Tuesday, December 31, 2013

How Different Can The Best Teaching Look?

"Is teaching an art or a science?" 
Well, that's a muddled question. For one, "science" and "art" are terribly abstract terms, and it's not clear what we mean by them. So we ought to stop asking this question. Instead, I'd suggest a better formulation of that question is
"Does the best teaching always look the same?"
You can tell that this is getting closer to what we mean to ask since the answer to this question has real consequences. If all teaching looks the same, then we might be able to (1) discover the recipe and (2) prescribe it or at least attempt to (3) convince people to follow the recipe.

What we're talking about here is convergence. We're wondering whether all good teaching converges, whether it all ends up being congruent.

But this question isn't quite right either, because of course all great teaching won't look the same. It's not going to have the same homework collection routines, and sometimes it's going to be in Japanese and sometimes it's going to be in Spanish. It'll almost never be in English har har har har.

We're not actually wondering whether great teaching always looks the same. What we're really wondering is
"How different can the best teaching look?"
Does great teaching always have the same core? Are kids always learning how to multiply via the Whatever Method? Can you learn the same amount in a classroom regardless of whether homework is assigned? Is note-taking always helpful?

I don't know the answer to this question. You don't either. Nobody knows the answer to this question.

But here are a bunch of arguments in support of the answer, "The best teaching can look very different."

  • Technique A works great in my classroom. But kids in someone else's classroom find Technique A infantalizing and they aren't willing to give it a shot. It's an interesting theoretical question to ask, "If those kids were more like Michael's, would Technique A be most effective?" but ultimately a meaningless one. You can't abstract the kids out of teaching.
  • To make that last argument a bit more explicit: different kids will end up needing different teaching.
  • There are no agreed upon goals for a math class. Some people think that the purpose of math class is to prepare kids for a career that uses math. Some people think that the purpose of class is to help kids get through math class and into college. Others want to prepare kids for a life of asking and answering interesting questions. Before we can converge on a definition of the best teaching, we need to converge on a shared purpose for math class. That aint going to happen, so we can't expect any sort of convergence on what best teaching is.
  • Let's talk directly here: Can we expect better results if we give kids lots of chances to problem solve in class, rather than use class for lecture? I would say, maybe. Even probably. But I can imagine a uniquely inspiring lecturer, combined with some very hard-working students who are dutiful on their homework. They get help from parents when they need it. They come in with questions, and the whole thing is a productive endeavor. An extremely productive endeavor.
  • "But for most teachers, with most kids, lecture won't work." Well, fine. But that's not quite what we're talking about. Instead, that's a pivot to a different question, "What should we be telling teachers to do?"
Like I said, I don't know the answer to this question. But I suspect that things can look very different and still be operating at peak awesomeness for kids. 


In case you're curious, here's the bit of writing that sparked this post. It's from an interview with George Saunders.
First, let me say that all of the above is true for me – I have no idea that those ideas are more widely applicable. A writer has to figure out what works for him/her and sometimes that bag of tricks is just that: a small bag, full of specially developed tricks that, even as he/she pronounces them (as one is called upon to do when teaching, in interviews, etc) seem crazy or overspecialized or dictatorial. That said...
You can go read the rest here

The post was also inspired by this twitter conversation, so I know that Ilana Horn thinks that the answer is "not so very different."

Sunday, December 22, 2013

Naming The World

The way we talk is deeply connected to how we think.

That's the best way I have to sum up what I've learned about teaching over the past few months. It's a simple statement with deep consequences. 

Consequence #1: Every piece of learning worth remembering needs a name. 

It's not important that the name be particularly descriptive. Names aren't usually descriptive. "Michael" doesn't describe my features in any way, but it is a hell of a lot easier to drop "Michael" in conversation than "that red-haired math teacher from NYC."

I almost always ask my students to come up with the names for these things, because it's fun, and that's how we end up with "The Feces Theorem" or "The Friday the 13th Theorem."

Consequence #2: Thinking is having a way of talking. Giving kids new ways to talk helps give them new ways to think.

By the end of our polygons unit we were able to go far beyond talking about this shape as "looking like a rocket". We were talking about pointiness, and the number of points, and which sort of points counted as sharp points, and we were able to talk about acute angles and obtuse angles and all sorts of other things. (Documented here.)

I put in all this work into helping my Trig kids see that the sin(45) is not 0.5 on the Unit Circle. We investigated the Unit Square, and it's "sine" didn't have this property either. This tends to be tough and mind-bending for my students, but the learning is slippery. You know how I can make it more effective, though? By introducing the language of "linear" and "non-linear", and giving them a way to talk about these functions and their properties. 

Having a concept is deeply connecting to having a way to talk about a thing. Not having a way to talk about a thing makes it harder to have a concept.

Consequence #3: You can't learn math from Khan Academy.

Learning from Khan Academy means learning alone, and that means never developing a language to talk about concepts. Which means that you probably aren't really learning them.

(Or do you believe in the possibility of a private language?)



Trying to implement Christopher Danielson's Orpda and Hexagons lessons sensitized me to the connection between language and concept development.

Naming Infinity is a really interesting book about intersections between mysticism and math that influenced some of my thinking here. Ditto on my philosophy courses in Philosophy of Language, in particular "Two Dogmas of Empiricism," which is arcane but I take the paper to be an attack on the hard distinction between knowledge and language.

The title of this post has been lifted from Bret Anthony Johnston's book.

Tuesday, December 17, 2013

Come Speak At The Global Math Department, And This Year Let's Get More Diverse

Have you checked out the Global Math Department? You should.

The Global Math Department is fantastic. Every week (Tuesday) a different math teacher leads a short session about something they're fascinated by. Sometimes it's great math, other times it's a teaching tip, and often it's just a tough question, unanswered and maybe unanswerable but tossed around by a bunch of really sharp and curious teachers.

This year, I'm on the planning staff of the Global Math Department. My job is to recruit speakers for our Tuesday sessions, and this post is a Call For Speakers.

With over a year of conferences under our belt, it's becoming clear that we have a diversity problem, i.e. there's a lack of diversity in our speakers, i.e. the vast, vast majority of our speakers have been white. After reading a great piece about how one conference did a better job recruiting people of color (thanks Max!), we're ready to make a concerted effort toward drawing our session leaders from a wider palette.



[Spread Widely]

Call For Speakers: Global Math Department

We're looking for teachers, consultants, teacher-leaders, writers, and all sorts of people interested in math education to lead sessions at the Global Math Department. Right now most of 2014 is unbooked, and we're looking to find speakers for the new year.

