Here’s what I was trying to say about diversity, Jews, and education

I hate the feeling of realizing that I’ve been unclear, but it happens. Here’s what I was trying to say about diversity, Jews, and education in this post and the comments.

Q: Why should educational spaces worry about being diverse?

A: I think because Black students currently are not served (even abused) by the school system. I think things would be better for these students if there were more Black educators in the system.

Q: Why? 

A: Because, one way or another, a lot of the problem is a systematic lack of empathy for Black students from non-Black people in education.

Q: Wait, but this is just about Black educators. I asked about diversity…

A: I know.

Q: So what about everyone else. If we care about diversity, we should care about other groups being present too.

A: Which other groups?

Q: All other groups.

A: Well I don’t think that we need that kind of diversity. I don’t think we need to work hard to include every group.

Q: But surely we do!

A: I don’t think so. For example, I’m (more or less) an Orthodox Jew, and I definitely don’t think that we need to work hard to increase the presence of Orthodox Jews in educational spaces.

Q: But isn’t that entirely different? It’s not like Orthodox Jews are systematically excluded from conferences or keynotes or panels?

A: Actually, I think that we are. It’s very hard to travel to conferences because of the religious restrictions. These conferences are not designed with Orthodox Jews in mind, and as a result it’s just hard to be at them. That’s one thing that explains why Orthodox Jews aren’t at these things.

Likewise, a lot of Orthodox Jewish teachers prefer to avoid getting involved with public schools, because of the difficulty of managing the holiday calendar and kosher food.

So I think that there is a sense of exclusion, but that this doesn’t matter.

Q: You’re saying that a group can be excluded, but that it shouldn’t bother us?

A: That’s right. Just because you’re excluding a group doesn’t, by itself, matter very much. Nobody should be worried about whether they are excluding Orthodox Jews, redheads, people who don’t like hot weather, etc. Just because a group is excluded doesn’t mean that it’s a problem.

It’s only a problem when it hurts people. As in the case of the exclusion of Black educators, and the way that ends up harming Black children. That’s why it’s important to be inclusive concerning Black educators.

Q: So you don’t want conferences to give you kosher food or whatever?

A: I mean I would gladly accept it, but I don’t think that anyone should worry about this. There is no moral issue with not being maximally inclusive to Orthodox Jews. I mean if you can do it, terrific, why not. But it just doesn’t rise to the level of an issue that anyone should worry about.

Q: Why do you keep talking about Orthodox Jews? Who cares?

A: I want to make talk of “diversity” somewhat problematic. Exclusion of a group is not necessarily something that should trouble us — that’s what I’m arguing. But I don’t want to pick on someone else’s group, so I focused on my own identity.

Diversity claims that all groups should be included, and that it’s a problem when any group is excluded. But, thinking aloud about my own case, I think that’s not true.

So instead I’m saying that it’s a problem only when there are bad results from the exclusion and lack of “diversity.” I believe this is the case for the exclusion of Black educators.

Q: Only Black educators?

A: No, but that’s all I’m going to talk about here.

Q: Why do you care about all this?

A: Because I think “diversity” is a problematic concept. I’ve never heard a fully satisfying answer as to why it’s important for a group to be diverse, in the sense of being varied in general. (This is something that Marian has done a tremendous job pushing at.)

Diversity tries to talk about difference in general, but I think diversity doesn’t get off the ground unless it’s specific. There is a specific case for why we should aim to increase the number of Black educators in education. For some other groups — such as Orthodox Jews –there is no such case, even though they are excluded from various educational spaces.

I think “diversity” as a concept lets us off the hook of talking about these specifics. But I think it’s important to talk about the specifics.

Q: I feel like you could have said that without the confusing stuff about Orthodox Jews.

A: I think you’re right. It’s just how my mind was working and I couldn’t snap out of it.

Q: But isn’t this still different? Orthodox Jews aren’t really targeted by society in the way others groups are.

