Making eight

I am writing a book. In the process of doing this, I come across homework assignments that parents find frustrating, and that they share on social media. These almost always get me thinking, and they frequently lead to math talks with my children.

This past weekend was one such instance.

Worksheet: "The whole is 8. One part is 8. What is the other part?"

Talking Math with Your Kids is not a place to hash out the details of whether this is a well written question, or whether this was an appropriate homework assignment for this child. We can discuss that on Twitter if you like, or through my About/Contact page.

Talking Math with Your Kids is about taking opportunities to have math conversations with our children. In that spirit, I share the conversation we had in our house.

Out of the blue, I asked Tabitha (7 years old) if I could ask her a math question. It was maybe Saturday afternoon. We had nothing special going on.

Me: Tabitha, can I ask you a math question?

Tabitha (7 years old): Yes.

Me: If I have eight things, and seven of them are in one hand, how many are in the other?

T: That’s not even a math question! That’s too easy!

Me: OK. But will you answer it anyway?

T: One.

Me: OK. What if I had five in one hand?

T: And you still had 8?

Me: Yeah.

She spent a few moments thinking.

T: Three.

I had a couple other questions, which I asked and she answered. The next day, I realized that I didn’t know how she knew that second one.

She was getting ready to brush her teeth on Sunday evening when I asked whether she remembered the previous day’s conversation. She did.

Me: How did you know it was three?

T: I counted.

Me: Like this? Five, then six, seven, eight?

T: Yeah. And that’s three. But actually, I kind of already had it memorized.

Me: Oh yeah? How did you memorize it?

T: Huh?

Me: Did you try to memorize it? When I want to memorize a phone number because someone told it to me and I don’t have a pen handy, I say it over and over to myself. Did you do that with 5 + 3?

T: No! I just have counted it out a lot of times.

Now, I should also mention that I asked Tabitha, If I had 8 things, and 8 of them were in one hand, how many would be in the other? She replied Zero without much hesitation. This If I have this many in one hand, how many are in the other formulation is probably less clumsy than the If this is one part, what is the other part? formulation on the original worksheet. But the intention is the same.

So What Do We Learn?

The kind of problem Tabitha and I were working with is called Part-Part-Whole. For young children, this is different from the standard “takeaway” problem because there is no “taking away”. I didn’t eat, lose, destroy or give away any of my eight things in these problems—I just have some in one hand and some in the other.

Because Part-Part-Whole involves a different way of thinking, it’s a good idea to practice some of these problems. It helps children to build a better understanding of addition and subtraction relationships if they see all the various ways these relationships appear in their worlds.

Tabitha herself pointed out an important principle of Talking Math with Your Kids: Many things that you hope to remember, you can remember by encountering them frequently. Tabitha has never sat down with flash cards to memorize her single-digit addition facts. Yet she is in second grade and is starting to feel confident with them.

She and I talked about familiarity—how maybe learning 5 + 3 is a little like learning the name of someone you see in your neighborhood. You don’t recognize the person as being the same person the first few times you see them. But eventually, if you see them frequently enough, you do recognize them, and you might introduce yourself. Pretty soon, you know their name. And if you just can’t seem to remember it? That’s when it’s time to drill yourself. That’s when you repeat the name over and over and over.

Starting the Conversation

Ask the questions I did. This is an easy conversation to have. If your child isn’t confident with addition and subtraction facts, ask about six in one hand instead of jumping to five in one hand. 

More broadly, look for Part-Part-Whole opportunities to talk about. This is an important interpretation of subtraction, and one that is often neglected. Examples include apples (Our fruit bowl has 8 apples—5 are red, how many are green?), pets (There are 8 pets on our block—5 are cats, the rest are dogs. How many dogs?), et cetera.

7 thoughts on “Making eight

  1. By coincidence we were looking at a natural extension of this idea today. [For the purpose of describing the problem in one sentence . . . ] the geometry problem below boils down to recognizing that if x + y and x + z are the same, then y and z must be the same.

    So, I don’t know what the concern was with the original problem, but it does seem to me that the idea in the problem is one that will occur in various forms in many different areas of math.

  2. The problem is that it’s “different”. People in general are uncomfortable with different. Different is somehow inferior to most people. Unfortunately, traditional education is a disaster right now and they’re hearing it from everyone involved so parents assume that this approach since it’s different must be the problem. Unfortunately, it’s not. I’ve learned more math myself teaching it and breaking it down for students like this than I did when I was in school.

  3. “The whole is 8. One part is 8. What is the other part ?”.
    Just what exactly is this supposed to mean? That the whole always consists of two parts?
    Since when did numbers have parts?
    What is the definition of “part”?
    Even if we are talking about 8 things, then the natural AND logical answer is “There isn’t another part”.
    If I want to see ways of creating 8, using adding, then what is wrong with

    and 8=8+0 for completeness sake.
    To call zero a part of 8 is going to lay the groundwork for a feeling that math hasn’t got a lot to do with real life, which is a crying shame. This feeling can arise at any stage, we should give reality achance at this level.
    To conclude “What a stupid question!”.

    • The problems with the terminology are part of the reason why Christopher’s formulation (I have some in this hand, how many in the other) is better.

      Mathsemantically, it is nice to have a version that extends naturally to zero. Splitting continuous measures (say a distsance) in two pieces allows a nice extension to fractional parts. I don’t have a great version that extends to include negative numbers and would be interested if someone has an idea for that.

  4. Reblogged this on math mom with a blog and commented:
    I love this type of thinking. As a math teacher for middle school I find that, sadly, many children did not have similar experiences with math growing up and do not have that pre-programmed understanding of numeracy. We do sort-of forget how math becomes a second language and many things we know and understand so well could be an outlandish idea to a small child.

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