Fundamentally Optimistic, Fiercely Critical

Reading much of the popular contemporary literature on race and racism written by men in this society, I discovered repeated insistence that racism will never end. The bleak future prophesied in those works stands in sharp contrast to the more hopeful vision offered in progressive feminist writing on the issue of race and racism. This writing is fundamentally optimistic even as it is courageously and fiercely critical precisely because it emerges from concrete struggles on the part of diverse groups of women to work together for a common cause, forging a politics of solidarity. 

[Source: "Killing Rage", bell hooks]

The Need For Non-Euclidean Geometry

I have argued against using whimsy to introduce complex numbers. You know what I mean by whimsy: "No square roots of negative? Let's invent them!" If you don't, go check out John and Betty.

One counter-argument, voiced in the comments and on twitter by MathCurmudgeon, is the Argument From Vi:

@mpershan "The way of mathematics is to make stuff up and see what happens." @vihartvihart
— Curmudgeon (@MathCurmudgeon) March 30, 2014

Historically, this is just not the way complex numbers were invented. They were invented out of need, in particular a need for real solutions to cubic equations.

But, no matter, maybe this sort of whimsical creation is just a regular practice of math, one that we could reasonably expect to resonate with our students. But where else is this sort of casual invention at play in the history of math? James Cleveland tentatively suggests that whimsy was responsible for the discovery of non-Euclidean Geometries.

But check out this 1824  letter from Prof. Gauss to a Taurinus who sent our cranky mathematician some of his geometric work:

"In regard to your attempt, I have nothing (or not much) to say except that it is incomplete. It is true that your demonstration of the proof that the sum of the three angles of a plane triangle cannot be greater than 180 degrees is somewhat lacking in geometrical rigor. But that in itself can easily be remedied, and there is no doubt that the impossibility can be proved most rigorously. But the situation is quite different in the second part, that the sum of the angles cannot be less than 180 degrees; this is the critical point, the reef on which all the wrecks occur. I imagine that this problem has not engaged you very long. I have pondered it for over thirty years, and I do not believe that anyone can have given more thought to this second part than I, though I have never published anything on it."

And now Lobachevsky:

"The fruitlessness of the attempts made since Euclid's time...aroused in me the suspicion that the truth...was not contained in the data themselves; that to establish it the aid of experiment would be needed, for example, of astronomical observations, as in the case of other laws of nature..."

Euclid's Postulates had been explicit for ages, and anyone could have just casually wondered what the world would look like if the fifth was false. That would be in the true spirit of mathematical whimsy -- "What if Euclid's Fifth Postulate was false?" -- but that's not what happened.

Instead, the existence of non-Euclidean geometries emerged from the need to explain why Euclid's parallel postulate couldn't be derived from the other four. Non-Euclidean geometries were posited out of the need to explain the failure of centuries of effort.

[Source: Euclidean and Non-Euclidean Geometries, Marvin Jay Greenberg]

Betty and John's Bogus Journey

[From: The Book of John and Betty]

"Wait, what?" Betty said. "Just make another number up? Seriously?"

"Just make another number up! Make another number up!" John was hopping in place now, giggling while he chanted.

"What does that even mean?" Betty said. "I mean, I guess you can do whatever you want in math, sure, but why would we make up an answer to an impossible question?"

"Curiosity!" John said.

"Curiosity? What sort of curiosity are we talking about here? You tried squaring 1 and -1 and you didn't get -1. Great. That's proves it: my question had no answer. It's a great argument, simple and beautiful, so let's move on."

"Sometimes in math we just need to take a chance, Betty. Maybe for once in your life could you learn to take a chance?" John stared at Betty. "We're having fun now, aren't we?"

"Fine then, let's make up another number," Betty said, pacing away from John. "But while we're at it, I've got some other numbers to invent. I mean, you can't divide by zero, right? Let's make up a number for that. Let's invent a number that stays the same when you add five to it, and another number that's bigger than itself. Let's just throw new numbers at every impossibility in mathematics and call it a day."

