The four pillars of IBL and my teaching philosophy

I’m up for contract review this year, and I’m also eligible for promotion, so I just completed my teaching narrative. I got real self-indulgent on this thing and ended up writing almost 3500 words (yikes; sorry, committee). There’s one section in particular that I thought would be useful to share.

In July I participated in an AIBL workshop and just enjoyed the hell out of it. Even though I’ve been an active learning devotee for most of my career, and thus have been practicing “big-tent IBL” for a number of years, I learned a ton and developed my confidence to go with a full-on, notes-only, student-presentation style analysis course this semester. I really can’t recommend AIBL workshops enough.

One thing that particularly resonated with me was how the workshop facilitators framed IBL in terms of the presence and interaction of four pillars. As soon as I heard them, I knew that they were going to be a really useful way to explain my teaching philosophy; as soon as I started writing, a lot of my attitudes and heuristics fell neatly into the framework. Maybe this will spark some similar thoughts for you in your own teaching practice and philosophy.

  1. Students engage deeply with coherent and meaningful mathematical tasks.
    • There’s a lot to unpack here. First, in a math class, we need to be learning math, so our tasks must be mathematical. What tasks “count” as mathematical? That is, what tasks honestly reflect the actual practice of working mathematicians? What tasks help students develop honestly mathematical habits of mind? My answers to these questions are always evolving, but in general, I try to focus on understanding over memorization, and on concepts rather than procedures. If there is a computation, I ask students what it means when they’re done.
    • Tasks must be rich enough to support deep engagement. This is another reason why I deemphasize tasks whose sole purpose is computation: a student can turn off their brain when doing such tasks, and therefore they’re not engaging deeply with mathematics while doing it. 
    • Tasks must also hold together coherently across multiple time scales. I try to help students see the connections between tasks they work on during one class session, or on one homework set. I also try to help students see the connections between tasks from September and December. This means that I have to create tasks that honestly support making such connections. One of the most wonderful things about mathematics is how deeply interconnected it is; designing tasks that help students see those connections is a way I can show them the wonder of a subject that sometimes looks quite dry from the outside.
  2. Students collaboratively process mathematical ideas.
    • This is not just a logistical statement about what happens in class on a given day; it is a statement about the general process of learning. To me, this means that if I am not providing time in class on a given day for students to collaboratively process mathematical ideas, then I am not providing them time to learn.
    • Four students can be sitting at the same table working on the same task at the same time without collaborating. So, tasks must be groupworthy; the physical space must support collaboration; and I have to help students learn to work together as equals.
    • I like that this statement is agnostic as to the source of mathematical ideas. It’s okay for me to introduce an idea I want students to think about — as long as I then give them room to process it collaboratively. Excellent teaching means moving responsively along a continuum between telling and discovery
    • This pillar implies a particular kind of caring and openness in the community of the classroom. Ideas are valued and examined, no matter what: whether an idea comes from a student or an instructor, whether it’s complete or a rough draft, whether it’s ultimately correct or incorrect, we work together to process it and learn from it. I work hard with students to negotiate norms and expectations that foster this kind of classroom community.
  3. Instructors inquire into student thinking.
    • First of all, this is my favorite part of my job. 
    • This is also a key part of my job. If I’m out to help students improve their understanding of mathematics, then I must diagnose their current understanding. So, if I am not providing room in class for students to express their thinking, then I have no hope of understanding it, let alone of helping them improve it.
    • Inquiring into student thinking helps students sharpen their thinking. Making students explain their thinking to me (and to other students) helps them see what they understand, solidify their understanding, and identify the precise things they’re still having trouble with.
    • This pillar helps inform my assessment philosophy: Assessment isn’t about giving points, it’s about understanding student thinking. So, I approach students’ work in the same way that I approach a conversation with that student: as an opportunity to understand their thinking and to help them sharpen it.
  4. Instructors foster equity in their design and facilitation choices.
    • I need to design equitable course experiences. For instance, I use open educational resources (OERs) whenever I possibly can, to help lessen the financial burden of education; I incorporate universal design principles to allow students multiple opportunities and multiple pathways to develop and demonstrate competence; and I carefully craft syllabi that are understandable and navigable.
    • I need to facilitate equitable course experiences in the moment. For instance, I help students (especially those from minoritized backgrounds) develop their mathematical identity and power by assigning competence (another reason to inquire into student thinking!); I work with groups to ensure that every student’s voice is heard; and I work seriously with campus disability resource centers to modify courses in order to support students with disabilities.
    • I need to know what equity means in the first place. (Gutierrez’s framework has been influential on my thinking so far.) I need to interrogate and address my own unconscious biases. I need to explore ways to dismantle oppressive systems, even when they are ones that have benefitted me.
    • I’m a proud gay man, and I’m out in the classroom, because I want to help LGBTQ+ students see that there are people like them who have fulfilling professional lives. I didn’t see many people like me when I was in college; more visibility of this sort would have been a great boon to me during my college years.
    • I’ve given a few examples of things that are currently in my toolbox, but it’s extremely important to me to continue to learn new ways to recognize and address inequity in my classroom.

Yesterday was a good day

It’s Friday afternoon, I’m pretty much caught up with grading, nothing is pressing, so it seems like a good time for a moderately self-indulgent blog post. I had a really good day in the classroom yesterday — no, you know what, scratch that, I felt like I kicked ass — and I think it might be useful for me to recap what happened and reflect on why I felt so good about stuff.

