So much bad data representation wrapped into a single image. Make sure to read the fine print!

]]>What if you reflect over a circle? I played today with the circle I understand best—the circle of radius 1, centered at the origin.

You still want the line segment to be perpendicular, which is tricky enough. But distance gets messy. Reflecting over a line means taking everything on one side of the line and matching it to something on the other side. Easy to do with two half-planes.

But with a circle? You need everything outside the circle to match up with everything inside the circle. The basic idea is a point outside the circle will match up with a point inside the circle, and that their distances will be reciprocals of each other. If the original point is 2 units from the origin, its reflection will be 1/2 unit from the origin. 3/4 matches with 4/3, 5 with 1/5 and so on.

Whether infinity and zero match up is open to interpretation and not important right now.

Here’s what this looks like.

Go play with the graph. Move the orange point around and start to get a feel for the relationship between the original and its reflection.

Now we’ll do two points and connect them with a straight dotted line segment. Each endpoint is reflected in the circle. (Play with it here.)

What does the reflection of the in-between points look like?

Imagine it. Sketch it. Then go see.

]]>In Chapter 3: Mathematicians Take Risks, Tracy invites us to follow her lead and take a risk in a manner similar to a student she documents in a classroom vignette. She suggests that we play with 10. I liked the idea, but was feeling pretty tapped out on risks related to 10, having explored that territory in great detail and complexity over the last ten years. So I read on with Tracy’s challenge in the back of my mind.

In a seemingly unrelated episode, Tabitha (9 years old) was doing some homework this week and needed to know the product 7 times 8. Spouse was offering a hint: *It starts with a five*. I asked Tabitha if she knew what 10 sevens is, which led to her expressing 7 x 8 as 70 – 14.

The next day I was thinking about Spouse’s hint. *It starts with a 5*. What math might there be in that relationship? Could I develop a way of knowing what a product *starts with*?

With Tracy’s voice in my head, I took a risk. I now take a bigger risk by sharing my ideas with you.

THE RULE FOR EIGHTS

If you’re multiplying 7 by 8, the product starts with a 5. This is two less than the multiplier, 7. The same is true for 8 times 8 (starts with 6) and 9 times 8 (starts with 7).

11 times 8 starts with an 8, though. If you look more deeply, you’ll notice that the difference between the tens digit (what the number starts with) and the multiplier increases by 1 for every group of 5 in the multilplier.

Try 22 times 8. 22 has four groups of 5, so 22 times 8 starts with 17 (which is 22-4, minus one more because 1 times 8 starts with a zero).

The opposite rule applies to twelves. Every five twelves, you need to ADD an extra unit to what the number starts with. 13 times 12 starts with 15 because there are two groups of five in 13.

There are exceptions. 25 times 8 has me stumped. It should start with 19, but does start with 20.

THE RULE OF SEVENS

This is much more complicated. You can get a good approximation by counting groups of 10 in the multiplier and adding three to the difference between starting digits and multiplier for each group of 10. But you’ll get a lot of exceptions.

71 works. Seven 10s, times three gives you 21, plus one is 22. 71-22 is 49, so we expect 71 x 7 to start with 49, and it does: 497.

But 78 fails. My rule predicts that 78 x 7 starts with 56, but it doesn’t. 78 x 7 = 546.

The failures seem to be predictable, though. So I’ll press onwards.

I haven’t yet decided whether 7 is different and difficult because of its 7-ness, or because of the fact that it’s 3 away from 10. The *three for every ten* bit makes me think it’s probably the threeness of seven that matters rather than the sevenness. But I don’t know that yet.

Please don’t answer these questions for me. They are mine. Explore them if you find them compelling. Don’t spoil my fun.

]]>The recipe calls for one cup of pumpkin from a can. A can of pumpkin contains about a cup and a half, and what are you gonna do with an extra half cup of pumpkin? So I always put in the whole can.

Today however, I was using pumpkin from the freezer. Last fall, I turned a couple of pie pumpkins into pumpkin puree and froze it for exactly this purpose. The bag of pumpkin puree I was working with contained two cups. So I made two batches, using the prescribed one cup per batch.

They are good, but they are not quite as good as the ones that have a cup and a half of pumpkin.

If you spend any time baking, you will surely run across claims that baking is different from cooking. Baking requires more precision and following of directions than other types of cooking, you’ll be told.

Similarly if you spend any time learning math, you will surely run across claims that learning math is different from other intellectual activities. Learning math requires precision and following of steps, you’ll be told.

These are lies. My deliciously moist pumpkin muffins prove that this is so.

This is what I love so dearly about Eugenia Cheng’s book *How to Bake Pi.. *She writes that in both math and baking, decisions have consequences. Sometimes these consequences are undesirable, such as bread that doesn’t rise or arithmetic that is inconsistent.

In math as in baking, you need to follow instructions carefully in order to achieve the known result. If I want muffins that are exactly like the ones in the original recipe, I need to use one cup of pumpkin. But here’s the secret we don’t let you in on: If I use the whole can of pumpkin, I will still get pumpkin muffins. They will be different ones, but they will still be pumpkin muffins.

