Constitution Day – Math and Politics

Sam: It’s a private poll. The press doesn’t have access to it… The only way they’d know what questions were being asked is if they were actually called by one of the pollsters and… Oh my god!

C.J.: Yes.

Sam: A reporter got called by one of the pollsters?

Josh: Wow. What are the chances of that?

Sam: The chances of that are astronomical.

Josh: We can calculate it. They sample 800 respondents…

C.J.: Would the two of you stop being amazed by the mathematics!

(2001). The Leadership Breakfast [Television series episode]. In The West Wing. New York: NBC.

 

Last month the faculty at my college received the following in an email from our Academic Dean:

 

Hello Faculty!

With Constitution Week this week (Sept 15-19), and Constitution Day approaching next Wednesday, Sept. 17, it’s time for all faculty to plan a small segment of their classes, relating it to any aspect of the Constitution.

We do this every year as required by law. Since I don’t teach classes that lend themselves to this kind of activity I have to be more creative.

This term I’m teaching College Mathematics, which is math for non-technical majors. This is the only math class that some of these students will take. It includes discussion of algebra, geometry, statistics, probability and financial math.

This week we’re covering statistics. Based on my firm belief that you can find math in anything,I did a search on ‘constitution statistics’. It led me to the blog Introductory Statistics. I found a post titled “A Statistical Look at the Amendments to the United States Constitution”. It included a table showing the proposed and enacted dates for all twenty-seven of the amendments. I copied the data into a spreadsheet:

 

Amendment Proposed Enacted No. of Months
1 9/25/1789 12/15/1791 26
2 9/25/1789 12/15/1791 26
3 9/25/1789 12/15/1791 26
4 9/25/1789 12/15/1791 26
5 9/25/1789 12/15/1791 26
6 9/25/1789 12/15/1791 26
7 9/25/1789 12/15/1791 26
8 9/25/1789 12/15/1791 26
9 9/25/1789 12/15/1791 26
10 9/25/1789 12/15/1791 26
11 3/4/1794 2/7/1795 11
12 12/9/1803 6/15/1804 6
13 1/31/1865 12/6/1865 10
14 6/13/1866 7/9/1868 24
15 2/26/1869 2/3/1870 11
16 7/12/1909 2/3/1913 42
17 5/13/1912 4/8/1913 10
18 12/18/1917 1/16/1919 12
19 6/4/1919 8/18/1920 14
20 3/2/1932 1/23/1933 10
21 2/20/1933 12/5/1933 9
22 3/24/1947 2/27/1951 47
23 6/16/1960 3/29/1961 9
24 9/14/1962 1/23/1964 16
25 7/6/1965 2/10/1967 19
26 3/23/1971 7/1/1971 3
27 9/25/1789 5/7/1992 2431

 

I emailed a copy of the spreadsheet to my students. I also sent a link to an article summarizing the contents of each amendment.

I have two hours each week with my students to cover that week’s topic. The rest of the class takes place online. The in-class session on Monday night sets the groundwork for the entire week. I have two major topics to cover this week:

  1. Using graphs to visualize data
    1. histograms (bar charts)
    2. pie charts
    3. line charts
  2. Measures of central tendency, ie mean, median and mode

With the amendment data, we used bar charts to see how long most amendments took to enact (one to two years). This led to a discussion of outliers. For example, the 27th amendment took over 200 years to enact. We constructed pie charts to get another view of the distribution and to confirm our earlier conclusion.

But there is also a time element built into the data – proposed and enacted dates. When you want to look at data that occurs over time, you use a line chart (or trend chart). In this case, we aggregated the amendments based on the half-century during which they were enacted. When you plot that into a trend chart, you get an interesting view of the last two centuries of the United States. You can then correlate the lines with social, demographic and political data.

You can find math in anything, if you look hard enough.

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#WeHateMath Blog Review – MathFiction

MathFiction is not strictly a blog per se but it’s unique enough that I felt it deserved a shout-out.  From the site description:

 

Do you like fiction and mathematics? Are you looking for a book or story that might be useful for the students in your math class? Are you interested in what our society thinks about mathematicians? Then you’ve come to the right place. This database lists over one thousand short stories, plays, novels, films, and comic books containing math or mathematicians.

