One consequence of this is that I am getting daily emails from people who read the piece and feel moved to comment. I do believe the Internet ought to facilitate dialogue. So I have been replying to these emails.

Sometimes, people leave a wrong email address in the contact form and they bounce back. So, in a show of good faith, I share with you a recent email and my reply. Perhaps Gavin will come back to the blog and read my reply. Perhaps he will not.

Anyway, here goes.

Gavin writes:

I do not know if you failed to do your research, but the number line is clearly part of Common Core, for instance:

“CCSS.Math.Content.6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.”

I am not sure how to comment on the article because you have banned commenting on them. I do not have a twitter and therefore cannot join the disucssion there. I hope that you’re not intentionally trying to cast a positive light on Common Core but are instead trying to give an unbiased account of it.

I reply:

Thank you for taking the time to read, and to write.

I just want to clarify that my claim was not that number lines do not appear in the Common Core. They do appear there, as you point out with your citation. You are completely correct.

But if I remember correctly, the worksheet in question was a second grade worksheet. My claim was this: “There is nothing in the Common Core State Standards that requires students to use number lines to perform multi-digit subtraction.”

I stand by that claim. Even the number line standard you cite in sixth grade doesn’t reference the number line as a way to understand multi-digit subtraction. Instead the spirit of that standard is to use the number line as a way to represent negative numbers (such as -9 or -1/2), and then to understand the coordinate plane. Simply put, if students are going to graph functions in algebra, they will need to work with number lines in earlier grades.

As for the comments thing…I was saddened to have to turn them off for that 5 reasons post. But I am committed to maintaining a reasoned and productive tone on this blog. The comments (both pro- and con- on the Common Core) were spiraling out of control and I simply did not have the time to manage them. It seems clear to me that people are able to comment on the piece as it gets shared on Facebook, but I don’t have access to the comments on other people’s shares so I cannot speak to their quality, and I am not responsible for them in the way I am when they are on my blog.

Finally, you can search my blog for “Common Core” and find that I have made some rather pointed critiques of some specific standards in the Common Core—including engaging and arguing with Bill McCallum (a Common Core author) on matters involving rates, ratios and unit rates. All on the record, and you would be welcome to join the conversation in comments on those posts. I have no interest in promoting CCSS. I do have an interest in making sure that critiques are honest and fair.

Tabitha (7 years old), Griffin (9 years old) and I walked to the local convenience store tonight. We had a math talk that I will describe in more detail on Talking Math with Your Kids soon.

In the meantime, I will excerpt a piece of that conversation here. It will give us some useful language and ideas.

Tabitha was using her own money to buy some hot Cheetos. She was under the impression that they would cost $1.35. While she waited in line, she had me verify that her 5 quarters and 1 dime matched this sum. I assured her that it did.

The Cheetos turned out to cost $1.49.

There were people in line behind her. This was a time to grease the wheels, not to slow down everybody else’s Saturday evening. So I told her to give the cashier 2 more dimes.

As she did so, I told her that she had given the man 20 more cents when he only needed 14 more cents, and asked her how much change she should get.

The cashier finished off the transaction. I stuck out my hand to grab her change (so as not to give away the answer to the question I was about to ask, and she was way more interested in the Cheetos anyway). We turned to leave.

I asked how much change she should get back. She seemed confused by the question. After going back and forth a couple of times, we settled on this question:

14 plus something is 20; what is the something?

Now we get to the question I pose to you, Dear Reader.

What is the goal of asking a child this question?

There are many possible goals, of course. I want to highlight two of these. I think that they stand in stark opposition to each other.

To get the child to say, “six”.

To get the child to think about number relationships.

Six is the right answer. I would like for her to be able to get there. But getting her to say, “six” is not the goal of the question for me.

Before I elaborate, I want to make clear that this is not a straw man argument.

Griffin piped up while Tabitha was thinking and asked, “How old were you last year?” The only thing that question had in common with mine was the answer. I have been in math classrooms where teachers offered these kinds of hints.

So not a straw man at all.

While the video below is supposed to be funny, it draws on this idea that the goal is to get the child to say (write) the answer.

No.

My goal in asking this question is to get the child to think about number relationships. I want Tabitha to think her way through to an answer. I want her to be able to say, “six,” yes. But I will be happy with a few productive wrong answers along the way because that will be an indication that she is thinking.

You see, options 1 and 2 above speak to very different ideas about how people get better at mathematics.

Option 1 speaks to the idea that fourteen plus something is twenty is a problem that has the same structure as many other problems (this plus that is something else) but that bears no other relationship to them.

Option 1 is related to a behaviorist view of mathematics learning—that we create associations between stimulus and response, and that learning is the formation and strengthening of these associations. With this view, fourteen plus something is twenty is a unique stimulus that requires a unique response: “six”. The strong version of this view would require me to tell her the answer, have her repeat the answer, and to make sure I ask her about fourteen plus something is twenty again in the near future in order to strengthen the bond.

Option 2, by contrast, speaks to the idea that learning arithmetic is about becoming familiar with number relationships. Option 2 suggests that fourteen plus something is twenty is not an especially important problem on its own, but that it provides us with a place to practice noticing and using relationships in order to strengthen our familiarity with these relationships.

