The hexagons are here! [#nctmnola]

Forgive the delay. Here are pdf files of the hexagons we built for use in my hierarchy of hexagons lessons. You should be able to open and edit them in Adobe Illustrator. Consider them CC-BY-SA.

Set 1 (pdf)

Set 2 (pdf)

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


Wiggins questions #3

Question 3

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.

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 \frac{2}{3} of a tank of gas in my lawnmower is a partitive division problem because I know what \frac{2}{3} of a tank can do, and I want to find what a whole tank can do. So performing the division 4\div \frac{2}{3} 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 \frac{4}{5}. Scaling from 4 to 5 requires multiplying by \frac{5}{4}. 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 \frac{2}{3} 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 \frac{3}{2}.

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,

4\div \frac{2}{3} = 4 \cdot \frac{3}{2}

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 -1 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.

Wiggins questions #2

Question 2

“Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

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.


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.

C_M = (r+h) \cdot (2\pi r)

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

C_M = 2\pi r^2+2\pi rh

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:

  1. The equal sign means that the two things on either side have the same value as each other.
  2. We often teach in ways that lead students to think that the equal sign means and now write the answer.
  3. 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.

Wiggins questions #1

Math folks online have been all atwitter (heh) about a recent post by Grant Wiggins on conceptual understanding in math. Within that post (which I have not read in its entirety for reasons to be explained later), he proposed a series of questions that we should offer students as a way of opening our minds to what conceptual understanding means in mathematics.

Max Ray expressed a wish for some math ed bloggers to answer these questions in writing. I am obliging. One question at a time. One per week. I have not read the post so as not to bias myself.

I reserve the the right to critique the questions along the way.

Question 1

“You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)

Fact families.

Division is defined in relation to multiplication. For every one multiplication fact, there are two division facts.

3•2=6 is matched with 6÷2=3 and 6÷3=2.

Zero is a special case. 0•2=0, 0•5=0, 0•a=0 for all possible values of a.

This is no problem for multiplication. But it is a problem for division.

0÷0=2 would be a fact from the fact families. 0÷0=5 is another one. 0÷0=a for all possible values of a.

That is, 0÷0 can equal anything. And if it equals anything, it actually equals nothing. So 0÷0 is undefined.

More generally, though, 2÷0, 5÷0 and a÷0 for all possible values of a are problematic. Let’s say we decide that a÷0=12 (and let’s say that isn’t 0, since we took care of that case already). Then the fact family tells us that 0•12=a. But 0 multiplied by anything is 0. So 0 can’t be 12. But it can’t be anything else either. So a÷0 is undefined.

Conclusion: We cannot divide by zero for two reasons.

  1. Division is defined in relation to multiplication, and
  2. Zero has a special role in multiplication: 0•a=0 for all values of a.

We can use intuitive notions to establish that division by zero is a strange beast, but we can’t really firm up why without these more formal mathematical ideas.

The future of math

Nothing important here. But in case you’re into this sort of thing, we had some fun on Twitter this morning imagining the future of math. Click on through if you like.


5 reasons not to share that Common Core worksheet on Facebook

You are browsing Facebook on a Sunday evening. Someone has shared a baffling piece of math homework that was sent home with their child. Accompanying commentary bemoans the current state and future trajectory of mathematics teaching and learning, and lays blame for this at the feet of the Common Core State Standards.

You are baffled by the worksheet too. You are about to click the Share button.

But here are five reasons that’s a bad idea.

1. Credentials are not a trump card.

Almost invariably, the parent who shares the worksheet cites a degree as a credential for the critique. “I have a Bachelor of Science in electronics engineering,” wrote the parent in the recent “Letter to Jack”.


The math you need to know to be an electronics engineer is different from the math you need to know to be a math teacher. I am quite certain that electronics engineers use math that I have not studied, and similarly I use mathematical ideas that they have not studied.

When my teacher friends watch the following video of my son Griffin, they tend to see that a number line would be the right thing to draw to capture his thinking, and they tend to know what is coming when he goes to solve the problem on paper. They tend to describe my son’s work as demonstrating competence or proficiency with subtraction, but suffering mechanical errors when solving with paper and pencil.

Non-teachers who view this video are less likely to see a connection to the number line, and they tend to consider Griffin’s knowledge of subtraction to be weak. 

The difference is that teachers have a different kind of mathematical knowledge from electrical engineers. Not necessarily more or less knowledge—different knowledge. This is because different mathematical knowledge is required to do their job.

Deborah Ball refers to this different knowledge as mathematical knowledge for teaching. It is what mathematics teachers know who are more successful in their work. This knowledge includes common errors with standard algorithms, as well as their sources [start at about 2:30 in this video]. It includes common correct, alternative ways of performing and showing computations.