Our Selection Process: This is the second full year of the Global Math Department. We have had sessions run by seasoned, big-name math education speakers, and we've also had sessions hosted by complete novices who have never ran sessions for teachers before. We don't accept every speaker, but when we don't accept proposals its not because of lack of experience. This is a friendly crowd, and if you're excited by something we'll probably want to offer you a session.

We Care About Diversity: Our process for recruiting speakers in the past mostly involved leaning on our friends or people that we knew from the online blogging world. In doing so, we accidentally ended up with primarily white speakers. This is a sort of self-propagating problem, because of the way our past speakers inevitably signal the sort of future speakers we're interested in.

This is trouble for all sorts of reasons. First, because there are people of color who haven't had the chance to have their voices amplified. Second, because our community itself has suffered from a lack of diversity of views and perspectives. Third, because our community has unintentionally mirrored the existing prejudices of the education establishment. This is at odds with our grass-roots effort to empower and connect teachers who are often silenced or ignored in their schools and departments.

With this new year, we're trying to break the cycle. It won't be easy, but we're going to reach beyond our comfort zone to find great speakers.

Nominations, Please!: Do you want to speak at the Global Math Department? Great! Do you have someone who you'd love to hear speak at the Global Math Department? Double-plus good. Let us know, and we'll do the sweet-talking for you and get your nominee into a speaker slot.

So, please nominate others, especially if they're outside of the math twitter/blogging crowd that we (the organizers) tend to run in.

Contact Us: Get in touch with us on twitter (@globalmathdept or @mpershan), or talk to Michael through email (mathmistakes-at-gmail works). You can also drop a comment to this post, and I'll make sure to follow-up.

Sunday, December 15, 2013

Why You Need A (Great) Curriculum, Not Just A Bunch Of Lessons

The above is an exercise from my favorite Geometry textbook.

You know when you're going to need to find the area of a 30 degree sector? In Trigonometry, when quickly knowing how to transform a central angle into a proportion of a circle is crucial for quickly moving between degrees and radians.

That's why you need a good curriculum, to start helping these kids get ready for Trigonometry a solid two years before they've even enrolled in the course. That sort of thoughtfulness is so, so hard to achieve while trying to make sure each individual class session is awesome.

Here's another bit of curricular thoughtfulness that amazed me this week.

Do you see it? (Actually, this picture is pretty small so you might literally not be able to see it.)
Question 1: Make an angle that measures 60 degrees.
Question 2: Explain how you know that this is a 60[degree symbol] angle."
This is the sort of little touch that makes this such a lovely 4th Grade curriculum. It's like, forget the progressive, weirdly controversial activities and lessons. Does your curriculum go the extra mile to make sure that kids can actually read mathematical notation?

I didn't even notice this genius move until a little 4th Grade girl asked me what that symbol meant in class.

If you're making your own curriculum -- using all your own worksheets, your own activities and assignments and everything -- you need to take a close look at some of the best work out there in curricular design. Every time I get cocky, I take a look at CME, CMP or TERC and I realize that they're thinking like ten steps ahead of me.

I'm not saying that a great curriculum's activities and tasks are better than mine. Sometimes they're great for my kids, usually I at least have to turn them inside out before using them. But for previewing, reviewing, and a general sensitivity toward common student pitfalls, I think it's going to be a long time before I can do better than these guys.

The Exploding City of Las Vegas, And An App That Helped

That's some pretty remarkable growth, Las Vegas. I asked you guys what questions you had after seeing this image, and a lot of you wanted to know what the population was now, or what it would be like in the future. Those are some great questions, and of course we could go down that route. There are a lot of great population modeling problems, and this could be a great population modeling problem.

Here's another great question: Does the green splotch grow faster, slower, or the same as the population?

We recorded opinions, arguments and predictions. Then we took rough estimates for the actual area of the green stuff in 1973 and 2000.

The problem with finding area is that Greater Vegas Township isn't shaped like a rectangle or anything else we know how to deal with. So we've got a blob, and we need a technique for finding the area of a blob. We talked about it a bit, and they mentioned that grids would be helpful, so I gave them what I had.

There were two different techniques that I remember seeing for approximating the area. Some students wanted to cut out the green splotches and impose it on some graph paper for a better estimate. (Good job, kids: a more fine-grained grid is going to get you closer and closer to the "true" area.) Others went about counting and chopping up the green parts to fill out the squares. (Maybe there were more techniques, but honestly this lesson was a few weeks ago and I've gone rusty on a few of the details.)

In the end, the kids ended up with their approximations. But we wanted to know how good our approximations were. Enter SketchAndCalc, a really solid app. SketchAndCal lets you import pictures and trace out the perimeter of an area, and it'll spit out the area. (A digital planimeter!)

So, I traced Vegas:

Using the info from the app, the area of Greater Vegas grew a bit slower than population. We had a great conversation about why this might be. Do cities tend to grow denser as they grow outward? Why would that be?

Anyway: this was a ton of fun. It's all posted over here, if you want the files.

Saturday, December 14, 2013

Improvement, Like So Many Things, Comes Down To What You Enjoy

For purely selfish reasons, I've spent a lot of time worrying about how teachers get better at teaching. That's lead me to write a series of hystericalish posts over the past couple years.

  • In July 2012 I argued that we need to find more "drills for teachers", whatever that means.
  • In January 2013 I said that taking on tough challenges was the key to continuous teacher improvement. I made a commitment to finding those tough challenges.
  • In May 2013 I reported that I'd picked a bad set of challenges, and that I was ditching them for new ones.
  • In August 2013 I made the case for writing your own lessons as a high-impact challenge for teachers.
I have a quick thought to add to this story, and it starts with "I was wrong."

I was wrong about how normal people become great at their craft.

I've got a long-term writing project that I'm working on, and it's absolutely terrifying for me to spend time with it. The chances of this project failing are high, and the product will almost certainly suck. It's absolutely crucial that I finish the project, though, because the only way that I'll ever get good enough to do this well is by practicing.

The only way that I can psychologically wrap my head around working on such a difficult, ill-fated project is by finding joy in the process itself. Any investment I have in the result is debilitating, since the result is almost certainly going to be crap. But if I'm enjoying the process then I've got a decent shot of chugging along.

In my thinking about how teachers become great, I've focused on finding routines that will artificially impose challenges and reflection. That makes some sense, since working through challenges is the way that we human folk get better at anything. 

But drills? Artificially imposed challenges? My imagined model for becoming great was the perfectly composed saint, the priest worshiping at the altar of his own self-control and diligence who forces upon himself the routines and habits that lead to greatness. And certainly such priests exist, toiling away and kept to task by some angel or demon that keeps them focused, maybe it's obsession or ambition or maybe it's just desperation.*

* Sorry for the ridiculous sentences. I've been reading a lot of Cormac McCarthy this weekend. He told Oprah that he prefers "short declarative sentences," so apparently Cormac McCarthy is not exactly the biggest fan of Cormac McCarthy.