A: That’s why I brought up the story about some guy on the subway yelling at me yesterday. I get this kind of abuse just from walking around NYC with my yarmulke on. It happens once or twice a year, but it’s enough to keep me on edge a lot of the time. It’s the sort of thing you don’t usually have to worry about if you’re just a person sitting reading your book on the subway.

I’m not saying it’s the same sort of abuse — anti-Semitism is just totally different in its nature than racism — but I think it’s fair to say that we are targeted, just for who we are and how we appear.

And that was the contrast I was trying to set up. You have two groups. Both experience prejudice, both are de facto excluded from many educational spaces, but in one case it matters and in the other case (my own) I strongly believe that it doesn’t. This flies in the face of the rhetoric surrounding diversity — that we need to be inclusive to all groups at all times. I want to make the case that we need to talk about specific groups and the specific costs of excluding them if we want diversity to make any sense at all.

And I suggest this is why there are so few good explanations of why diversity matters.

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When and Why Diversity Matters

Whenever I think about diversity — about whether workplaces, conferences, panels, etc. need to be diverse — I think about Orthodox Jews.

Given who I am and where I’m from, it’s natural for me to think this way. The “suburban shtetl” is a cliche but it describes something real. The (Modern) Orthodox world I grew up in really is a culture apart from the rest of the world. Sometimes, when I want to scandalize my students, I let slip a detail or two about what my high school was like. Like that hours went from 7:30 AM (prayer) to 8:30 PM (when evening Talmud study wrapped up), or that we ate three meals a day at school, or that we didn’t even start studying “secular” studies until 3 PM. (Pity my teachers!)

And despite a few boring intra-Jewish technicalities (we’re members of a decidedly non-Orthodox synagogue) it is still my life now. You can tell this as soon as you see me, because I wear my yarmulke everywhere I go.

So when I think about diversity, I use my own situation to help think things through.

Wherever I go in education, I’m usually the only Orthodox Jewish person around. (Assuming that I count as Orthodox, it’s complicated, etc., almost no one cares about this hedge so I’ll stop making it.) This is particularly true when I’ve traveled places for conferences or other edu gatherings, as it’s ridiculously annoying to travel while keeping traditional Jewish dietary practices i.e. kosher. The details truly are boring so suffice it to say that restaurant eating is, in general, out of the question and it’s always a scramble to find food when in e.g. Oklahoma or downtown Denver.

There is a lot of other stuff too that can conflict with conferences. For example the Jewish holidays or fast days, or anything that ever happens on Saturday, etc., etc.

And wherever I go, I also experience a lot of…I’m not exactly sure what to call it. Heckling. Shouting things at me. Stopping me on the street. On the subway today some guy was having a bad day and he walked past me and shouted, “Even the Jewish guy isn’t paying attention! What’s wrong with you? Does God like that?”

Right, right, back to diversity. So the thing is that there are very few Orthodox Jews who are part of these educational conferences or gatherings. And whenever I think about the value of diversity for an educational space I ask myself, is there a diversity problem with Orthodox Jews? Is there an imperative to make sure more Orthodox Jews are part of these spaces? Should more Orthodox Jews be in positions of power and influence? Is it a problem that more Orthodox Jews aren’t serving on panels or keynoting?

I think the answer is absolutely not. No. There is no diversity imperative that is relevant for Orthodox Jews in education.

The thing is that this immediately would contradict a lot of the reasons people give for valuing diversity. So if you say that you value diversity because you want to make sure your conference is open to everybody, then you have to really start worrying about Orthodox Jews. Or if you say that you want to learn from diverse people or diverse cultures, because different people have different perspectives…well, Orthodox Jews have a different perspective. Nothing special about Orthodox Jews as a group here, obviously, but that’s the point.

And another thing: it’s fair to say that Orthodox Jews have been systematically kept from attending these conferences, in the sense that (a) they often happen on Saturdays and (b) in places where it’s hard to get kosher food. So if the reason to value diversity is to undo systematic exclusion, Orthodox Jews should be part of the discussion.

But, again, I don’t really think that’s right.

