Other uses for tokens

April 15, 2019

People at my school have been talking about Specifications Grading lately. One aspect of it is the idea of creating a token economy to create an artificial scarcity on resubmissions (this is both to lessen the instructor’s grading load and to give the student an incentive to give an honest effort on resubmissions).

Tokens might be a solution to a problem: philosophically, I want a student’s grade to be based on the mathematics they know. However, there are behaviors that correlate to success, and it would be nice to give an incentive to do these things (attendance and participation are two common examples). Instead of giving grades for things like attendance, perhaps we could give tokens to reinforce “good” behaviors?

We have been thinking about ways that students can “mine” tokens. Here are some examples:

  1. Students can get one token for finding a mistake in your course notes/solutions/whatever.
  2. In an IBL classroom, there may be a problem that is unsolved for a while. You can place a “bounty” on that problem by offering a token to any student who successfully presents a solution.
  3. Students can get a token if they send a picture of themselves working in a study group outside of class.
  4. Students can get a token for posing a particularly interesting conjecture.

If you have enough of these, you might even start with an artificially low number of tokens to require them to do these things.

Of course, there are drawbacks to this. Starting with a low number of tokens would likely disadvantage less organized students who might need the most help with resubmissions. I suppose that I should also say the words “intrinsic” and “extrinsic” here, too.


  1. What other ways could students mine tokens?
  2. What other disadvantages does mining tokens have?

Our Value

April 8, 2019

Robert Campbell and I were discussing why/if we are valuable to the students, and how we could maximize our value to the students (we are probably really talking mainly about nonmajors here). We started with the agreement that almost all students are not going to remember the details of the mathematics from our courses for more than a few weeks after the final exam. Calculus students might retain some broad notations of what derivatives and integrals mean (or not), but most students do not seem to remember the antiderivative of 1/x.

This is disputable. For instance, I am certain that my students know a lot more mathematics—including the details of things akin to the antiderivative of 1/x—than a seventh grader. So at the very least, there is probably some cumulative effect. However, let’s take this as an axiom: students do not remember details from our course.

Given that, how valuable are we? I teach about 50 students a semester right now, and I am guessing that I only make a large difference in 2–3 of those students lives. That is not bad, but it also suggests that my courses might not be working too efficiently. Is there a better way of doing this? If my goal is to positively affect students’ lives (in a mathematical way, broadly defined), then I have a much better success rate with advising and undergraduate research.

For instance (this is only a thought experiment), perhaps I should spend almost no time planning and grading for my courses. I can just follow the textbook, cook up a lecture for each class day (which does not take very long for 100-level courses), and just grade three exams over the course of the semester. This would cut down on my planning significantly. Then I could put the time saved into high-impact activities like advising and engaging in student research.

This is not something that I think is a good idea, and it is certainly not something I am planning on doing (I enjoy course planning too much, at the very least).

So this is something I have been thinking about. I suppose it boils down to a couple of questions.

  1. What is the real purpose of my job?
  2. What parts of my job make real progress in achieving the real purpose?
  3. What parts of my job do not make much progress in achieving the real purpose?

Once these questions are answered, how do I spend my days to best do my job? I suppose that this is similar to the idea that IBL can give students a transformative experience. In an extreme case, IBL might give students a transformative experience without teaching them much content. Is this a win? I was fortunate enough to be able to have lunch with David Walmsley this weekend, and he said he heard of an example where a real analysis professor had the students discover the students discover (with very little guidance) the definition of limit. These students owned the notation of limit when they were done. But it took half of the semester to define limit. Is this a win? I think the answer is probably “yes,” but I am not certain.

Your job is to answer the questions about in the comments. I thank you in advance for justifying the existence of my job.

More on “Finishing Times”

April 5, 2019

I wrote last week about the value (for me, anyway) on defining a time when I can stop doing a task. I have another example: unsubscribing to junk mail.

