## Which Poster Doesn’t Belong?

(Cross-posted from Talking Math with Your Kids)

Two and a half years ago, I was developing Which One Doesn’t Belong? (before Stenhouse had signed on to publish it). I went on a tour of elementary classrooms to talk with K—5 students all around the Twin Cities about these collections of shapes. I learned a tremendous amount of course, and much of that learning went into the Teacher Guide (which Stenhouse convinced me needed to exist).

I learned a lot, and I also noticed something.

Most of those classrooms had some form of shapes posters on the walls. Triangles, rectangles, squares, and rhombi were proudly and prominently displayed so that students would be surrounded by correct geometry vocabulary. Most of those shapes posters had something important (and unfortunate) in common with the shapes books in the school library and in the children’s homes.

There were rarely squares on the rectangle poster. All of the triangles were oriented with one side parallel to the ground, and most of them were equilateral. Sometimes the shapes had smiley faces. You and I know that a triangle is still a triangle, no matter its orientation. I can assure you not all elementary school children know this. While the vocabulary is good on your standard shapes poster, the math is not. (I decided not to link to examples—you can do your own search and report back if you find my claims exaggerated.)

This summer, Stenhouse is helping all of us to fix this. You can now preorder Which One Doesn’t Belong? shapes posters.

They come as a set of eight, with an insert in the spirit of the Which One Doesn’t Belong? Teacher Guide to help you facilitate student thinking and classroom conversation as they hang in your classroom.

1. Which SQUARE doesn’t belong?

2. Which RECTANGLE doesn’t belong?

3. Which RHOMBUS doesn’t belong?

4. Which HEXAGON doesn’t belong?

5. Which TRIANGLE doesn’t belong?

6. Which POLYGON doesn’t belong?

7. Which SHAPE doesn’t belong?

8. Which CURVE doesn’t belong?

These posters are filled with good mathematics. Consider the triangle poster on top of the pile. The triangle in the lower right is the only right triangle. The one in the upper right is the only equilateral triangle. The one in the upper left is the only isosceles triangle (or is it? do equilateral triangles count as isosceles?) The one in the lower left is the only one you can’t build out of the triangle in the lower right. Students will notice side lengths, angle measures, orientation, composition and decomposition, and more properties of triangles. Some will complain that not all of them are triangles (“too pointy” or “doesn’t have a bottom”). These posters let you and your students sit with—and play with—these ideas over a period of weeks or months.

So as you plan your back-to-school classroom organizing and decoration, I hope you’ll consider making space on your walls for these posters. And I definitely hope you’ll share your students’ ideas here and on Twitter.

Available for pre-order now. They’ll ship in early August.

## The New Basics

I gave a talk last week.

This talk had a click-bait title, for which I am apologetic. The New Basics: Arithmetic and Algebra with 21st Century Tools.

Sorry, not sorry.

Here is the planned text of that talk. It should bear more than surface, but less than perfect, resemblance to the actual text of that talk, which exists nowhere but in the minds and memories of me and the attendees.

THE NEW BASICS: ARITHMETIC AND ALGEBRA WITH 21st CENTURY TOOLS

Friday 11-12 in 301

What do students need to know about arithmetic and algebra, and how can digital tools support their learning it?

I’ll show you our last task first. The place where we’ll end up in 45 minutes or so. You have a 35-foot wide parking lot and three 1-foot wide dividers to place at regular intervals. The size of the intervals is controlled by what you type in this box. The question we’ll consider is “What is the idea thing to type in this box?” or “What is the best answer here?” Keep that in the back of your mind as we work together this morning.

My talk today has four chapters; each anchored by a task. Each task is facilitated by 21st Century tools. I would not have been able to do these tasks as we’ll do them in my middle school classroom in the year 2000.

But first a prologue. I feel like most slides featuring storm clouds are metaphorical. These are literal.

One day when my daughter was 7, we were standing at the window at the top of our stairs, watching storm clouds roll in.

We were in a thoughtful mood, so I asked her if I could ask her a math question. She consented and I asked her how many tens are in 32? She said 3. I asked her how she knows and she counted 10, 20, 30 on her fingers. A moment passed and she said “There are ten tens in a hundred, though.” But how many in 200? I asked. “Twenty” she said. “Whoa” I said, and she said, “Yeah.”

Another silent moment passed and she offered this simple observation.