"Oh, come off it, you big stickler," John said moving his beautiful golden hair off his ears. "You're completely ruining our afternoon, just like you ruin everything."

"Well as long as you're at it why don't you invent a number to fix our ruined afternoon you ridiculous elf."

And suddenly trumpets sounded and Heaven opened and Lord Math Teacher descended upon the White Room where John and Betty fought. And lo a chalkboard appeared as well as chalk and John and Betty were verily impressed.

And the Lord Math Teacher spake, saying: "John, Betty, you are both loyal servants, and I wish to praise you for your great attitudes and overall grit. But let Heaven and Earth bear witness: there indeed is a difference between these numbers you wish us to invent, oh Betty, and that which John has invented. Know it: John's number will lead to no further contradictions, while your numbers lead to contradictions, which is basically a no-go in mathematics, for contradictions are impure in the eyes of the Lord."

"Lord Math Teacher? Lord?" Betty waved in the Lord's direction. "Excuse me? I have a question?"

"Umm, yes Betty."

"How do we know that inventing a square root of -1 won't lead to contradictions?"

"An excellent question, my daughter!" Lord Math Teacher replied. "Yes, and unfortunately proofs of non-contradiction are sort of dicey but I swear upon the heavens that Model Theory has some of the answers. I am Lord Math Teacher, and I hope that answered your question."

And then from the skies a bell rang and the Lord departed.

And this answered all of Betty's questions and she was entirely satisfied with John and complex numbers for the rest of her days, verily.

"It's More Important To Defend Stupid Then To Correct It."

It’s pathetic, and it’s why the internet is such a broken, stupid, aggravating place. It’s why nothing ever changes. Because nobody who has any clout, visibility, or stature is willing to risk their position at the party. Because it’s more important to defend stupid then to correct it. But there’s no reform possible; if any of them find this post, they will ask the crowd, “should we take this criticism seriously, crowd?” and the crowd will soothingly reassure them. It’s all very elegant. And it’s exactly these people who will complain the most about the stupidity and the brokenness. They create the conditions they say they hate, and they live in them, and they deserve to.

This is Freddie deBoer explaining why he left a comment calling a paragraph "monumentally stupid" and why he hates calls for "being nice" on the internet. Temperamentally, I fall into the "why not be nice?" camp, but it's disturbing to think of that habit as a norm whose existence serves to protect my status on the internet.

Then again, there's probably a difference between people who get paid by publications to write things on the internet and some nobody teacher trying to think through his teaching life with friends. I'll stick to being nice until I grow up.

(By the way? That back and forth between Chait and Coates is great.)

Learning Complex Numbers, With One Eye On Elementary School

From Children's Mathematics:

The thesis of CGI is that children enter school with a great deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing understanding of the math of the primary school curriculum. Without formal or direct instruction on specific number facts, algorithms, or procedures, children can construct viable solutions to a variety of problems. Basic operations of addition, subtraction, multiplication and division can be defined in terms of these intuitive problem-solving processes, and symbolic procedures can be developed as extensions of them.

The book Extending Children's Mathematics applies the CGI framework to fractions. They recommend that "Equal Sharing" problems serve as the basis for a kid's fractions knowledge:

In contrast to Equal Sharing, problems that have children identify a fraction based on a given number of parts or divide a shape into a specified number of pieces offer few opportunities for children to make connections between fractions and whole numbers. Even though such problems are concrete in that children can see and maybe even manipulate the shapes, they are not situated in any meaningful context -- such as the process of sharing food with friends -- that would allow children to use their understanding of number to create and make sense of the resulting parts as quantities.

The rest of the book serves as a really masterful show of just how far you can run with "Equal Sharing" problems. The authors turn these Equal Sharing problems inside out. You can change what we know and what we don't -- are we looking for the amount per sharer? the number of sharers? the numbers of shared things? -- and you can change every relevant number. The authors show how these subtle changes in problem design drastically change the way that children see and try to solve them. They show what sorts of numbers and unknowns lead to the most productive insights and conversations about equivalence, ordering, adding, common denominators, decimals, and all things fractions.

