Archive for the ‘Uncategorized’ Category

The New Proposed Curriculum Fails

May 10, 2017

I told you a couple of weeks ago about how I was nearing the end of a 4-year process on building a new curriculum. We had the vote last week, and we lost: the faculty decided to reject the proposed curriculum. We lost by five votes (if we had only changed three people’s minds! Actually, I am not sure if I would have wanted the curriculum to pass by one vote—I don’t want 50% of the people unhappy).

This is disappointing, but the people have spoken. There may be a tiny bit of hope for the curriculum, though: I talked to several people (at least three, which would be enough for it to pass) who wanted more details about a separate, but related, distribution requirement that will be decided in the fall. So it is possible that the Faculty Senate will decide to resurrect the curriculum after the distribution requirement is settled, but there is no guarantee of that.

That is the bad news. The good news is that a lot of my time has just been freed up over the next couple of years.

New General Education Curriculum

April 24, 2017

I haven’t been writing much up until now because I have been in the middle of creating a new general education curriculum at our school. This is year-four of the process, so we have put quite a bit of work into this. We are in the middle of a five-day discussion period, followed by a five-day voting period on whether to adopt the proposed curriculum. Our current general education curriculum is a pure distribution requirement, where students take

  • a two-semester writing seminar in their first year
  • one Mathematics course
  • one Natural Science course
  • one Social Science course
  • one Fine Arts course
  • two Humanities courses
  • two Theology courses (I am at a Catholic school)
  • one Ethics course
  • one Gender course
  • one Intercultural course
  • one Experiential Learning course
  • a capstone course within the major

The Gender, Intercultural, and Experiential Learning courses can be double-counted with other courses; that is, a course could count as both Humanities and Gender.

There are many reasons why our Senate decided to explore changing the curriculum. Historically, very few people were happy with the way our current curriculum was decided upon in 2005ish (I was not at my current school then). Additionally, many feel that these courses occur in isolation, and the students are not given any opportunity to see how they are related.

Here is a link to the website of our proposed curriculum. This was a huge amount of work, and roughly 1/6 of our faculty ended up working on it in some capacity. It is safe to say that this curriculum is no individual’s ideal curriculum, but rather a result of considering 300 faculty members’ needs and desires for their students.

We are happy with this curriculum, although I have no idea how the vote will go. Many people have come out both in favor of it and many people have come out against it. If this does not pass, we (my school, not me in particular) will have to spend a couple of years revising our curriculum to get it up to snuff (many of the current outcomes are unable to be assessed, for instance).

Galois Theory

March 22, 2017

I am lucky enough to be teaching a course on Galois Theory that only has five students in it. The set-up of the course is this: we are working out of Pinter (which I really like due to its ease of reading, great problems, and price of $12; I don’t like that he defines subgroups/subrings/etc in a weird way and is sloppy with defining variables), and students present problems from the textbook to each other.

This will be obvious to people who do IBL, but: holy cow do you get a sense of what students understand and what they don’t. A large part of this is the small class size, but this IBL-like format helps. What students understand (or not) is often not what I would expect.

Because I only have five students, I am doing oral exams. The format is this: I give them four problems the week prior to the exam. Students are allow to work together to figure out the answers. They come to the oral exam having written one of the four problems up nicely in LaTeX. The oral exam starts by the student choosing one of the remaining problems and explaining it. I then (randomly) pick one of the two remaining problems. The session ends with me presenting a problem they haven’t seen before (I try to make these easy enough that the student should know what to do immediately). We will do these exams four times during the semester, and they take 30 minutes per student.

Once again: this testing format makes it crystal clear to me what (and how well) students understand things, and it is clear how I should adjust the in-class work based on the information I get from the exams.

I am wondering if this could scale: could I give a class of 25 students, say, 4 questions, have them write up one, and then have them write the solutions to a subset of the remaining problems? I like that this is a learning opportunity for the students, since they get to learn from each other in an exam situation, but they are still individually accountable. However, I am wondering how much is lost if the exams aren’t oral.


Teaching Real Analysis

March 16, 2017

I am teaching real analysis for the second time in the fall, and I am excited about it. I used Stephen Abbott’s Understanding Analysis when I taught it in Fall 2011, and the students and I both loved it.

My one issue with it, though, is that I would rather do more with metric spaces (Abbott works with sequences in the real numbers as a foundation for the course); I found that I would often draw pictures of R^2 on the board to illustrate ideas relating to distance, and I would like to leverage this slightly more.

I am sold on the idea of using Abbott: it works ridiculously well for my flipped classroom, the students love it, and I am already familiar with it (I am hoping to stop completely redesigning every course I teach from scratch). Here are my ideas for incorporating metric spaces more:

  1. Just follow Abbott’s book as is, and forget about using metric spaces.
  2. Start the semester by looking at Abbott’s brief chapter on metric spaces (in Chapter 8), let students know that we are mainly going to be using it for examples in class, and they are not very responsible for knowing it (perhaps I might give challenge problems where they generalize results in terms of metric spaces, but not every student would need to do that).
  3. Supplement Abbott with a cheap textbook (roughly $10) on analysis like Rosenlicht.
  4. Supplement Abbott with something like Kaplansky’s text on metric spaces ($30).
  5. Supplement Abbott with something like Keith Conrad’s notes on metric spaces (free).

