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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.

Awesome stuff 1: Curvahedra

This is Awesome Stuff Other People Have Made Week on Overthinking My Teaching. So much great stuff to share.

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.

Or will the MTBoS Blogbot be unable to notice a self-referential blog title?

Happy kindergarten surprises

I was at a table full of kindergartners playing with triangles, trapezoids, and concave hexagons. We were building and chatting. They wanted to know if I wanted to hear them sing in French. I said yes. The sweetest two minutes ensued as this table’s Frere Jacques spread to the next and then the next, and soon the whole classroom was singing and doing math.

A few minutes later I made this.

Photo May 14, 11 23 56 AM

And then I went scrambling for my notebook.

Yup. Sure enough. (a-b)^2+4(ab)=4(\frac{1}{2}ab)+c^2 is equivalent to the Pythagorean Theorem.


The sequence machine

The fun we have had with a Lakeshore Learning Multiplication Machine in our house is well documented. Not once has that fun been based in the machine’s original purpose, and I am here once again to report to you on a new off-label use for these things.

You should know that Lakeshore Learning makes a machine for each of the four basic operations: addition, subtraction, multiplication, and division. You should further know that they got the structure of the subtraction and division machines wrong (more on that at end of post). So today, I’ll focus on the addition and multiplication machines.

These have the same format as each other: 9 rows of 9 buttons. Each button labeled on top with m+n or m×n  accordingly, in the mth row and nth column. The button pops up when you press it; down when you press it again, like a ball-point clicky pen. On the front of the button, visible only when popped up, is the corresponding sum or product. Lakeshore Learning views them as mechanical flashcards and nothing more.

2016-03-09 10.17.30

I convert them to Pattern Machines by covering all the numbers with colored vinyl. These are a ton of fun at home and in classrooms—great for patterning, counting, making pixel pictures, etc.

Now I am playing with a Sequence Machine. I have covered all of the tops of the buttons on a Multiplication Machine.

2016-03-09 10.19.24

Now you can generate number sequences, without being distracted by the multiplication facts. Above, you see a familiar sequence—the squares.

Things quickly get more complicated, though. If you read each of the sequences below from left to right, ask yourself What comes next?What would be the 10th or 23rd term in the sequence? and What is the general relationship between the term number and its value (i.e. What is the nth term?)

There are many more sequences to be made on these; many more questions to explore.

I’m working on a follow up question about the relationship between the sequence generated by the same button patterns on Sequence Machines built from an Addition Machine, and from properly redesigned Subtraction and Division Machines.

What is a properly designed Subtraction Machine? you ask? I’m glad you did! Such a thing would have mn or m÷n on the button in row m, column n. This is not how Lakeshore Learning makes them. They make them like this:



Find what you love. Do more of that. [#tmc15]

Here is the text of the keynote I planned to give at Twitter Math Camp. Actual product may have varied substantially in content (but not in spirit) from the typed original.

If you’ve never seen me talk in a large group you’ll have to imagine the energy, cadence and passion you see in my ShadowCon talk brought to this longer form.

Thanks to the Twitter Math Camp organizing committee for inviting me to talk, and to the whole community for supporting my work over the years. It means a lot; I hope to return the favor many times over.

Lisa Henry’s introduction:

Christopher Danielson teaches and writes in Minnesota. You may know him through his documentation of his children’s mathematical antics on Talking Math with Your Kids, through his exhortations not to share bad “Common Core” homework assignments on Facebook, or through his shapes book Which One Doesn’t Belong?

Of course you may not know him at all. But if you do, you’ll recognize that he holds little sacred besides the responsibility we take on in this profession to foster the growth of young minds.

He is currently working on a teacher guide for Which One Doesn’t Belong? to be published by Stenhouse in the spring. He encourages each and every one of you to promote the heck out of Common Core Math for Parents for Dummies. Most of his time this summer is devoted to bringing Math On-A-Stick—a new event to support children and caregivers in informal math activities—to the Minnesota State Fair, and he continues to work with Desmos on developing online networked classroom activities such as Polygraph and Function Carnival.

This is a very American talk about teaching. From what I’ve learned about teaching in other countries with robust educational systems—Singapore, Finland, Japan, Germany, and so on—the U.S. is unique in its tradition of sink-or-swim for teachers.

We equip new teachers with a modest set of tools and experiences, and we say Do the best you can with what you’ve got!

At the policy level, we understand that this is a disaster. But in the American fashion we try to legislate and standardize our way to improvement. We issue pacing guides and measure fidelity to adopted texts. And of course we measure teacher quality by testing students in an effort to standardize learning.

In this sense, the message of my talk today is a very American one. My message is this: Find what you love. Do more of that.


Viewed one way, this is advice to teachers trying to survive and to serve their students well in the era of NCLB and high-stakes testing. (There are probably several such folks in our midst today.)

And that will be a valuable takeaway, but it’s not the heart of my talk. The heart of my talk is more forward-looking and hopeful.

Sending minimally prepared teachers into the field, leaving them to figure it out on their own, and then evaluating whether they have—these things we do well in this country. If you believe that quality comes from sorting out the bad apples, then we’ve built a good machine for this, and the major impediment to improving it is the unions. I assume that this is familiar rhetoric.

What we don’t do well is orient and induct teachers to a community of professionals. We don’t structure our communities to draw on the diverse strengths and passions of its members. This is something that I understand those other nations I mentioned do much better than we do.