“Asking ‘How many tens are in 30?’ is like asking ‘How many ones are in 2?'”

The trouble with place value is that a claim such as this one sounds simple and obvious but is in fact deep and profound. You can learn to name the places without noticing or being able to use the underlying structure. This is a talk about the structure of numbers. We will look at a bunch of tasks together that are designed to bring out number structure.

Here’s an example of a task that attends to number structure. How many eggs in each photograph? How did you count them?

Here is an example of a task that does not attend to number structure.

What prevents attending to number structure?

• Random order
• Time constraints
• Reporting percent correct
• Absence of representations

This is the bulk of my 10-year-old daughter’s experience using technology for math in school, and I think it’s safe to say that resonates with many children and parents.

Not all, of course, but I have no reason to believe my daughter’s experience is either unique or rare. But if we step back a little bit and look at the situation with a critical eye, we’ll notice that we tend to have computers teach children to do what computers themselves do better—they are teaching children to compute. And there is value in that.

But in the words of Papert, we are using the computer to program the child.

But if the sum total of the outlay of millions of technology dollars in our public school system is somewhat less distasteful arithmetic instruction, then we need to revisit our priorities.

Papert wanted the child to program the computer; a worthy mission about which someone else is surely speaking at this conference. I’ll take us in a different direction.

I want to share with you some ideas for using computers to open conversations in classrooms. Conversations that allow for deep and rich exploration and observation. I’ll share with you examples of instruction that uses new tools to teach things that computers don’t do well, and that are essential to using mathematics in whatever fields our children end up pursuing.

CHAPTER I: HOW MANY?

We’ll focus on multiplicative structure because it’s the key to so many things—place value, algebra, two- and three-dimensional measurement.

The thing I have learned in recent years about multiplicative structure is that children’s experiences lay the foundation long before they study multiplication formally in third or fourth grade. We can and should build on this foundation. Here’s one way to do that.

There are some here who know my Twitter habits; I’ll ask you to bear with us for a moment while I catch the rest of the folks up. One of the key ways that educators use Twitter is chats. A chat is typically a scheduled thing; every Thursday at 8PM Eastern time is an elementary math chat, for example. Someone organizes the conversation by feeding questions to the group, People respond to questions and respond to the responses. By the end of an hour, many subconversations have added up to a semi-cohesive whole. Some people participate actively; others just read along. There are no requirements or memberships.

The technology for organizing all of this is the hashtag. #elemmathchat is attached to every tweet in the conversation and you set your TweetDeck to follow that hashtag. Because chats are organized with a hashtag, they tend to spill outside the bounds of their scheduled times. Once I know who tends to attend #elemmathchat, I might also attach that hashtag to a question or observation I happen to have on a Monday morning in the hopes that one of the regulars will notice it and we can have a conversation. There is a middle school math chat. There is a terrific diversity and equity conversation that doesn’t call itself a chat, but which all educators need to know about and read along with—#EduColor.

I’ll take a brief moment to address my white colleagues directly. #EduColor is a terrific place for us to do a lot of listening before speaking, and it is a place that requires an open mind and an open heart.

At one point I began to play with the idea of increasingly specialized chats and began to amuse myself this way. I have been fortunate to have made one of these stick. #unitchat.

I started with situations where units were missing or wrong, but have progressed to using it mostly to point out situations involving ambiguous or multiple units.

Here’s where I am now with this. Let’s look at a few images together, and each time I’ll ask a simple question—How Many? If your instinct is to ask How many what? in return, then you’ve got the idea behind the task. This is about counting, and paying careful attention to what it is that you’re counting.

Shoes–Two shoes, one pair. When we can express the same count in two different units, we’re building multiplicative structure. For every pair, there are two shoes. A pair is a group; a unit. It’s a thing you can count.

Playing with units has opened my eyes to the importance of groups in mathematics. Groups drive multiplication. Groups drive place value. Groups drive the distributive property. When you group groups, you notice iteration which is foundational to many areas of mathematics, not least chaos theory and fractals. In short, time spent studying groups in our world provides an experience base for the abstraction of mathematics that Cathy Fosnot and others have called mathematizing. As human beings, we make mathematics in our minds, and we need something to make mathematics out of—one of the important raw materials for this manufacturing process is lived experience. (Plug Malke’s session.)