 

The list is maintained by Alex Kasman, who teaches in the Department of Mathematics at the College of Charleston. I came across this site when I was doing some research for a blog post about using calculators in math class. I was trying to remember an Isaac Asimov short story that I had read which described a future where nobody is taught arithmetic any more since it’s all performed by machines. (“The Feeling of Power” [1957]) A brisk search of the Intertubes brought me to MathFiction and I was so impressed that I bookmarked it for my classroom tool kit.

You can browse through the material at your leisure or, if you have something specific in mind, use the search page, where you can either use the site-specific Google keyword search or power-search by author, title, summary, medium, genre, topic or motif. Entries are also rated by both math content (1 to 4) and literary quality. (You can search on these fields as well.)
As both a nerd and a math teacher, I have a feeling that I’ll be spending a lot of time on this site. So head on over there and tell Alex I sent you. (I’m trying to start a thing.)

#WeHateMath Book Review – How to Think Like a Mathematician

It’s not often that you find a math book that opens with a joke. I mean, literally in the opening, just before the preface:

Question: How many months have 28 days?
Mathematician’s answer: All of them.

(Okay, it’s no ‘man from Nantucket’ but I got a chuckle out of it.) I also found one of my favorite footnotes:

….use or like mathematics are considered geeks or nerds*

* Add your own favorite term of abuse for the intelligent but unstylish.

This was just in the preface (ProTip: always read the preface, kids!).

The author, Kevin Houston, teaches at the School of Mathematics at the University of Leeds. His dry British wit is evident throughout this very readable text. But this isn’t just a math-oriented joke book. It’s a solid collection of mathematical ideas and skills starting with sets and functions through proof techniques and equivalence relations. Houston wraps up with a discussion of how to recognize true mathematical understanding.

References

Houston, K. (2009). How to think like a mathematician: A companion to undergraduate mathematics. New York: Cambridge University Press.

#WeHateMath A Better Way to Teach Math?

Ever since I started teaching math, I’ve been trying to figure out why so many students (and non-students) seem to have problems with it and what I could do about that. With that in mind, I’ve been trying different techniques to see which ones are effective.

The first one was what I call the “Age of Aquarius” technique. It usually has me saying things like “Math is the secret language of the Universe” and “We’re all made of math” in an attempt to engage student imaginations. Unfortunately these work best with an audience that is already receptive and math classes are generally full of people who want to be anywhere else. As a result, saying these things makes them think that either you are high or you’re trying to make your job sound more interesting than it is.

I call the second method “Eat Your Vegetables”. This consists of explaining how learning math is good for you, usually by citing the Stanford Medical School study that showed improved brain function from the very act of learning math. In other words, even if you never use this stuff in your real life, it’s the mental equivalent of CrossFit. This also fell flat, being a bit too abstract for a group that just wanted to get through the class with a minimum of effort.

I decided to tackle this from the other end. I wanted to find the source of this distaste for math. I’ve always felt that hate is a fear-based emotion so if I can lower the general anxiety level in the room, I should get better responses.

One technique I used was to use tools to handle the mechanics of problem-solving. For example, whenever convenient I encouraged the use of calculators once the math portion of the problem had been processed. (If you understand the math well enough to explain it to a machine, this is a good thing.) For statistics, I showed how to use spreadsheets to quickly and easily observe the effects of different sample data on results as well as how to create different types of charts to visualize the numbers. For probability, I brought in decks of cards. I used a drawing program to sketch out problems and scenarios.

I also tried to minimize the use of jargon during class. While it’s useful to have a common vocabulary, I wanted students to pay more attention to the relationships and patterns than to worry about coming up with the correct terminology. We introduced terms as needed but it was okay if they had to use a few more words to explain what they were talking about or if they had to draw a picture or diagram.

This was the first term I used these techniques so it’s early days. But my last class session was today and overall I feel quite positive. Several students have remarked that this was the best math class they’ve ever had and my retention rate was pretty good. For math classes, it’s not unusual for half or more of registered students to withdraw before end of term. Out of an initial class of twenty one, I lost just seven and of those, four withdrew before the start of the term.

All in all, a good start. I’m teaching two sections of the same class next term so I’ll continue to refine my techniques and report my data here.