The thing I need to do if Tabitha is struggling with fourteen plus something is twenty is very different if I choose option 2. I need to think about what related problem is likely to be easier for her than this one. I need to think about how to help her make progress.

Here, the most likely productive direction (based on what I know about her, and about her mathematics learning experiences) is to ask:

Do you know this one? Fifteen plus something is twenty.

She probably knows that five is correct here. This is because she has counted by fives many times. Once she establishes that fifteen plus five is twenty, she will likely be able to reason that fourteen plus six is twenty. Fourteen is one less than fifteen, so the other addend must be bigger to get the same sum. She wouldn’t say it that way, of course, but she can think that way.

She can think that way for two reasons: (1) it is natural for children to think this way, and (2) this sort of thinking has been modeled, supported and encouraged.

In short, I and her teachers have taught her in ways that support powerful mathematical thinking.

What we see in the video above does not support that. While I (mostly) get the joke, it is not so far from the truth. This is precisely what goes on in many classrooms and homes. The parent does not ask the child what he is thinking. The child has gotten the message that there is a right way to perform the computation, and that it involves the 4 turning into something else. The whole thing is a mess and it is very very true.

It is too true.

Everything about that interaction needs to change. Everything.

But really, if we change one thing we’ll be on our way to changing everything.

It is a big change, of course.

We need to stop worrying about the child says, “six”. We need to start worrying about how (and whether) the child is thinking.

A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

This is a classic example demonstrating the danger of applying procedures without thinking. The quotient can be expressed either as 31, remainder 3; or as . Neither of these answers the question, though. According to unspoken principles of table renting, we will probably need 32 tables.

Of course, I can imagine a student thinking like a caterer and building any of the following arguments:

We need 31 tables (or fewer) because 5% of people on a typical guest list do not show up.

We need 31 tables because if everyone comes, several will be young children who will sit in their parents’ laps.

We need 31 tables—if everyone shows up, we can just stick an extra chair at each of three tables.

We need at least 35 tables: No one wants to sit on the side where they can’t see the band playing at the front of the room, so we need to allow for fewer than 8 people at each table.

Et cetera.

I would argue that we need to teach in ways that do two things:

Allow/force students to interpret their computational results in light of the context (there is a CCSS Mathematical Practice standard about this), and

Focus students’ attention on the role the computation plays in answering this kind of question. Why are we dividing? and What does the quotient mean? are the kinds of questions I have in mind here.

You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

Oh dear. If anyone on the Internet has had more to say about dividing fractions than I have, I am unaware of who that is. (And, for the record, I would like to buy that person an adult beverage!)

Unlike the division by zero stuff from question 1, this question is better tackled with informal notions than with formalities. The formalities leave one feeling cold and empty, for they don’t answer the conceptual why. The formalities will invoke the associative property of multiplication, the definition of reciprocal, inverse and the multiplicative identity, et cetera.

The conceptual why—for many of us—lies in thinking about fractions as operators, and in thinking about a particular meaning of division.

1. A meaning of division

There are two meanings for division: partitive (or sharing) and quotative (or measuring). The partitive meaning is the most common one we think of when we do whole number division. I have 12 cookies to share equally among 3 people. How many cookies does each person get? We know the number of groups (3 in this example) and we need to find the size of each group.

I can mow 4 lawns with of a tank of gas in my lawnmower is a partitive division problem because I know what of a tank can do, and I want to find what a whole tank can do. So performing the division will answer the question.

2. Fractions as operators

When I multiply by a fraction, I am making things larger (if the fraction is greater than 1), or smaller (if the fraction is less than 1, but still positive).

Scaling from (say) 5 to 4 requires multiplying 5 by . Scaling from 4 to 5 requires multiplying by . This relationship always holds—reverse the order of scaling and you need to multiply by the reciprocal.

putting it all together

Back to the lawnmower. There is some number of lawns I can mow with a full tank of gas in my lawnmower. Whatever that number is, it was scaled by to get 4 lawns. Now we need to scale back to that number (whatever it is) in order to know the number of lawns I can mow with a full tank.

So I need to scale 4 up by .

Now we have two solutions to the same problem. The first solution involved division. The second solution involved multiplication. They are both correct so they must have the same value. Therefore,

There was nothing special about the numbers chosen here, so the same argument applies to all positive values.

We have to be careful about zero. Negative numbers behave the same way as positive numbers in this case, since the associative and commutative properties of multiplication will let us isolate any values of and treat everything else as a positive number.

Please note that you do not need to invert and multiply to solve fraction division problems. You can use common denominators, then divide just the resulting numerators. You can use common numerators, then use the reciprocal of the resulting denominators. Or you can just divide across as you do when you multiply fractions. The origins of the strong preference for invert-and-multiply are unclear.

This has made the rounds on the Internet, and it has angered lots of folks in education. And rightly so. Because there is no learning going on in that video.

But those teachers are being trained to deliver that sort of instruction to students in classrooms. Go ahead and search EDI or whole-brain teaching. You’ll see these very techniques being promoted as good practice.

So, is it how people learn, or is it not?

Tip of the cap to David Wees for reminding me that the parallels are not necessarily obvious.

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