But normal people can't spend their lives motivated by obsession or ambition or anything else. Normal people don't do things that they hate. They do things that are fun, or interesting, but mostly fun.

The best long-term strategy I can see for continuously getting better is for the process of improving to be fun. If I want to get better at teaching, it's got to be fun for me to do so, because that's the only way for me to stare down the abyss of my current craptitude and the probability of my own immediate failure. That's going to look different for different people, because we've all got different tastes. I enjoy planning lessons, so I spend a lot of time on that. You like giving feedback, so you spend your time on that and you get great at that. I hate it, so I suck at feedback and am decent at curriculum.

I feels a bit silly.  To have spent a year thinking about how to become a great teacher and landed on "Enjoy the process" seems, like, duh, but there it stands.

Monday, December 9, 2013

"Many people enjoy readings cut-and-pasted from reflective emails."

I'm betting that Danielson is wrong on his cut-and-paste theory, but I figure we'll give him a shot.

Earlier this year, I tried to implement Prof. Danielson's hexagon sequence in my Geometry class, and it basically failed. I tried again in my 4th Grade class this past week, and it's been going really well. Part of that is the age difference, but the other part is what I wrote to the kindly Professor about.

I sent him this email tonight. Enjoy, and keep the fantastic comments coming.*

* Kudos to the crew responsible for a great conversation on the last two posts. I'm looking at you, Denise, Chris Painter, mrdardy, Sue, Gregory, Megan, Justin, Christopher, Mike, EdRealist, David and Teresa. You guys rock.


Hey Professor, this email is too long, but I wanted to share some exciting hexagon moments with you.

Another fun day of hexagons today. After a slow start last week, things have been clicking with the 4th graders over the past few days. In particular, I was surprised by a bunch of coolness that happened today, and I wanted to share it with Prof.

Teresa called her shape "Squashed" because it looked like a squashed hexagon. Cool. Rather than directly ask students to clarify their thinking, I asked them to make little "What's My Rule?" puzzles and had the class try to discover and express the rule. Someone articulated a version of the "squashed" definition, so I turned a squashed shape on its side (so that it was now very narrow), and asked whether it was squashed.

Unanimously, no.

Today I cut out a bunch of shapes (squares, triangles, various hexagons) and got all of the students sitting in a circle on the floor. The idea was to force them to observe the shapes from different perspectives, and to build the notion that our classifications ought to be invariant of viewing perspective.

I put down the shapes in the middle of the circle, one at a time, and asked them to speak up if we already had a name for this shape, or if it was a new one. I accidentally stumbled onto a really great question, because the hexagons that I presented were all similar, but non-identical to, some of your hexagon pieces.*

* I put those to good use as well. Each student got one hexagon piece handed to them, and that was their shape. I was eager to make this more about individual perspective than anything mediated by the group, so I wanted each kid to own a different hexagon.

This, more than anything else, helped drive us to some clarity in our terminology.

I meant to cut a shape that looked a lot like the three-pronged shape from your set, but I accidentally altered it a bit. When I put it in front of the kids, a lot of them started giving it a new name. I asked whether any of our old names applied to it. Someone said it looked like a rocket. Others said it didn't. Then a bunch of stuff clicked for me.

I realized that Lily, who gave us the "rocket" in the first place, owned the shape. I asked her, as the originator of the term, whether this new guy was also a rocket. She said that it also had three points, and then I realized that this was our data. This was the fact, end of the story. So I turned to the rest of the class and said, "Hey, that's really interesting. We weren't sure whether it was a rocket, but Lily said that it counted because it had three points." Then -- thank god for this -- a kid pointed out that my shape had four points. So then this became our puzzle. "Yeah, you're totally right. This does have another point. So how come it looks like a rocket to Lily?"

The big, big reason why this flopped so hard for me earlier was because I was completely unable to find the constraints on the problems involved in this context. If the kids have complete ownership over all these observations and terminology, where's the reality pushing back on their views? What are the problems to solve?

Over the past two sessions I feel as if I've had a breakthrough. The move is to make the particular visions of individual students into problems that the rest of us have to solve. We become students of our various ways of seeing things. The way our friends see the world is non-negotiable, and it's the constraints that push our language to more precision.

Sunday, December 8, 2013

Nobody Appreciates Your Lack of Confidence, Even When They Say They Do

I started at a new place this year, and things were generally going really well in the first few weeks. Except for one class, where there was some rockiness. I was trying some weird stuff, and the kids said that they wanted the normal math class experience.*

* When the kids are saying class isn't going well, they're almost always right, but almost always wrong about the reason why. (Credit to Neil Gaiman, who said something like this about readers.)

Anyway, I've got a lot more colleagues and administrators in this new school, and they wanted to know how my first few weeks of classes were going. I told them the truth. I mentioned how it felt like a lot of these kids felt unhappy, and when I just came out of a bum lesson I told people that it was a bum lesson.

The result: a lot of administrators visited my classroom.

Forgive me, but I'm fairly self-deprecating. Lately I've been thinking too much about how this way of carrying myself has affected my career.

When I see conventional success, I tend to find a certain kind of confidence, what my grandmother or rabbi might call chutzpah. Our writers of note are those with answers, so rarely those with questions. There's no room for doubt in a keynote address. Your most popular posts are those with opinions, clearly stated and forcefully held.*

* (OK, here's a strong opinion: David Foster Wallace is overrated partly because of his almost divine levels of chutzpah. He wrote a book about infinity riddled with mathematical errors, and nobody bothered to send it to a mathematician for an edit?)

I'm trying to carry myself with a bit more confidence. When I write, I try to write with a bit more magnitude and direction. (Like a vector!) When people around school ask me how things are going, I don't stutter as much. Things are going well, thanks for asking, though we're all working hard, amiright?

But I'll never really be able to shake who I am, I don't think. I'm always going to be giving myself and my work a hard time. And I've got a feeling that people like me end up in the second row of success, professionally speaking.

Friday, December 6, 2013

What's wrong with being too easy?

A great math activity can't be too hard. That makes sense to me. If I don't think that I'll be able to do the activity, then what's the point of even trying?

A great math activity shouldn't be too easy. If a problem is too easy it holds no interest to us. This is a claim that's all over the comments of Dan's recent post.

But, why not? I mean, we do easy things all the time. Why would a math problem become less interesting if it's too easy.
"Because people have an intrinsic love of challenges."
Really? Is our perception that something would be challenging (but doable) sufficient to get us interested in an activity? Aren't most of our worksheets doable but challenging? And why would some sort of activities be more interesting than others? Do they appear more challenging? More doable? What's the theory?

