Math, Joy and Borders

For the past few weeks, with the Julia Robinson Mathematics Festival, I have been doing math work with different groups of 5 to 17-year-olds who have been separated from their parents at some border, transported to the South Bronx. From there they are bused to a downtown office where they are interviewed by an advocate as part of the next stage in their journey. For 90 minutes I work with kids I will never see again, each of whom, by definition, is a trauma survivor. Yet for those 90 minutes the ugly reality pursued by a morally bankrupt regime is put on hold and the universality of the relations between numbers and shapes is there for us to work with, together to see patterns, solve problems and discover insights we didn’t know we had the power to find.

I will really never know anything else about these too young children, but they have helped me see again why it is a civil right and a human imperative that every person deserves the opportunity to learn mathematics with understanding.

Thus, for example, the set of real numbers is infinite, and each real number can be associated with a unique point on the number line. It is an illustration of the truth from the world: when one more is added the total is one greater, whatever the quantity is called, however it is expressed. There is no value great enough that there can’t be one more (or small enough that there can’t be one less) And further: There are an infinite number of values between any two other values, only bound by our ability to measure.

The number line fixes the relationship between all possible values, and allows us to name new values by relating any two values in different ways – the sum of two values, the difference of two values, the product of two values or the quotient of two values (and later exponentiation). These relations just are.

In the same way, a triangle retains its shape and area no matter how it is slid, flipped or rotated. The relationships never vary, the forms conserve their shape. We can depend on them as perhaps we can depend on little else these days.

So for 90 minutes the children put aside their worries and fears. We face each other across a table and play Spit, challenging each other to run up and down the number line with the cards we are dealt, one more or one less without dispute, laughing when we are too late to a pile.

A smile lights up a face when a cat finally appears out of the 7 pieces of tangrams– five triangles, a square and a parallelogram.

A particular figure often needs a precise arrangement of pieces, yet this cat’s tail can be made with either the parallelogram or the two small triangles. Which should it be? Figuring that out, on your own time, with the opportunity to try and regroup, is a certain kind of freedom born from dependability.

What are all the ways you can organize 5 same-sized squares, each piece sharing at least one full side with another? There are 12 such ways. Can you make them all? Then can you use the resulting pentominoes to build a rectangle?

I haven’t yet succeeded. But with a colorful puzzle template it is still a challenge to flip, rotate and turn the pieces into a rectangle. We all have the power to turn 12 different shapes into one gorgeous rectangle and are pleased together when we do so.

What makes Mathematics joyous is not turning lessons into planning for a party or a game that disguises memorization of facts. What makes mathematics joyous is the continuing discovery of the eternal nature of the underlying relationships that are there for all of us to discover, the relationships that are the same for me and for you, no matter on what side of the border you happen to be.

Making practice more perfect

The ground was rocky where my two-year-old granddaughter and I walked in the woods, looking for blueberries, and there were so many treacherous roots and branches in the way as we searched. Yet my granddaughter persevered, falling, getting up again, practicing what was already easy for me, navigating the forest floor. As I watched her I thought about engineering professor Barbara Oakley’s article on girls and math practice in the August 7 issue of the New York Times.

I have been a math educator in NYC public schools for 25 years, both as a classroom teacher of elementary school children and as a math coach for elementary school teachers. Girls and mathematics, children and mathematics, is something I think about a great deal.

A huge majority of elementary school teachers are women. All of them had mathematics in elementary school, middle school and high school, and many of them find mathematics teaching, if not terrifying, then the least rewarding part of their teaching day.

Ms. Oakley posits that one reason for this is that these former elementary school students found easy success in reading and writing, and—compared with themselves—a harder time with mathematics. Or, rather, they soared above their male counterparts in reading and writing but were only even with them in mathematics. She says that the basis for success in mathematics is practice, which boys get more of than girls, because girls choose things at which they are superior, rather than equal, on which to expend their energy. Therefore, Ms. Oakley says, parents and educators need to make sure their girls and girl students practice mathematics.

To this point I am intrigued and wondering about what significance this has for action in the classroom. I have seen myself that in Kindergarten classes there are often more girls than boys in, for example, the play-acting corner, while more boys than girls are building with blocks. I have considered that we would be better off as a society if boys had more experiences building relationships and girls more experience building structures. But what does it mean to practice mathematics?

The author’s answer is to liken practice to chords and scales, and to disparage the emphasis on conceptual understanding. As someone who learned to play the piano without a conceptual understanding of the relationship of notes to each other, I don’t think separating conceptual understanding from developing neural pathways, both physical and mental, is helpful.