I mean it, the book is astounding. And I can't think of any similar treatment of any high school topic.

---

What would it mean to do something in the spirit of the CGI treatment of fractions for complex numbers?

Skeptic: "Well, you can't really do that. This level of math is entirely different from primary school. Listen. The CGI people say it straight away: children enter school with a great deal of informal or intuitive knowledge of mathematics. That's fine for things like multiplication and fractions. But nobody has informal knowledge of complex multiplication. Why? Because nobody knows that complex numbers exist before we show them that they do!"

---

Quoth the skeptic, "Nobody knows that complex numbers exist before we show them that they do!"

What are complex numbers?

• If complex numbers are an algebraic extension of the reals, then students have never pre-formally experienced complex numbers.
• If imaginary numbers are the square root of negative numbers, then students have never pre-formally experienced complex numbers.
• If complex numbers are solutions to polynomials, then ditto.

And, of course, complex numbers are all these things, but they are other things as well. 

• From a geometric perspective, complex numbers are points in the plane
• From a transformations perspective, complex numbers are rotation/dilations (amplitwists? roliations?)
• From an algebraic perspective, complex numbers are rules that describe transformations 

---

What's great about those "Equal Sharing" problems is that they can be solved both efficiently/formally and inefficiently/informally. 

5 kids are sharing 8 bananas. Boom. Everyone gets 8/5. That's how I'd solve it. But a kid might take her time. She'd start assigning individual bananas to kids. And then maybe she'd split the remaining bananas in half. And then maybe do that again. And then maybe she'd think, shoot, how do I share one banana with five kids? And then she'd say that every kid gets 1 banana and 1/2 a banana and a fifth of 1/2 a banana.

We're looking for problems that can be solved both efficiently and inefficiently, both formally and informally. 

Do such problems exist for complex numbers? Problems that are rich, with lots of possibilities, that kids can solve without complex arithmetic but better -- faster, cleaner, more reliably -- with complex arithmetic? And problems that could serve as the basis of a lot of complex arithmetic?

I'm betting that the answer is totally "yes."

This post is already far beyond the attention span you can expect on the internet, so I'll wrap things up with some questions.

• What other problems fit the bill: questions that can be solved both efficiently with complex arithmetic, and inefficiently with the normal set of skills you can expect a high school student to have?
• How far can you run with these "rotation rules" questions? What are the different strategies that students will use?
• What other high school topics are waiting for the elementary school treatment?

A Brief Note On Internet Collaboration (Or: It's not about collaboration, it's about collaborators)

When people think of collaboration on the internet, they tend to think of something like the hive. The hive is Wikipedia. The hive is massive. The hive is all about lots of people coming together to break barriers and increase access and share information and beat the experts and everybody's wrong but together? everybody can be right.

The hive is posting an editable doc on twitter and asking for contributions. The hive is making a lesson-sharing site and asking users to upvote their favorites to the top of the stack. It's about asking hundreds of followers all around the world for feedback or suggestions.

This has worked excellently for me when seeking a dentist, an mp3 player. Professional collaboration? Not so much.

What is amazing about the internet for me is becoming friends with people and asking them if they want to work on something together. (Or reverse that order, if you please.)

The hive doesn't work for making a great lesson or doing some math or discovering something new. For that you need lots of hard work and a great partner in crime, two if you're lucky.

My advice for fellow #MTBoSers: figure out what you're interested in, figure out what other people are interested in, and email them to see if they'd like to think together. That is just the most fun thing to do.

Help Wanted: The UMass Calculus Readiness Test

I teach a Precalculus class. I'm very interested in properly preparing my students for Calculus.

Do you think that this Calculus Readiness Exam gets it right?

(What about this one? Or this one? What does your test look like?)

Thanks in advance.