Drop a comment, and we'll hash this out.

Sunday, December 1, 2013

Blogging > Twitter

This is just to say that I spent November tweeting less and blogging more, and it made being on the internet more productive for me than it had been in a long time.

We read poems, short stories and novels differently, and one of the the many reasons why is because of length. What we expect from a piece of writing depends crucially on how long it is, and for good reason. Writing is hard, and we expect people to write something that's roughly as long as it absolutely needs to be. A novel, presumably, couldn't have been a short story. I'd suggest that much of the power of short poems comes from their brevity. Their length announces a sort of immediacy and clarity that ought to come as revelations. (A haiku isn't the sort of thing that's supposed to need argument or evidence.)

All of this to say is that, much to my past frustration, it's very hard to be subtle on Twitter, because the brevity of any tweet makes anything you say come off as a proclamation. That's good for a gal or guy with a lot of confidence, but I found myself just making people angry on Twitter with (what I thought were) speculative comments. These days I find Twitter most helpful when (a) I want to proclaim! or (b) I have a question.

Some more reasons why I prefer blogging to twitter for working out ideas:
  • Tweets disappear, blog posts stick around. 
  • Every comment on a post is worth ten replies to a tweet.
  • Blogging is consistent with me not constantly being near the internet. Twitter supports some of my worst internet habits.
Of course, Twitter is good for many things, and its brevity is a great deal of what makes it great. And, of course, you should do whatever you want to do. But November has been a very happy month for me.

Wednesday, November 27, 2013

Out of Class Interventions - Never Look Back

Andrew wants to spark a conversation about intervention strategies that work, and I've got something small to share.

In my teaching life so far, "intervention" has always meant "a time to meet with a kid outside of class." For me, that always seemed to be basically a waste of time. What can I do for a kid in forty minutes that I couldn't do in two months?

I'd use SBG. I'd say, look, you've got seven standards that you haven't mastered. Let's do two a week for the next month. Let's meet on Monday during lunch, and I'll tutor you in those skills. Let's reassess on Thursday. And every once in a while a kid would pull it together, but most of the time he would stop coming, or he wouldn't be able to study on his own, or he would still be getting lost on the new material as he's reviewing the old stuff...

Last year I basically begged people on twitter to show me a better way, and Frank Noschese sent me a document that made a small, but important difference in the way my interventions went. The most important part of that doc was the second line of this table:

After reading this, I immediately stopped going over old material with kids, and instead spent our time prepping them for the upcoming week's lessons. 

The theory is simple. In a weekly session, it's usually unrealistic to help a kid learn large swaths of material that they're struggling with. But it is totally realistic to help a kid understand tomorrow's class. That just requires a little bit of foresight and the careful selection of examples. And if the kid gets Tuesday's class, then they've got a decent shot at Wednesday. And we can build an area of strength for this kid, and that will be our start.

I don't want to paint too rosy a picture here. By the time you've got a regular intervention with a kid, it's often going to be rough going. Still, looking ahead worked much better for me than looking back.

Tuesday, November 26, 2013








Tuesday, November 19, 2013

Is Your Own Math Work Shareable?

(Lots to disagree with here, kids. Get excited!)

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math? Here are some possibilities:
  • Kids care more about answers than about the underlying reasoning
  • Kids are lazy
  • Kids don't know how to write clearly about math
Without a doubt, these factors all play their part. But I think that there's something else going on.

Here are a few pages from one of the notebooks that I do math in:

There's a ton of scribbling, some diagrams, some arrows, all of which make a ton of sense to me right now but would take a significant amount of work for anyone to decipher.

I tend not to do math in a way that would make sense to anyone else. The question is, should I? What would I gain by presenting my notes in a cleaner, more legible way?

I can see two reasons for writing this up in a cleaner way. 
  1. For sharing my thoughts with others.
  2. For clarifying my own thoughts.
Let's immediately dismiss the first possibility as unrealistic. These aren't new discoveries, and they aren't even new ways to think about old problems. This is me trying to understand some ridiculously well-trodden math. Why would I share this?

The second possibility is a more serious one. And, look, I'm numero uno in line for the "Writing Helps Me Think More Clearly About Things" rally. (I'm getting clearer on these ideas as I'm writing it right now which is fairly meta.) But writing a clear statement about math is only occasionally what I do when I want to get clearer on a mathematical idea. When I want to test my learning, sometimes I rederive my result. Sometimes I try to tackle a new, but related, problem. Sometimes I take a walk and think about it.

What's behind my own tendency to skip the writing-up process when I'm doing my own math? As before, it might be laziness, it might be a lack of skill. Upon reflection, I think that it mostly has to do with my preference for problem solving. I typically figure that if I don't understand something, then eventually it'll get in the way of my ability to solve problems. Sure, I could prevent that by trying to sort everything out now, but that wouldn't be nearly as much fun as trying to solve another problem. "I'll understand everything with the depth that it currently needs," seems to be the principle by which I usually operate in my own math work.

So, here are my (somewhat loaded) questions:

  • What does your own math work look like? How often do you create something that could be shared with others as part of your learning process?
  • Should I be changing the way that I do math in my notebooks?
  • Should we be asking students to practice math in a way that differs from our own?
  • Does writing about reasoning typically solve a problem for the teacher or the student? 
  • What exactly is the value of writing one's reasoning in math, as opposed to articulating it in speech or in thought?
Lots, lots to disagree with here. Please do! Let's get closer to the truth together.

Sunday, November 17, 2013

Research Review: "Interest -- the Curious Emotion"

Inspired by Dan, I've been struggling to come up with a framework for what people find interesting. In making a list of all the things that I found interesting over the course of a day, I noticed that I tended to ask questions that were relatively easy to answer. This made me think that the ease of answering a question is a crucial factor in how interesting it is.

This morning, I'm reading some actual psychological research on this whole issue. This literature review is called "Interest -- the Curious Emotion," , and it's by Paul J. Silvia. Here are some highlights, lightly edited for blogability:
"In my research, I have suggested that interest comes from two appraisals. The first appraisal is an evaluation of an event’s novelty–complexity, which refers to evaluating an event as new, unexpected, complex, hard to process, surprising, mysterious, or obscure...The second, less obvious appraisal is an evaluation of an event’s comprehensibility."
In other words, whether we find something interesting is jointly determined by its newness and its comprehensibility. What's "comprehensibility"? It's a sense of whether a person has the "skills, knowledge, and resources to deal with an event."