As a (more-or-less) Orthodox Jew, I encounter a world that was not made for me, or people like me. There are professional opportunities or conferences that I have to pass on — I am excluded from them. And yet I also think that, basically, this holds no moral weight when it comes to diversity.

That’s because there isn’t anything valuable about diversity, in general. There’s always the question of who and why. So let’s get concrete and very specific about this for a second.

There are very many Black children in schools. There are not very many Black teachers in classrooms, compared to the number of Black people in the country. There are good reasons to want to increase the number of Black teachers a Black student encounters. Besides this, Black educators have been less likely to rise to influential positions within education. This is because of plain-old prejudice, geographic segregation, higher levels of poverty, more things too, and further this is a bad situation, because better decisions would be made if the system had marginally more sympathy for Black students.

Contrast this with the case of Orthodox Jews: there are very few Orthodox Jews in the public school system, further they are on the whole not educationally mistreated, there is no particular imperative for increasing the number of Orthodox Jewish educators and there isn’t any reason to think things would be better if more Orthodox Jews had power in education.

My point is this: exclusion is not always a big deal.

I mean, sure, if you can easily not exclude people, go for it. But it’s really not a big deal. Everyone — every conference — excludes people. If your conference is in California, it excludes poor people who live far away. If it’s on a Thursday, it excludes people who hate Thursdays. If it’s anytime or anywhere outside of a major Jewish area on a weekday, it excludes Orthodox Jews. Look, it’s tough. What can you do?

Exclusion only becomes a problem if…it’s a problem. Diversity is only an imperative if there is some imperative reason why you need to include a group of people. There is no such imperative for my own group, but there is for Black educators.

(There is also likely an imperative for the inclusion of other groups too. My point is that it needs to be considered on a group by group basis, that’s the only way the whole thing holds together.)

The whole language of diversity seems to me an elaborate way to beat around the bush. It’s popular precisely because it allows us to talk about race without talking about race, but the whole thing only holds together if we think about racism in frank terms. Whenever I think about my own group, the value of diversity in education just slips away.

Here’s how it is, I think: Orthodox Jews have very little power in education, and that seems about right. Black people should have more power in education. That’s why diversity matters.

Interesting Paragraphs

Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

From this, by Thony Christie. I didn’t know about this.

It reminds me of something else that I think about all the time, which is the article from 1920 announcing the creation of NCTM:

About ten years ago an organization known as the National Council of Teachers of English was formed. Today it has a membership of 5,000….

During the same period high school mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses, and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. The individual teachers and local organizations have made a fine defense to be sure, but there could be no concerted action. Finally, the American Mathematical Association of america came to the rescue and appointed a committee to study the situation and to make recommendations. Already two valuable reports have been issued and others are in preparation. The pity of it is that this work, wholly in the realm of the secondary schools, should have to be done by an organization of college teachers. True they have generously called in high school teachers to help, but the fact is that it remained for the college people to initiate the work. They could do it because they possessed a live, vigorous organization.

To help remedy the existing situation the National Council of Teachers of mathematics was organized last spring at the N.E.A meeting held in Cleveland.

Math has a far less secure foothold in the curriculum than you might think. Historically speaking, at least.

Mathematicians Behaving Badly

I write this post in a state of mild panic. New York Public Library has made a margin call on When Genius Failed: The Rise and Fall of Long-Term Capital Management, and I still have a few pages left in the epilogue. Holding on to this asset is costing me, I’m losing equity, etc., etc.

Having now rushed to finish the book over the past few days, I can’t recommend it highly enough.

The story, in short, is this: Long-Term Capital Management was a hedge fund created by “geniuses.” These included two university mathematicians who could make legitimate claims to being key creators of contemporary mathematical finance. (They won Nobel Prizes for this while working for the fund.) It included key disciples of these mathematicians. There were PhDs and Ivy League degrees up the wazoo.

They quickly made a bajillion dollars. They made this money by developing mathematical models that could help them identify pairs of items that were mispriced, relative to each other. They then bet that those prices would converge. They bet this over and over gain in many, many different situations. They always won.

Then, they didn’t. They very quickly lost it all and had to be bailed out by the NY Fed and a consortium of banks.

