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.

Reporting Grades in SBG

January 25, 2017

I mostly have liked the course management software I have used (Moodle and Canvas), but both are pretty terrible when it comes to keeping track of grades in an Standards-Based Grading system. I have mostly kept the my grades in a spreadsheet, which does all of the calculations that I want it to, but the students then do not have access to their grades. I tried using Canvas to report grades in Spring 2016 and Fall 2016, but Canvas will not do the calculations I need it to (I just posted the raw scores to Canvas, and I gave the students the logic to figure it out); I had to keep a separate spreadsheet to do everything I needed.

Neither of these made me happy, because I want my students to have access to their grades (if only to check for mistakes I have made), but I also want a single place to put my grades. My solution was inspired by Drew Lewis, who created a PERL script to send his students email updates of their grades directly off his spreadsheet. If I were more computer-savvy, this probably should have been an obvious solution, but I am very grateful that Drew pointed out what I could not recognize on my own.

I am more familiar with Python, so I wrote my own script (included below). Once I have the code written, I go to a command line (I use Linux), type “crontab -e” to edit my crontab, and type (without the quotation marks) “14 3 * * 2 /usr/bin/python 118-S17/Grades/118EmailGrades.py” to send an email at 3:14 am (the 14 3) every Tuesday morning (the “2” in “14 3 * * 2”). The “/usr/bin/python” says to run the program “python” and input the file “118-S17/Grades/118EmailGrades.py.”

Below is the code. It seems to work, but there is one issue that I am ironing out: I am only allowed to send five emails at a time when I tested it. I am pretty sure that this is a limitation on the server’s end, since I am sending the messages to only a couple of email addresses (all mine for the test runs). My (ugly) hack, which worked on Tuesday, is that I broke up my code so that each program only emails 5 students. I welcome troubleshooting ideas from those who know about this stuff, although I suspect that I could just try the single program and it would work, since I am not actually going to email to the same email address more than once for my class.

Here is the code. Note that indentation matters A LOT in Python, so be careful if you cut-and-paste.

import openpyxl
import smtplib
import email
import time

#This gets the spreadsheet the grades are in.

#Here I am getting each 'sheet' of the spreadsheet.

#I put my password in my spreadsheet, since that is supposed to be more secret than this code is.  I put it in Cell AA1 of the "Roster" sheet, and this gets it out.

#I am logging into my email server here.

#I want to put the date in the email, so I am getting it here.

#The From email address and Subject of the email will be the same for every student; I put today's date in the Subject for the students' convenience.
subject="Math 118: Grade Update for "+todaysDate

#Put the last row you want to check prior to the +1  

#Put the rows you do not want to check (because they are blank or because the student dropped) in the list below.

#I have to hardcode the range, and I skip rows 28 and 29 because they do not contain student data.
#The commented out for loop is just to test so that I am not flooded with emails.
for rowVar in range(2,NUMBEROFROWS):
	&nbsp#This just skips the blank rows that I hard-coded into the exceptions.	
	if rowVar in EXCEPTIONS:
	#This gets the student's first name and email, and I print the first name so that I can see who received an email (I get an email update once this program runs).
	firstName=rosterSheet.cell(row=rowVar, column=3).value
	print firstName
	toEmail=rosterSheet.cell(row=rowVar, column=4).value
	#I am going to put together the body of the message in several steps, storing it in the 'text' variable each time.  This is just the saluation of the email.
	text="Dear %s,\n\nBelow is your weekly grade update for %s.  If the semester ended today, you would receive a grade of %s.  Of course, I fully expect your grade to go up, since the semester is not yet over."   % (firstName,todaysDate,todaysGrade)
	#Here I am getting the summaries of their grade components and putting it in the text.		
	text+="\n\nBelow are your current letter grades for each of the components of your semester grade.  Your grade is determined by the lowest of these, so you should focus on the component with the lowest grade; see the syllabus for more details.\n\nQuiz Grade: %s \nGateways Grade: %s\nTeam Project Grade: %s\nIndividual Project Grade: %s\nSelf-Regulated Learning Reflections Grade: %s\n\n" % (quizGrade,gatewaysGrade,teamProjectGrade,individualProjectGrade,SRLGrade)

	#Here I am giving them the next two things they should be studying to improve their grade; the logic in the spreadsheet figures this out.	
	text+="The two Learning Outcomes you should focus on next are %s and %s.  At the bottom of this email is a list of the number of times you have demonstrated each of the Learning Outcomes.  Please check this over to see that it is correct, and be sure to email me if you find a mistake.\n\nHave a great day!\nBret\n\n\n" % (firstMissingQuiz,secondMissingQuiz)

 	#Next, I am just going to loop over the raw data for each Standard and print it out at the end of the email.  This is so they can check to make sure that their records agree with mine.	
	#Put the number of the column corresponding to your last learning outcome prior to the +1
	#Again, I am hardcoding the column range for my spreadsheet.	
	for columnVar in range(2,NUMBEROFCOLUMNS):
		text+="%s:  %s\n" % (labelCode,numberOfMarks)	

        #Here I just format the final message, addding a subject header to my 'text' variable.  Then I send the email.	
	message='Subject: %s\n\n%s' % (subject,text)	

#I log out of the email server.

My return

January 18, 2017

This is a quick post to announce that I intend to start blogging again. After several years of being really good, I overcommitted to things last week. I also overcommitted to things this semester, but I am trying to build several things into my schedule anyway.

Here are the ways that I am overcommitting.

  1. My school is completely re-doing our general education program. I have been third-in-command of the effort, and that took enormous amounts of time last semester. This work will continue this semester, but I think it will be less work.
  2. I am assistant direct of our honors program this semester. Our goal is to consider re-shaping Honors according to what they learned from program review.
  3. I am in charge of program review for the Department of Mathematics this semester.

I will write more about this in coming weeks.

More Social Media (for now)

May 4, 2016

The end of the semester is here, and my thoughts are turning to planning next year’s courses. This means that I will likely be on social media a bit more for the planning phase. The feedback is invaluable, I love hearing other people’s ideas, and I miss you all!

That said, the norm for me is now going to be “no social media.” It was really nice to not have to cede a small portion of my brain to wondering about it, checking it, and wondering when it is going to get its next hit of dopamine. I think the on-going plan will be to be on social media (really, Google Plus) for May and part of June, and off for the rest of the year.