The context

I’m currently teaching a summer semester DATA 220. This is our version of your usual intro-level service stats course that’s required by whatever departments across the university. Ours is taught using a nice set of R modules designed for RStudio to handle computations, which frees us up to talk more in class about using your human brain for the process of meaningful inference. Our summer semester is 8 weeks long as opposed to the usual 15 or 16, which means that the course is highly compressed, even though we meet in three-hour blocks rather than the usual two-hour blocks. I’ve got 12 students enrolled in the course, approximately 10 of which are going to show up on any given day. I also have a colleague sitting in on the course just to learn some statistics. Students sit at hexagon tables in groups of two or three, and there’s three big whiteboards at the back of the room that I mostly use as student space.

The plan

The topic for yesterday’s class was t-tests of two means — both paired means and independent means. My plan was to focus on how to decide whether data are paired or independent, then look at an example that would allow us to discuss checking conditions and interpreting a confidence interval. This is pretty representative of the kinds of things I like to spend time on: human brain stuff that you need to think about before interacting with a computer.

Part 1: A student question

At the beginning of class I asked if people had any questions from previous classes. The main question that emerged was in relation to a recent assignment I gave them about interpreting confidence intervals. In particular I asked them, when we say “we’re 95% confident that the true population whatever is between blah and blah”, what do we really mean? Percentages are always of something, so 95% of what? This is a hard question, even for like actual scientists (see 1 2), and the right answer is quite technical*, so students always struggle with this. One of the R modules I linked above contains a pretty nice discussion of the issue — in particular, it gives examples of three or four common misinterpretations of the confidence interval — so I told students to be sure to read that when they were composing their answers.

So, a student asked about this part of the assignment, and this question launched us into a long, profitable discussion about confidence intervals. I decided to bring in another one of the problems on the assignment, which I’ll just screencap here:

Exercise 3.6 from Introductory Statistics with Randomization and Simulation

We read through the discussion in the R module together, and then I asked students to illustrate the misinterpretations in the R module by seeing which of the statements in Exercise 3.6 was similar. (The statements labeled (a) and (c) are incorrect for reasons discussed in the module.) This went really well, and it showed students how useful the discussion at the end of the R module actually was — by necessity, it’s quite a technical discussion, and I think a lot of people read it with their eyes glazed over before we dove into it together in class.

I then called back to another exercise on this assignment: “A survey found that 52% of U.S. Twitter users get at least some of their news from Twitter, with a confidence interval from 45.8% to 58.2%. Does this survey give statistically significant evidence that more than half of U.S. Twitter users get some news from Twitter?” This is an example of a very common way that people will lie with statistics, either intentionally or unintentionally: they’ll report a point estimate without acknowledging a margin of error. I asked students to do a one-minute paper (one of my favorite little active learning techniques!) writing a better sentence reporting the result of this survey: still understandable to a layperson, but more statistically responsible. We then combined ideas to make a good sentence that the whole class agreed on. I don’t remember exactly what we came up with, but something like, “Probably somewhere between 45% and 58% of U.S. Twitter users get some of their news from Twitter.” I do remember that they insisted on the “probably.”

This whole discussion was so great. I really enjoy the affordance that the three-hour block gives me to dig deep into things I didn’t necessarily plan to do. We got to talk about the extremely relevant issue of people lying with statistics, I got to insist on precision in language, we saw the relevance of class resources, we built connections between various problems on an assignment, and we left with everybody feeling more solid in their understanding of confidence intervals. (Also, I felt pretty secure in my own content understanding; this has not been a given in previous statistics classes since I’d never actually taken one before I started teaching them. 🙂 )

Part 2: Paired vs. not-paired activity

So next we moved into the planned portion of class. Before class, I’d given the students some reading from the book about paired vs. independent samples. However, I follow our textbook in hating the term “independent samples” because of the immediate conflation with “independent” as in “not associated,” so I called them “paired” and “not-paired” data. In the reading quiz (on Canvas and due one hour before class), I asked students to take a stab at defining paired vs. not-paired data, and give examples of both.

The first thing we did in this segment of class was another one-minute paper where I had them recall in a sentence or two their answers to the reading quiz, and then compare their sentences with the other people sitting at their table. After they’d had some time to discuss, we talked as a whole class about this, and got some words out, including “correspondence” and “groups,” that would be useful later. I told them that my point with this activity was that it’s hard to give a definition in words, and I wanted to introduce another tool, which is to look at the shape of the dataframe:

Paired and not-paired data frames. Tables from ISRS.

Looking at the dataframe, it’s much easier to see the difference: in paired data, each case has values of two numerical variables; in not-paired data, each case just has one numerical variable (that we’re interested in), but the cases fall into two groups.

I then rehashed an activity I’d used in a previous semester: I asked each group to pick their favorite examples of paired and not-paired data someone came up with on the reading quiz, draw a dataframe for their example on one of the back boards, and use their drawing to explain why the data are paired vs. not-paired. This was a fun activity. People got super into it. There was a lot of argument within each group about what the variables should be called and what each case was (which was exactly my point: in order to decide whether the data is paired or not paired, you need to think hard about what a case is, and what the variables are). Eventually, each of the three groups had a couple of dataframes drawn up.

At this point, I suddenly thought about an activity I’d done in a different course: a mini poster session. Kinda making this up on the fly, I told students that someone from each group besides the person who had the marker while drawing the dataframes had to stay and be the presenter, and then the other members of the group would go and visit the other groups. Not to toot my own horn too much here, but honk honk, this was awesome. I was really happy with how this activity forced everyone in the group to be accountable for the group’s work. As soon as I described what was happening, I saw a couple of students who had previously been less-involved get nominated to be the presenters, and then start asking their groupmates really seriously about how to explain their examples.

