Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults *can* see that, of course, but we are more likely to think “not a polygon”.

On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of *not being a polygon*.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use *vertex* instead of *corner*. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of *vertex* or not).

Now kids can work on stating exactly what makes a *vertex*.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape *is* *a square* when your head isn’t tipped, or whether it only *becomes* *a square *when you tip your head. He confirmed that it’s the latter.

• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• *Diamondness* is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.

Even the unshaded ones that had come before *could have been colored in*, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.

Tagged: elementary math, geometry, polygons, shapes ]]>

I created a shapes book for all ages. The digital version is free for now. Details are in this post over at TMWYK.

Tagged: book, geometry, shapes ]]>

- Lines
- Parabolas
- Rational functions
- Hexagons

The hexagons will be familiar to long-time readers of this blog.

I have run the parabolas version in College Algebra, and the hexagons version in my Ed Tech course. It was a huge hit both times—lots of conversation happened both electronically and out loud in the classroom. It’s a ton of fun.

I am especially pleased with the rational functions version. It makes for challenging work—even among the mathematically astute Team Desmos in recent trial runs.

Read the Desmos blog post on the matter if you like.

Tagged: desmos, functions, game, hexagons, polygraph ]]>

Griffin (10 years old, 5th grade) had been doing some multiplication in class today. Somehow his scratch paper ended up on our couch.

Here is one thing I saw.

Naturally I wanted to ask the boy about it. He consented.

Me:I see you were multiplying 37 by 22 here.

Griffin(10 years old): Yeah. But I got it wrong so I did it again with the lattice.

Me:How did you know you got it wrong?

G: I put it in the answer box and it was wrong.

It turns out they were doing some online exercises. There is an electronic scratchpad, which he found awkward to use with a mouse (duh), plus his teacher wanted to be able to see their work, so was encouraging paper and pencil work anyway.

I was really hoping he would say that 37 times 22 *has* to be a lot bigger than 202. Alas he did not.

Anyway, back to the conversation.

Me:OK. Now 37 times 2 isn’t 101. But let’s imagine that’s right for now. We’ll come back to that.

G: Wait. That’s supposed to be 37 times 2? I though you just multiplied that by that, and that by that.

He indicated 7 times 2, and then 3 times the same 2 as he spoke.

Me:Yes. But when you do that, you’ll get the same thing as 37 times 2.

A brief moment of silence hung between us.

Me:Whatis37 times 2?

G: Well….74.

Let us pause to reflect here.

This boy can think about numbers. He got 37 times 2 faster in his head than I would have with pencil and paper. But when he uses the standard algorithm that all goes out the window in favor of the steps.

THE STEPS WIN, PEOPLE!

The steps trump thinking. The steps trump number sense.

The steps triumph over all.

Back to the conversation.

Me:Yes. 74. Good. I like that you thought that out. Let’s go back to imagining that 101 is right for a moment. Then the next thing you did was multiply 37 by this 2, right?

I gestured to the 2 in the tens place.

G: Yes.

Me:But that’s not really a 2.

G: Oh. Yeah.

Me:That’s a 20. Twotens.

G: Yeah.

Me:So it would be 101 tens.

G: Yeah.

I know this reads like I was dragging him through the line of reasoning, but I assure you that this is ground he knows well. I leading him along a well known path that he didn’t realize he was on, not dragging him trailing behind me through new territory. We had other things to discuss. Bedtime was approaching. We needed to move on.

Me:Now. We both know that 37 times 2 isn’t 101. Let’s look at how that goes. You multiplied 7 by 2, right?

G: Yup. That’s 14.

Me:So you write the 4 and carry the 1.

G: That’s what I did.

Me:mmmm?

G: Oh. I wrote the one

Me:and carried the 4. Yeah. If you had done it the other way around, you’d have the 4 there [indicating the units place], and then 3 times 2 plus 1.

G: Seven.

Me:Yeah. So there’s your 74.

This place value error was consistent in his work on this page.

Let me be clear: this error will be easy to fix. I have no fears that my boy will be unable to multiply in his adolescence or adult life. Indeed, once he knew that he had wrong answers (because the computer told him so), he went back to his favorite algorithm—the lattice—and got correct answers.

I am not worried about this boy. He is and he will be fine.

But I want to point out…I *need* to point out that this is exactly the outcome you should expect when you go about teaching standard algorithms.

If you wonder why your kids (whether your offspring, your students, or both) are not *thinking *about the math they are doing, it is because the algorithms we (you) teach them are designed so that people do not have to think. That is why they are efficient.

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

And we shouldn’t kid ourselves that this is only an issue in the elementary grades when kids are learning arithmetic.

**Algebra**. The quadratic formula is an algorithm. Every algebra student memorizes it. How it relates to inverses, though? Totally obfuscated. See, we don’t have kids *find *inverses of quadratics because those inverses are not functions; they are relations. If we *did *have kids find inverses of quadratics, they could think about the relationship between the quadratic formula:

and the formula for the inverse relation of the general form of a quadratic:

**Calculus**. So many formulas (algorithms) that force students not to think about the underlying relationships. If we wanted students to really think about rates of change (which are what Calculus is really about), we might have them develop a theory of secant lines and finite differences before we do limits and tangent lines. We might have Calculus students do tasks such as Sweet Tooth from Mathalicious (free throughout October!). There, students think about *marginal enjoyment *and *total enjoyment*.

**On and on**.

This is pervasive in mathematics teaching.

The results are mistaken for the content.

So we teach kids to get results. And we inadvertently teach them not to use what they know about the content—not to look for new things to know. Not to question or wonder or connect.

I’m telling you, though, that it doesn’t have to be this way.

Consider the case of *Talking Math with Your Kids*. There we have reports from around the country of parents and children talking about the ideas of mathematics, not the procedures.