This was happening in the ’90s. The people involved with LTCM were (if I understand correctly) early pioneers of using quantitative models to capitalize on the mathematical theories of modern finance. I know that this previous sentence is pretty vague, but my point is that this is a story about mathematics and mathematicians.

While reading about these mathematicians, I recognized an arrogance that I’ve experienced in mathematical cultures waaaay too often:

Hilibrand finally offered to dispatch Scholes [Nobel Prize winner — MP] to give the bank a lesson on option pricing, but Pflug was too smart to go head-to-head with the guy who had invented the formula. “You can overintellectualize these Greek letters,” Pflug reflected, referring to the alphas, betas, and gammas in the option trader’s argot. “One Greek word that ought to be in there is hubris.”

What hubris did Pflug divine? The partners were not arrogant in their mannerisms or even in their speech; it was more deep-seated. It was the arrogance of people who had been to Harvard and MIT — of people who really believed that they were more intelligent than others. “Do you know why we make so much money?” Greg Hawkins once asked an old friend from Salomon. “It’s because we’re smarter.” Once the Hawk even tried to lecture a colleague’s wife about molecular biology, her longtime specialty. “You’re full of shit,” she finally replied.

In particular, it seems to me this is partly the arrogance of believing that math is truly about everything. I see this in books that claim math gives you the tools for being logical, in general, or in somehow not being wrong — again, in general.

I also see it in the belief that math can give students general-purpose reasoning skills, ready to be deployed willy-nilly in any context, whatsoever.

It’s something that comes up in the belief that models can be usefully deployed on any dataset, which is one way that LCTM dug its own grave:

Characteristically, Meriwether encouraged the firm to explore new territory. Even at Salomon, the troops had always sought to extend their turf…In retrospect, such moves had been baby steps, not bold new departures. But the partners’ experience — to them, at least — seemed to belie the adage that it is dangerous to try to transport success to unfamiliar ground. Trusting their models, they simply rebooted their computers in virgin terrain.

This is an attitude I even saw in something like Cathy O’Neill’s Weapons of Math Destruction. I don’t have the book in front of me (so apologies if I get this a bit wrong) but there are various moments when she suggests that the same algorithms that are currently deployed oppressively could simply be turned towards more socially useful causes.

I don’t think that, on reflection, O’Neill would deny that deep knowledge of particular context matters, but a reader of her book could be forgiven for thinking that it’s just a matter of where you aim your weapon.

(Hannah Fry’s Hello World does a better job thinking aloud about how deep knowledge of context is necessary for making mathematical models work. So, for example, medical experts might use a trained AI to help identify likely patches of cancerous cells. But the human is the crucial bit here; AI is the tool.)

I’m rambling now, so let’s wrap this up. When Genius Failed: I recommend it. And in general I recommend looking at finance as a major site in the contemporary mathematical landscape. If we want to understand what math is in the world these days, we have to look at finance.

Smoothing the path from simpler to more complex systems of equations

We played with Venn Diagram puzzles, and now my students had a better sense of what it means to be a solution to a system. So, onwards! But I noticed that my 8th Grade curriculum jumped pretty quickly from questions like this…

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…to questions like this.

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To bridge that gap, I paired an estimation question and a worked example activity.

The purpose of the estimation question was just to make sure everyone was ready for the example. I wanted to remind kids that each of these equations would produce a line, and I wanted to remind them that their existing techniques wouldn’t work to provide a precise solution.

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I was hoping that there would be a nice example activity as part of the Algebra by Example collection. Sadly, nah, it jumps into too much, too quickly for my group. So I was forced to make my own (doc here):

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I find it really necessary to add a “Ready for more?”-type question to these, as some kids really need time to study the example carefully. A good reminder that it all works together — a good extension for fast-finishing students is an accommodation for my slower-finishing students.

I ran the example activity using the routine that I described here.

What makes this bloggable, for me, is that it shows how various things can work together. A worked example, an estimation question, connecting graphical and algebraic representations — there are people who advocate for any one of these, but they only really work for me when carefully aligned.