Here's how he thinks his theory plays out in an art museum:
"Consider, for example, a group of college students meandering through the campus art museum. Some of the students find the modern-art gallery interesting: The works strike them as new, different, and unusual, and—thanks to a few classes in art history—they feel able to get what the artists are trying to express. But most of the students, such as the students forced to attend as part of a class assignment, do not find the modern-art gallery interesting. The works strike them as unusual but also meaningless and incomprehensible: They do not know enough about this art to find it interesting."
Silvia's "comprehensibility" is a lot like what I meant by "ease." If you push me, I would have to admit that the questions that I asked myself weren't all easy, but I had high confidence that I would be able to answer them. I'm happy to drop my "ease" for Silvia's "comprehensibility."

Has he done research in math? Yes he has:
"[Subjects] spent more time viewing complex polygons than they did viewing simple polygons."
Anyway, the paper's really worth checking out, and it's chock-full of citations to a really cool body of research. Here's one line worth remembering:
"New and comprehensible works are interesting; new and incomprehensible things are confusing."

Wednesday, November 13, 2013

Things That I Found Interesting Today, Ctd.

Following up on yesterday's post, here are some questions that I found myself wondering on the way to work:

On leaving my building: What sort of symmetry is there on the building's door?

On the subway: For what percent of a year have my wife and I been married?
In the elevator: The weight capacity is listed as 2500 lbs or 13 people. How much are they assuming that each person weighs?

In all these cases I was intrigued to figure out the answer, and I felt satisfaction when I did.

(In the elevator on my way home I also saw this, booo NYC.)


What's remarkable about these questions is how easy they all are for me. They are not very challenging. And I really like math! I do it in my free time. I teach it. That makes me for sure in the top 1% of mathy people on the entire planet. And I know way tougher math than lines of symmetry, percentage conversions and division.

You might have thought that lots of questions occur to me over the course of the day of varying difficulties. But that's not what's happened for me over the past two days. There seems to be an intimate connection between the questions that strike me as interesting and the questions that I'm capable of answering successfully.

I'm essentially asking questions that I've seen asked before.


To temper this a little bit, some tough questions did occur to me later in my day. I'm doing some work with a 7th Grader as part of an independent study, and she's really into infinite series. While preparing for our time together, I asked myself whether I could measure the speed at which a sequence of decreasing fractions approaches zero. That was tough for me, and I didn't know how to take it. I tried to directly compare consecutive elements in the sequence, but that didn't really seem so informative.

I gave up really easily. And then I caught myself, and then I tried again. And then I gave up again when it wasn't really yielding anything.

Why did I find this question interesting? I think, for a moment, I thought that it would be easy to answer, and that I would have a bit more valuable knowledge under my belt. When it turned out that I didn't have easily accessible knowledge, I more or less gave up, and then it took all of what I know about learning to get back to work. And then I still gave up.

I have two take-away thoughts: (1) I need to work harder and (2) Questions rarely stay interesting if the road forward is unclear.


Danielson says that questions are evidence of learning. Maybe part of the reason why is that we usually ask questions when we're fairly secure that we could figure out the answer.

Or not? Again, thanks for kicking around these ideas with me while I try to make sense of it all.

Tuesday, November 12, 2013

Things That I Found Interesting Today

[This is a very, very tentative post. You should consider this a formal invitation to rip it apart in the comments, but, yeah, I want to put a little bit of distance between Future Michael and this thing.]

Here is a partial list of questions that I found myself thinking about today:

  • Had I gotten any emails or tweets after I turned my computer off?
  • What's the best way to understand conjugacy classes?
  • What sorts of things do we find interesting? What sorts of things do people get curious about?
  • What would my students do if I gave them a period of free-choice math?
  • What do the people look like in the subway car running parallel to mine?
  • What's the song that's coming out of that classroom?
  • When it snows, why does it harder to see the tops of tall buildings than the middles?
  • What was my wife's day like?

Dan Meyer has been thinking about what makes pure math tasks interesting, likable or enjoyable. I think that this is going to push him to a general theory of engagement, and he's asking folks to describe what makes their most likable pure math tasks so interesting and enjoyable.

This is worthwhile, but I think that reflecting on what interests our students will only take us so far. The problem is that we have so little access to what our students find interesting. It's hard for us to get into their heads.

On the other hand, it's really easy for us to get into our heads. Here's what I suggest: carry around a pencil and paper with you for the next few days, and every time you find yourself curious about something, mark it down. Then, after a few days, try to understand what sorts of things you find interesting. These can be math things, or they can be non-math things.

Based on the sorts of things I found myself curious about today, I'll toss out a couple early conjectures:
  • I almost always find myself curious about questions that I'm actually able to answer. I almost never find myself really curious about a matter that there is a low chance of me figuring out. 
  • I find myself most interested in questions whose answers are rare or uncommon. I suspect that this is the reason why I don't find easy questions interesting; it's because I perceive their answers to be common, cheap and readily available to others.
  • You can usually predict how interesting I'll find a question by asking two further questions: (a) How difficult will it be for me to figure this out? (b) How valuable is the answer of this question to me? (This value often comes in the form of other people being impressed with me.)
I'm not especially confident in my tentative ideas, but we'll see if they hold up as I pay closer to attention to the things that I find interesting.

Monday, November 11, 2013

Exponents Without Repeated Multiplication

The Problem With Exponents Education 

Kids have trouble learning exponents. In 5th and 6th Grade they regularly multiply the base and the power instead of performing exponentiation. In 9th Grade they think that negative exponents must yield negative results, and in 11th Grade they struggle to figure out what in the world a fractional power might mean. And, if you push them hard enough, they'll often reveal a tendency to still multiply the base and the power.

What is the source of all these troubles? One answer is that exponents are introduced to kids solely as repeated multiplication

You might not find this troubling. You might say, "Hey, that's a fine mathematical definition. After all, isn't multiplication just repeated addition?"

In math we deal in abstractions, and abstractions are best understood from multiple perspectives. Sometimes it's helpful to think of multiplication as repeated addition. But is 3.4 times 5.7 really best understood as repeated addition of 5.7 some 3.4 times? Instead, you might think that there are times when area is a good model for multiplication. Other times call for scaling as our model. Arrays are often helpful. The number line is often useful.

The point is that, in multiplication, we have multiple models that help us glimpse aspects of what is essentially an abstraction. For thinking about the Distributive Property, you want the array model. But arrays are less helpful for decimal multiplication, and there you really want to be talking about area with your students. For multiplication of negative numbers, you want transformations of the number line. Different models for different moments.

We have one way of thinking about exponents, and it's not terribly effective all on its own. We need more models for exponentiation, and we need to think about replacing "repeated multiplication" as our students' first exponents model.

