Tag Archives: subtraction

The latest “Common Core” worksheet

You have seen this on Facebook.

Original (Click to enlarge)

Ugh what a mess.

Please share the annotated version widely.

I’ll say what I have to say (comments closed) and move on. If you wish to discuss further, hit me up on Twitter or pingback to the blog. Want to talk in private? Click the About/Contact link up top.

Also, Justin Aion—middle school teacher extraordinaire—wrote up his views on the matter. You can read them over in his house.

Here goes…

The intended answer

Dear Jack,

You only subtracted 306 from 427, not 316. You need to subtract another 10 to get the correct answer of 111.

Sincerely,

Helpful student

The purpose of this task

I cannot say whether this was the right task for this child at this time because I do not know the child, the teacher or the classroom.

I can say the following:

  • Analyzing errors is a useful way to encourage metacognition, which means thinking about your thinking. This is an important part of training our minds.
  • The number line here is a representation of a certain kind of thinking—counting back. The number line is not the algorithm. The number line records Jack’s thinking. He counted back from 427 by hundreds. Then he counted back by ones. He skipped the tens. We can see this error because he recorded his thinking with a number line.
  • Coincidentally, the calculation in question requires no regrouping (borrowing) in the standard algorithm, so the problem appears deceptively simple in its simplified version.
  • This task is intended to help students connect the steps of the standard (simplified) algorithm with reasoning that is based on the values of the numbers involved. Why count back by three big jumps? Because you are subtracting 300-something. Why count back by six small jumps? Because you are subtracting something-something-6. Wait! What happened to the 1 in the tens place? Oops. Jack forgot it. That’s his mistake.

So what?

The Common Core State Standards do require students to use number lines more than is common practice in many present elementary curricula. When well executed, these number lines provide support for kids to express their mental math strategies.

No one is advocating that children need to draw a number line to compute multi-digit subtraction problems that they can quickly execute in other ways.

The Common Core State Standards dictate teaching the standard algorithms for all four arithmetic operations.

But the “Frustrated Parent” who signed that letter, and the many people with whom that letter resonated, seem not to understand that they themselves think the way Jack is trying to in this task.

Here is the test of that.

A task

What is 1001 minus 2?

You had better not be getting out paper and pencil for this. As an adult “with extensive study in differential equations,” you had better be able to do it as quickly as my 9-year old.

He knows with certainty that 1001 minus 2 is 999. But he does not know how to get the algorithm to make that happen.

If I have to choose one of those two—(1) Know the correct answer with certainty based on the values of the numbers involved, and (2) Get the correct answer using a particular algorithm, but needing paper and pencil to solve this and similar problems—I choose (1) every time.

But we don’t have to choose. We need to work on both.

That’s not Common Core.

That’s common sense.

[Comments closed]

Guess the temperature

Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, this means we get to study integers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

Cheese-pepperoni crackers

It was snack time in the Triangle household the other day. Tabitha decided she wanted crackers with pepperoni and cheese, baked briefly in the oven.

She asked for eight. I was skeptical that she was that hungry. I was worried about wasted food. I asked her about this.

Tabitha (five years old): I promise to eat a little bit out of each cracker.

Me: Wait. Say that again.

T: I’ll eat some of the crackers, and the ones I don’t want to eat all of, I’ll eat some out of each one.

Me: So you want eight, and you’re not promising to eat all of them, but you’re promising to eat part of each one?

T: Yeah. I promise.

I relented. And I got out the baking sheet and 8 crackers.

Me: Can you put those crackers out in nice rows?

T: What kind of rows? Three of three?

As we’ll see shortly, she was meaning to ask how long each row should be—should each row have three crackers? She did not seem to be asking also about the number of rows. Although I didn’t know that yet and was prepared to be dumbfounded.

Me: Whatever you can do. You choose. Make them the same size, though.

T: I made three rows and there’s two left. I can’t do it.

Me: Oh. So what are you gonna do instead?

T: Two of two?

That is, she is going to make rows of two, since rows of three didn’t work. She worked busily at this for about a minute, chatting to herself about her work.

Me: Now. How many pieces of pepperoni do you need?

T: Eight.

Me: How do you know that?

T: Cause there’s eight crackers.

Me: Oh.

T: But first you need the cheese.

Me: No! It’s pepperoni and then cheese goes on top.

I cannot explain my need to adhere to this principle. But I do stand by it.

Me: Here you go. Here’s ten pieces of pepperoni.

T: What?!?

Me: Are there gonna be any leftovers after you put them on there?

T: No. I’ll put two on two.

Here, she means to say that she will put two pieces of pepperoni on each cracker. Again, I am prepared to be amazed. I am thinking that she might mean that she’ll put two pieces on two crackers, and one piece on the others. She does not mean this.

Me: What? What do you mean?

T: Two on two.

Me: Show me. I don’t get it.

She proceeds to put two pieces of pepperoni on each of the first few crackers. The result is one of the funniest things she has seen in days.

T: Like that! [Laughs loudly and long]

Me: Keep going. I want to see how this works out.

T: But if there are, I’ll eat them…This is funny, isn’t it? Putting two pieces of pepperoni on each one?

Me: You gonna have enough?

T: Uh oh. No, no, no. Different idea. I need eight, sorry.

I did two and I only had three left. I mean I had two left.

This is pretty difficult to parse if you were not there. She has covered one of two columns of four crackers with two pieces of pepperoni each. This leaves her with two pieces of pepperoni in her hand, which she notices is inadequate for doing the same to the remaining column.

Me: So you need eight exactly?

T: And there’s a whole row left!

As a developing authority on the topic of oneI would like to point out the various units Tabitha is keeping track of in this work. She is counting (1) crackers, (2) pieces of pepperoni, (3) the number of pieces of pepperoni per cracker (a unit rate), (4) the number of crackers in each row (another unit rate), and (5) the number of rows of crackers. This is cognitively complex stuff.

Me: I understand. I totally understand. So you have ten pieces…

T: Oh! I’ll put eight…I’ll put one on each one and then eat the rest…and then eat the leftovers.

Me: How many leftovers are there gonna be?

T: Oh. I don’t know. Oh. I have an idea.

She removes the pepperoni pieces from the crackers, stacks them and counts in a whisper.

T: Ten…I mean no! One! I mean two! Two.

Me: Two.

T: I can eat two.

Me: See if this works out.

Tabitha proceeds to match the crackers with pepperoni slices.

T: Why don’t they call this cheese-pepperoni crackers?

Me: We could call them cheese-pepperoni crackers.

T: It’s cheese and crackers if there’s no pepperoni, so it should be called cheese-pepperoni crackers.

She finishes up.

T: I was right, two! Look. There’s two left.

Me: What are you gonna do with them?

She shoves both in her mouth dramatically and laughs.

And here is the result; ready for cheese and then the oven.

20130126-120205.jpg

This is what not to do

Oh my, do I love this video.

But seriously. Don’t do this.

Please.

Thanks to Approximately Normal for the find.