What Complex Numbers Are

A complex number is not a solution to a previously unsolvable equation. It's not a formal addition to our mathematical system, without meaning but with numerous application. It's not a square root of a negative number.

A complex number is a point.

And, of course, a complex number is all those other things too. But the entire story makes the most sense if a complex number is just a two-part number, a point in the plane. The fact that it can be all those other things is just a wonderful surprise.

Discussion invited!

"Why We're Addicted To Online Outrage": Agree or Disagree?

As a result, when a politician utters a barely outdated cliché, or the slightest impolitic word, we no longer hear it as a faux pas or mere insensitivity. Instead it becomes the latest menacing incarnation of the evil we oppose. Micro-aggression is no longer “micro” at all, but the very real appearance of Patriarchy, or Anti-clericalism, or whatever evil you most fear. If your ideological hearing aids are tuned correctly, a gaffe becomes a threat, returning you to witch-trial-era Salem or the Vendée before the massacre.

Worse, this kind of hypermoralized politics has some serious implications for how we look at governance and power. As C.S. Lewis once wrote, “Of all tyrannies a tyranny sincerely exercised for the good of its victims may be the most oppressive.” In other words, if we are simply doing good in the world, and our enemies evil, then there’s no limit to the power we ought to acquire. What a charming fantasy that can be.

Holiday is right to be concerned that our capacity for real outrage is dulled by the sort of “outrage” that we perform, or fake, or convince ourselves to feel in our self-regard. But we should consider the possibility that fake-outrage is popular precisely because it is an indulgence that requires so little from us. Fake outrage allows us to hide within the mob, to feel righteous without doing much of anything, to suffer like martyrs from words not spoken to us. If we subtracted all the outrage porn tomorrow, most of us would continue to do what we already are doing about the Syrian refugee crisis, or faraway famine, or unjust war: nothing.

Go check out the whole piece. I'd love to know your thoughts about this piece, particularly if you disagree.

(Not to be coy, but I really don't know my feelings about the piece. I think, in the past, I would've agreed heartily with it, but now I'm not so sure.)

Where Creativity Comes In

Sam Shah, who was there from the start, reflects:

I want to make it so that kids see math as an artistic and creative endeavor... I am now pretty good at coming up with deep and conceptual approaches to mathematical ideas. And I’m okay at promoting mathematical communication. And I’m transitioning to having kids do groupwork all the time, to learn from each other — so I am not the sole mathematical authority in the room.

But all of that said: I don’t think I teach math in a way to shows how it is an art form, how deeply creativity and mathematics are intertwined.

I was just hanging out with some third graders. A kid asked me, "What's 6 times 8"? I said, "Dunno. What's a good problem to start with?"

She decides to start with 5 times 8. Clearly, she's thinking that she'll start by counting 5 eights, and then add one more eight at the end. But -- and this was the really cool part -- she decides instead to count 8 fives. She quickly ends up with 40, and then she returns and adds on an eight and lands on 48.

Isn't that a beautiful bit of thinking?

I don't have an answer to Sam's question -- duh -- but I know that creativity and math can be intertwined when kids learn their operations. What are the conditions for creativity in a math class? Dunno, but I have one trick that works with some reliability: Take some problem that could be solved efficiently and formally and give it to kids that only have inefficient ways to solve it.

E.g. Ask kids to find the area of a trapezoid without a trapezoid formula.
E.g. Ask kids to equally share 13 donuts with 5 people.
E.g. Ask kids to approximate the speed of a car from its distance graph

The gap between inefficient and efficient techniques is the sweet spot for learning, as far as I'm concerned. The hardest areas for me (us?) to teach are the ones that can't be solved by our kids using inefficient techniques, and its a fundamental problem of curriculum to ensure that kids have those tools at each moment of new learning.