After the poster session, I had them look at an exercise that’s going to be on this week’s homework:

Exercise 4.14 from ISRS

I assigned each group to one of these three scenarios. The group chat went pretty quick, but the class discussion afterwards was really interesting: group (a) talked about how, depending on how you interpret the sentence, their scenario could be either paired or not-paired. This made me so happy to see that my students were comfortable with ambiguity and weren’t pushing for “but what’s the riiight answer???”.

Part 3: Sheep heart attacks example

The next thing on our docket was an example: data from an experiment testing the use of an embryonic stem cell treatment on 18 sheep that had a heart attack. (I didn’t previously know that sheep could have heart attacks, but I guess they have hearts, so sure, why not.) We had a quick discussion about whether the data would be paired or not-paired, and then moved into checking conditions for using the t-test.

One of the conditions as we’ve set them up is to ensure that the sample is less than 10% of the population. (This is part of checking independence of observations.) At this point a student said, “I’m not sure about this one, because, what’s the population? Is it just sheep in this farm, or what?”

I love this question because it is absolutely a human brain question. This is absolutely the kind of question I want my students asking. This question opened a door for us to have a really good discussion about the relationship between sample and population: what do you, with your human brain, think is a reasonable broader population for your particular sample to generalize to? We threw out a bunch of ideas (sheep in this barn, sheep on this farm, sheep in this city, sheep in this state, sheep in the US, sheep in the entire world, whatever particular breed of sheep, etc. etc.), and then I had them rate how comfortable they felt generalizing our sample of 18 sheep to these various different populations. I concluded this bit by saying, my point is that there is no right answer to this question (to which a student said, “Well, why didn’t you just say that at the beginning?!”) — there is only a right thought process.

We wrapped up this example by looking at a confidence interval (that I got from RStudio and am not particularly interested in the computational details, thanks) and using it to answer our important real-world question: did the treatment work?

So much great stuff happened in this segment. A student felt comfortable asking a great question that spurred us into an important discussion about a human brain part of statistics. We were able to tie back to previous understandings of confidence intervals. Students made good real-world conclusions and backed them up with meaningful statistical evidence.

Part 4: Work time

We finished up with some time for students to work on their term group projects. I wandered around and helped answer questions that came up. This was also super fun. Everybody was very engaged in their own little world of personally-interesting data. Even when class time was officially up, and I packed up to go home, two students were still working together on an interesting thing they’d found. Nothing warms my grubby little soul more than students getting really interested in something and ignoring the time. 🙂

Some umbrella thoughts

Beyond all the individual blow-by-blow reasons, I had some sort of overarching thoughts while reflecting on why I felt like this class went so well. Really, they boil down to that I felt capable. Because I now have a good toolbox of active learning techniques that I’ve developed over the last few years, I was able to deploy interesting activities at useful times. I remixed some old stuff and threw in some new stuff and improvised like crazy. Also, because this is now the third time I’ve taught DATA 220, I feel pretty solid in my content knowledge, even for tricky technical points. I feel like I have a reasonable handle (that will only get better) on what things students will find confusing, and on what the high-leverage ideas are for breaking confusion. What’s more, I now have a depth of experience that allows me to identify that a class went super well, and the reflective tools to think critically about why.

Ultimately, I’m writing this blog post to a future myself, because I know there’s going to be a future day where I feel way less capable than I did about this class session, and this will be a good reminder for me:

Some days, I really love my job. 🙂

* Were we to take a whole bunch of random samples, and correctly construct a 95% confidence interval from each one, then we could expect approximately 95% of those confidence intervals to capture the true population parameter. (See, that’s tricky.)

Make your own whiteboards for collaboration

Okay, so, you may have seen pictures or videos of people doing cool stuff with medium-size whiteboards in their classes, and by medium-size I mean not the big one that’s on the wall, nor the tiny paper-size ones, but the ones that are a couple of feet square, big enough for a couple of people to simultaneously write on, but small enough to fit on a table or at least balance on a couple of desks. Here’s some of my students doing concept maps on these whiteboards at the end of Calculus II:

I like the hell out of these things because they are collaborative spaces. You don’t even have to tell people that they’re supposed to work together on this thing: the physical size of the board just kinda implies that it’s the obvious and natural thing to do. Students almost can’t help but work together when the space looks like this. If there’s one thing I love, it’s tricking students into doing something cool in such a way that it’s not evident that it was a trick. 🙂

The basics

So where do you get these things? Turns out that it’s super cheap and easy to make yourself a classroom set of these out of tileboard. You can buy a 4′ x 8′ sheet of tileboard at your local home improvement warehouse store, whether it be the orange one or the blue one, for about $15.

4′ x 8′ is big. You’ll need to cut it down to a more manageable size. I suggest that you cut it into 6 pieces by cutting the 4′ side in half and the 8′ side in three — then each of the resulting pieces will measure (about) 24″ x 32″. (Of course you could also make eight 2′ x 2′ squares.) Since you probably don’t have your own table saw, you will probably want to have this done for you at the home improvement warehouse store. Here’s a big ol’ blog post about how you can do this.

That’s it! You’re done! You now have a bunch of collaborative-size whiteboards, yay!

