Category Archives: Uncategorized

On Belonging, and Not

I didn’t invent Which One Doesn’t Belong. I’ve told the story of how I came to the idea many times (notably in the teacher guide to my book), but the origins lie here:

Mine isn’t even the first book with this title.

Two editions of my book Which One Doesn't Belong? and a copy of a 1970's era Sesame Street book with the same title. This older book has a drawing of Cookie Monster in front of a plate with three chocolate chip cookies and a toy ball.

It does appear to be the first whose intention is to invite and welcome diverse perspectives—to take all true ideas about ways to categorize and distinguish, and to call them all “math”.

As with all creative endeavors, I built on the work of others. I learned from it, polished and adapted it, and I offered it up to my colleagues in the hopes that they would use these ideas to invite learners into the intellectual community of mathematics. I want Which One Doesn’t Belong? to widen the range of people who feel that they belong because their ideas matter. I want Which One Doesn’t Belong? to provide a frame for helping others to see the world from new mathematical perspectives, to ask new questions, and perhaps even to help the world see new mathematics.

In everyday life, the meaning of belonging is simple. Dirty socks don’t belong in the refrigerator. A basketball shoe doesn’t belong with a hiking boot. A ball doesn’t belong on a plate of cookies.

It is in this sense that mathematicians adapted the informal language of belonging to the mathematical register as a foundational concept in set theory. 0 either belongs to the set of natural numbers, or it does not (although which of these is true depends on whom you ask). \frac{1}{2} belongs to the set of rational numbers. A square belongs to the set of all polygons.

These two senses of the term belong are what make Which One Doesn’t Belong? a powerful tool for drawing learners into mathematical conversation.

Occasionally in my journeys, I have heard from someone concerned about another sense of belonging—that of students belonging in classrooms, schools, or even in mathematics itself.

I know some things about not belonging in this sense. I don’t belong in public schools—my sense of time and my priorities are very ill-suited to my long-term survival there. I am too interested in the complexity of learning simple ideas to belong in many college math departments, and too interested in the abstractions of mathematics to belong in most education departments. I am too intellectually impulsive and impatient to belong in higher education more generally. (Have I mentioned that I got the idea for and developed the first draft of Which One Doesn’t Belong? in a day? That within five minutes of first visiting the Alphabet Forest, I determined to build the math version of it?)

Perhaps in my Which One Doesn’t Belong? work, I have had some affinity not just for those mathematical outsiders whose ideas (such as the geometry of diamonds) are not valued in textbook mathematics. Perhaps I have also identified a bit with that triangle in the upper left, who has the wrong number of sides to belong on the opening pages of this book, but who does have the right orientation (for now….it is balanced rather precariously).

Four polygons in a 2 by 2 array.
Upper left: equilateral triangle balanced on a vertex
Upper right: square balanced on a vertex
Lower left: non-square, unshaded rhombus balanced on a vertex
Lower right: non-square rhombus resting on a side

It is in this third sense of belonging that people sometimes raise concerns. Are we inadvertently sending a message to students that, in the various ways they are different from their classmates, they might not belong? The most honest answer I have is that I don’t know.

We do know that children receive all kinds of implicit and explicit messages about whether they belong in this classroom, this school, doing this mathematics.

In skilled hands, Which One Doesn’t Belong? offers opportunities to unpack those messages—to question and reject them. Opportunities to find alternative ways of conceiving of self and community, as far more nuanced, complex, and important than the sorting of socks or of triangles.

I am also sure that in clumsy or malevolent hands, Which One Doesn’t Belong? could cause real harm. Is that harm alleviated by replacing the language of belonging with the language of differences or uniqueness? I don’t know.

When people have expressed concerns to me in the past, I have had two go-to messages. First is that we must never use children as the constituents of our Which One Doesn’t Belong? sets. Second is that you should absolutely adapt these ideas to work best in your context. Use the parts of this work that will build a welcoming and diverse community of mathematicians in your classroom; modify as necessary, and please report back to the wider community so that we may learn along with you.