For me, the group assembled here, together with the ones who would be here if they could, and many more of the teaching professionals we interact with—whether regularly or sporadically—for me, this group is a community. A community that grows and changes in response to the contributions of its members. It’s not a community that agrees on everything—no community can while remaining honest, open and vibrant. Instead, disagreements offer healthy opportunities for the community and its individual members to grow.

So the hopeful vision in my message (Find what you love. Do more of that.) is that in identifying where your heart is in this profession, you can strengthen your voice and focus your efforts as you contribute to and help shape this community. What our larger American educational system does poorly—foster a professional community that grows and responds to the diverse strengths of its members—the MTBoS does quite well.

So I put two big questions in front of you:

What do you love?

How can you incorporate more of that in what you do? (In your classroom and in your community)

When I ask, What do you love? I don’t want to hear that you love…

Rectangles or stats or 3-Act Lessons or Spirals or Technology or Groups.

I want you to dig deeper. Those things embody what you really love. Whatever you are truly passionate about is bigger than these things. If you can say why you love rectangles or stats or whatever, you’ll be closer to the kind of thing I have in mind.

So now I’ll tell you what I love and how that—in the context of what I have to do—helps to guide my teaching and my contributions to our community.

This is so nerdy. I really hope this is a safe space for this.

I love ambiguity.

The spaces between the certainties are much more interesting to me than the certainties themselves. Ambiguity can provoke wonder, surprise, reflection and clarification.

I’ll share with you how I incorporate this love in the work I do in the classroom and in our community. Let’s start with a video.

This video lacks ambiguity.

Children are smarter than this.

Children can handle ambiguity, which is what I have found so appealing about the Which One Doesn’t Belong framework. (Megan Franke and Terry Wyberg mentions)

In case you aren’t familiar with it, I’ll give you the rap I give kids.


Let me tell you what children say in response to this richer set of shapes. 


[do that]

As they talk about these things, I summarize, paraphrase, probe, review and restate. By the time we’re done, we have a list of properties of shapes. Which of these properties are they supposed to use in deciding which one doesn’t belong? Which of these properties are important? Which ones matter? It depends.

Here’s another set of shapes. 


Sometimes ambiguity comes from studying new objects for which we don’t have a repertoire of vocabulary.

Which properties are important? Which ones matter?

We don’t know yet when we’re looking at a new class of objects. But when we do agree that a property is important, we can name it.

So all that vocabulary which is associated with geometry—that vocabulary isn’t important on its own. Instead it points to important properties, or to properties which somebody has deemed important.

(A non-math example: at some point, it became clear that this growing collection of math teachers online needed naming. The community’s historians are still working on the lineage of the phrase mathtwitterblogosphere and the abbreviation mtbos, The point is that the name exists because there was a thing that needed naming)

Here is what kids do with the ambiguity of the spirals.

[say what that is]

One last WODB example.


The shape in the lower right is the one that provokes discussion. (get to vertices)

Returning to the theme of what you have to do…

In my capacity as a College Algebra teacher, I have to teach rational functions.

Lots of rules and vocabulary and certainty are associated with standard textbook treatments of rational functions. There are asymptotes (vertical, horizontal, oblique…), rules to go along with locating these. There are zeroes and intercepts and symmetries and on and on…

So I played Polygraph: Rationals with my students.


The design of Polygraph is that it puts you in the position of needing to describe to somebody else what you see before you have a shared vocabulary for it—as with WODB. But now importance has an implicit definition. A property is important if it’s useful for distinguishing between functions, and for helping your partner to do that too.

At the beginning of the game ,we shuffle the functions to suggest to you that location in the grid isn’t important. But maybe you think it is. So you ask about it. And you’re likely to get burned.

Orientation, by the way, is important in this context. The difference between 1/x and -1/x is one of orientation.


So orientation doesn’t matter in plane geometry, but it does matter in coordinate algebra.

And the graph on the left is a function, but the one on the right isn’t.


Orientation matters.

But these are both squares.


(Tangentially related…How far can you turn a parabola and have it still be a function?)


I’ll finish with an example from the community. Meg Craig cares about kindness and empathy. She wrote about this on her blog recently. She urged us all to remember that We as teachers are all trying, to the best of our ability, to have students reach the best of their ability.  If you haven’t read this post, you need to.


That kind of thing has a lasting impact on our community. Just the other day, I got worked up about fences. 


There is so much wrong here. Unambiguously wrong, and I called it out. Then I was thinking later that day. OK. That was angry Triangleman. What could kinder gentler Triangleman do?

Well, what do I do in my classroom? What do I do at home with my children? In both of these contexts, I educate patiently. I accept people for where they are and I try to help them see new perspectives; to think differently about things than they do right now.

What does that look like on Twitter? I don’t know. It really is such a great medium for ranting. But I do know that the reason I’m asking myself this is that MathyMeg brought her strengths and passions to our community.


I told you to find what you love, and to do more of that. I told you that I love ambiguity. Maybe you’re the opposite of me. Maybe what you love is certainty in mathematics. How can you help your students appreciate and understand the unique nature of mathematical truth (different from all other disciplines)?

Maybe your heart is with the beauty of geometric forms, or the rhythmic regularity of patterning. Maybe you love how statistics can inform us as we strive to make equitable decisions in an unjust world.

Truth, beauty, regularity, fairness…

Each of these is a more important grounding for your classroom perspective than a pacing guide or textbook sequencing. But none of them is antithetical to these either. What you love can be found in what you have to do.

So my message to you is simple. Name what you love—be explicit about what makes your mathematical heart sing; what resonates in the depth of your teacher soul—and look for it in every corner of your professional life. Share with your students and share it with us, your colleagues and your community.