Take the counting sequence. There is no doubt that it is important to memorize the sequence, so that the mental pathway is so smooth and well lit that no effort is needed to find or navigate it. But consider the different task confronting a speaker of English (or Spanish or German) and a speaker of Chinese (or Japanese, or Quechua). Children speaking English routinely consider that they must learn the order of 100 numbers names before they start over with “one hundred one, one hundred two,” etc. But Chinese children learn the names of 10 numbers, and because their language mirrors the base 10 system that the whole world uses to organize large and small quantities, they start repeating those names with 11. In English the equivalent Chinese name for 11 would be ten one (or one ten one). 26 is named two ten six, and so on. You want to add 24 and 35? No need to rename twenty and thirty as fifty – instead two ten and three ten is called five ten, and 4 + 5 is another 9, for five ten 9.

The quantity denoted by the word twelve is always and forever one more than the quantity denoted by eleven and internalizing the eternal nature of this relationship is key to navigating mathematics, but demystifying the word used to denote the quantity 1 ten 2 (twelve? What’s up with that?) and connecting it to its place in the base 10 structure we all use, goes a long way toward making it possible to connect the sequence memorization with the organizational structure that names all numbers in terms of groups of ten and not yet tens.

Mathematics is all about relationships. Experience (practice) with those relationships is necessary. But it isn’t separate from the urge to gain insight into the structures that make mathematics so beautiful.

The Common Core standards in mathematics have gone a long way toward helping teachers see the relationships between (formerly discrete) topics for themselves so they can help students make the connections that will ensure the most robust development of neural pathways.

Mathematics asks the questions, What is the pattern? Will this always be true? Can you convince us of your thinking? If we ask these questions about observed patterns in elementary school girls and boys can dig in and strive to prove or disprove, and finally generalize, with over and over again experiences. For example:

In first grade asking “Is it always true that we can find ten inside of every addition problem? (8 + 7 is the same as 10 + 5, which we already know is 15. What about 9+4? 17 + 6)?”

In 4th grade, Can every fractional amount be expressed as the sum of unit fractions? (1/7 + 1/7 + 1/7 is three 1/7s (3/7) just like 1 + 1 + 1 = 3 ones. What about 13/15?)

In 3rd grade, Is it ever possible to have an odd product when one of the multipliers (factors) is even? Why or why not?

Imagine conjectures like these debated in mathematics class, arguments grounded in the experience of many examples. Imagine the resultant internalization of relationships, each relationship opening doors to others, like cresting a rise in the woods, which then reveals further vistas to explore, practice and embrace.

It’s the discovery of your own power to understand these connections and then make more of them that makes mathematics like wanting to learn to walk unaided in the beauty of the woods, even though you aren’t very good at it yet.




Yelling At Each Other (In Math Class)

“Math is actually fun now,” Joshua said to me, as he and his math partners skipped back to their classroom. “We get to yell at each other!” Any teacher would like the first part, right? But the second part…

The second part was the key to everything.

They weren’t yelling at each other. But they were talking, about math, with genuine emotion. They were excited. They were drawing out diagrams showing how 1/3 of 24 looked just like 24 ÷ 3, connecting new concepts about division with fractions to earlier work with division of whole numbers, and trying desperately to convince each other—without calculating—that 4.3 x 0.99 would result in a product less than 4.3.

These three students hadn’t particularly cared for math. They forced themselves through lessons, listening for what to do so they could complete their worksheets. They hadn’t even been friends before this, and never would have chosen to work together. So much had changed.

“Math joy” gets a lot of play these days, for good reason. To grow mathematicians, we need students to be motivated to do the work of learning mathematics. We hope they see it as a fascinating topic, filled with knowable relationships. What a beautiful realization to see that you can figure new things out. The power of reasoning inspires joy.

But the “yelling” was also pretty important. Joshua and his two partners were well matched. No one of them held greater status over the others. Each was recognized as someone who could share ideas worth considering and as someone worthy of being convinced. Even better, they grew confident enough to demand that math make sense—that it match the structures they understood already.

Watching this small group work together has been a real pleasure. It’s the best evidence for the kind of math instruction I believe in. The funny thing is that most of the time I had the weird feeling like I was doing nothing to make it happen. But that’s only true if you see the teacher’s role as the holder of knowledge. My job here was more important than that: I made the partnership, chose the tasks, briefly framed questions, and stepped back. Every once in a while I might remind them of what they already knew. But it was their job to prove their ideas correct or not by connecting other ideas they had about the structure of mathematics.

