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.

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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]