients and, just as your student did, add them together at the end to get the complete quotient.

This is serious stuff. And the more you get her thinking this way, I suspect the less dependent she may be on her fingers. However, some work to help her strengthen her familiarity with multiplication facts (through playing games?) would also be useful, I suspect.

As a remedial math teacher who constantly runs in to problems caused by low familiarity with multiplication facts–what would you suggest as a way to practice them? I am doing weekly fact practice to help kids see [hopefully] improvement in their accuracy and time, but I know that it not sufficient for actually helping them internalize the facts.

Most of the resources I have found are for 3/4/5 grades, but I have 7th and 8th. It’s hard to find something that looks like it would engage them.

Any thoughts/suggestions?! 🙂

I’ll kick things off, but I’m new to teaching multiplication. If you ever want to chat more about this stuff, feel free to drop me a line by twitter or email.

I’m a big fan of mental math for building multiplication knowledge. The Distributive Property is key to much of this mental math, which means that the mental math is an especially good context for drawing out the Distributive Property.

I’m also a fan of quick, daily practice. When I was teaching remedial classes last year, I would usually kick off the first few minutes of class by announcing some mental arithmetic problem, waiting a few seconds, and then tossing a rubber ball to one of the kids. They’d catch the ball, they’d try to answer the question, we’d pass the ball back, ask another question, etc. It would usually take under 10 minutes. It was good review, and a good way to break up a 45 minute period for kids with something like 30 minute attention spans.

I’m also a big fan of making the mental math reasoning explicit. I’ll do this with my 4th and 5th Graders now. I’ll give them something like 13 times 7 and they’ll explain how they got it. Sometimes it’s by breaking 13 into 10 and 3. Sometimes it’s by doing 11 times 7 and adding 14. Whatever — there are a few strategies, and it’s important to have those articulated.

I can imagine a variant of this with bigger numbers. Like, flash a multiplication problem (e.g. 3,284 x 11) on a slide for 5 seconds. The person who gets the closest answer wins a Something. That would create a need for some rounding and estimating.

Not that I’m a math teacher, but I have a son who seriously didn’t enjoy practicing multiplication facts. We enjoyed thatquiz.org because they time the responses and it’s fun to try to beat your times. That one is short answer, you have to type the number. And I liked freerice.org because it’s quick reaction multiple choice. The idea is quick twitch drills, they do it fast for a couple of minutes and then move on, but you do it every day for a while and slowly it seems in. You can ask their parents to do it with them.