Three Different Meanings of Mathematics

September 12, 2014

I “overheard” an exchange on social media that can be summarized like this:

Person A: I teach mathematics using an IBL-style.
Person B: I could never learn mathematics that way, even though I am good at mathematics.

I spent a lot of time thinking about this exchange, and I have found it helping me immediately in several ways. I guess this means that I might ramble a lot in this post. [Edit 20 minutes after first posting: this is probably not new to most people, and I have had similar thoughts before. But this was a bit of an epiphany for me for some reason I cannot explain.]

First, while Person A and Person B are both talking about “mathematics,” I think that they mean two different things. In fact, I think that there are (at least) three meanings for the term “mathematics” with respect to teaching.

The first meaning is what I call “application of existing mathematics.” [Edit 20 minutes after original post: mathematics that is often described as "procedural" belongs in here, although I suspect there might be more]. This comes in two flavors: the application to mathematics, and the application to outside fields. In a stereotypical “traditional” mathematics classroom, this is what is mostly meant by “mathematics.” For example: in a calculus class, finding the derivative of x^2+\sin x is an application of several existing bits of mathematics (the Power Rule, the Sum Rule, etc) to a mathematical problem to get the answer. And almost any sort of word problem fits this description.

The second meaning is what I call “understanding existing mathematics;” I think a lot of people would say this is about understanding concepts. In a Peer Instruction class (at least, in a PI class that operates in a similar way to how I do PI), this is what is mostly meant by “mathematics.” For example: in a calculus class, asking students how many tangent lines can be drawn at the point (0,0) of f(x)=|x| might be an example of that. To answer this, students need to understand the existing notion of tangent line to do this. Another example would be getting students to understand the \delta-\epsilon definition.

The third meaning is what I call “creating new mathematics,” or “doing mathematics” (when I say “new mathematics,” I mean that it is genuinely new to the student, not new to the entire community of mathematicians). I imagine that this is mostly meant by “mathematics” in a good IBL classroom. Students need to engage in the actual process of how mathematics is done by mathematicians, which includes dead ends and wrong answers (but also includes successes).

[Disclaimer: I am not trying to put a value judgment on these three meanings, although I am probably failing given that I am using the term "do mathematics" for one particular meaning. But I do happen to think that all three are extremely important. I also am probably talking in absolutes more than I should; please insert your own nuance.]

So it seems to me like that conversation actually was:

Person A: I teach students how to create new (to them) mathematics using an IBL-style.
Person B: I could never learn how to apply existing mathematics that way, even though I am good at applying existing mathematics.

I am guessing that Person A does teach students how to apply existing mathematics, but that it is secondary (or tertiary) to teaching students how to create/do mathematics.

Questions:

  1. Do any seasoned IBL instructors want to comment on the accuracy of my claims?
  2. Am I missing any other meanings?
  3. Anything else?

Doceri vs. Explain Everything

September 4, 2014

I create a lot of screencasts for my classes. I have evolved to mainly using screencasts to provide solutions to quick questions, of which I have roughly 400 this semester. Because of this, I can save a lot of time if I can import my PDF file of quiz questions to my screencasting software so that I do not need to re-write the questions.

It is not convenient for me to create videos at work. I have a Linux box in my office, but it is a bit unreliable for screencasting, and I do not have complete control over it to make it reliable. For instance, I had screencasting in my office figured out a year ago, but now my Wacom Bamboo tablet has stopped working. I do not have the permissions (I don’t think) to fix this, since we have a central Linux administrator (I also don’t immediately have the know-how to fix this tablet issue, although I think I could figure it out).

Another alternative is to use a Windows machine elsewhere on campus, but I don’t really like leaving my office.

Instead, I decided to start screencasting from my iPad at home after my family has gone to sleep. This has a number of advantages: there is comfortable furniture, I can see what I am writing on the iPad (as opposed to the Bamboo tablet), and there are tasty snacks.

The main issue me was deciding which screencasting app to use. I have toyed around with Doceri previously because it is free, but I was concerned that it did not support importing PDF files. I had heard great things about Explain Everything—it supports but I was wary of committing to a $2.99 price tag. I am merely a consumer of both Doceri and Explain Everything; neither company has paid me anything to write this post.

Because I have 400 videos to make, I decided that importing my PDF quiz file was important enough to spend $2.99. The file imported well, and I was able to create a couple of screencasts.

But the problem came in when I started uploading the files to YouTube. It was taking Explain Everything roughly 10 minutes to upload a two minute video. Because I have 400-some videos to create, it is simply unacceptable to spend 500% of the time I spend creating the video in uploading the video.