  • The graph/equation piece is an important bit of non-procedural knowledge.
  • The estimation bit makes sure students understand what a type of problem is asking.
  • The example helps students focus on a useful procedure, and the prompts to explanation make sure they understand it.
  • The “ready for more?” challenges students while giving me more time to help those who need it.

Today was a day when every piece mattered for me.

How to make worked examples work

I’ve written a few times about how I use worked-out examples (and mistake analysis) in my teaching:

The resource I’ve returned to again and again is Algebra By Example, which is really well-done. (And get excited for Math by Example, their expansion pack for 4th and 5th Grade!)

I realize that there are two things that I haven’t yet shared: some research on how to design these things well, and my routine for using these materials in my teaching.

The Research

Whenever you get a bunch of people talking about research in education, somebody eventually mentions how sad it is that research in education usually happens without the input of educators. Wouldn’t it be grand if researchers and educators were more collaborative?

Yes! This indeed would be grand, and it should happen more often. But sometimes it does happen, and those partnerships can produce really wonderful things. Algebra by Example is the result of one of these partnerships.

The project was led by Julie Booth of Temple University, and Booth has written a very nice piece in an NCTM journal about the design process her team uses and the research behind it:

McGinn, K. M., Lange, K. E., & Booth, J. L. (2015). A worked example for creating worked examples. Mathematics Teaching in the Middle School, 21(1), 26-33.

Booth’s piece is very nice. It cites a more research-y piece that more carefully trawls and summarizes the worked example literature, and that is also an interesting (though more research-y) read:

Atkinson, R.K., et al. (2000). Learning from examples: Instructional principles from the worked examples research. Review of educational research, 70(2), 181-214.

Though it’s tough to summarize these pieces, here are a few important principles when designing example tasks:

  1. Try to put everything in one place…
  2. …but also try to visibly separate different steps.
  3. Switch up correct examples with incorrect work.
  4. Ask kids to explain what’s going on.
  5. Ask kids to try it on their own.

Try to put everything in one place. That’s a NO for splitting the problem and the solution with a not-so-useful line of text. Keep words at a minimum; ideally, you can see the whole example at once.

Worse:

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Better:

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It’s useful (according to Booth’s research and also me) to switch up correct with incorrect examples.

Can you spot the error?

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I like the way this incorrect work has some good thinking (I like the regrouping idea) and that it also clearly distinguishes each step.

From experience, I’ll say that people new to writing examples often try to do too much with the space they have. The goal really is to eliminate all but the most important thinking from the example, and it’s OK to use multiple examples to get at different aspects of the thinking. Overstuffing one of those little boxes can be a sign that it’s time to break this up into two smaller problems.

So much for the visual design of these materials, which is really a huge subject in of itself.

Some of the most interesting bits of those reviews are about self-explanation. Here’s a bit from the more research-y review:

Research on explanation effects suggests that self-explanations are an important learning activity during the study of worked examples. Unfortunately, the present research suggests that most learners self-explain in a passive or superficial way.

“Passive or superficial”: YES! All too often this is how students interact with an explanation (or with feedback).

So how do examples help at all, given the tendency of students to just passively say “OK fine yep I get it” when they see an example? I use a routine to really make example activities hum.

The Routine

Whenever I use an example activity with students, here is my routine.

  1. Get ready: I show the problem, covering up everything besides the problem itself. I usually use a document camera, so I mean I literally cover it up with my hand. Sometimes I ask students to begin thinking about how they’d approach the problem before I reveal the student work.
  2. Read: I ask everyone to read the student work to themselves. I ask students to offer a quiet thumb to let us know that they finished reading. I usually tell students to put a thumb up if they’ve read each line, even if they don’t understand everything fully yet.
  3. Discuss: I assign partners (really I’ve already done this) and ask students to discuss the example until each person understands every line of the student work. (I focus my class on spreading understanding, not on solving problems.)
  4. Explain: Every Algebra by Example problem has prompts for student explanation. When I make my own materials, I always include such prompts. It’s at this stage in the routine that I make sure everyone tries to answer the prompts. Then I usually call on a student to explain.
  5. You Try: This is where we help students try it on their own. There’s a whole art to picking good problems for students to try here — they should change the surface-details in some way, while keeping the underlying ideas constant. This is where we try to keep kids from making false generalizations, and it’s another level of protection against superficial understanding.