What’s that you say? You want some bonus pro tips for getting even fancier?! Well YOU’RE IN LUCK, because here’s some

Bonus pro tips!

So, now you have whiteboards, but a whiteboard without markers is sad. Get yourself a bunch of cheap food storage containers. Pack each one with five or six markers in different colors and a small eraser or two. These containers are awesome because they stack in each other pretty well and it makes it way easier to hand each group of students a little pre-packaged bucket of markers.

Here’s my favorite whiteboard-related pro tip that works just as well on your big wall-mounted unit as it does on these collaborative ones. The big problem with whiteboards is when they get all smeary and gross and make everything illegible. If this happens to you, clean the whiteboards really well with some of that Expo cleaner or Windex or whatever, and then bust out the secret weapon: Turtle Wax. Smear a thin layer of this magical paste on with a microfiber cloth, rub it in real good, let it dry for 24 hours, and then buff it back off with a (clean!) microfiber cloth. Your markers will now erase like an absolute dream. Bonus points for doing this when your whiteboards are brand new and haven’t had time to get all stained and gross yet.

We can create positive student ratings of teaching. Here’s (maybe) how.

Notable academic Twitter person Terry McGlynn recently wrote a blog post arguing for changes to the structure of student ratings of teaching (SRTs) that would make them more useful. While I entirely agree with the premise that SRTs as currently constructed aren’t particularly useful, I’m not sure I’m on board with the rest of the argument. In particular, I got to wondering along two parallel tracks:

  • Do I believe that Terry’s proposed changes will produce a more useful set of data for the evaluation of teaching effectiveness?
  • If indeed I do believe that this data is more useful, useful for who?

Some background and positioning

It’s important to me to start with some background here so that it’s clear who I am and where I’m coming from. I’m relatively new faculty — this is my fifth year out of my Ph.D. in mathematics education — in my first year at a new job in the math department at Westminster College. (Standard disclaimer that this blog is my opinion and not theirs etc. etc.) I’m an out gay man, and I think it’s important for my students to see examples of successful Gays In STEM™, so I don’t hesitate to talk about my husband in any situation where a spouse might come up in conversation.

I had a particularly bad experience with SRTs two years ago, about which I will probably blog eventually, but suffice it to say that I experienced a significant instance of homophobic bias. This caused me to dig into the (substantial!) body of research on SRTs; I’ve not yet found much redeeming about them. We know that quantitative scores exhibit statistically significant biases along the lines of race and gender, but also along disciplinary lines and lines of seniority. What’s more, we also know that SRTs are not correlated with measures of student learning, but that they are correlated with the availability of cookies. Universities and organizations including Ryerson, Oregon, USC, and even the AAUP have come to the conclusion that they can’t legally be used in tenure and promotion decisions (because if a professor from an underrepresented background was denied tenure on the basis of SRTs, they could quite reasonably sue the pants off their university for using racially-biased data). In short, student ratings of teaching are a dumpster fire.

You may have noticed by now that I’m not using the more common term “student evaluations of teaching,” and maybe you’re wondering if that’s on purpose. Reader, it is: calling these things “evaluations” ascribes to students particular skills and bodies of knowledge that they by definition don’t have. Students are not trained evaluators of teaching, and so we can’t reasonably call any data they produce evaluations. However, I think it’s fair to call this data ratings or feedback.

I’m not asking for the complete abolition of SRTs; I do think it’s important to give students a voice. However, I’m asking for us to think harder about what this data is and isn’t good for, what we can and can’t meaningfully conclude from it, and if we can’t come up with a better way of hearing student voices, for some meaning of the word “better”.

Will Terry’s changes produce more useful data?

So that leads naturally to my first question. Certainly, one way we could operationalize “better” is as meaning “more useful.” Terry argues that a particular set of changes would produce useful SRTs. Let’s take a look.

Terry proposes that we should ask students “unambiguous questions that reflect explicit performance criteria.” For instance, we might ask, “Was the instructor late to class on a regular basis?” or “Did the instructor use disparaging language about a student in the class?” or “Was your instructor present at posted office hours?” These kinds of questions do seem like an improvement. With apologies to John Hodgman, specificity is the soul of data, and at least these questions are way more specific than the kind of open-ended “how did this class go” questions that too often populate SRTs.

My question, though, is whether these questions are specific about the right things. Terry is arguing for questions assessing whether “instructors are meeting baseline performance criteria.” Which, sure. Let’s detect the “derelict tenured professor who fails to do their job at the minimum level expected of them.” But how useful is this really? How many “derelict” people are we truly going to detect? And for (I’m willing to venture) the vast majority of instructors who at least minimally care about their teaching responsibilities, what do we learn from this data?

Useful for who?

It doesn’t seem to me like these questions are useful for anybody but administrators looking to detect the worst offenders. Without getting too Marxist about this: as labor, I’m in general not stoked about handing more punitive tools to capital. I don’t know what kind of positive changes most instructors can make based on the responses to questions like the ones Terry is suggesting.

Can’t we make something that’s positive? Something that’s useful to instructors? For the vast majority of us who at least minimally care about our teaching responsibilities, can’t we make something that helps us actually improve our teaching skills?

A lot of people argue that SRTs as presently constituted do help them improve. When I talk to people about how SRTs are a dumpster fire, I often hear this: “Well, I’ve really learned a lot from reading my SRTs. I used to do [X] but then students gave me the idea to do [Y] instead and I started doing it and it went really well!” Which, fine, this happens. I’ve gotten good ideas from SRTs myself. But here’s a question I can’t help but ask whenever I hear this story: is [Y] a good thing to do, or is it just a popular thing to do? Was [X] a bad thing to do, or was it just unpopular? Too often, students’ perceptions of what’s good for their learning are diametrically opposed to what science tells us is actually good for their learning (another reason we should be deeply skeptical of student-produced data of teaching effectiveness).

