# My work on the AMS Teaching & Learning Blog

I don’t know why I didn’t think to tell you this earlier, but: in 2019 I joined the editorial board of the American Mathematical Society’s Teaching & Learning Blog, and I’ve written several pieces for it. I’m extremely proud of each of these, and would like to share them with you.

• Some thoughts about epsilon and delta (August 19, 2019) is a deep dive on student difficulties with a notoriously challenging definition from calculus. I got pretty scholarly and read a bunch of research for it, but the core of the post is a discussion of challenges faced by specific learners I’ve known, one of whom is my own self. I also include a brief history of this definition.
• The things in proofs are weird: a note on student difficulties (May 20, 2020) is a meditation on the nature of the objects we use in proofs, and the difficulties students have in getting used to working with objects with this strange nature. I again got pretty scholarly and read a bunch of research. Nonetheless, it includes an extended riff on Abbott and Costello’s Who’s on First?
• A K-pop dance routine and the false dilemma of concept vs. procedure (August 18, 2020) is a… ok let me back up. People used to fight about whether conceptual or procedural knowledge was more important. I think we’ve more or less reached a place in the public conversation about math teaching where there’s an official public consensus that conceptual and procedural knowledge are both important and are mutually supportive. But just because we all can say these words doesn’t mean we’ve necessarily fully reconciled the impulses behind that older fight. For example, in spite of firm intellectual conviction that this view is correct, I have a bias toward the conceptual in my teaching, in the sense that I have a strong tendency to assume any student difficulty is rooted in a conceptual difficulty. This bias is really useful a lot of the time, but sometimes it can lead me to misdiagnose what a student needs to move forward. Anyway, so one day I was learning a BLACKPINK dance and the learning experience just really eloquently illustrated both the advantages and disadvantages of that exact bias. Hopefully you’re intrigued!
• The rapid expansion of online instruction, occasioned by the pandemic, has forced academia to contend with the limits of the control that its usual physical setup allows it to exercise over students’ movements and choices. One place this manifests very clearly is in the setting of timed tests, which are historically proctored in person. Remote proctoring: a failed experiment in control (January 19, 2021) is my heartfelt contribution to the pushback against the Orwellian trend of turning to “remote proctoring” (where the student is surveilled in their home during tests) to try to claw back the lost control, rather than accepting that the game has changed and rethinking assessment from the ground up, as the situation demands.
• Three foundational theorems of elementary school math (November 22, 2021) could have been titled, “The logical structure of elementary school math is actually extremely beautiful and intricate, and I want everyone to pay more attention to this.” It’s a love letter to three closely related facts from elementary school math that I think often don’t get their due, making the case that they deserve to be thought of as theorems. I discuss proofs (including some relevant student work) and connections. (If any long-time readers of this blog are still here in 2021, this post is a distant but direct descendant of this post I wrote nearly 12 years ago, when I was a baby blogger.)

I also solicited a piece from Michael Pershan, which I am also extremely proud of:

• What math professors and k-12 teachers think of each other (November 18, 2019) is Michael’s synthesis of and meditation on an informal survey he ran, canvassing math educators teaching in schools and universities about what they think about the differences in the shape of math education at these different levels. Michael’s characteristic thoughtfulness is on full display here, and it’s all with an eye toward how we can collaborate effectively. I love it.

# Re-invitation

Dylan Kane’s recent post about prerequisite knowledge has me wanting to tell you a story from my very first year in my very first full-time classroom job, which I think I’ve never related on this blog before, although I’ve told it IRL many times.

It was the 2001-2002 school year. I taught four sections of Algebra I. I was creating my whole curriculum from scratch as the school year progressed, because the textbook I had wasn’t working in my classes, or really I guess I wasn’t figuring out how to make it work. Late late in the year, end of May/early June, I threw in a 2-week unit on the symmetry group of the equilateral triangle. I had myself only learned this content the prior year, in a graduate abstract algebra course that the liaison from the math department to the ed department had required of me in order to sign off on my teaching degree, since I hadn’t been a math major. (Aside: that course changed my life. I now have a PhD in algebra. But that’s another story for another time.)

Since it was an Algebra I class, the cool tie-in was that you can solve equations in the group, exactly in the way that you solve simple equations with numbers. So, I introduced them to the group, showed them how to construct its Cayley table, and had them solving equations in there. There was also a little art project with tracing paper where they drew something and then acted on it with the group, so that the union of the images under the action had the triangle’s symmetry. Overall, the students found the unit challenging, since the idea of composing transformations is a profound abstraction.