And guess what? When you learn math that way, it’s actually fun.

Problem Solving for Life (and Not Just the Test)

Testing season is upon us, and along with it comes anxiety. Have we taught all the necessary content? Do students remember what they learned at the beginning of the year? There is a temptation to forgo deep math learning and cram for the test, to review content with lots of direct teaching and practice tests. Avoid the temptation! Teaching for memorization won’t lead to long term growth!

For learning to stick, kids need to make deep and lasting connections within structures of mathematics. We want to keep our math communities vibrant, our instruction deep, and our math periods intact. If students are persevering and working to make sense, then these skills will help them be successful on all sorts of measures, including standardized tests.

Instead of traditional test prep, we can help students revisit and review pivotal standards while growing their habits as thoughtful problem solvers. The Math Collective has put together problem sets and a routine that make this work easy and accessible for teachers. This routine allows you to engage in quick conferences that push on the Standards of Mathematical Practice. For example, you could ask a student who is stuck, “What is this problem asking you?” This allows them to re-read, re-state, and re-think — habits students should be using every time they encounter problems.

How about those students who rush to answer questions by operating on numbers without really understanding the context? They might create a beautiful model and accurate solution, but they added instead of multiplied. We can honor their thinking and encourage them to strive for precision. We could say, “This is not the solution to this problem. It is a solution to another problem. What can you do to make sure your solution answers the question the problem is asking?”

In summary, we can make this season powerful for students and remove the anxiety of cramming in content for the sake of the test. We can hold tight to our rich math communities and strengthen the value we place on the habits and dispositions of thoughtful problem solvers.

Check out the many resources in our eight-week problem solving routine here: Grade 3, Grade 4, and Grade 5.

How Long is Too Long?

by Kim Van Duzer, Math Coach, PS 29K

“Keeping kids on the rug” is kind of a thing in elementary schools.  Since the dawn of, well, rugs in elementary classrooms, teachers, administrators and coaches have been pondering the length of time that is appropriate to keep kids in the meeting area for a lesson.  How long is too long? Should a math lesson be 8, 10, 13 minutes long? We seem to universally agree that 20 minutes is too long. Yet I doubt there’s a teacher I know, myself included, who hasn’t kept kids on the rug for 20 minutes or more. There’s just so much to teach!

A typical day in my classroom could easily look like this:  The kids gather on the rug for the math lesson.  Everyone is settled and attentive.  I start out with a little warm-up, kids are engaged, we’re feeling good.  I move into the big idea for the day.  Mild fidgeting ensues. We’re about 8 minutes in. Maybe a turn and talk will help, I think to myself.  After the turn and talk, I want to show them what one solution path might look like.  But I want to show it with Unifix cubes first.  Noah and Eric have started poking each other, and Alysia is fully reading a book from the classroom library at this point.  “Alysia, can you make 18 with these cubes?” I ask, trying to pull her in. I’m sweating a little bit. We’re at like 12 minutes right now and I haven’t even done one example with them yet.


We work through one problem with the cubes and I think they can see it. I think like 12 of them can see it. But there are 14 other kids in the class! I’m not sure if they can see it.  Let me try one more with the cubes, I think to myself (now arriving: 15 minute mark).  After the second example, I feel like I’ve gotten maybe four more kids on board.  At the same time, at least five more kids are completely checked out.  Alysia has moved on to Chapter Two of the book after her brief stint as “18 maker.”  I am definitely sweating.  

“OK guys, let’s just take a quick look at the problem set together,” I say.  They take out the workbooks they’ve been dutifully sitting on for 20 minutes.  We read the directions out loud.  About two kids total are paying attention at this point.  I am 100 percent sure that they’re not going to know what to do when they go off.  At the same time, it’s been 22 minutes, which is officially WPTTGOTR (Way Past Time to Get Off the Rug).  “OK, go on back to your seats and get started,” I mutter in defeat.

I know why all of this happened, and happened repeatedly, in my classrooms.  It’s a very simple, very honorable reason:  I wanted to make sure they learned.  