What if, instead, we asked students about the incidence of specific evidence-based practices? Or, heck, since we’re skeptical about students as evaluators for all the very good reasons discussed above, why don’t we ask instructors themselves?

This is precisely the approach taken by the fine people at the Carl Wieman Science Education Initiative, who have developed the excellent Teaching Practices Inventory. This is a shortish (10-15 minutes) self-reflection that instructors can complete at the end of the term. It’s a structured way for instructors to think hard about what they do in their classrooms, and to think forward about how they can incorporate more evidence-based practices in their teaching. Speaking for myself, I’ve learned way more, and made way more substantial changes to my teaching, from this kind of structured reflection than I ever will from students complaining about the same three good-for-them-but-unpopular things on my SRTs for the rest of eternity.

Don’t worry, we’re not going to leave the students out of the fun. CWSEI researchers have also developed really good student surveys that get students to report the incidence of evidence-based practices — and thus also just maybe get students to see that there’s a gulf between evidence-based practices and their preconceptions about good education. And students’ perceptions of their learning experiences are important data — if instructors think they’re implementing evidence-based practices, but students don’t see them, then that’s a good sign for the instructor to rethink their implementation.

I don’t think these instruments are silver bullets. I fully expect that students’ implicit biases will continue to manifest in any quantitative instrument we ask them to fill out, and so we really need to keep thinking hard about how to incorporate this data into more holistic evaluation of teaching effectiveness. And self-report instruments are of course subject to people manipulating them, or not taking them seriously. But, dang, isn’t this at least a good start? We can create a system of student ratings of teaching that is positive, and that helps instructors teach better. And if that’s not manifestly the purpose of SRTs, then what (or who) are they actually for?

Let’s learn Lagrange multipliers by hiking

So here’s the problem: You want to optimize some function f(x, y) subject to the constraint g(x, y) = k. The method of Lagrange multipliers says that you can do this by solving the equations \nabla f = \lambda \cdot \nabla g. But wait, why?

Here’s how I like to think about this. You’re hiking on Mount Glass, a weird transparent mountain whose elevation above the “floor” is given by f(x, y). Down on the “floor”, someone has painted a path given by the constraint g(x, y) = k. You can look down through the mountain (because it’s transparent) to the “floor” below to see where the path is drawn, and you can’t leave the path — your shadow always has to land right on the path painted on the floor. (It’s noon, so your shadow just goes straight down.)

In this weird scenario, what’s the highest you get?

To answer this question, we’re going to exploit one of my favorite facts about the gradient vector: it points in the direction of fastest increase. In this context, that means that \nabla f is the vector that points straight up the hill from where you are standing right now. (Note that it does not necessarily point at the peak of Mount Glass!)

We’re simultaneously going to exploit my other favorite fact about the gradient vector: it’s perpendicular to level curves. (These two facts are equivalent.) In particular, it’ll be useful to us to note that \nabla g is always perpendicular to our path, because our path g(x, y) = k is nothing but a level curve of the function z = g(x, y). (Note that this is a different function from the one that describes the height of Mount Glass!)

For the sake of intuition, let’s make some assumptions: let’s say that g(x, y) = k is a simple closed curve which we’re traversing clockwise, and that \nabla g always points in (that is, to your right as you walk around the path).

Okay, off we go on our hike. Hold your right arm straight out from your body. If you look at the shadow of your arm, you can see that it’s perpendicular to the path painted on the floor. Thus, per our assumptions, this is \nabla g.

Pretend you’re on an uphill portion of the path. Here, you can increase your elevation by just moving forward. This means that “straight up the hill” is going to be a little bit forward — in particular, your arm isn’t pointing exactly that way.

Now pretend you’re on a downhill portion of the path. Here, you can increase your elevation by walking backward. Again, this means your arm isn’t pointing “straight up the hill” — that would be a little bit behind you.

Compare this to the situation where you’re at a high spot on the path. You can’t get higher by going forward *or* backward. If you wanted to get higher, you’d have to step off the path, and you’re not allowed to do that — “straight up the hill” is in precisely the wrong direction for you to do anything with it. Now look at your arm — it *is* pointing straight up the hill. (Or maybe straight down. Same deal.)

Now let’s bring this home by putting some math on it. “Straight up the hill” is precisely \nabla f; again, your arm is pointing in the direction of \nabla g. At a high spot, your arm and “straight up the hill” are in the same direction. That is, \nabla f and \nabla g are parallel. A great way to think about when two vectors are parallel is that one of them has to be a scalar multiple of the other — in other words, \nabla f = \lambda\cdot\nabla g.


Some words about clickers

Recently a friend of mine sent an email to a list I’m on, asking for advice for teaching a 200-student class in some kind of non-boring way. In particular, she asked about clickers, and since I’ve used them before, I wrote some words back (at like 6am in Germany because of jet lag; I’m here for ICME). I decided to edit those words into this blog post for more people to read.

A few years back, I taught a 200-ish person calculus for life sciences class (Math 121/122 at SDSU). I used peer instruction with clickers and it was great; there’s no other way I could have gotten the level of engagement I wanted out of a class of this size. I’ve also done a fair amount of training / support for instructors who want to try peer instruction with clickers for the first time.