In subsequent years, I mapped out the whole course in more detail beforehand, and once I introduced that level of detail into my planning I never felt I could afford the time to do this barely-curricular-if-awesome unit. But something happened, when I did it that first and only time, that stuck with me ever since.

I had a student, let’s call her J, who was one of the worst-performing (qua academic performace) students I ever taught. Going into the unit on the symmetry group, she had never done any homework and practically never broke 20% on any assessment.

It looked from my angle like she was just choosing not to even try. She was my advisee in addition to my Algebra I student, so I did a lot of pleading with her, and bemoaning the situation to her parents, but nothing changed.

Until my little abstract algebra mini-unit! From the first (daily) homework assignment on the symmetry group, she did everything. Perfectly. There were two quizzes; she aced both of them. Across 4 sections of Algebra I, for that brief two-week period, she was one of the most successful students. Her art project was cool too. As I said, this was work that many students found quite challenging; she ate it up.

Then the unit ended and she went back to the type of performance that had characterized her work all year till then.

I lavished delight and appreciation on her for her work during that two weeks. I could never get a satisfying answer from her about why she couldn’t even try the rest of the time. But my best guess is this:

That unit, on some profound mathematics they don’t even usually tell you about unless you major in math in college, was the single solitary piece of curriculum in the entire school year that did not tap the students’ knowledge of arithmetic. Could it be that J was shut out of the curriculum by arithmetic? And when I presented her with an opportunity to stretch her mind around

• composition of transformations,
• formal properties of binary operations,
• and a deep analogy between transformations and numbers,

but not to

• do any $+,-,\times,\div$ of any numbers bigger than 3,

she jumped on it?

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. – Dylan Kane

Acknowledgement: I’ve framed this post under the title “Re-invitation”. I’m not 100% sure but I believe I got this word from the Illustrative Mathematics curriculum, which is deliberately structured to allow students to enter and participate in the math of each unit and each lesson without mastery over the “prerequisites”. For example, there is a “preassessment” before every unit, but even if you bomb the preassessment, you will still be able to participate in the unit’s first few lessons.

# “Gifted” Is a Theological Word

Quite the juicy convo on Twitter:

Hard not to reply with every thought I have, but I want to keep the scope limited. One idea at a time.

In some sense, I work in “gifted education.” Big ups to BEAM, my favorite place to teach. This is a program that is addressing the intellectual hunger of students who are ready to go far beyond what they are doing in school. I have profound conviction that we are doing something worthwhile and important. (NB: to my knowledge, BEAM does not use the word “gifted” in any official materials, and most BEAM personnel do not use it with our kids out of growth mindset concerns.)

It is also true that I myself had a very different profile of needs from my peers at school as a young student of math. I taught myself basic calculus in 5th grade from an old textbook. I read math books voraciously through middle school, and in class just worked self-directedly on my own projects because I already knew what we were supposed to be learning. I am not mad that I didn’t have more mathematical mentorship back then — my teachers did their best to find challenges for me, I appreciated them both for that and for the latitude to follow my own interests, and in any case things have worked out perfectly — but looking back, at least from a strictly mathematical point of view, I definitely could have benefited from more tailored guidance in navigating my interests.

In this context I want to open an inquiry into the word “gifted” as it is used in education.

I hope the above makes clear that this inquiry is not about whether different students have different needs. That is a settled matter; a plain fact.

The subject of my inquiry is how we conceive of those differences. What images, narratives, stories, assumptions, etc., are implicit in how we describe them. In particular, what images, narratives, stories, and assumptions are carried by the word “gifted”?

This question is too big a topic for today. Today, I just want to make one mild offer to that inquiry, intended only to bring out that there is a real question here — that “gifted” is not a bare, aseptic descriptor of a material state of affairs, but something much more pregnant — containing multitudes. It is this:

What do I mean?

A gift is something that is given; bestowed. My nephews recently bestowed on me a set of Hogwarts pajamas, fine, ok, but when we speak of “giftedness,” you know we are not discussing anything that was bestowed by any human.

By whom, then, is it supposed to have been bestowed?