Money is a factor, so I don’t want an expensive supplement.

I am mainly looking for comments like “It is a bad idea to try to integrate metric spaces with Abbott” or “It is a good idea, and here is the perfect source.”

What toys would you buy?

March 6, 2017

Let’s suppose you had $1000 to spend to help you become a better teacher. What would you purchase? Please answer in the comments—your numbers do not need to total $1000, but nothing should cost more than $1000.

Here is my short list (as usual, no one paid me to mention these):

What else do you recommend? You can either list things that you have already purchased or things that you wish you could purchase.

Examples of Math Circle Activities for Young Kids

March 3, 2017

Joss Ives ask for a list of my Math Circle activities for young children. What Joss asks for, Joss gets, so here is a blog post on the activities. Also, my wife wants a list of the activities we did in January and February, so I get to kill two birds with one stone with this post (Note: my usual communication with my wife is not via this blog).

Procedure: We meet once per month in a room at the public library. I usually run two 30-minute sessions, with three to seven kids attending each session. Their parents are rarely in the room, although it happens on occasion.

I stole most of these ideas, mainly from Math from Three to Seven by Alexander Zvonkin and Math Circles for Elementary Students by Natasha Rozhkovskaya. I also steal ideas from Talking Math with Your Kids by Christopher Danielson. I am trying to cite every place where I stole ideas, but I am sure I am forgetting. Please contact me if I should be giving you credit.

Disclosure: I have met Christopher and I am friends with Natasha, but no one has paid me to link to anything in this blog post (or any others).

Here are the activities.

Day 1
1. I gasked questions like “Are there more geese than birds?” and “Are there more women than moms?” I wanted them to start thinking about set containment.
2. I showed them pictures of, say, a bunch of dolphins (and also cats) that look similar (I drew them). I asked, “Is it true that there exists a dolphin with a ball?” “For all cats, the cat has an umbrella?” The purpose is to get them thinking about quantifiers.
3. I showed them a geoboard. I put a geometric design on the left half, and they had to do the mirror image on the right half.

Day 2
I taught them to use Base Ten blocks with the help of a puppet named Yachel. I told them that Yachel is afraid of the number “ten,” so he does not to see ten of anything. The kids organized the Base Ten blocks into groups of ten so that there would be only one big group, rather than ten small groups (so Yachel wasn’t afraid). They kept doing this until all of the blocks were organized. In the end, I asked them to guess how many blocks there were (something like 1287), and I showed them that they can actually tell exactly how many of them there are by just counting how many big groups of each type they made.

I don’t know how much the kids learned about the Base Ten number system, but Yachel was a huge hit; my kids still treat him like he is one of the family.

Day 3
I read them a book called something like 5 Cats, in which the cats categorize their family members into different groups (3 are male and 2 are female; 1 is black, 2 are white, and 1 is calico; etc). I brought hula hoops, and then I asked questions so that the kids could sort themselves (“Stand in that hula hoop if you are wearing something blue today.”). This was a bit of a flop.

Day 4 (March 2016)
This was a Pi Day celebration. I found a bunch of circular lids of different sizes, and cut up a bunch of pipe-cleaners so that they were the length of the diameters of the lids. Then I hot-glued googly eyes on them, and called them “Diameter Worms.” I made up a story about how Diameter Worms find circles to live in, just like hermit crabs. The Diameter Worms need to have a circle that is exactly the right size for it. Then I asked them to figure out how many Diameter Worms can lie end-to-end around the outside of their circle home. First, they made a prediction, then they actually wrapped the worms around the home. Of course, everyone learned that “a little more than three worms” could fit around, regardless of the size.

I don’t know how much they learned about pi, but it started a Diameter Worm craze in my son that lasted for several months.

Day 5
1. More work on subsets and quantifiers, as we did in Day 1.
2. I did something with 3×3 patterns, but I don’t remember what.
3. I gave students cut-out polygons and scissors, and they had to do certain challenges. For instance, cut quads into 2 triangles, or cut a quadrilateral into 2 quadrilaterals, or cut a quadrilateral into a triangle and pentagon.)

Day 6 (11/2016)
1. We did Danielson’s Which one doesn’t belong?
2. We played Nim on graphs, which is a game developed by Marie Meyer and me for her senior thesis.

Day 7 (12/10/2016)
We played (in an unstructured way) with Base 10 blocks (there were only two kids that week).

Day 8 (January 2017)
1. Danielson’s How many? to get at the idea of units.
2. We talked about “Doot Aliens.” When Doot Alien A touches Doot Alien B’s nose (which makes a “Doot!” sound), both Aliens disappear and some Doot Alien C reappears in its place (A-Doot-B always results in the same C, although B-Doot-A might not result in C). There are Ghost Aliens such that A-Doot-Ghost yields A, as does Ghost-Doot-A. I asked them: “What happens when one Ghost touches another Ghost’s nose?” They didn’t come up with the answer, but a couple of them have asked me about Doot Aliens (without me prompting) since then.
3. I gave them a bunch of statements like, “The sky is blue” and “There are seven people in the room,” and I asked them whether each statement was true or false. Then I asked about “This sentence is false.”
4. More of Danielson’s Which one doesn’t belong?