NB I’m saying “peer instruction with clickers;” think about this phrase like one of those big German compound nouns that are all mashed together. Maybe I’ll abbreviate this as “PIwC.” Clickers are just a platform, and you have to have a pedagogy in mind to run on said platform. (In particular, if you’re just going to use clickers as a way to ensure attendance, students will chafe.)

In case you haven’t heard of it before, peer instruction (PI) is a clicker-based pedagogy invented by Eric Mazur, a Harvard physicist. The idea is that your clicker questions are mostly conceptual (Mazur calls them ConcepTests), and you run them in a four-step process:

  • first an individual vote, which makes students commit to one particular answer (an important metacognitive skill),
  • then some time for small-group discussion with your neighbors during which you’re supposed to come to a consensus, which makes students verbalize, listen to, and critique each others’ reasoning (an important metacognitive skill),
  • then a second group vote, where you’re supposed to vote the same as your neighbors,
  • and then some whole-class discussion, where you can again make your students exercise their verbalizing, listening, and critiquing skills.

NB that one of the big upsides of this process is the practice and (hopefully) development of metacognitive skills. This is something that I think a lot of classroom practices aren’t particularly strong in, which makes PIwC a valuable addition to your toolbox.

Within this broad framework, there’s room for a lot of interesting variation. This is another thing I really like about PIwC – even though it looks pretty rigid, there’s lots of room for you to tweak what you’re doing on a per-course, per-class, or even per-question basis in service of whatever your particular aims are. Here’s some of my favorite tweaks:

  • Most clicker software will display a histogram of student responses, and you can choose to show students the histogram after the first vote or wait until after the second. Most often, I show the histogram right after the first vote. I do think you should show the histogram at some point, for student self-evaluation purposes. (Oh look, more metacognitive skills.)
  • There’s lots of fun questions you can ask to provoke debate (and more metacognition 🙂 ) in students’ groups:
    • “Ooh, interesting, big split between A and C, why do you think that is?”
    • “Seems like everyone is pretty convinced B and D are both wrong; how do you know?”
    • “Why might a reasonable person think the answer is B?” (NB that there are probably good reasons for B, given that you carefully chose this question to have reasonable distractors.)
    • “What would the question have to be in order for the answer to be D?”
    • “How can we decide between A and C?”
    • “Remember that we just decided that X is true; what would that mean for this question?”
  • It’s often worthwhile to have a little whole-class discussion before the re-vote.
  • Wander around the room during the group discussion and try to overhear interesting things, then ask if you can pick on them for the whole class discussion.
    • “Alice said something interesting to her group that I asked her to share. Go ahead, Alice.”
  • I told you above that the questions are usually conceptually-focused. Maybe use some moderately procedural questions, but I’d lean away from anything that’s just number-crunching. There’s a happy middle ground of questions in which conceptual misapprehensions will reveal themselves in procedural mistakes; I’m sure you can think of many of these in whatever class you’re teaching.
  • I’ve even done a thing where the right answer was not given as an option in the original vote, to provoke some productive (and amusing) frustration. 🙂

It’s also worth noting that there are probably, floating around on the internets, good banks of ConcepTests for whatever your class is. Also check your textbook; many publishers provide ConcepTests or something like them in their online supplemental materials. So this is nice because you don’t have to start from scratch, and even if you don’t like the questions you find, you can use them as templates or inspirations for developing your own.

Of course you will have people who will disengage during the discussion process, or who will have off-topic conversations with their friends. Wandering the room will control this to some extent, but there’s just nothing you can really do to ensure that all 200 students are on task all the time. It’s the real drawback of large lecture classes. Womp womp. Don’t pull out your hair trying to control this behavior; it’s not really worth it. Whatever minimal gains you make in terms of time-on-task will generally, in my opinion, be canceled out by equivalent gains in student grumpiness. It’s an easy way for you to be perceived as autocratic and heavy-handed, and that carries a real danger of damaging your classroom community.

So those are some tips that I think will help you get started. Please feel free to ask any other questions you might have, and I’ll do my best to answer them.

[RUME 2016] Borrowing from linguistics: Metonymy and linear transformations

Here’s part 2 of my “borrowing from linguistics” series.

So like I said in yesterday’s post, Rina Zazkis’s talk grabbed my attention because of some previous work I’ve done. (Some day I will bother to make my own personal website and link to things there, but that day is not today.) My coauthors on that paper, Chris Rasmussen and Michelle Zandieh, had conducted these interesting interviews on students’ construal of similarity between linear transformations and functions like from high-school algebra. These interviews were wide-ranging, including such topics as injectivity, surjectivity, invertibility, and composition. I focused in on the bits about invertibility and composing a function / linear transformation with its inverse.

One of the tasks from those interviews was to predict what you’d get when you compose a function with its inverse, and then to predict what you’d get when you compose a linear transformation with its inverse. When I started digging into this data, I noticed something unexpected: all ten students initially said you should get 1 in the function case — i.e., that f(f^{-1}(x)) should be 1. This seemed weird, and it drove the investigation that turned into this paper: why did everyone make this incorrect prediction??

One of the things that ended up being salient was that many of these students predicted that a linear transformation composed with its inverse should be the identity matrix (i.e., that \mathbf{T}(\mathbf{T}^{-1}(\mathbf{x})) = \mathbf{I}). This seems less weird and more understandable. The place where this comes from, we ended up arguing in the paper, is a metonymy.