So I went back to Doceri. It turns out that there is a very easy work-around for import PDF files in my situation. I can open my PDF quiz file in Dropbox, take screenshots (press and hold the power button, then press the home button) of the questions I want to do, and then I can import the screenshots easily into Doceri from my Pictures app.

The great part: Doceri takes about 10 seconds—rather than 10 minutes— to upload a two minute video. This has worked extremely well—I was able to create 35 videos in two hours last night (as opposed to the roughly eight videos I would have been able to create with Explain Everything).

I was so happy with Doceri that I paid them $4.99 to remove the watermarking on my screencasts.

[Edit: Andrew Stacy and Dale Buske reminded me that I meant to write about the Explain Everything Compressor. This is a $15 app for a Mac (not the iPad) that does the compressing for you so that you can continue to make screencasts on the iPad while the Mac compresses. I was very close to purchasing it, when I decided to give Doceri another chance (Robert Campbell was very encouraging here). The bottom line: I get to save $15 and avoid having to use two machines by going with Doceri. Additionally, I found some reviews saying the compressor was mediocre, and I didn't want to spend $15 on something that doesn't work well.]

I hadn’t read anything about this being an issue with Explain Everything; I imagine that it might be because Explain Everything has greater editing capabilities, so it stores more information. But this is not a feature creating quick and dirty screencasts.

Assessment Idea for Calculus I: Near Final Draft

August 18, 2014

Sorry about the two month hiatus—Dana Ernst sucked me into a great research project about games with finite groups.

I previously wrote about my plan for calculus I. Basically, it is this:

  1. I give the students a list of learning goals. These are much finer than I have done in the past, which means that there are many more of them.
  2. I give students quizzes in class.
  3. For each quiz question, the student solves the problem as best as she can.
  4. Here is the important part: after solving the problem, the student reviews her work and determines which learning goals she has met.
  5. She indicates exactly where she met each learning goal. If she does not claim a learning goal, she does not get credit for the learning goal.

This basic idea has not changed; I have decided to go for this to see how it works. I have made a couple of changes since last time, though:

  1. I change my learning goals (see below for a list).
  2. I am only requiring that they demonstrate mastery of each learning goal four times, rather than the six that I previously had. There just is not enough time to assess that much, considering that I try to give my students at least twice as many attempts as is required. I am able to cut from six to four by scaling down homework: I previously required at least three demonstrations on a quiz and up to three demonstrations on homework, but I have changed this to requiring at least three demonstrations on a quiz and up to one demonstration on homework.
  3. I change my quiz template to include a margin on the left side. This is where students will write their code for each achieved learning goal. They then need to circle exactly where the learning goal is met, and connect that circle to the code. This should make the quizzes easier to grade and easier to read (less messy). I think that I am not going to require that this be done in a different colored pen, either.

I think that is mainly it. I have included drafts of my learning goals and syllabus (sorry for being three weeks late on this, Robert) below. Please see my previous post to get an idea of what students will do with their quizzes.

As always: feedback is welcome.

View this document on Scribd
View this document on Scribd

Assessment Idea for Calculus I: Feedback desperately wanted!

June 25, 2014

I am planning an overhaul of Calculus I for the fall. I used a combination of Peer Instruction and student presentations in Fall 2012, and I was not completely happy with it.

So I am starting from scratch. I am following the backwards design approach, and I feel like I am close to being done with my list of goals for the students. Here is my draft of learning goals, sorted by the letter grades they are associated with:

View this document on Scribd

I previously had lists of “topics” (essentially “Problem Types”). These lists had 10–20 items, and tended to be broad (e.g. “Limits,” “Symbolic derivatives,” “Finding and classifying extrema”). This list will give me (and, I hope, the students) more detailed feedback on what they know.

This differs from how I did things in the past, in that I used to list “learning goals” as very broad topics (so they weren’t learning goals at all, but rather “topics” or “types of problem”). Students would then need to demonstrate their ability to do these goals on label-less quizzes.

The process would be this:

  1. A student does a homework problem or quiz problem.
  2. The student then “tags” every instance of where she provided evidence of a learning goal.
  3. The student hands in the problem.
  4. The grader grades it in the following way: the grader scans for the tags. If the tags correspond to correct, relevant work AND if the tag points to the specific relevant part of the solution, the students gets credit for demonstrating that she understands that learning goal. Otherwise, no.
  5. Repeat for each tag.
  6. Students need to demonstrate understanding/mastery/whatever for every learning goal n times throughout the semester.