I have had one of my email addresses for 20 years, and it has a lot of junk mail that isn’t exactly spam, but isn’t wanted either. Each week, I have roughly 100 of these messages from probably 30 different sources. I get overwhelmed when I think of having to unsubscribe from them all, so I have to wade through 100 emails each week (I only check this account weekly).

My solution: each week, I am going to unsubscribe from three emails. This will take me two minutes, and it doesn’t overwhelm me. In ten weeks, the account should be clean.

Assigning Finishing Times Prior to Starting Tasks

March 25, 2019

I made a realization about motivation when I was working with my kids: there is a lot of power in assigning the finishing time of a task prior to doing the task. That is, it can be really helpful to know when you can stop because you have done enough. I give you five examples.

  1. What prompted this realization is an effort I made to make my house cleaner. My wife and I have been trying to get our kids to pick things up for years without much success. We finally found a system that works. We have clean-up time every night (that I remember to have clean-up time, which hasn’t been often lately. This is the flaw in the plan). The fundamental rule is: you can’t clean up anything after the time is up. So we remind ourselves of the fundamental rule before we clean, we pick a random integer x from \{2,3,4,5,6\}, we set the timer for x minutes, and then we clean until the timer goes off. We then stop cleaning, I throw each kid into the air as high as I can (trust me—they find this fun), and we go on with our lives.

    Our house gets cleaner, and there are minimal complaints from the kids.

  2. My wife is good at cleaning things; I am not. She often asks me to do things like clean out our file system (with her help). I normally avoid this—I hate it. However, about two months ago she asked me to spend thirty minutes cleaning up our file system with her. This was no problem for me—I didn’t have the avoiding behavior at all. We cleaned for thirty minutes, and we basically did everything that we needed to do with the file system.
  3. This is an important feature of Robert Talbert’s approach to grading. One important point is that you split up the grading into 15 minute chunks (e.g. “I will grade four students’ papers”, or “I will grade one question from the midterm”), and you just work on one 15 minute chunk at a time. Once you have done a chunk, you have no obligation to move on for a while—you can do something else. This helped me a lot.
  4. I have pre-defined times when I work. I leave the house at 7 am, and I am back home by 5 pm. I do not work past 5 pm (except for semi-rare special events). I work hard all day in part because I know that it will end.
  5. My college has a great program to education faculty and staff on diversity issues called “Becoming Community.” There are about 10 events per semester. One thing that is kind of genius is that they give a “certificate” to anyone who attends five events over the course of the year. I do not care about the certificate—I am not one to put such things on my wall (although I might make an exception for this one, since it might be helpful if students saw it). However, I am convinced that I am attending more events because of the certificate. Again, the certificate itself doesn’t matter to me; what matters to me is that they defined “enough” (as “five events”).

    I am interested in this stuff in general, but I do not want to go to an event each week (I have too many other things that need to get done). If they did not have the certificate series, I probably would have gone to three of these events. Because they told me that five is “enough,” I am likely to go to six or seven. There is some psychology here—I know that I am going to fail at making all of them, but I know that I can make five.

I have found in all of these examples that it is really useful to let yourself know when you can quit without feeling guilty. Now I need to exploit that more. This is probably well-known among psychologists—let me know in the comments what psychologists can tell us about this. Basically, I think that I (we?) are sometimes afraid that a task will go on forever, and it is useful to know that it will end (and soon!). Also, I know that this is similar to the part of GTD where you determine the next “action” of a “project.”

There is one added benefit: I sometimes work longer than I need to. That is, I might grade two chunks of assignments instead of stopping at one chunk when using Talbert’s method (note: my kids never continue cleaning after the time is up, but that is fine!). As my dad often says: “The hardest part of painting a house is opening up the first can of paint.” It is much easier to start if you think that don’t think about having to paint the entire house.

Productivity (from Talbert)

March 18, 2019

Robert Talbert, being awesome as always, is writing about productivity. Talbert writes that GTD may have saved his life. This is true for me in the sense that GTD-type plannng has allowed me to keep a social life during extremely busy times at work. I do not use according-to-Hoyle GTD, but rather a version of it that has evolved that works for me.