You know the answer — by God. Or if not God, then by “Nature,” the Enlightenment’s way of saying God without saying God.

When we say a child is “gifted,” we are declaring them to have been selected as the recipient of a divine endowment. Each of these words carries a whole lot of meaning extrinsic to scientific description of the situation — selected; recipient; divine; endowment.

When we use this word in contemporary educational discourse, we usually aren’t consciously evoking any of this. Nothing stops a committed atheist from saying a kid is gifted. Nonetheless, I don’t think it can really be avoided.

Why I say this is how easily and quickly the full story — selected, recipient, divine endowment — becomes part of the logic of how people reason about what to do with a student so labeled. To illustrate with a contemporary slice of pop culture, the 2017 film Gifted, starring McKenna Grace, Chris Evans, Lindsay Duncan and Jenny Slate, hinges on the question of what is a family’s obligation to its child’s gift? How can a bare material state of affairs create a moral obligation? — but being chosen as the custodian of a divine spark on the other hand, it’s easy to see how to get from that to something somebody owes.

So, this is my initial offer. I’m not saying anything about what to do with this. For example, I am not evaluating Michael’s assertion that “giftedness is true.” I’m just trying to flesh out what that assertion means — to call attention to the sea of cultural worldview supporting the vessel of that little word.

# Education and Markets (reblog)

Ben Orlin kills it on the complete incoherence of the notion that public education can only be improved by increased exposure to market forces. This is something I’ve been brewing thoughts on for years, and Ben says pretty much everything I would want to say, except with his signature drawings and economical word use in place of my epic and probably gratuitous verbosity.

While everything he says is gold, I will pull out one point I want to amplify:

The difference between businesses and schools is that nobody cares if most businesses fail.

# Reality Check: Does Anybody Care What Andrew Hacker Thinks?

So I’m just trying to figure out who, if anybody, cares what Andrew Hacker thinks about math education.

This is an earnest question.

At least since 2012, when he had an opinion piece in the NYT, he has been going on about how we should stop requiring “advanced” math, from algebra up, in schools. He was in the NYT twice again recently and has a new book out about it.

Now, of course both math educators and mathematicians are going to “care” in the sense that it annoys us. We are spending all this time trying to figure out how to improve students’ appreciation for and understanding of algebra etc., and out comes this dude talking about “scrap that whole project.” I remember there being at least a few responses in the MTBoS back in 2012, although Dan Meyer’s and Patrick Honner’s are the only ones I remember specifically. (Dan had some more fun with it a few years later.) And I was moved to write this reading mathematician Evelyn Lamb’s piece in Slate responding to Hacker’s book. (Dan’s responses succinctly summarize Hacker’s lack of imagination. Mr. Honner sees algebra in what Hacker wants to replace algebra with. And if you want to get more into the details, go read Lamb’s piece, it’s great.)

But annoying math teachers and mathematicians is definitely not the same thing as being remotely relevant. I mean, he is suggesting to do away with required algebra precisely at the point in history when, between the Common Core, the increasing quantity and stakes of standardized testing, and the incessant press handwringing about international competitiveness,[1] it seems to me that math, including advanced math, is more centrally ensconced in the curriculum than it’s ever been. (In my lifetime anyway. Possible historical exception of the immediate post-Sputnik era.) Is anybody taking him remotely seriously?

Yes, he has coverage in the New York Times and the Chronicle of Higher Ed. This doesn’t answer the question. He’s being intentionally provocative and succeeding in getting a rise. Is anybody taking his proposals seriously?

*****

Postscript: Let me indulge myself to go ahead and give you my take, just for the record. Disclaimer that I haven’t read his book. I’m going on the 3 NYT pieces.

At the level of fundamental goals, he is upset about the fact that so many Americans have traumatic experiences with their math education, meanwhile graduating without basic numeracy needed for citizenship, and he wants to do something about it. I’m not mad at this, nor could I be.

I’m also sort of delighted that being a public intellectual counts for enough that this 86-year-old Queens College political scientist can mouth off in the NYT whenever he wants. I hope that when I’m 86, the NYT still exists and I can mouth off in it whenever I want.