But aren’t there other ways that kids can learn, beyond sitting on the rug for a lesson?  I knew that there were.  I just had to let go of some control over the learning, and have some faith that learning could happen even when I wasn’t right there to see it.  At first, I tried just telling myself something I wanted to believe, but didn’t totally believe yet:  that the lesson is just one opportunity for kids to learn, not THE opportunity for kids to learn.  I experimented with giving kids lots and lots of time to work on an interesting and appropriately problematic task. I walked around. I conferred. Kids were engaged. Kids were learning. I tried structures like partner coaching that provide kids time to work independently and talk about their thinking in small groups and partnerships.  I overheard conversations kids were having when they didn’t know I was listening. They were actually talking about math.  I started to believe it.  I didn’t need to keep them on the rug until they got it.  I needed to give them something to think about, and let them go think about it, with tools and partners and good problems, for a long time.

There is some anxiety that comes with letting go of control.  I admit that it can feel a little scary to send kids off the rug, knowing that they don’t totally “get it” yet.  So much of our self-worth as teachers is tied up in students understanding.  But there is also great freedom and relief in admitting that I, the teacher, am not the sole nexus of learning. Learning can happen even when we’re not there to witness it.  If we believe this is true, our work as teachers can be directed towards creating opportunities for kids to learn in all of these different ways — from us, yes, but also from rich, meaningful tasks, from studying the work of their peers, from having time to consider new ideas alone, and from having time and space to explore and grow ideas with their math communities.  

Rookies and Coaches and Mathematics, Oh My…

By Erika Salzman, Math Coach

When a student finds joy in math learning and can express that joy to 31 peers, you know you have succeeded as an educator. Some 4th graders in my school recently experienced Partner Coaching for the first time. The goal of this teaching structure is to have students explore mathematical ideas without teacher direction. They work independently, as part of a small group, and as part of a partnership. Although the teacher confers with the students throughout the process, the teacher does not give a “lesson” in the traditional sense. You can learn more about partner coaching on our webpage, in Teaching Structures.

I was inspired by the student’s reflections at the end of the hour. When asked what they liked about the Partner Coaching experience, one boy expressed his joy in working with his coaching group: “I liked getting together with my group to come to an agreement about our solutions. We didn’t all think the same way and it was helpful to see how someone else drew a picture of the problem. It really helped me understand.“

Another boy, an ELL student said, “I felt good when my coach helped me understand some of the language in the story problem. I know what few means, but in this problem it said fewer and I didn’t know that word.”

One girl enthusiastically shared about helping her rookie. “I liked how it felt to show my partner my solution. I worked hard with my coach group and after we finally came to a consensus, I was able to share my new understanding with my partner and it felt great because he had a lot of questions.”

Watching my students create their own understanding of mathematics is a powerful teaching tool. I can tell by their actions and reflections that my students feel empowered when given the opportunity to investigate and work to understand what each other is thinking. I love when children are the active participants and the teacher’s role is decentralized. I once heard an educator say that kids should leave school exhausted from  a day of hard thinking and discussion, rather than teachers leaving exhausted from so much talking and teaching.

A rose by any other name.. .

by Kate Abell

My heart sinks every time I hear someone complain about “Common Core Math.” To talk about Common Core Math is akin to talking about the “debate regarding climate change.” There is no debate about climate change. There is only the undisputed data showing the correlation between human activities, rising carbon dioxide, and higher temperatures worldwide.

And there is no such thing as Common Core Math. There are only the relationships between quantities, which serve humans in their attempts to solve problems, and the trajectory of sense-making, which humans pass through as we move through childhood and become adults.

The Common Core for Mathematics is unprecedented in the history of US education as a tool to make visible the patterns and structures that allow us to manipulate relationships among values. It is written to highlight the progressions people need to experience in order to develop their mathematical thinking. No matter where you are on the mathematical trajectory the Common Core makes explicit the connections between things you already know and things you are just learning about. That is the way all humans make sense.

I am sorry that, due to political pressures, New York State has adopted a new set of standards. But I am heartened by the fact that the Next Generation Learning Standards are heavily indebted to the Common Core, despite lack of attribution. And, it is clear that those involved in the revisions worked hard to make the standards user friendly. Most of the clarifications seem mathematically helpful.

I also welcome the intention of the NYS Education Department to provide training in the new (“new”) standards for teachers. But it is important that we understand this as  help in deepening understanding of the through-lines of mathematics, not training in a brand new set of standards.

In the meantime, we can get on with the work of understanding our students’ thinking and acting accordingly,  by continuing to study the progression of ideas laid out by the writers of the Common Core—now with a name and format that make them feel safer.


Read the Math Collective’s analysis of the Common Core vs New Generation Learning Standards  and find out what impact the changes might have on your school or grade curriculum.