Consider the case of Kristin (@MathMinds on Twitter), a fifth grade teacher, and her student “Billy”. Billy made an unusual claim about even and odd numbers. She followed up, she shared, we discussed on Twitter. Pretty soon, teachers around the country were engaged in thinking about whether Billy would call 3.0 *even* or *odd*.

But standard algorithms don’t teach any of that. They teach children to get answers. They teach children not to think.

I have read about it. I have thought about it. And tonight I saw it in my very own home.

Tagged: 10 years old, algorithms, Griffin, lattice, mathematical thinking, multiplication, standard algorithm ]]>

You can either shed a tear or do something about it. (or both)

If you choose the latter, join me and a whole bunch of others at EdCamp Math and Science MN.

It is free. It takes place Friday, October 17 (during MEA weekend).

See you there.

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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.

Best wishes and thanks again for writing.

Christopher

Tagged: ccss, ccssm, common core ]]>

Audience is parents, and this may appear in the title (*Common Core Math for Parents For Dummies* is the working title). It goes for the big picture in each of the grade levels, K—8.

The *For Dummies *format is pretty rigid but there will be no mistaking authorship. A few sample section headings (and the grades where they will appear) to whet your appetite:

1st grade.Saying bye-bye to key words

1st grade.Understanding the importance of ten

2nd grade.Why units matter

2nd grade.Place value

2nd grade.More about place value

2nd grade.Seriously. Place value.

4th grade.Multiplication: Whatisit and why not just memorize the facts?

5th grade.Standard algorithms: Doing things “the old-fashioned way”?

6th grade.Dividing fractions—More fun than you’d think!

6th grade.Area: It all goes back to rectangles

8th grade.Congruence and similarity: Two kinds of sameness

Catch you all later. I have some writing to do!

I’ll keep you posted.

]]>Here is my first stab at the genre, from this spring’s NCTM/NCSM conference in New Orleans. The others who presented that day are all worth watching. You can get the complete list, links and a bit more context from The Math Forum, which hosted the talks.

Enjoy.

Tagged: ignite, math forum, nctm, video ]]>

To be able to read something like José Luis Vilson’s recent post, or Mia McKenzie’s recent post, without feeling defensive.

A modest goal, for sure. But a necessary one, and one that will allow us to move forward.

Each of these posts is by a Black (/Latino) writer, with teachers as (at least) part of the intended audience, and each calls out racism in schools. (And sexism—for which I have an equivalent goal for my male colleagues—it shouldn’t be hard to reread this post replacing *race* with *gender *wherever it appears.)

When white people read this writing, there is an instinctive reaction that begins and ends with *Not all white people*. That is the defensive response I hope we can do away with.

Here’s the problem with that response: Racism is not about white people’s understanding of the nuances and varieties of white people. It is about the lived experience of people of color.

“Not all white people” is a racist response.

“Not all white people” denies the experience of the writer.

“Not all white people” cuts off further conversation about race.

This leads me to a second claim.

*Refusing to discuss race* is a racist act.

There is a certain brand of white liberalism, for example, that believes *noticing** race* to be a racist act. This view makes it impossible to talk about race.

In such a climate, asking a colleague what he knows about Somali culture in a quest to better understand a classroom incident is called into question as an act of racism because some white people engage in the same behaviors, and therefore there should be nothing to ask about. In such a climate we cannot speak of the vastly differential racial demographics of developmental math courses and College Algebra courses at the college level. To do so is seen as racist. Because—after all—we give the same placement tests to everybody.

Now a question for my white colleagues: *Why is “racist” that rare varitey of action that we allow the power to define us?*

We can live with duality in other areas of our lives: *I did/said a ___ thing, but this does not make me a ___ person.*

I have done many stupid things in my life, and I accept the potential for doing more stupid things in the future. Yet I am not a stupid person. I am comfortable owning that something I did was stupid. I can wish that I hadn’t done that stupid thing. But I don’t let the stupid thing define me.

Furthermore, it is OK to talk about how stupid something I did was, and the goal in talking about it is to ensure that I don’t do something that stupid again—or at least to eliminate this particular brand of stupidity from my repertoire.

But we treat racism differently. We pretend that only racists do racist things. (Again, do only stupid people do stupid things?) Therefore, we cannot own our racist actions. If we admit that we have done, thought or said something racist, we *become* racists.

This mindset—this inability to speak of our racist actions; to name them (even the inadvertent ones) as racist—keeps us from being able to talk about our mistaken ideas and actions. But talking about them would help us to avoid perpetuating and repeating them.

You don’t need to own the racism of your fellow white people. You don’t need to identify as a racist because someone else has done something racist, nor even because you have.

No.

You need to (I need to) honor the experiences of others. When a racist incident is brought to your attention, you need *not* to explain that “not all white people…” or that *you *have not experienced this. Doing so puts the focus back on you as a white person (which, again, is a racist act; and which, again, you—I—can own as an act without needing to own the title *racist*).

See, you don’t need to explain the experience of others away. Instead you need to listen. You need to acknowledge that racist acts are committed in the world, and that our goal is to reduce and ultimately to eliminate their incidence. Pretending—through denial or through silence—that racist acts do not exist is itself a racist act. Pretending—through denial or through silence—that racist acts have no relevance is a racist act. Pretending that racist acts can only be committed by people who are racists through and through—this is not an effective means to the end.

I understand that my goal is modest: Reading accounts of racism, written by people of color, without becoming defensive. But we have ample empirical evidence that the goal has not yet been attained, and it is clear to me that moving forward to really dealing with racism is impossible in its face.

Achieving this goal allows us to listen.

And listening—to our own hearts, and to the hearts and experiences of others—is where learning begins.

]]>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.

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