But on to the merits themselves. I think he sees a real problem but I, like Dan, think he lacks imagination about both (a) what math education could be, and (b) what math is for – he doesn’t get it as a domain of human inquiry, or an intellectual inheritance, just as a tool, so he applies a utilitarian standard to it I’m sure he’d never apply to history, or art, say. But also, (c) I think he misdiagnoses the problem if he thinks removing required algebra (and up) will solve it. Algebra isn’t the first point in the curriculum where massive numbers of American children jump ship emotionally. This is already happening with fractions, and may begin much earlier. Hacker would never propose to take fractions, or even more fundamental stuff, out of the curriculum, since it’s obviously (even to him) part of the “citizen mathematics” he champions.[2] Doing a good job teaching math is a problem to be solved, not avoided. Finally, (d) I think he would probably puke if he really thought through the antidemocratic implications of a general public without advanced technical literacy while all the contemporary centers of power – finance, info tech, biotech, etc. – are technocratic and growing more so. He sometimes argues that you don’t need to know algebra to learn to code. Depends on what you are coding I suppose, but in any case this is beside the point. Sergey Brin does know algebra. Jamie Dimon does too.

Update 4/5: Sam Shah sends this graphic of number of people on feedly who “saved” or “favorited” Hacker’s Feb. 27 piece (which is about what he wants to replace algebra with):

Update 6/28: Patrick Honner wrote something relevant on the Math for America blog back in May – When It Comes to Math Teaching, Let’s Listen to Math Teachers.

[1] I’m only lumping these three things (CCSSM, high-stakes testing, and international-comparison handwringing) together from the point of view that all three seem to me to be moves in the direction of consolidating the consensus on the centrality of math in contemporary American education. I do not have them confused with each other and I don’t feel the same way about each of the three at all. For exmaple, I basically dig the CCSSM but (as any regular reader of this blog knows) I do not at all dig high-stakes testing.

[2]Patrick Honner’s post points out that Hacker’s notion of “citizen mathematics” almost surely involves algebra as well…

# And, a teacher shortage…

An obvious observation –

The last 6 or 7 years in public education policy seem to have been characterized by the following trends:

1) Tying teacher evaluation, hiring, firing, and teacher pay to student standardized test results.
2) Relatedly, using value-added measurements in making these decisions.
3) School closings and state takeovers.
4) Using VAM in making decisions about those too.

I.e. Stressing all the adults who work in schools the f*ck out.

Also:

5) Subcontracting to charter networks.
6) Direct funding cuts.

I.e. divesting from education as a public trust.

All in all, these trends, spearheaded by the US Dept. of Ed. under the leadership of Arne Duncan, but with numerous assists from other folks, representing both public and private interests (being in NYC, I’m lookin at you Mike Bloomberg), seem to me to have an obvious common theme:

Making public schools shittier places to work.

Recently, both the NYT and EdWeek have reported a national teacher shortage as enrollment in teacher training programs has dropped precipitously for several years in a row. Even TfA is having trouble recruiting.

Motherf*ckers, what did you think was gonna happen?

….

….

….

….

(Cynical voice at back of head: Ben, you are so effing naive. That’s what they wanted to happen. What better excuse to hire un-credentialed people to teach poor children? Me: No! I don’t believe it!)

Addendum (5/9/16): It came to my attention at some point that there was some debate last summer about the validity of NYT’s and EdWeek’s coverage. Is/was there really a national teacher shortage this fall, or were there certain districts with a shortage and others with a surplus? Michael Pershan had some tweets about this. So, let me just acknowledge this debate. This post was a quickly-fired-off response to seeing talk of a shortage in two major press outlets, after several years of running workshops with young teachers and feeling awe for their willingness to stay in the game even as working conditions have become shittier. If there is a shortage, I’m not surprised. If there’s not, then, let’s hope it stays that way. Go young people! That is all.

# “You have made us the enemy. This is personal.”

Just caught this from the Washington Post blog 2 months ago. Word.

# Some Followup on “A Note to My Fellow White People”

If you were interested, challenged or otherwise engaged by my Note to My Fellow White People, I have come across a bunch of other things recently you will be interested in:

Here is the other video he refers to in the video:

Also a propos is this recent opinion piece in the NYT by Ta-Nehisi Coates.

I was talking in general about white people receiving feedback about race, but several people who commented took it (very reasonably) in the direction of how to have conversations about race in the classroom. In which case I have the following strong book recommendation:

High Schools, Race, and America’s Future: What Students Can Teach Us About Morality, Community, and Diversity

I am cross-posting my review of this book on goodreads.com:

Full disclosure: the author of the book is my dad. The high school featured in the book is the one I both attended and taught at.