Below are three examples of how this might be done on a quiz. The first example is work by an exemplary student: the student would get credit for every tag here (In all three of the examples, the blue ink represents the student work and the red ink indicates the tag).

View this document on Scribd

The second example has the same work and the same tags, but the student would not get credit due to lack of specificity; the student should have pointed out exactly where each learning goal was demonstrated.

View this document on Scribd

The third example (like the first) was tagged correctly. However, there are mistakes and omissions. In the third example, the student failed to claim credit for the “FToCI” and the “Sum/Difference Rule for Integrals.” Because of this, the student would not get credit for these two goals (even though the student did them; the point is to get students reflecting on what they did).

Additionally, the student incorrectly took the “antiderivative of the polynomial,” which caused the entire solution to the “problem of motion” to be wrong. Again, the student would not get credit for these two goals.

However, the student does correctly indicate that they know “when to use an integral,” could apply the “Constant Multiple Rule for integrals,” and “wrote in complete sentences.” The student would get credit for these three.

View this document on Scribd

I like this method over my previous method because (1) I can have finer grained standards and (2) students will not only “do,” but also reflect on what they did. I do not like this method because it is more cumbersome than other grading schemes.

My current idea (after talking a lot to my wife and Robert Campbell, and then stealing an idea from David Clark) is to require that each student show that he/she can do each learning goal six times, but up to three of them can be done on homework (so at least three have to be done on quizzes). I usually have not assigned any homework, save for the practice that students need to do to do well on the quizzes. This is a change in policy that (1) frees up some class time, (2) helps train the students on how to think about what the learning goals mean, (3) force some extra review of the material, (4) provide an additional opportunity to collaborate with other students, and (5) provide an opportunity for students to practice quiz-type problems.

My basic idea is that I will ask harder questions on the homework, but grade it more leniently (which implies that I will ask easier questions on the quizzes, but grade it more strictly).

I have been relying solely on quizzes for the past several years, so grading homework will be something that I haven’t done for a while. I initially planned on only allowing quizzes for this system, too, but it seemed like things would be overwhelming for everyone: we would likely have daily quizzes (rather than maybe twice per week); I would likely not give class time to “tag” quizzes, so students would do this at home (creating a logical nightmare); I would probably have to spend a lot more time coaching students on how to tag (whereas they now get to practice it on the homework with other people).

Let’s end this post, Rundquist-style, with some starters for you.

  1. This is an awesome idea because …
  2. This is a terrible idea because …
  3. This is a good idea, but not worth the effort because …
  4. This is not workable as it is, but it would be if you changed …
  5. Homework is a terrible idea because …
  6. You are missing this learning goal …
  7. My name is TJ, and you are missing this process goal …

Summer Plan

June 11, 2014

My family and I agree that things work best when I work pretty strict hours—I work 7:45 am to 5 pm during the school year. I do very little work at home. However, I need to do a lot of prep work during the summer to make this possible. Because of this, I work a lot in the summer (we allow for 6 weeks of vacation for the year, so the default mode for the summer is “work”), although my hours are now 8:15 am to 5 pm.

Here is my plan for the summer:

  1. Take care of all of the annoying paperwork-type-stuff that needs to be done. This includes some work that I do every summer: updating my CV, updating websites, and reading and summarizing course evaluations. I also have some jobs that are particular to this summer, such as determining which mathematics courses should be considered for transfer credit at some neighboring colleges. (I am happy that I have already done this entire item).
  2. Do some reading about redesigning general education requirements. My college is considering restructuring these requirements, and my main goal for the summer is to try to determine (along with my other committee members) some sort of reasonable process for this. Fortunately, this is paid work (mostly).
  3. Plan my geometry (and prob/stats/graph theory) course for elementary education majors for the fall. This is also done, largely because I taught this course in the spring. I kept detailed notes (I am grateful I did this), and I mainly updated this course by building in more feedback. In particular, I wrote all of my quizzes for the semester, created solution videos for each quiz, and updated my examinations.
  4. Plan my calculus course. I am planning on using Team-Based Learning, which I learned about from Eric Mazur in this video. Again, planning includes (in chronological order) creating all learning goals, creating all assessments, and creating all class activities. When the semester comes, my main task will be briefly reviewing the plan, adapting that plan based on the students’ needs, recording what actually happened (and how I might improve things next time), meeting with students, and grading.
  5. Do research. I have 3–4 papers that I need to write up, and I hope to re-start work on two projects that have been on hold for too long.

Finally, one benefit of working during the summer is you can be amazingly productive. I am often the only person here, and I can be very productive in such an environment.