Start With Squaring and Cubing

In the early grades, students should be able to talk (talk!) about squaring and cubing a length, and that's it. You don't worry about the notation. You don't talk about anything besides lengths. You don't talk about powers. You don't do anything besides squaring and cubing lengths, and getting kids used to that language.

You might kick that unit off with a visual pattern such as this one:

You make sure that kids can draw the next step. You ask them what the 7th picture looks like, and you ask them how many bricks are in that picture. You use these problems to give them practice with multiplication and you use the language of "squared" persistently.

Then you toss this in front of them:

4th Graders will find these problems much more challenging. They'll have trouble drawing 3D models, and you should teach them to sketch drawings of cubes, an important Geometric skill that's often never taught to kids explicitly. And, though you start by asking them how many cubes are in the fourth picture, pretty soon you're asking them "What is 4 cubed?"

And, if your kids are anything like mine, they will absolutely not figure this out by multiplying 4 by 4 by 4. Instead they'll start trying to calculate 16 times 4.

This difference is subtle enough that you might dismiss it, but you shouldn't. 16 times 4 is an entirely different conceptual model for cubing than 4 times 4 times 4 is. You and your class should make the relationship between "Something cubed" and "That same thing squared" explicit by the end of your study. 

This, and nothing else, should be kids' first exposure to exponents.

The Next Models

Of course the Geometric model of exponentiation will hardly suffice for kids in the long run. But it's the foundation, it's where we start. And what we're going to do for kids is extend the operation of squaring and cubing to other powers,to fourth and fifth and seventh powers. To do this, we'll need another model for exponentiation.

For extension to other powers we'll rely heavily on Geometric series, our second model of exponentiation. If we want to, we can even be fancy and explicitly connect the old model with our new one:

And, again, the spoken language that we use is pretty much the point here. We're going to start talking about the picture after the cube as the "Fourth Power of Three." More generally, we're going to talk about powers as entries in these patterns.

What's going on here is that we're moving toward a third model, which is a recursive definition of exponents, with squaring and cubing as our anchors. We'll define "3 to the fourth power" for students as multiplying "3 cubed" by 3. After all, isn't that how we moved from 3 squared to 3 cubed back in our earlier work?

Still not convinced that the recursive model is a worthwhile investment? Think ahead to how hard it is to convince kids that negative exponentiation doesn't have to produce negative results, and think about how helpful these patterns will be for that.

Of course, now that you have extended exponents beyond squaring and cubing, you might find it interesting to revisit that earlier model with your kids. Can we make sense of raising a length to the first power? Can we make sense of raising a length to the fourth power?

We have three models -- Geometric, Geometric Series, and Recursive Definition -- and we haven't said a word about repeated multiplication. So where is repeated multiplication in all this? It's festering in the classroom. Some kids have figured out that you can skip steps in the recursive definition using repeated multiplication. Kids have shared this as one of a few computation strategies for doing exponents calculations. (Others are skip counting to follow the Geometric Series or using exponent properties to take shortcuts, e.g. 3^5 is 9 x 27.)

And, then? Then you make it explicit. You talk about the repeated multiplication definition, because that's a really important model for exponentiation also. In particular, it's one way to see that negative exponents often create fractional values. (Though, as evidenced by student errors, it's not a particularly effective tool on its own.) Seeing exponentiation as repeated multiplication makes simplifying expressions easier. (But, again, this is an area that is currently riddled with student errors.) So we should teach repeated multiplication to kids explicitly, and this should happen some time in (how about) 6th Grade.

But the foundations for exponentiation need to be laid several years earlier in a child's education if we want to really help our kids avoid all those pesky, Algebra-killing errors in high school.

Friday, November 8, 2013

Yarmulke Tales!

If you've ever seen me in person, you know that I wear a yarmulke on my head. Here's how that's impacted my life, lately:

  • One of my 4th Graders pulled on my arm during dismissal. I looked down. "Mr. Pershan! Mr. Pershan!" She pushes a tiny girl in front of me, and there's pen doodles all over her hands and face. "This is Sarah. She's not Orthodox, but she's Jewish. And she's really weird."
  • During lunch duty, some kid who I don't know wished me "Shabbat Shalom!" on his way out of lunch.
  • A bum threw a penny at me on the subway.
  • A 2nd Grader who I don't know (standing in line, waiting to get into the music room) asked me why I was wearing a yarmulke on my head. I shrugged. He said, "It's because you're Jewish!" Bingo!
  • A 3rd Grader who I teach ran up to me before class. "I KNOW WHAT YOU BELIEVE" he said as he pointed and grinned at me. I asked him what he meant. "I KNOW WHAT THE THING ON YOUR HEAD IS!!!!!"
OK, and this last story is from the beginning of the school year. I had to miss a day for a Jewish holiday (Sukkot) and I told my 4th Graders that they could expect a sub the next day. This was on the second week of school, so I told them that I was a religious Jew and that the thing on my head, etc. Then I told them if they ever wanted to ask me questions about it that they should totally feel free to. 

Yeah, of course they had some questions. "Do you always wear it?" "Do you wear it when you shower?" "What happens if you take it off?" "Is it something that gets passed down from generation to generation in your family and it's actually really old?" (No.)

I had been teaching at a Jewish school for the past three years, so none of this ever happened. It's weird, but pretty adorable.

Tuesday, November 5, 2013

The Double Ferris Wheel

It's really hard to find models and contexts for Unit Circle Trigonometry. Like, really tough. The one go-to that everybody uses is the Ferris Wheel, which is great, but it's practically all that we have.

"Oh, no!" you'll say. "What about all that astronomical stuff? What about the percentage of the moon that's visible on a given night?"

Well, two things. First, why would you want to model the percentage of the moon that's visible with a sinusoidal function? If I really want to know what the moon's going to look like on January 17, 2015, then I'm just going to subtract a bunch of 29.5 day intervals from 1/17/2015 until I land back on my data.[1]

The second problem is this:

The moon's visibility isn't sinusoidal. Then again, of course it isn't. If it were really sinusoidal, then its orbit would be circular.[2] 

What is a Trigonometry teacher to do? Practically nothing interesting in the world shows truly circular motion. (Oh, pendulums?) And even the things that do show circular motion are rarely worth modeling.

We're stuck with Ferris Wheels. So, find cooler Ferris Wheels.

We watched the video, and I asked them to graph height vs. time on a Post-it.

I tossed these under the doc camera, and we narrowed down our options. I chose two at a time and asked the kids to compare them, which usually resulted in us throwing out one of the graphs. When we got stuck, I suggested that we separate the two wheels and figure out an equation that determines the motion of each. After some thinking, a kid suggested adding the equations together. I asked her to show us what she meant, and she produced this:

This lesson was fun, tough, and a genuine context for some Daily Desmos-style sinusoidal modeling.