Moral of the story: You may not want to look to an electrical engineer as your primary resource for the current state of math teaching.

2. It is probably misinterpreted.

Homework time can be stressful. This is not new to Common Core. 

Parents are trying simultaneously to be helpers and enforcers. When a child does not understand what appears to be something simple, tempers can flare

We parents are not at our most rational at these times, and this may prevent us from fully understanding the goal of the task. 

When Frustrated Parent wrote to Jack, he committed two important errors of misinterpretation: (1) He assumed that Jack’s method was being taught as a preferred algorithm for subtraction, and (2) He assumed that something unfamiliar to him must be complicated.

These are totally understandable. I do not hold Frustrated Parent in contempt for his frustration.

I am simply asking the rest of us to resist sharing without asking critical questions.

The strategy shown on this particular worksheet is counting back

This is how many people count change: You gave me $20.00 for a $3.18 item, so you get $17 minus $0.18 in change…that’s $16.90 minus $0.08…$16.82. 

The number line Jack drew captures this thinking. 

The number line could have been improved by (1) arrows on the arches, and (2) smaller jumps to suggest that Jack is counting by a small number (as it is, the relative sizes of the pictured jumps on the number line suggest Jack is counting by 10s or 20s to the left of 127—as Frustrated Parent notes in a later Facebook post).


This worksheet was not about getting students to use the number line as an algorithm. It was about having students try to understand the thinking of someone else.

This may have been a bad worksheet—but not for the reasons cited when people share it on Facebook.

[For the record, in this case Jack’s error was forgetting to subtract the 10 in 316. He counted back three hundred, then six ones. The result is that his answer (indicated on the left-hand end of the number line) is too big by 10. He gets 121. The correct answer is 111.]

3. It is probably not “Common Core”.

There is nothing in the Common Core State Standards that requires students to use number lines to perform multi-digit subtraction. In fact, standard 4.NBT.B.4  requires students to “Fluently add and subtract multi-digit whole numbers using the standard algorithm”.

The standard algorithm, of course, is what the frustrated parent suggests that Jack use.

4. Anecdotal evidence is not research data.

While parents who share these worksheets in frustration will make claims such as the old way worked for me, the research evidence is quite strong that the old ways did not work.

Changes being made to American mathematics teaching are (1) very slow to take root, and (2) based on years of American and international research on student learning. 

5. Teachers need our support, not our scorn.

The frustration and anger of a parent who is struggling to help their struggling child is completely understandable. Any parent who claims they have never been frustrated at homework time is living in (a) a fantasy world, (b) denial, or (c) both.

But when we widely share the product of others’ frustration online, we amplify the anger. Ultimately, classroom teachers are the targets of this anger, as they are the public face of the education system. As a group, teachers work very hard with limited resources. They are called upon to equalize the inequities our society creates, and to offer not just equal educational opportunities, but equal educational outcomes to all children.

Now—more than ever—teachers need our support, not our scorn.

What to do instead

If you are Frustrated Parent, you can write a level-headed note to the teacher. It might look something like this:

Dear Ms. Crabapple,

I worked with my child on this problem tonight. Neither of us could figure out what is going on with the number line. You can see the work we did together, but we did not know how to write an explanation to Jack. We are confused. Please help.


Frustrated Parent

If you run across the work of another Frustrated Parent online, please consider asking someone about it before sharing it as evidence of the decline of American mathematics education.  Some possibilities:

Ask a teacher friend. You probably have at least one on Facebook. 

Ask on Twitter. There are many eager-to-help math teachers who follow the hashtags #mtbos and #mathchat—sincere questions asked on those hashtags will get sincere answers and offers of help.

Ask on a website. The Mathematics Educators stack exchange is a new resource for people to ask and answer questions related to teaching and learning mathematics. Anyone is welcome to ask a question, anyone can answer, and everyone votes on the quality of the answers so that you can easily find the best ones.


The New York Times published a piece on “Common Core homework” in July. I wrote a response to it that clarifies a critique in the article about dots. I invite you to read there for more information.

Note: Things got far beyond my ability to curate in the comments, so I needed to turn comments off. I would be more than happy to take up the dialogue on Twitter or through a pingback to your blog. You can also contact me if you wish to discuss further—Hit the About/Contact link at the top of the page.

Second note: I will curate and organize the major threads of the comment discussion in the coming days. In the meantime, I have sequestered the existing comments as the discussion threatened to overwhelm the point of the initial post. I have not deleted them.

A quick Fox TURD for you

A Truly Unfortunate Representation of Data (T.U.R.D.) from Fox News.

Posted without comment.


Thanks to David Radcliffe for passing it along on Twitter.