Wnen I left the public school classroom in the year 2000, I was not able to take photos such as these and bring them to the attention of my students. The math I saw in the world stayed in the world and could not come into my classroom. Similarly, the math they saw in the world stayed out there. Now when I visit classrooms, they have projectors attached to computers. We can all see exactly what I saw. We can have a shared experience with an image with no more effort than opening Dropbox on my phone and computer.

Similarly, social media allows me to bring the mathematics other people see into my classroom. That avocados picture is adapted from one that someone sent me through Twitter a few years back.

CHAPTER II: REVERSE NUMBER TALKS

Here are 13 wooden circles. Imagine for a moment that this is a workshop not a session, and imagine that you can reach out and touch these 13 circles. Make or imagine some interesting arrangement with them. On a piece of scrap paper, or in your notes, or on your iPad, sketch the arrangement you see in your mind.

Share with a nearby partner who hopefully has made a different interesting arrangement. Record it and continue. Keep going for a few minutes. DO NOT change your number. Your goal is to see your number in as many ways as possible.

Here are some possibilities.

21st century tools. I have a couple of things to say here. One is that it matters that these are circles. I can show rows and columns with circles, and I can also show 7 as a hexagon with a center. I can’t do that with the plastic square tiles that were in my classroom in 1999. But if I have access to a laser cutter, I can turn \$2 worth of walnut into 20 circles over the course of 15 minutes. The other thing I’d like to say here is that 21st century tools complement older ones, but need not always replace them. Yes, children enjoy brightly colored screens and virtual worlds, but they also enjoy things are subtle, beautiful, and tactile. A view of the world holding that children require electronic stimulation in order to be engaged is a cynical one.

CHAPTER III: LUSTO’S DOTS

Knowing the structure exists allows you to look for it.

Here’s a pretty thing my colleague and friend Chris Lusto made recently.

He used javascript to make the original, probably, and I don’t know that. But I do know how to use Desmos, so I made a version in the calculator. The link is on the blog. Feel free to use it, adapt it, have fun with it.

How many dots? How do you know?

There are 12 dots, and I noticed quickly that there’s a point where those 12 dots are pretty clearly separated out as two sets of six. Quickly make a list of other ways you know of making 12.

Let’s start with 3 times 4. 12 is 3 times 4, and I’ll think of this as three groups of four. Let’s watch and I want you to shout when you see three groups of four. And we could look for four groups of three, and six pairs, and 7 plus 5, etc. There is something really satisfying to me about finding a structure I know should exist, but which isn’t revealed at first glance.

CHAPTER IV: CENTRAL PARK

On screen 5, what do you want to type in that box? Two alternatives: a number, and an expression.

What we’ve learned from teachers is that if structure isn’t something you’ve studied, you can’t move beyond guessing and checking. But guessing a number and adjusting it doesn’t generalize. Structure generalizes. An eighth grade teacher with a classroom full of guessers can do her level best to remediate with additional experiences with structure. Number talks are useful tools in high school classrooms, and I recently spent a delightful 10 minutes with a College Algebra class noticing the associative property in the language we use to describe the avocado picture. But structure starts earlier. Like language, it needs to build over time through repeated use, incidental noticing and deliberate exposure.

[Closing remarks and a brilliant summary occurred here]

## TURD pizza

A Truly Unfortunate Representation of Data accompanied a recent article on British pizza toppings.

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

## Reflecting over a circle

If you’re reading this blog, you probably know something about reflections in geometry. You have a point (purple in diagram below) and a line (solid). If you reflect the point over the line, you get a new point (orange) on the other side of the line, the same distance as the original point. The segment connecting the two points (dashed) is perpendicular to the line of reflection. (Link to live graph.)

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.

## Please don’t solve this problem for me

I’ve been reading Tracy Zager’s Becoming the Math Teacher You Wish You’d Had. (Full disclosure: Tracy is my editor at Stenhouse, which published Which One Doesn’t Belong?)

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.

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.

## Muffins, math, and the lies we tell about both

I made my favorite pumpkin muffins this morning, for the first time in quite a while, but I made them differently today.

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 $(x+1)^2=x^2+1$, 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 $(x+1)^2=x^2+1$, then when $x=-1$, $1=2$. If $1=2$, 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.

## Book recommendation(s)

I have many wonderful books to recommend.

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.

1. You have to commit the facts and techniques of arithmetic to memory before you can use them to think, and to do real math.
2. 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 AnswersMike 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.