This year I find myself just going nuts with Ferris Wheels and rides. We've already studied square-shaped tracks and rides, inspired by this lovely visualization:

What's next? There are a lot of rides out there, but many of them seem to be versions of this Double Ferris Wheel. Maybe the next step is to get weirder. Like, what sort of ride would have this height graph?

I'm running out of rides. Ideas? 


[1] In other words, all you need to model is the periodicity of the moon's cycle. There's nothing that pushes people or students to pay attention to its sinusoidal nature.
[2] In class we model the visibility of the moon, and I tried to escape these problems by asking "Is the curve of the moon's visibility sinusoidal or not? Is it like our Ferris Wheel's motion, or is it a different pattern?"

Saturday, November 2, 2013

The Six Acts of a Mathematical Story

Act One: Introducing The Central Conflict

"What's the most ridiculous looking book I can walk out of here with?" I asked. The librarians were incredibly helpful. And I lugged this enormous dictionary back to my classroom and hid it in a filing cabinet.

I told the 4th Graders that I had a surprise for them. In the time it took for me to pull this monstrosity out of my filing cabinet and drag it to the middle of the room, the kids already started asking how many pages long it was. I didn't say a word. I just went to the board and started recording guesses.

Act Two: Confronting The Central Conflict

The first round of guesses were all in the 10,000s, so I took a folder and stuck it after the 100th page. I told them where I'd placed the folder, and we took another round of guesses. The kids seemed split on whether this data made their initial guesses too high or too low. (You can see the lines connecting the initial guess and the second guess.)

That was interesting, so I moved the folder to the 500th page and took a final round of guesses. You can see those in the third column.

Act Three: Resolution of the Central Conflict

[Insert picture of last page here. I'll get it when I get back to school.]

The dictionary ends on page 3,210, and we marched that page around the classroom. (Wikipedia says that it has 3,350, though I don't exactly know how they're getting that number.)

Act Four: Those Clips That Sometimes Show Up After The Credits?

"What would be some interesting math questions that we could ask about this book?"

"How thick is each page?"
"How many words are defined in the dictionary?"
"How many words are in the dictionary, including everything?"
"How many letters are there in the dictionary?"
"How many pounds of ink went into that dictionary?"
"How many atoms are in the dictionary?"
"How many of those Arabian Nights books on the desk would fit into the Big Book?"
"How many words are on a typical page of the Big Book?"
"How many words are there that are not defined in the dictionary?"
"How many pixels is this on a computer screen?"
"How much room is there on each page?"
"How long would it take to read the entire dictionary out loud?" [My question.]
"How many words long is a typical definition?"

The bell rang, and the kids went home, or to some other class, or whatever it is 4th Graders do when they're not in my classroom.

Act Five: Falling Action? Rising Action???

Everybody got a dictionary page. I photo-copied them and handed them out.

How many words are defined in the entire dictionary? What information are we going to need to answer that question? If we could figure out how many words are defined on a typical page, then we could do some crazy multiplication to settle the bigger question.

There's our line-plot. Most kids figured that, based on the data, either 84, 85 or 86 was the typical number of words defined on a page. 

(One girl -- the duck, actually -- insisted that because she had counted 143 bold words on her page, that this was the typical number. I think that she just didn't like the idea that her counting was somehow in vain. Either that or she was burned out from Halloween.)

Act Six: The Sixth Act

There are around 276,606 words defined in the dictionary, though this is tough multiplication for my kids to do.

I'm having trouble finding out exactly how many entries there are in the 1934 Second International Edition, but Wikipedia reports that the Third International Edition "contained more than 450,000 entries, including over 100,000 new entries and as many new senses for entries carried over from previous editions." So that should mean no more than 350,000 for the Second Edition, right? Not bad, kids.

With the rest of class we tried to knock off a few more of their questions. Did you know that linguists estimate that there are roughly 1,000,000 words in the English language? We figured out (roughly) how many words are not defined in this dictionary.

We also had a fun conversation about how to figure out how thick each page of the dictionary is.

[Six Acts]

There you have it. This was a ton of fun, and I certainly recommend checking your local library for a book as silly as this one. I'd love to help those without access to a physical copy to play with this. I uploaded some of my pictures to 101qs, but please let me know if you think of certain pictures that would make this problem more useful to you.

Wednesday, October 30, 2013

My Love Letter To Orpda

This is my love letter to Orpda, an invented number language in Base 5 that I learned about through Christopher Danielson. Half of what I want to say about Orpda is "Go read Danielson's post!"

(A quarter of what I want to say is "Go read Weltman's ba-na-na post!" That leaves me a quarter to talk about what actually happened in my 4th Grade class.)

(The last quarter is sort of long.)

We start with this:

Then we draw another red circle and ask kids: "What do we call this many things?"

What's the point of this?

The teaching assumptions here are worth making explicit. Sometimes students think that complex things are simple. Often that's just because they're used to it, not because they actually get it. So, how do you reveal the complexity that kids are just ignoring?

A nice solution is to shift to a similar, but unfamiliar context. Since the kids aren't used to it, the complexity becomes clear. Since it's similar, you can make connection to the familiar context.

(This is basically Danielson's fade-away jump shot. See here, here, here and of course Orpda.)

(That and the hair. The hair is also Danielson's fade-away jump shot.)

So, here's why Orpda is a solution to a teaching problem of mine. I'm hanging out with some very confident 4th Graders. They have a decent sense of whole-number place value. But they can't, for example, explain why the "add a zero when you multiply by 10" thing works. And we're going to put a lot of pressure on their place value understanding when we work with multiplication and division. They need to understand, in a deep way, that place-value involves grouping.

So? We make it unfamiliar, and then we connect it to the familiar.

What to expect from the kids

Here's what my kids came up with for representing the next number in Orpda:
  • $ + #
  • @@@@@
  • ##@
  • Invent a new symbol for that many dots.
All of these options were considered. I tried to put pressure on these choices by asking them to represent higher numbers. So I drew 24 circles on the board. I asked the kid who offered @@@@@ how they'd represent this many circles. They folded. 

Another kid went up to the board and wrote some sort of multiplication problem using the defined symbols. A second kid offered a new suggestion, one that we hadn't seen yet: #%, or "at-percent." 


I put pressure on these using different numbers and I point out confusions with their number systems. They don't come up with place-value, which is interesting. After a day (or two?) of discussion, I suggest that we group stuff.

I happen to throw up a slide with ducks on it as we're discussing grouping, so the kids decide to name this new number a "flock." 