Using the Algebra by Example materials, the routine flows from left to right:

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It’s not so easy to design these prompts, either! Here are some suggestions, from Booth’s piece:

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If you’re in the mood for a bit of practice, you might think about how you’d fill in the rest of this example task:

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Example analysis works especially well as a feedback routine. There was a problem that kids especially had trouble with? Write an example that focuses on how to improve, and then follow the routine with some time to revise and improve the quizzes (or a re-quiz, or etc.).

All of this — the examples, the prompts, etc. — is worth your time, I think. It’s a lot of fun for kids to puzzle out how someone else is thinking, and the format allows us to really focus on a whole idea. It’s a bit of a myth, I think, that these types of problems are boring.

It’s also a myth that this kind of math is boring. It’s not, because there’s something to be understood here. And fundamentally, understanding is interesting.

Multiplication Resources for 3rd/4th Grade

As with all topics, the stronger your main curricular resource is the better. There have been years when I used a heavily modified version of Investigations for 3rd/4th Grade, but now I don’t find it as useful.

I assume that most teachers do have some main curricular resource for math, so here are the things that I use to supplement:

There are a lot of people with strong disagreements about how to teach multiplication. As in a lot of areas of my teaching, I see myself as being one of those annoying people that accepts each side’s critique of the other. I aim to teach mental strategies while also teaching facts for memory and skills for automaticity. I will very clearly explain both mental strategies and written procedures, though not as early as the traditional math crowd would prefer.

I don’t know. Part of this certainly has to do with the sort of place where I teach. The kids are for the most part not poor, I’m not under state test pressure, and the kids don’t have grades. My goals are for kids to know their stuff and be prepared for future coursework, but I also want them to have fun, and I don’t feel as if those goals are in tension in my current situation. (At other places that I’ve taught, though, I have felt those sorts of tensions.)

I do think that it can be a big mistake to focus only on strategies, as I’ve seen kids left behind on their fact knowledge when I taught that way. I wrote at length about this in Teaching Rachel.

I’m sure other resources will come to mind and I’ll try to add them to this post. Feel free to drop your own favorite resources in the comments.

Isn’t “roots of unity” a nice bit of mathematical phrasing? I think so. Nice job, mathematicians.

[Mathematical content warning: roots of unity]

Over the weekend I was reading and practicing with problems from Learning Modern Algebra. I came across these:

Let n be a positive integer and let \zeta = e^{\frac{2\pi i}{n}}.

1. Establish the identity

x^n - 1 = (x - 1)(x - \zeta)(x-\zeta^2) \cdot \cdot \cdot (x-\zeta^{n-1})

2. If x and y are integers, show that

x^n - y^n = (x - y)(x - \zeta y)(x-\zeta^2 y) \cdot \cdot \cdot (x-\zeta^{n-1} y)

3. If x and y are integers, show that

x^n + y^n = (x + y)(x + \zeta y)(x+\zeta^2 y) \cdot \cdot \cdot (x+\zeta^{n-1} y)

I enjoyed thinking about these from a geometric perspective.

Most of the work of the first identity is being done by the definition of \zeta and the very notion of a root of unity. What are the solutions to x^2 = 1? to x^3 = 1? x^9 = 1? x^{anything} = 1?

Some quick facts, in case you’re bravely reading this extremely sketchy blog post without already being familiar with roots of unity:

  • x^n = 1 always has n solutions, no matter what n is.
  • 1 is always a solution.
  • -1 is often a solution.
  • Besides for 1 and -1, all the solutions are non-real complex numbers.
  • If you graph them in the complex plane, all the solutions are on the radius 1 circle away from (0,0).
  • The points are always equally spaced from each other on that circle (which is all that $latex.
  • Another way of saying the previous point is that \zeta = e^{\frac{2\pi i}{n}} provides one solution, and then the rest of the solutions are powers of \zeta.
  • This guarantees that the solutions of x^n = 1 will form a regular n-gon.