THAT SAID.

This is a beautiful book. The author is a (white, Jewish) professor of philosophy at a university. The book chronicles his venture into teaching a class about race and racism at his local racially diverse public high school. It offers a model of what a functioning, productive cross-race conversation about race and racism can look like, in an era where (depressingly) this is still a rarity. It makes a case for the civic value of integrated public education in an era where we seem to be forgetting that education even has a civic purpose.

It belongs broadly to the genre of teaching memoirs, along with books like Holler if You Hear Me. But two related features distinguish it in this genre:

(1) The author is a serious scholar. Unsurprisingly, then, the content of the course he taught features heavily in the book. So this teaching memoir also functions, with no cost to readability, as a scholarly book about race. (As an aside, I am very proud of him on the readability front. It was a real stretch for him to write a book whose style didn’t place a technical burden on the reader, and it took a lot of rewrites, and help from his editor, but he totally pulled it off!)

(2) The genre is characterized by taking students seriously as moral and psychological beings. That’s one of its strengths as a genre as a whole. But this is the first book I’ve read that takes students equally seriously as intellects. The author often writes with plain admiration for his students’ ideas. This may be my favorite feature of all. Developing students as minds is, after all, the point of education. So it strikes me as surprising that it’s so rare for a memoir about the lived experience of teaching to give such loving attention to what those minds produce.

# A Note to My Fellow White People

I haven’t talked openly about race or racial difference on this blog before, but I actually think about it a lot. One of the most damning legacies of our racist history has been systematic libel against the minds of black and brown children (and adults for that matter). Meanwhile, in our culture, math is the ultimate signifier of intelligence. So the math classroom has heightened power, both to inflict injustice and to rectify it. Given this, plus the diversity of teachers and students, a comfortable cross-race conversation about racial matters is a must for the profession. In the spirit of contributing to that conversation, I offer

A Note to My Fellow White People

Guys, we have to chill out a little. It has to be possible for somebody to say to you, “that was ignorant,” or “that was racially offensive,” or even “that was racist,” without you flipping out, getting offended or defensive, or needing to be reassured you are not a horrible person. It’s not a good look, on any level: it’s not dignified, and it makes it impossible to have a productive conversation about race across racial lines.

I was at a cafe a couple months back trying to get some schoolwork done when I found myself distracted by a profoundly uncomfortable conversation at the next table. There was a white man in his early 50s and two black women, one close to his age and one closer to mine. They seemed to be sharing a familiar and friendly meal. Things started to go south when the man admitted to being afraid of a young black man on the street. The younger of the women said something to the effect of, “you might have work to do on that.”

Her tone was warm: she wasn’t being accusatory but rather seemed to be offering her words in the spirit of holding her friend to a high standard. But the man immediately became anxious, although his face and words were all smiles and jokes. His first response was that white people make him more uncomfortable than black people, as though he could re-establish his lost racial coolness with sufficiently loud declamations of prejudice against white people.

The women weren’t having it. “You’re being ignorant against white people now.” I interpreted their response as saying, “you can’t get off the hook with this diversionary tactic.” But he kept trying. His anxiety was as audible to me as a fire alarm, even when he wasn’t talking. I tried to concentrate on my math but I couldn’t get anything done.

Things stayed in this state, a tense, anxious impasse overlaid by a thin layer of too-eager conviviality and jokes, for about 20 minutes, till they got up to leave, no noticeable progress having been made in the conversation. At this point the man, in that same overly-eager joking tone, almost-but-not-quite-explicitly asked for reassurance that everybody was still his friend. They gave him the reassurance. On their way out, the younger woman leaned over to my table and apologized for her “ignorant friend.”

I’m not telling you this story to put the man down or call him ignorant. I don’t remember the context of the conversation and I don’t have my own opinion about it. Also, I think in all likelihood he’s a completely nice and decent person, and so are the women.

The point of the story is the man’s intense anxiety at being put on the spot racially, and the way that anxiety dominated both the conversation and its goals (so that what started as an attempt to raise consciousness was aborted, and turned into a reassurance fest), and the social and public space (so that the younger woman felt the need to apologize to a neighboring table).