I have learned a tremendous amount from the opportunities I’ve had to work with a wide range of audiences—to listen to children, adolescents, parents, and teachers tell me which one doesn’t belong and why. I have also learned a lot from teachers and others in talking about this work. I look forward to seeing how these ideas grow and change with the wisdom and experience of the community.

I look forward to seeing what Which One Doesn’t Belong becomes, and what it inspires.

Posters Posters Posters! (plus a prize drawing!)

I am working on my next book from Stenhouse, a counting titled How Many? I took a little time away from that this summer to work on a set of shapes posters. I designed them, Tracy helped me refine them, and the Stenhouse production team made them beautiful.

Poster images

Most of the shapes posters available to teachers are very very bad. (Seriously…go ahead and do a quick image search on shapes posters….I’ll wait.) Everything is in standard orientation; the triangles are mostly equilateral; the rectangles never are, and on and on.

The Which One Doesn’t Belong? shapes posters are mathematically correct, and they also give everyone in the room something to think about. Why is there a square on the rectangle poster? Is a heart a shape? What about a spiral? Does a curve have to be curvy? Does biggest count as a property? These are the kinds of questions I meant these posters to elicit, while still serving the noble purpose of being a visual reference for important geometry vocabulary.

I am very excited to see these posters out in the world. If you’ll help me spread the word that they exist (and where to find them), I’ll do two things: (1) Express my heartfelt gratitude, and (2) Put your name in a drawing for one of two classroom math play sets.

There are three ways to enter:

  1. Tweet a photograph of these posters in the wild to the #wodbposters hashtag.
  2. Design your own Which One Doesn’t Belong? poster and tweet a photo of it to the #wodbposters hashtag.
  3. Send me a note through the About/Contact page on this blog right here, and include this set of characters: #wodbposters . (This is essential as I’ll search my inbox for it when assembling the drawing.)

Here are the rules:

  • One entry per person
  • You may tweet as many times as you like on the hashtag (In fact, please do!), but together those will count as a single entry. #unitchat
  • You may have someone else tweet or write on your behalf. This is totally fair.
  • Entries end at 11:59 p.m. Central Time on November 1.
  • The drawing takes place using a spreadsheet and random.org on November
  • Winners will be notified by whatever means they entered (I’ll tweet you if you tweeted; email if you submitted through the contact page), and we’ll connect at that time to work out a shipping address.
  • Worldwide participation is welcome.
  • A classroom math play pack includes Tiling Turtles, Spiraling Pentagons, Curvy Truchet Tiles, 21st Century Pattern Blocks, and maybe something else, depending what prototypes I’m playing around with when I put the packs together.
  • If you have already tweeted a photo of the posters in the world, you probably didn’t use the #wodbposters hashtag, which is how I’ll search for them when the drawing comes to an end, so you’ll need to do that if you want to be entered.

Got questions? Hit me with them in the comments!

 

Good White People

Let me tell you a quick story.

A number of years ago, I was for the first time working with an immigrant population that was new to me. I had questions. I asked those questions of a colleague who had been working in this school for a number of years, hoping that he might have insights for me.

Instead, my every question was met with “Why are you asking this about this population when you might also see this behavior in the majority population?” with an insinuation that my even asking questions involving race and culture could be seen as indications of racism.

There is a certain brand of white liberalism that believes discussing race to be racist.

This is not healthy.

Here is a book recommendation: Good White People by Shannon Sullivan.

My major takeaway from this book is that if we don’t talk to our children about race, someone else will.

Do you know which white people can be counted on to talk about race to our children?

Right.

The folks who turned out with torches in Charlottesville last weekend will be more than happy to talk to our children about race.

It is important that white people with love in their hearts do so too. Just as we can talk about math with our children without always having the right answers, we can talk about race with our children without always having the right answers. Just as our children don’t have to get every math idea correct the first time they encounter it, they don’t have to nail every nuance the first time they encounter it either.

These are challenging times. This is an important book. White people need to read it.

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.

Avocados–15 (half avocados); (15 halves) avocados; half of 15 avocados. Multiplication is commutative and associative.

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

[Here’s a link for readers to play along if you like.]

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]

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.

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.