I love everything that Robert wrote, and I just want to add/paraphrase two things:

  1. The subtitle of the original GTD book is the art of stress-free productivity. I don’t hear much of stress-free part of the system often, but this is the big win for me. Even if I did not actually do more during the course of my day, my stress levels are much lower using a GTD-like system. This aligns well with Robert’s great point that we are humans who should be treated as such, and we all deserve to have less stress.
  2. My other point might seem to contradict what Robert is saying, but I don’t really think so if one considers it carefully. My other point is this: productivity systems allow me to create meaningful work, and more of it. One of my great joys in life is designing classes, and I have more time to do so because of the productivity systems. How is this? Because I spend less doing stuff I don’t like (e.g. going through email). So: assuming that I spend a constant amount of time on work (which I do—this is one of the key components of my productivity system), I spend more of that time on work that is meaningful to me and less time on work that is not. This means that my life is better.

I find that productivity systems are like having kids. You can imagine how much you might love your hypothetical kid prior to actually have the child, but you will actually underestimate the love you feel for your child. Similarly, you can probably imagine how much nicer a productivity system would make your life, but you are probably underestimating how much nicer it would actually be by quite a bit.

Being Chair

March 15, 2019

This is my first year as chair of my department. The consensus is that being chair sucks, and one only should do it when it is “your turn.” However, I have enjoyed my time as chair this year for the following reasons:

  1. I am learning a lot about how the school runs.
  2. I am learning a lot about how the department runs.
  3. I am learning how to effectively run meetings (we had a meeting yesterday, and I learned a couple of key things that will help us out in later meetings).
  4. I enjoy interacting with people outside of Mathematics more frequently.
  5. I am forced to think about what I think is important for the department (and then set the meeting agendas to reflect this).

The best part of being chair, however, is that I can help students in ways that I didn’t think would be possible. I am able to arrange things to get students into classes when are full-but-not-too-full. If the class is too-full, then I have usually been able to give them an alternate solution that works roughly as well. I find that I am doing a lot more advising, which is a part of the job I really enjoy; students come to the chair by default if they are a non-mathematics major who needs help thinking about mathematics.

Of course, there are parts of the job I dislike, but (so far) the benefits have outweighed the costs. I hope that this continues.

I am writing this because I typically hear mathematicians grumbling about having to be chair (at schools other than mine, too), and I want to provide some balance to the discussion.

One more thing on Weekly Writing Homework

March 11, 2019

I was on spring break, so I didn’t blog last week. Here is a quick one to make up for that.

I forgot one important thing about Weekly Writing Homework: I can give students as much or as little help on the problem as they need. For some students, this means that I can tell them how to do the entire problem (since the point is not the matheamtics, it is the writing!).

This is the only venue that I have seen where the students are highly motivated to do the work (it counts toward their grade), yet I have no problem telling them the solution. This has been a great tool for students’ mathematical learning, too.

Exam Format

February 28, 2019

I had (exactly) two good ideas from last year. I already wrote about Weekly Writing Homework, which I am pretty sure I have never heard of before (but I am also pretty sure this is a new idea). My second idea is one I am certain that I have heard before, but I don’t remember where. It is an exam format that worked really well for me—students came back after the semester ended and told me that the exams really helped them learn.

Here is the basic outline— you can adjust as needed.