Now I don’t fail to have empathy for him. If you are a white person with a modicum of sense and decency, you know that you are the beneficiary of an unjust history. (Shout out to Louis CK.) Just knowing that you’re benefiting is already a little uncomfortable to begin with. Feeling like you might be participating in that injustice can make the discomfort acute. I’ve been there many times.

But, guys, we’ve got to get it together! It is necessary to learn how to be with that discomfort and still function. First of all, the story I just told you is about a grown-a** man! Trying to prove how un-racist you are, and then needing to be coddled and preened so that you know the trouble is past, is unbefitting of the dignity of an adult. So is any other response aimed at removing the source of your discomfort rather than tolerating it – throwing a fit, acting defensive or offended, etc. Shouldn’t we aspire to some grace here?

Secondly, it makes it impossible for the conversation to advance! If we want to avoid participating in injustice we have to be willing to tolerate the possibility that we already are participating. Otherwise how will we learn what to avoid? In the anecdote I’ve recounted here, the man’s anxiety shut down the ability of the conversation to make any progress. He was blessed with friends who were willing to hold him to a higher standard and he was too busy freaking out to get the benefit of that! The bottom line question is, would you rather spend your time and energy proving how un-racist you are, or would you actually like to learn how to make the world better?

All of this puts me in mind of a much more public incident. In 2009, Attorney General Eric Holder gave a speech at the Dept. of Justice Black History Month program in which he said that Americans are afraid to talk about race and called upon us to do better. Multiple commentators immediately jumped down his throat.

Thereby proving his point.

The Attorney General made an effort to hold the nation to a higher standard. At the time, we didn’t react with grace or manifest any interest in growing.

Featured comment

Aiza:

IMO the best thing white teachers, or any teachers who find themselves teaching classes of black/brown students can do is to constantly hold their students to the same high standards they would hold their own biological children to. Giving these kids a high standard education is one of the few ways to equip these kids to deal with racism.

# Wherein This Blog Serves Its Original Function

The original inspiration for starting this blog was the following:

I read research articles and other writing on math education (and education more generally) when I can. I had been fantasizing (back in fall 2009) about keeping an annotated bibliography of articles I read, to defeat the feeling that I couldn’t remember what was in them a few months later. However, this is one of those virtuous side projects that I never seemed to get to. I had also met Kate Nowak and Jesse Johnson at a conference that summer, and due to Kate’s inspiration, Jesse had started blogging. The two ideas came together and clicked: I could keep my annotated bibliography as a blog, and then it would be more exciting and motivating.

That’s how I started, but while I’ve occasionally engaged in lengthy explication and analysis of a single piece of writing, this blog has never really been an annotated bibliography. EXCEPT FOR RIGHT THIS VERY SECOND. HA! Take THAT, Mr. Things-Never-Go-According-To-Plan Monster!

“Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks”, by Denisse R. Thompson, Sharon L. Senk, and Gwendolyn J. Johnson, published in the Journal for Research in Mathematics Education, Vol. 43 No. 3, May 2012, pp. 253-295

The authors looked at HS level textbooks from six series (Key Curriculum Press; Core Plus; UCSMP; and divisions of the major publishers Holt, Glencoe, and Prentice-Hall) and analyzed the lessons and problem sets from the point of view of “what are the opportunities to learn about proof?” To keep the project manageable they just looked at Alg. 1, Alg. 2 and Precalc books and focused on the lessons on exponents, logarithms and polynomials.

They cast the net wide, looking for any “proof-related reasoning,” not just actual proofs. For lessons, they were looking for any justification of stated results: either an actual proof, or a specific example that illustrated the method of the general argument, or an opportunity for students to fill in the argument. For exercise sets, they looked at problems that asked students to make or investigate a conjecture or evaluate an argument or find a mistake in an argument in addition to asking students to actually develop an argument.

In spite of this wide net, they found that:

* In the exposition, proof-related reasoning is common but lack of justification is equally common: across the textbook series, 40% of the mathematical assertions about the chosen topics were made without any form of justification;

* In the exercises, proof-related reasoning was exceedingly rare: across the textbook series, less than 6% of exercises involved any proof-related reasoning. Only 3% involved actually making or evaluating an argument.

* Core Plus had the greatest percentage of exercises with opportunities for students to develop an argument (7.5%), and also to engage in proof-related reasoning more generally (14.7%). Glencoe had the least (1.7% and 3.5% respectively). Key Curriculum Press had the greatest percentage of exercises with opportunities for students to make a conjecture (6.0%). Holt had the least (1.2%).