Let’s digress a bit and recall what metonymy is. For our purposes, metonymy is a literary device whereby a thing is called not by its name, but by the name of one of its parts*, or by the name of something that is associated with it. For instance, you can use “Hollywood” to refer to the American movie industry, or “the press” (i.e., the printing press) to refer broadly to the journalistic endeavor. (These examples are gratuitously stolen from the Wikipedia page).

What’s the metonymy that’s going on here? It’s actually very mathematically sophisticated: Every linear transformation from \mathbf{R}^n to \mathbf{R}^m can be represented as a particular m \times n matrix. (What’s more, every matrix represents a linear transformation, so that Hom(\mathbf{R}^n\mathbf{R}^m) is in fact isomorphic (as a vector space) to \mathbf{M}_{m\times n}! This isomorphism is canonical as long as you’re working in the standard bases for \mathbf{R}^n and \mathbf{R}^m.) We make use of this fact all the time: we usually write \mathbf{T}(\mathbf{x}) = \mathbf{A}\cdot\mathbf{x} whenever we’re discussing some transformation.

Now here’s the metonymy: we tend to speak of the matrix \mathbf{A}, which represents the transformation, as the transformation — for instance, it’s not unusual to say that \mathbf{A} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} is a CCW rotation by an angle \theta. However, it would be more accurate to say that \mathbf{A} represents such a transformation. For mathematically sophisticated people, this little bit of playing fast and loose is not really problematic, and in fact it enables fluency that is a mark of mathematical sophistication. (If I need to calculate the composition of two transformations, I can quickly and fluently pass into thinking about multiplying the two matrices, and then quickly and fluently pass back into thinking about the resulting matrix as representing the resulting transformation.) However, for our students, it can cause trouble, as I’ll detail below.

Let’s think about composing a transformation with its inverse: \mathbf{T}(\mathbf{T^{-1}}(\mathbf{x})) = \mathbf{A}\cdot(\mathbf{A}^{-1}\cdot\mathbf{x}) = \mathbf{I}\cdot\mathbf{x} = \mathbf{x}. (Note that this calculation sort of implicitly invoked the isomorphism discussed above.) Not so bad, right?

But now let’s really buy into this metonymy and try that again:

\mathbf{T}(\mathbf{T^{-1}}(\mathbf{x})) \leadsto \mathbf{A} \cdot \mathbf{A}^{-1} = \mathbf{I}.

Hm. That’s slightly wrong. And, as I’m sure any of you who teach linear algebra will immediately recognize, lots of students do this. It’s not a big deal as long as you remember to “un-metonymize” and stick that \mathbf{x} back on the end, and mathematically sophisticated people do this fluently. However, if you’re not super aware that this is a metonymy, and you just think that the matrix is the transformation, then this could lead to problems.

In particular, it could lead you to say something like, “if I compose a linear transformation with its inverse, I ought to get the identity matrix,” which is indeed something that a lot of people in our interviews said. And again, this is wrong, but maybe only slightly wrong. If you compose a linear transformation with its inverse, you get a transformation that is represented by the identity matrix, i.e., the identity transformation \mathbf{T}(\mathbf{x}) = \mathbf{x}.

So I think this is another example of a productive borrowing from linguistics: the word “metonymy” is probably a good one to have in our conceptual toolboxes when we’re teaching linear algebra. I’m not going to say that telling students about this whole metonymy thing is going to 100% keep them from making this mistake, but I think it might help. At the very least, it gives us a conceptual label for this tricky subject that doesn’t require an understanding of isomorphisms of vector spaces.

Anyway, with this train of thought in mind, it’s sorta unsurprising that people say that they should get 1 if they compose a function with its inverse. 1 is kinda like \mathbf{I} (in that they are both the multiplicative identity in their respective rings), so maybe this new “fact” that comes from thinking metonymously in linear algebra is influencing an old thing they used to know — which leads us to backward transfer, the subject of a future blog post!


* Some sticklers will probably here insist that a part-whole relationship is a synecdoche rather than a metonymy. I prefer to consider synecdoche a particular kind of metonymy. If this bothers you, then feel free to mentally alter my terms; I don’t think it will impact the argument.

[RUME 2016] Borrowing from linguistics: Rina Zazkis and the superscript -1

So, to begin with, let me introduce this sequence of blog posts tagged [RUME 2016]. I was recently awarded a travel grant to the ICME conference coming up in Germany. Part of the requirements for this award is that I’m supposed to help disseminate stuff I learn at that conference, so I decided to practice / build readership of this blog by blogging about RUME talks I found particularly fun and interesting. I’ve got basically three blog post topics (which may span multiple posts) in my brain, with a fourth that may come later.

The first talk I’d like to blog about is Rina Zaskis’s talk “On symbols, reciprocals, and inverse functions.” (Igor’ Kontorovich is a coauthor on this talk but could not make it because Auckland is v. far away.) This talk immediately grabbed my attention because I wrote a paper some time ago about students’ construal of similarity between functions and linear transformations, and in particular, the ways they think about the inverses of each of these things. More on this later (in a forthcoming post or two).

Dr Zazkis gave students a scripting task, in which she asked students (pre-service secondary teachers) to extend this imaginary (but completely plausible!) interaction:

T: So today we will continue our exploration of how to find an inverse function for a given function.

S: So you said yesterday that f^{-1} stands for an inverse function.

T: This is correct.

S: But we learned that this power (-1) means 1 over, that is, 5^{-1} = \frac{1}{5}, right?