Mutt revisited

June 4, 2014

This is a short story about why it is nice to blog; your comments helped me realize what I actually wanted to do. I wrote last week about how I was unhappy with Mutt. Summary: I tried to run GMail through the email program Mutt, and the result was a really slow email program.

Because of several comments by different people, I realized that

  1. I have a slight concern about how Google respects my personal privacy; this is not a huge concern for me, though, and it would not be enough to make me switch.
  2. I have a huge concern about my students’ privacy, and I have been concerned about GMail for a while. Your comments helped me realize that Mutt could be a solution.

Because of a conference relating to my biological children’s educations that I attended last weekend, I realized that I really want to learn more about Linux. So Mutt gives me a chance to do this.

So now I have three reasons to change, and a colleague (Michael Gass) gave me the most elegant solution: use Mutt without GMail. That is, I now use Mutt and POPMail to get mail directly from my school’s servers. Now, Mutt is as fast as it should be.

Today is the first day that I have Mutt up and running, although I have been reasonably happy with it so far. But if I run into problems, I might switch back. One potential problem is that I check my email mostly on my iPad at home, so I need to figure out something to do there (although it could be that my ssh app will work just fine with Mutt on the iPad). A possibly related problem is that getmail is not deleting the emails once they are fetched from the school’s servers (even though I have ‘delete=true’ in my .getmailrc file). This drives me crazy because I crave empty inboxes, although this may be a solution to my iPad problem; I can access Outlook via the school’s webpage or some other app, and the messages will all be there.

I suppose there is a good chance that I will change my mind again next week.

Mutt vs Gmail

May 29, 2014

I really wanted it to work. I really did.

I experimented for a couple days with using Mutt for email. I love the ability to compose emails with Vim, and I had heard that it was lightning fast. If I am honest with myself, my interest in Mutt was also related to the fact that I aspire to be a Linux geek (which I cannot claim to be right now).

So I configured Mutt to get mail from my Gmail account. That way, I could still get all of the benefits of Mutt at work while still enjoying the ease of access to Gmail everywhere else (I usually use the iPad to check email at home).

I wanted Mutt to work. I really wanted Mutt to work. But it didn’t. The problem was that it was muuuuuuuch slower than Gmail. When sending or archiving messages, Mutt took probably 5 to 10 times longer than Gmail did (Mutt takes maybe 2 seconds, whereas Gmail is near instantaneous. I recognize that 1998 Bret would be ashamed of 2014 Bret for caring about this small of a difference). Since I can mostly use Gmail without a mouse, I have come to the conclusion that Mutt is not worth it for me.

Of course, there is always the possibility that I screwed something up while configuring Mutt; let me know if you think that I did something wrong to cause this.

The Importance of Feedback

May 22, 2014

My semester is ended, and now is the time to write some post-mortem entries into this weblog. The first idea is something that is probably obvious, but I over-thought it. I have been been putting more of the course’s assessment at the end of the semester lately, thinking that that is when students are most prepared to do well.

And I am correct, but I took it too far. I did not give my students enough regular feedback during the first part of the semester this spring. My education students actually pointed this out to me—I realized that they were correct as soon as they said it (it also reinforced that they are pretty on top of education issues). Fortunately, I get to teach that course for education majors again this fall; I will make things right this time.

Additionally, I am working on ways of getting students immediate feedback. Clickers are one way of doing this, but I also might have students start grading their own quizzes (I would provide a couple of solution keys and a marker for them) and doing more computer-graded stuff.

Speeding Up Videos

May 14, 2014

I had my 20 elementary education majors produce video projects. These were all due at the end of the semester, which means I have to grade them all now. Each student produced roughly seven 4-minute videos, which means that I have roughly 10 hours of video watching to do as part of my grading.

Or do I? I downloaded this app for my Chromebook, and I have been watching the videos at twice or triple the usual speed (depending how quickly the student naturally talks), cutting my time watching videos to 3–5 hours with no loss of grading quality.

I am very grateful for this app today.

Undergraduate Reseach: Jump Before Looking

May 7, 2014

I talked about my plan for undergraduate research last week. This week, I invited my linear algebra class to join a research team I am forming.

The class is roughly half sophomores and half first-years. They have had calculus and linear algebra. My plan is to come up with a research question based around either finite fields or group actions on cyclic groups. I feel like I have some questions at the appropriate level that have come up in my own research, although I cannot explicitly state them right now. I had better be able to by next fall if any students decide to join the research team.


Follow

Get every new post delivered to your Inbox.

Join 591 other followers