Day 9 (February 2017)
One-cut hearts to celebrate Valentine’s Day (plus, one-cut stars to celebrate Betsy Ross).

Day 10 (March 2017)
I am working a new Pi Day activity, but I haven’t thought of it yet.

Use Mathematics for the Social Good?

February 23, 2017

I am going to keep this short today: I am really excited about Moon Duchin’s plan to create an army of expert witness mathematicians for gerrymandering cases. This is going to be a summer class at Tufts, with other courses planned for Wisconsin, North Carolina, Texas, and San Francisco. I am really interested in doing this, but I want to educate myself more on gerrymandering first.

How many other such volunteer groups are there? I can think of:Statistics without Borders. I thought there was a similar one for “Operations Research without Borders,” but I can’t find anything on it.

Can anyone think of other organizations?

Inquiry-Oriented Instruction

February 15, 2017

I was part of a grant last semester to implement a set of teaching materials that has been refined over the last decade. The materials use a teaching method called inquiry-oriented instruction, which I would say is a subset of inquiry-based learning (IBL). I used these materials in my abstract algebra class, although there are materials for both linear algebra and differential equations, too.

A very brief description is “intuition comes before definitions.” The materials introduce quotient groups by discussing Even and Odd integers, which students could easily see is a group at that point (using rules like “Even + Odd = Odd”). Once they got familiar with the idea that we could have sets of elements make up a group, we slowly backed our way into the definition of coset. It was pretty impressive to see students very naturally come up with definitions—having the right prompts helped a lot.

As part of the grant, I went to training at North Carolina State to use the materials. I also had funds to have student video my class, which will be used to analyze how well instructors who were not involved with the development of these materials can implement them.

We also used the class video as part of a weekly online working group. The purpose of this group was to prepare us, both in terms of pedagogy and course materials (not everyone was an algebraist), to teach the class. We discussed the purposes of the prompts, talked about what was going well and poorly, and watched video of each others’ classes. I found this immensely helpful.

I would use these materials again (in fact, I am planning on using the linear algebra materials next year). My sense is that my students had an abnormally good grasp of the definitions; previous students have struggled to understand what a coset means, for instance. My focus for the next time I use the abstract algebra materials is to work harder on the technical proofs—I think that my students did better on writing proofs than the previous time I taught the course, but not by a lot. Still, I think that the gains in intuition were worth it.

Links to the abstract algebra, linear algebra, and differential equation materials can be found here in the middle of the page.

Unschoolers’ Math Circle

February 8, 2017

Last week, I told you about a Math Circle for Teachers my colleagues and I created. I simultaneously created a math circle for a variety of homeschooled kids known as “unschoolers;” this math circle is really just a math circle that I created for my kids.

My kids are 5- and 7-years old, and finding appropriate problems for them is more difficult for me than for the teachers. Fortunately, Zhvokin and Rozhkovskaya have written great books from which to steals problems. I also supplement them with activities from Christopher Danielson, as well as activities I used with my mathematics for elementary education majors (made age-appropriate).

I have had a good turnout so far. We have been meeting monthly for 1.5 years now, and I usually get 7–12 students per session (we usually do two groups of 5ish, since I am not the best at “classroom management”). We use a room at the public library.

This is has been a lot of fun, and it has been remarkably easy to set up (though my wife, who is much more socially connected than I am, rounded up the kids who are not related to me). It is also an interesting task for me to think about what mathematical ideas are important for 6-year olds to know, and then to design a lesson that gets at it that is fun and educational.

Teachers’ Math Circle

February 1, 2017

I started a Math Circle for K-12 teachers last year with three of my colleagues. Roughly, a Math Circle is just a place where people get together and work on interesting math problems. So far, it has been a wonderful experience. I got to fly to Denver to get some training, and we have had a great time putting it together.

We have started off by focusing on 6–12 teachers, and we have had only a tiny bit of success. We have seen a total of four different teachers, with two of the teachers being dedicated regulars (and a third possibly joining them now). This could be a little weird, with a 2-to-1 professor-to-teacher ratio, but it has not been. The sessions have been a lot of fun, and the actual dynamic is that one of the professors leads the session and everyone else acts as a student (and the leader of the session is often a student, too, since s/he also often has not thought too deeply about the problems). We have been meeting 3–4 times per year.

Our budget so far has been $0, although we have tried to get several grants. The National Association of Math Circles has been supportive, though, even sending us a Math Circle starter pack. We hope to get some money to provide dinners for the teachers eventually. We are able to offer them “continuing education” credits, which helps them renew their teaching licenses (these don’t cost us anything; we just get a little help from the chair of our Education Department, who needs to sign them).

The Math Circle has been a fun and interesting experience with a shockingly low start-up cost and time investment. Let me know if you have questions about starting one.