The difference between baking and math is that you nearly always see the natural consequences of the decisions you make, while we structure most people’s experiences with math in ways that hide those consequences.

If you leave out the sugar, your muffins will not be delicious. They may have structural problems as well. You notice these consequences and you tend to try to figure out what went wrong.

If you claim that , the only consequence is that someone else tells you that you are wrong—whether teacher, tutor, or back of the book. In Cheng’s book, she describes treating math exactly like baking. She pushes her students to consider the natural consequences of their claims. If , then when , . If , then there are going to be lots of troubles later on.

Go read her book. Then make some pumpkin muffins. And please don’t listen to Chris Kimball; the man is a total killjoy.

]]>Not least my own. If you’re reading this and do not own a Teacher Edition for *Which One Doesn’t Belong?* I need you to fix that situation before reading further. Because clearly you’ve found value in the words and ideas on this site, and the book is a better, more focused version of those ideas.

But I’m not here today to push my book. No, I’m here to get another thing on your reading list. First a prelude.

People (well *some* people, who are not me anyway) like to carry on about differences between elementary school math and high school math. When they do, they fall into one of two camps.

- You have to commit the facts and techniques of arithmetic to memory before you can use them to think, and to do real math.
- You can play around with the early, intuitive ideas of math, but when you get to algebra (or trigonometry, or calculus, or whatever) the game changes and you need to be told stuff. Then you need lots of practice with what you’ve been told.

That these two make opposite assumptions about elementary math, and also about secondary math should suggest to us that people are just choosing to look at things from a perspective rather than describing the true nature of the discipline.

Whatever the eventual fate of the Common Core State Standards for Mathematics, hopefully one of its lasting legacies will be an understanding that mathematics as a discipline is more than content, and that this should be represented in school mathematics. Where this appears in the standards is the *Standards for Mathematical Practice *(SMP).

*Aside…true story: I once spoke at length to a producer at a semi-famous radio show as she did background on the question How Much Math Should Everyone Know? 20 minutes of conversation before it became clear that this person hadn’t read the SMP. Andrew f-ing Hacker was going to be on the program assailing the state of mathematics teaching in this country and the producer hadn’t read the Standard for Mathematical Practice. I was livid. But back on task now…*

An important thing to know about the Standards for Mathematical Practice is that they pertain across all levels of mathematical activity. These are K—12 standards, but they describe the kinds of activity that distinguish math as a discipline.

When I wrote about the Standards for Mathematical Practice on this blog, and in the Dummies book, I took the easy way out. I addressed their spirit without going into all eight of them in depth.

But I’m here today to tell you that Mike Flynn has taken the high road and done the much more challenging job of treating each of these standards right.

Flynn structured his book *Beyond Answers* around the Standards for Mathematical Practice—one chapter per standard. He illustrates each with a vignette from his own life demonstrating the utility of the practice in his own daily life, with vignettes from classrooms, and with clear writing demonstrating the very real and important mathematical work of which young children are capable.

On its surface, this is a book about elementary children doing mathematics. But it’s really a book about *people* doing mathematics. If you’re a secondary or post-secondary teacher, and you read this book without seeing important connections to the work that you do, I’ll buy you a cup of coffee so we can talk about that.

Ultimately, my own critique of the SMP was about them being too numerous to remember, and about them overlapping in ways that make it difficult to communicate their individual importance. I still have those critiques. Flynn doesn’t convince me (nor does he try) that this is the perfect set of such standards. But he doesn’t need to do that.

These are the standards we have. They resonate at all levels of mathematical activity, and in *Beyond Answers, *Mike Flynn shows convincingly that young children’s mathematical work is not fundamentally different from that of older students. Mathematics as an intellectual discipline is alive and well.

First up is a Kickstarter campaign from Edmund Harriss, who makes beautiful and brilliant things. His coloring book Patterns of the Universe was part of the Summer of Math. He invented a new spiral.

Anyway, Curvahedra is a system of pinwheely spirally things that attach to each other in a simple and clever way, and which then allows you to build strange and interesting 3-dimensional objects. He brought samples along to Twitter Math Camp this past summer, and they are clearly quite wonderful.

If you are able, you should throw 10 bucks or more into the pot over at Kickstarter so you can play along when they ship next spring.

]]>I will keep the source anonymous by request.

Consider these two year-end summaries, both for the same child. One is by the child’s regular classroom teacher, and the other is by the child’s math teacher. This is a school where children are shuffled for math instruction.

[Child] has worked hard in third grade to increase her literacy skills in becoming a life-long reader and writer. She’s a mindful reader when choosing books of personal interest and challenge level. She reads with superb confidence and stamina. As a reader and writer of nonfiction, [Child] researched an animal of interest and published a book that demonstrated expertise of the topic. She crafted nonfiction text features that reflected her individual voice. She read and analyzed poetry—then applied the tools of a poet to craft individual poems. Wow! You’re an amazing reader and writer! Way to go, [Child]!