In Praise of Ilana Horn and Assigned Groups

The Incredible Ilana Horn writes,

Skeptics might protest linking participation and status. “Some students are just shy,” someone might say. That is true. Likewise, students learning English often go through a silent period or may be self-conscious of their accents. Our goal with reluctant speakers is to design ways for them to comfortably participate more than they are perhaps naturally inclined to do. Strategies such as small-group talk first or individual think time may help build the confidence of shy or nervous speakers. 

I was a skeptic until, like, two weeks ago. Here's my learning story, which I'm posting for the benefit of myself and others who wish to teach people who are like myself:

1. At first, I was skeptical of assigning groups for group work because I didn't see much bang for my buck in planning groups ahead of time, and I really didn't like making up groups in class. My kids worked in small groups anyway, or they worked alone. They had choice, and how many things in class do kids get to have control over? Definitely: don't assign groups.

2. Two weeks ago, I was working with some Shell Center materials (oh my god their stuff is good) and they recommended giving kids a sort of activity before the activity for assessment reasons, and then assigning groups based on interesting combinations of approaches to that initial activity. I had a bit of planning time, so I figured I'd try something new and give assigned groups a shot.

3. It went great. I saw completely different student dynamics in the small assigned groups then I saw in the groups that kids usually chose for themselves.

4. Then I remembered that Fawn had blogged about using a random group assigner, and that I had once seen a Henri Picciotto piece on group work. I remembered that the Incredible Ilana Horn wrote a book on it. So I decided that this was finally time for me to try this again and figure this out.

5. I started randomly assigning groups in class all the time. When I wanted kids to go over an assignment, like a quiz or homework, instead of marking it I assigned random groups. Or I'd set kids up with a set of individual work, and then bring them together to discuss in assigned groups. Or I'd break out of a group conversation into assigned groups to make progress on some debate. This helped the pacing of class tremendously. I also really liked maximizing the number of kids who were talking.

But that's not what was most striking. Since kids weren't working with the people right next to their seats, their regular social patterns were broken. The girls were talking with the boys. The quiet kids were talking with the most talkative and eloquent folks. Weak students were with stronger students.

I thought this was amazing. Consider my 4th Grade class, where the girls and the boys never choose to work in mixed-gender groups. How does that social dynamic play out over a year? If boys and girls never talk math with each other, the boys can end up totally clueless about the skills and talents of the girls in the class. The girls never end up figuring out how their abilities stack up to the boys'. And now play this out not over just one year, but a whole school life.

Swap in gender for any other significant student quality -- outspokenness, race, ability level -- and I saw the same sort of productive behavior emerge in assigned small groups. By lowering the costs of participation we give low-status kids a chance to speak up, and high-status kids a chance to hear them.

Do the kids like it? Not always. It's uncomfortable. If it was their choice then...well, they'd choose the groups that they usually choose. But they get it. (Also, on a practical level, there's no polite way to object to an assigned group without dissing the folks you've been assigned to. So there hasn't been much protest, despite the discomfort.)

So now we have individual work, whole-group conversations, you-choose group work, and I-choose group work to switch between in class, and it's making a huge change in the way my classes feel.

My Response to Mathematicians Complaining About The State of Math Education

Hmm, so with your total curricular freedom to show kids the true beauty of real mathematics, university math courses must be the best taught and most loved math classes that kids have to take and not total wastelands of learning that everyone hates because their professors just endlessly lecture.

Wait, seriously? You guys are still lecturing?

Go on. Tell me more about the crisis of k12 math education.

(See here, here, here, wherever else math professors roam.)

Free Teaching Materials That You Need To Know About

Take to the airwaves, people! There are some excellent resources that are free and available for you and your kids!

From the people that brought you the Math Assessment Project...

...and the authors of that thing that Dan Meyer said that you should read...

...comes four amazing books that look like the 80's! Free to download!

Seriously, these books are a treasure trove. "The Language of Functions and Graphs" is just fantastic stuff. That's one of the books you can grab. The others are "Design a Board Game", "Problems with Patterns" and "Numeracy Through Problem Solving."

Seriously, these are some of the best problem-solving resources available for free right now.