A picture is worth a lot of words, via Wolfram MathWorld, and possibly violating their reuse permissions:

rootsu.gif

That first identity really is just summarizing the roots of unity situation. If all these powers of \zeta are the solutions of x^n = 1, then these \zetas are the roots of x^n - 1 = 0 and so each of those powers of \zeta is a factor of some polynomial that is equivalent to P(x) = x^n - 1. Multiply all those factors together, and you get back the original polynomial:

x^n - 1 = (x - 1)(x - \zeta)(x-\zeta^2) \cdot \cdot \cdot (x-\zeta^{n-1})

The second identity is the one that I enjoyed thinking about as the result of scaling the entire unit circle by a factor of y. So being on a circle of radius 1, the circle is now of radius y. Which means that each of the roots has been multiplied by y, so that the roots around the circle are now y, \zeta y, \zeta ^2 y, ..., \zeta ^ {n-1} y.

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But if y is now a root of the equation, instead of x^n - 1 = 0 it must be x^n - y^n = 0, and likewise each the roots arranged around the circle will make y^n when raised to the nth power. And this is just what the second identity is saying:

x^n - y^n = (x - y)(x - \zeta y)(x-\zeta^2 y) \cdot \cdot \cdot (x-\zeta^{n-1} y)

The third identity is just what happens when you replace y with -y, though this replacement is only relevant when the power of n is odd and we get a negative (and so we end up swapping from subtraction to addition):

x^n + y^n = (x + y)(x + \zeta y)(x+\zeta^2 y) \cdot \cdot \cdot (x+\zeta^{n-1} y)

My textbook reports that the above factorization represents “an early attack on Fermat’s Last Theorem.” Feel free to link me to a readable explanation of what this early attack consisted in!

***

I find all of this so interesting, both mathematically and sort of meta-mathematically. I love the historical transition of algebra from “hey let’s solve some equations” to “hey here’s this entire metaphysics of structures whose names will give you absolutely no sense as to what they represent.” (I’m thinking of rings, ideals, modules, fields, domains, etc.) How did this happen? Where are the motivations?

Anyway, the above is a small part of that story.

Teach at BEAM this summer in NYC, LA or at one of their residential programs

I’ve taught at BEAM NYC twice. I’ve had so much fun being part of their work. Here’s how they describe what they do:

This summer, change the lives of underserved students with exceptional potential in math. We are looking for people who love math and working with youth to create a community of deep thinking and mathematical joy.

They’re hiring for this summer, and you should strongly consider making BEAM part of your summer plans.

Here is some info about teaching at their non-residential programs in NYC (where I’ve taught) and LA:

Staff responsibilities run weekdays June 26th – July 24th in Los Angeles, and weekdays July 2-August 12 in New York City…Both part-time (teaching an hour and helping for an hour of study hall) and full-time (teaching two courses per day) positions are available. The salary ranges from $2000-$5800 depending on course load and experience.

The non-residential programs are for 6th Graders, going into 7th Grade. If you teach full-time, it’s a full day of work that includes helping kids work on their problem sets after each class.

The residential program is for 7th Graders, going into 8th. Maybe you’re interested in teaching there:

The program runs at two college campuses in upstate New York with dates July 4-July 31 and at one college campus in Southern California with dates to be determined, starting in mid-June and ending in mid-July. You will have some flexibility in deciding which site you prefer. The salary is $5000 for faculty, and $3300 for junior faculty.  In addition, food, housing, and transportation are all provided.

There are a lot more details here. I am enthusiastic about my experiences teaching at BEAM, and this is really a situation where classroom teachers are needed and can make a difference. So, please, spread the word and share this opportunity with good teachers everywhere.

I’m happy to answer questions about my experiences and working at BEAM in the comments.

Update: I forgot that you can read about BEAM in the NY Times, here.