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.

]]>

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 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 ]]>