T: Right.

S: So is this the same symbol, or what?

T: …

(I think this is such a brilliant task.)

So here’s three ways you could sensibly answer this question:

  1. The group theory approach: In every group, every element g has an inverse g^{-1} such that g * g^{-1} = e, the identity element. So then these are totally the same thing; the only difference is what’s the group and what’s the operation (the group of rational numbers under multiplication vs the group of all functions under composition). Unfortunately, it’s probably not the best idea to drop some group theory on some high-school students, so we should probably explore other approaches.
  2. The context-dependent approach: The common symbol \Box^{-1} means different things depending on what’s in the box (e.g., a function vs. a number). This smacks of rule-based thinking and obscures the legitimate connection between inverses, so I don’t like it twice over.
  3. The middle-ground approach: The common symbol \Box^{-1} means slightly different things depending on what’s in the box, but there is a relation between these slightly different meanings. I’m calling this the middle-ground approach because it seems to bring out this relationship without invoking all the machinery of group theory. This is probably how I would choose to answer the question, should it come up; I’d probably talk in some amount of detail about how we could consider both things instances of some more generic idea of inverses. I think we’d all pretty much agree that this is a better way of explaining the relationship, even though it may be difficult right now to articulate why.

But Spencer, you’re thinking, the title of this blog post is something about linguistics, and this is just a bunch of math. You’re right; now it’s time to borrow some linguistics words to give better descriptions to ways #2 and #3 above.

The words Dr Zazkis chose to borrow were homonymy and polysemy. These are good words that do exactly the work that we’d like them to do. Here are some definitions I synthesized from various google results:

Homonymy: the relation between words with identical forms and sounds but different and unrelated meanings. Example: “river bank” vs. “savings bank” vs. “bank shot” vs. “bank of interview questions.” (Yikes!)

Polysemy: the relation between words with identical forms and sounds but multiple, contiguous meanings. These meanings emanate from a central origin, and they form a network such that understanding any one meaning contributes to understanding any other meaning. (The wiki page on polysemy is v interesting.)

It’s probably clear where this is going now: way #2 above is understanding the different \Box^{-1}s as homonymous, and way #3 is understanding them as polysemous. This fits so super well: we could certainly call 5^{-1} “five-inverse” just like we call f^{-1} “f-inverse,” and there is absolutely a central origin for all these words (i.e., the group-theoretic construct of inverses).

What I really like about this, pedagogically speaking, is the network-y bits of the definition of polysemy I gave above: understanding one kind of inverse will help you understand another. Calling both 5^{-1} and f^{-1} “inverses” helps us recognize and talk about both the similarities (in both cases, the one thing “undoes” the other thing) and the differences (the operations are different) between the two cases. What’s more, I think this is precisely the way in which way #3 feels better than way #2. Look how much mileage we got by borrowing some ideas from linguistics!

I’ve got two more post ideas lined up that build on this idea of borrowing interesting things from linguistics. The first borrows the idea of metonymy to talk about inverses of linear transformations; the second borrows the idea of backward transfer (from people who study second-language learning).

I’ll close this post with this lovely quote Rina Zazkis presented, from Henri Poincaré:

Mathematics is the art of giving the same name to different things.

What I did when my grading load became completely unmanageable (because of course it did)

So, uh, then this happened, about 6 or 7 weeks into the semester:

I’m not gonna sugarcoat this one: please pardon my French when I say that I got fuckin’ buried in rewrites. I was getting super burned out with all the grading I had to do (in addition to, y’know, all the other normal teaching stuff) and it was having a legitimate negative impact on how I felt about my job. “Unsustainable” is a gross understatement.

I got some good advice from people which you can see in the @replies to that tweet. In the end, I came up with a couple of ideas that I thought I could implement mid-semester without inciting a massive student revolt complete with torches and pitchforks. I decided to survey my students to see what they thought of the ideas I came up with, and to see if they had any ideas that could work better than whatever I thought of. I also threw in a couple of questions to see how they felt about the labs and my grading scheme in general.

The responses I got were actually pretty insightful, and I took to Twitter to get some help thinking about them. Here’s the first tweet of my 15-tweet rampage, if you want to follow along and see the (really good) feedback I got. I’ll highlight a couple of tweets in particular. A student came up with a really good idea that I liked more the more I thought about it:

(Ignore the tweet labeled (5/n). I don’t know how to get it to not show up. What you should be looking at is the (6/n) one.)

This is the idea I ended up going with. I like this idea for three reasons in particular:

  1. Pragmatically, I’m pretty sure it’s going to reduce my grading workload, which was kinda the point. Certainly it’s going to add to my workload to create solutions, but (a) I’d rather create one solution key than write the same comment on 15-20 student writeups, and (b) because I’m using the CLEAR Calculus labs, some solutions are already available and will just need slight modifications.
  2. It allows me another opportunity to model expert thinking and expert writing in mathematics, and in a form that students are going to be motivated to pay attention to.
  3. It makes students practice metacognition by examining and explaining what they did wrong, and that’s going to be super beneficial for their learning. (And actually, this has been a reasonably easy sell, because we already did a couple of exam rewrites this way; I told students, hey, I think you learned a lot from these rewrites, don’t you?, and they mostly agreed.)

I’ll also add that I’m open to further rewrites after the second one, but I’m going to make students meet with me before a third resubmission, because if they’re still getting something majorly wrong after looking at the answer key, there’s a big problem that I need to help them solve.

So — on to solution writing, and I’ll update as events warrant.