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 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.

]]>He taught me to write.

I didn’t really have anything to say yet. But I learned grammar, structure, passion, and the value of just getting words on the page from Mr. Webb.

I draw on those skills in everything I do professionally; whether it is curriculum writing, blogging or conversing.

My words flow more easily. My ideas are more clear. My thinking is better. I owe these things to him.

Thank you for that, Mr. Webb.

]]>All measures of center hide variation.

This is what makes them useful, and it is what makes them problematic.

Using the mean makes zeroes a problem in grading. Wildly divergent values (such as a zero in a gradebook) will greatly affect the mean. It is hard to argue that 2 A’s and a zero is the same as consistent D work. Yet this is how the mean plays out.

But going too far down this road will only lead to critiques of the whole system of grading students at all. I find that system to be indefensible and counterproductive. I have made my peace with it, and I try to do as little harm as possible with the responsibility I have to assign grades in my work.

All of which is to say, it is not* using the mean *that leads to a poor measure of achievement. It is *mistaking quantitative measures for accurate ones* that leads to a poor measure of achievement.

Tagged: average, data, mean, wiggins ]]>

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.

Tagged: division, wiggins, word problems ]]>

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea *is.*

In recent years, I have come to an understanding of why *repeated addition* is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it *is*.

You see, any rich enough mathematical idea has multiple meanings. What *is *subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it *takeaway*? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an *equivalence class* resulting from the equivalence relation,

if and only if

Is that what a fraction *is*?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. *Hours* times *miles per hour* gives *miles. *Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: means *A* groups of *B *(I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). *A*, in this interpretation, is expressed in one unit. *B* is the unit rate (*things per group*). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”

Why *do* we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—*one-to-one correspondence*. One number word for each object, and one object for each number word.

The answer is that mom really did *not* count the corner meatball twice. The first time, she counted the *meatball* to establish that each row has 4 meatballs. The second time, she counted the *rows*. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or *What does this have to do with mathematics?*)

Twister (on sister site, *Talking Math with Your Kids*)

Tagged: array, multiplication, wiggins ]]>

One sixtieth is the biggest of these, as the others are between one and two one-thousandths (or between one one-thousandth and one five-hundredth).

Then, in descending order, we have .002, .00156, .0015 and .001.

It is common that students will order the decimals this way:

.00156, .0015, .002, .001

I cannot predict where a person who does this will place one sixtieth.

Nonetheless, when we read these decimals as *point zero zero one five six,* we encourage students to ignore place value, and we encourage the misapplication of whole-number rules to the right of the decimal point.

We really do need to use place value language for decimals in classrooms. *One-hundred-fifty-six one-hundred-thousandths.*

Much, *much* more about this and related ideas in the Triangleman Decimal Institute posts from last fall. Short version: learning decimals is * WAY* more complicated than most people have any reason to imagine.

Tagged: decimals, wiggins ]]>

Set 1 (pdf)

Set 2 (pdf)

Shout out to former students Jen Carlson, Nadaa Hassan and Brenna Magnuson for collaborating on these.

Hexagons by Christopher Danielson is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Update: Below is the current complete set, with added hexagons from former students Ruth Pieper, Brandon Schwab and Mona Yusuf.

Tagged: geometry, hexagons, hierarchy of hexagons, nctm, nctmNOLA ]]>

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.

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.

When dividing by a fraction, partitive division means that we know the fractional part of a group we have, and we need to find the size of a whole 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.

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.

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.

More on partitive fraction division here.

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.

Tagged: division, division of fractions, fractions, wiggins ]]>

This question is a strange one. It really isn’t how I would define problem solving, and I certainly wouldn’t include *equality* as a major component underlying problem solving.

Nonetheless…

I suppose he is getting at the idea that expressing equations in equivalent forms sometimes reveals different details of a problem.

For instance, I have created a new measure for cylinders: the *circumradial measure*. You add the radius and height. Then multiply this sum by the circumference.

In exploring this measure, one might end up restating this formula in equivalent terms, as:

This is more recognizable as a formula for surface area of a cylinder. The form of the equation affects how we think about the relationship it expresses.

**What does the equal sign** **mean?**

This is an important question. There is lots of research about it (CGI folks have worked on it, for instance). Three quick points:

- The equal sign means that the two things on either side have the same value as each other.
- We often teach in ways that lead students to think that the equal sign means
*and now write the answer*. - You can’t really understand much about algebra with the conception that (2) fosters. You need (1).

Finally, there are deep ideas underlying the equal sign. *Equivalence *is the mathematical way of talking about *sameness*. Stating the meaning of *sameness* precisely in mathematics turns out to be tricky and interesting work, and is a foundation of modern algebra.

Tagged: equal sign, equivalence, wiggins ]]>