  1. Assign the students four problems one week before exam date. These are problems that could be on the exam.
  2. Students can work together to solve the problems. In fact, I encourage students to work together to solve the problems.
  3. Each student picks one of the four problems to write up prior to the exam in \LaTeX. I make it clear that this write up cannot be done with other students—students can develop the ideas together on, say, a blackboard, but the students need to separate and write up the problem alone.
  4. The student comes to the exam and hands in the \LaTeXed solution immediately.
  5. The student gets a copy of the in-class exam. Of the remaining three problems the student did not write up, I assign them one to do for the in-class exam (My exam instructions are roughly: “If you did not submit a \LaTeXed version of Problem 1, do Problem 1; otherwise, do Problem 2.”). Since they have already discussed the solution with other students (hopefully), they just need to re-create and write down the answer.
  6. So one problem was written in \LaTeX and one was assigned to them for the in-class exam. This leaves two of the original four problems; students need to pick one of the remaining two to do.
  7. I also give the students a simple problem that they did not see before.
  8. If there is enough time and it is appropriate, I might ask students to state some definitions, along with examples and nonexamples. (h/t Robert Campbell for this).

I like that I can give students challenging problems, and they can rally to do them; if one of the problems is too hard for a student, they might not need to do that problem on the exam. Students learn a lot from working together, but they also know that they will be held individually accountable.

Can you see ways this can be improved? What are the flaws?

Weekly Writing Homework

February 20, 2019

I had two ideas that worked really well last year. One of these is “Weekly Writing Homework.” In courses that are writing-heavy (real analysis, introduction to proofs, abstract algebra, linear algebra), I assign a weekly assignment every week that is only graded on the writing quality. In particular, I do not grade it on the mathematics at all—a student can get a perfect score on the assignment even if all of the mathematics is wrong. This has allowed students to rapidly improve their writing skills—my students consistently create a good-looking proof after roughly four weeks. This means that I only get pretty-looking proofs for two-thirds of the semester.

I have struggled to get students to write well on these proof-intensive courses previously. Weekly Writing Homework is designed to isolate the skill of writing. I use Specifications Grading in these courses, and the Weekly Writing Homework is only graded on these specifications. An important facet of this is that the student should be able to judge whether they have met all of these specifications.

The specifications were largely stolen (very slightly adapted) from here, which was written by Anders Hendrickson.

This homework is graded in the usual Specifications way, Pass or NotYet. Students have to spend a token to resubmit, but they get full credit for the resubmission (I usually give the entire class a free pass for the first assignment, since they do not understand how seriously I take the Specifications at that point).

This was a low cost way of rapidly improving their writing skills—grading takes roughly 30 minutes for a class of 20 after the first couple of assignments. I am certain that Anders’s specifications were the key—they provided the specific instruction on how to write well. This is one of the huge strengths of Specifications Grading—it is a method of instruction as well as a method of grading. In the past, I typically would not explicitly tell students what good writing is. The Specifications give my students the explicit instruction they needed.

I should blog more

February 15, 2019

I have been gone for a long time—it has been about a year-and-a-half since my last post. I want to start blogging weekly. I am writing this to hold myself accountable, although I understand that my plan for accountability might backfire. (although I don’t think that this is really an identity goal).

For now, let me just state how I hope my next evolution as a teacher will be: I want to introduce more play in my classes. I feel like I have taken a lot of the fun out of my classes (particularly in one of my courses for elementary education majors, which is terrible), and I don’t think that this is so great for learning.

I am starting this semester. I am teaching an Introduction to Proofs course for the first time this semester, and I have baked more play into the structure. I was strongly considering using some jiblm.org-style notes, but these seem too rigid to me (and I am trying to make things less rigid). This course has modest content goals, and the focus is on proof-writing. I figure that students can practice proving all sorts of things, so I am going to try to have them work on “play” problems as much as possible (I am also using Dana Ernst’s excellent notes to supplement these play problems).

One example of a “play” problem is a subtraction game that I gave my class at the beginning of the semester: two players start with n pennies between them, and take turns removing pennies for the pile. Each player needs to remove exactly one or exactly two pennies each time. The player who takes the last penny loses. What is the winning strategy?

This has elements of play, in that it is literally a game. This is good. However, it is not as open-ended as I would like—I know where the students will end up. I hope to give the students more open-ended problems as the semester goes on (I have a list, but I do not want to publish them yet).

As I plan my courses for next year, I will make play a priority.