Then we count, out-loud. (This is so important!) We count: "At, hashtag, dollar-sign, percent, flock, flock-at, flock-hashtag, ..."

Then, we get stuck. After a little bit of discussion...

... Super Flock!

Random Thoughts

This is such a rich environment to play around in. Here are a few stray thoughts, or things that I learned as I was doing this:
  • Language matters. I didn't properly realize until diving into this lesson how there are two number languages, operating side by side: the written and verbal representations of number. These are obviously deeply related, but the kids literally couldn't figure out how place-value would help them until we had named the number a flock. That was huge, and it sensitized me to the way that our spoken language is deeply connected to our conceptual understanding.
  • (I actually screwed this up for a while at the beginning by asking the kids "What should we call five things in Orpda?" As a very sweet 4th Grader explained to me, this was a stupid question because Orpda doesn't have the number five. In fact, that's the whole point of this freaking exercise. Thanks R! Saved my ass there.)
  • Once the kids landed on calling the next number a "flock," the question was how do we represent it in writing. A kid helpfully drew a pair of underpants on the board and labelled it "Super Flock", ala Captain Underpants. Ha ha, laugh it up kid.
  • There were all sorts of computational problems that I created for kids once we had the number system. These questions all pushed on parallels between Orpda and our number system. So I created problems that required carrying in Orpda, or number patterns like @@, ##, $$, %%, ____. 
  • We also spent a good chunk of class time thinking about questions such as, "What number is like 99 in Orpda?" Or "What's like multiplying by 10 in Orpda?" Or "What number is like 11 in Orpda?" These were all tough and fun.
  • A random interesting note: the move from 2 digits to 3 digits was harder than I thought it would be. I thought that, after we had figured out what number came after %, that it would be easy to figure out what comes after %%. Actually, no. It was only when we explicitly drew parallels to our own number system ("What's the biggest two-digit number in our system?") that we were able to figure that out.
  • @!!! is a Super-Duper Flock, in case you were wondering.
Anyway, do Orpda with kids. It's a blast, and it really deepened my kids' ability to talk about place-value. I anticipate coming back to it periodically as the year goes on whenever we need to lean heavily on place-value in a sneakily complex way.

Monday, October 28, 2013

Someone Is Wrong on The Internet; "Bad at Math" Edition

This article -- which is showing up every 15 seconds in my twitter feed right now -- is not very good.

We believe that the idea of “math people” is the most self-destructive idea in America today.
The most? Really? I'm not going to respond to this, but if you want to defend this line I'll see you in the comments.
So why do we focus on math? For one thing, math skills are increasingly important for getting good jobs these days—so believing you can’t learn math is especially self-destructive. 
This line is repeated a few time in the article. The idea that there's a shortage of STEM workers shouldn't be taken for granted.
While American fourth and eighth graders score quite well in international math comparisons—beating countries like Germany, the UK and Sweden—our high-schoolers underperform those countries by a wide margin. This suggests that Americans’ native ability is just as good as anyone’s, but that we fail to capitalize on that ability through hard work.
Yeah, it's either hard work or sub-par teaching or inequality or anything else. Sloppy reasoning.
Different kids with different levels of preparation come into a math class...The well-prepared kids, not realizing that the B students were simply unprepared, assume that they are “math people,” and work hard in the future, cementing their advantage. Thus, people’s belief that math ability can’t change becomes a self-fulfilling prophecy.
Great! So there's no genetic advantage. It's just...an incredibly robust advantage in skills, psyche and mindset.

What sort of comfort is this supposed to provide a kid? "The reason why you suck at math is because your parents didn't do math with you, and because you fell into a self-destructive cycle of behavior. And now you're very far behind."

It's just replacing one deterministic story with another, in my opinion.
A great deal of research has shown that technical skills in areas like software are increasingly making the difference between America’s upper middle class and its working class. While we don’t think education is a cure-all for inequality, we definitely believe that in an increasingly automated workplace, Americans who give up on math are selling themselves short.
Hear that working class? Stop selling yourselves short! You're contributing to income inequality, guys, so cut it out.
We think what many of them are afraid of is “proving” themselves to be genetically inferior by failing to instantly comprehend the equations (when, of course, in reality, even a math professor would have to read closely). So they recoil from anything that looks like math, protesting: “I’m not a math person.”
It's not obvious to me that when kids say that they're not math people that they exclusively mean that they are the sort of people that are bad at it.

Often when kids mention it to me it's in the form of an apology while they're asking me for help. As in, "Hey Mr. P, I'm really sorry but I'm totally not a math person and this isn't making sense to me." In that context it can't possibly mean "I can't get better at high school math." There it's just providing a description, something like "I'm not good at math and I've never really been good at math, and I'm pretty slow when it comes to understanding this stuff." Say what you will about kids having that attitude, but it's not the same as a deterministic view that they can't handle high school math.

And don't some kids just mean "I don't like math" when they say "I'm not a math person"? That's what I mean when I say "I'm not a vanilla person."
We see our country moving away from a culture of hard work toward a culture of belief in genetic determinism.
Not any time soon, you aren't.


I'm pretty confident that the piece is pretty sloppy. I'm less sure about all of this, though...

The core argument of the article is that your level of success in high school math is determined by the amount of work that you put in. This is largely true -- no disagreements from me about the value of hard work. But since high school students are children, what it comes down to is blaming children from not working hard enough. And that seems unfair to me. Your typical 5th Grader can't be expected to break through the sorts of disadvantages that put them behind in math through hard work. That's an unfair burden to place on a child.

I see articles like this all the time. They trumpet hard work as the cure to so many of society's evils, and the key to personal redemption. Once you commit yourself to a mindset of hard work, you've unlocked the secret to success. We lament the death of hard work and rediscover it in science, ignoring the reality that it's part of our society's cultural backbone. And -- here's the clincher -- by putting hard work on a pedestal and acting as if we've just discovered it, we let the culture of hard work off the hook for being part of the very social problems that we're lamenting.

If the culture of hard work has been around for centuries, then how come there's all this inequality? How come students don't realize that hard work will help them get better at math?

I buy what Dweck says, to an extent. (There's other research that complicates her rather clean story, but never mind that for now.) I mean, who's going to disagree with the idea that hard work helps you get better at something? (I've never met anyone who doesn't believe that.) But it's important to recognize that the notion that hard work will solve many of our social issues is an old one in this country. It's part of our culture, and it might even be part of our problems.

Take us home, Shamus Khan!
Which means that these are tremendously unequitable institutions. But instead of explaining their position relative to their social advantage, their membership within an upper class, these elites explain it by their hard work and individual skills.
Check out his book Privilege, for real.