[Child] has made progress in math this year. She should work on learning her basic subtraction and multiplication facts over the summer. I have enjoyed having her as a student! Have a wonderful summer, [Child]!

I understand that these are different teachers, and I understand that [Child] probably spent quite a few more hours with the literacy/home teacher than the math teacher. But I don’t think those are the major things at play here.

The difference in the richness of the description of [Child]’s current literacy and growth from the description of [Child]’s mathematical knowledge and growth reflects the ways we view these two subjects in American schooling.

Throughout my own children’s elementary schooling, I have seen deep and rich attention paid to their learning to read and write, and surface, fact-based attention paid to their learning math.

Literacy is about power and beauty and self-actualization. Math is about memorization and speed. These are the ways we represent these subjects to parents, teachers, and children.

Tagged: girls, literacy, parents, reading ]]>

A few minutes later I made this.

And then I went scrambling for my notebook.

Yup. Sure enough. is equivalent to the Pythagorean Theorem.

Tagged: kindergarten, triangles ]]>

My first thought was, “Send them all to New York to visit the Museum of Math!” but this was off by a couple orders of magnitude.

As the conversation continued, it became clear that they weren’t seeking advice so much as someone to make it happen. So I said yes.

I am spending three Thursday mornings, and one afternoon, with these fourth graders. Today was day 1.

The theme of the residency is scale. We are playing with small versions of big things and big versions of small things.

A few favorite moments from today:

When asked to share a big version of a small thing, one girl said “Horses”. I pressed her to state her meaning. “If you had a map with stables on it, the horses in those stables would be really small, then when you went to the stables, the *actual* horses would be really big.”

Ladies and gentlemen, I give you the big math idea of *inverse!*

I thought *the horses on the map are small versions of the big real-life horses.* But she was very clear that her experience was *small horses on the map, then see the big ones*. The small-to-big relationship isn’t just the opposite of the big-to-small one; it is its own relationship. These two relationships are inverses—each existing on its own, but with a special connection to each other.

I cooked up a little *Which One Doesn’t Belong? *set in preparation for our work.

*Which One Doesn’t Belong? *never disappoints. (Student/home version and Teacher Guide coming this summer from Stenhouse, by the way!)

We noticed all the things I had hoped for, and more. And then afterwards a girl came up to me to make her case that we weren’t being totally precise about our description of the upper-right image. If—as we claimed—the shape in the upper right is composed of four of the upper-left triangles, then the big triangle wasn’t exactly the same size as the one in the lower left because the triangles have outlines which are not infinitely thin.

I brought in many small laser cut triangles of these seven types:

I gave them time to play with these triangles. One student said she knew what we were going to do with them. So I asked her what that was, and she replied that we were going to see which ones could fit together to make other ones. This was not the plan, but was behavior I was eager to encourage.

She asserted that the pink and the black make the red.

This was a detour worth five minutes, so we took it. Arguments were presented pro and con. The major pro argument was based on the *close enough* principle. Con arguments were of two flavors: (1) put the red underneath and you’ll see some red peeking out from underneath, and (2) the long side on the pink plus black shape is not straight, while it is on the red one.

The main question I wanted to get to—remember that our focus is *scale*—was *Which of the triangles in our set will do what the upper-right shape in our Which One Doesn’t Belong? set does?* Which of our triangles can you make into a larger version?

**Spoiler Alert!**

All triangles do this. But these fourth-graders don’t know that. And because they don’t know that, they got to feel a little thrill of success when they found one that did.

And of course they produced some evidence that the relationship we’re investigating is a challenging one.

This is what we had on the document camera at the end of one of three sessions this morning.

HOLD THE PHONE! LET’S LOOK AT ONE OF THESE CLOSE UP!

Do you see? All the others use four triangles to make the bigger version, and this one can too. But this can scale up to make a bigger version that uses only two of the original!

Of course there is a part of my math-major brain that knows this about isosceles right triangles, but it’s a wonderful wonderful thing to have pop up unexpectedly in the middle of fourth-grade math play.

Overall, a delightful morning of math. We got to only a small fraction of what I’ve got chambered so we’ll pick up where we left off next week. I’m hoping I can get them to build one of these.

Either way, I am thankful for the opportunity to play math with this group of kids. They are creative, enthusiastic, curious, and delightful. Their teachers have been very welcoming and open to the intellectual chaos I began to unleash today.

I chose a set of triangles that would have interesting variety and some discoverable properties.

**Purple**: 3-4-5

**Pink**: Isosceles obtuse

**White**: Isosceles right

**Red**: 30-60-90

**Light blue**: One-eighth of a regular octagon

**Black**: Equilateral

**Dark blue**: One-fifth of a regular pentagon

I also made some yellow obtuse scalene triangles, but they are missing so they didn’t make the trip. Within these classes, these triangles are all congruent. Each class has at least one side that is one inch long.

Tagged: elementary math, residency, scale, similarity, triangles ]]>