The authors conclude that mainstream curricular materials do not reflect the pride of place given to reasoning and proof in the education research literature and in curricular mandates.

“Expert and Novice Approaches to Reading Mathematical Proofs”, by Matthew Inglis and Lara Alcock, published in the Journal for Research in Mathematics Education, Vol. 43 No. 4, July 2012, pp. 358-390

The authors had groups of undergraduates and research mathematicians read several short, student-work-typed proofs of elementary theorems, and decide if the proofs were valid. They taped the participants’ eye movements to see where their attention was directed.

They found:

* The mathematicians did not have uniform agreement on the validity of the proofs. Some of the proofs had a clear mistake and then the mathematicians did agree, but others were more ambiguous. (The proofs that were used are in an appendix in the article so you can have a look for yourself if you have JSTOR or whatever.) The authors are interested in using this result to challenge the conventional wisdom that mathematicians have a strong shared standard for judging proofs. I am sympathetic to the project of recognizing the way that proof reading depends on context, but found this argument a little irritating. The proofs used by the authors look like student work: the sequence of ideas isn’t being communicated clearly. So it wasn’t the validity of a sequence of ideas that the participants evaluated, it was also the success of an imperfect attempt to communicate that sequence. Maybe this distinction is ultimately unsupportable, but I think it has to be acknowledged in order to give the idea that mathematicians have high levels of agreement about proofs its due. Nobody who espouses this really thinks that mathematicians are likely to agree on what counts as clear communication. Somehow the sequence of ideas has to be separated from the attempt to communicate it if this idea is to be legitimately tested.

* The undergraduates spent a higher percentage of the time looking at the formulas in the proofs and a lower percentage of time looking at the text, as compared with the mathematicians. The authors argue that this is not fully explained by the hypothesis that the students had more trouble processing the formulas, since the undergrads spent only slightly more time total on them. The mathematicians spent substantially more time on the text. The authors speculate that the students were not paying as much attention to the logic of the arguments, and that this pattern accounts for some of the notorious difficulty that students have in determining the validity of proofs.

* The mathematicians moved their focus back and forth between consecutive lines of the proofs more frequently than the undergrads did. The authors suggest that the mathematicians were doing this to try to infer the “implicit warrant” that justified the 2nd line from the 1st.

The authors are also interested in arguing that mathematicians’ introspective descriptions of their proof-validation behavior are not reliable. Their evidence is that previous research (Weber, 2008: “How mathematicians determine if an argument is a valid proof”, JRME 39, pp. 431-459) based on introspective descriptions of mathematicians found that mathematicians begin by reading quickly through a proof to get the overall structure, before going into the details; however, none of the mathematicians in the present study did this according to their eye data. One of them stated that she does this in her informal debrief after the study, but her eye data didn’t indicate that she did it here. Again I’m sympathetic to the project of shaking up conventional wisdom, and there is lots of research in other fields to suggest that experts are not generally expert at describing their expert behavior, and I think it’s great when we (mathematicians or anyone else) have it pointed out to us that we aren’t right about everything. But I don’t feel the authors have quite got the smoking gun they claim to have. As they acknowledge in the study, the proofs they used are all really short. These aren’t the proofs to test the quick-read-thru hypothesis on.

The authors conclude by suggesting that when attempting to teach students how to read proofs, it might be useful to explicitly teach them to mimic the major difference found between novices and experts in the study: in particular, the idea is to teach them to ask themselves if a “warrant” is required to get from one line to the next, to try to come up with one if it is, and then to evaluate it. This idea seems interesting to me, especially in any class where students are expected to read a text containing proofs. (The authors are also calling for research that tests the efficacy of this idea.)

The authors also suggest ways that proof-writing could be changed to make it easier for non-experts to determine validity. They suggest (a) reducing the amount of symbolism to prevent students being distracted by it, and (b) making the between-line warrants more explicit. These ideas strike me as ridiculous. Texts already differ dramatically with respect to (a) and (b), there is no systemic platform from which to influence proof-writing anyway, and in any case as the authors rightly note, there are also costs to both, so the sweet spot in terms of text / symbolism balance isn’t at all clear and neither is the implicit / explicit balance. Maybe I’m being mean.