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.

#WeHateMath Apples v. Oranges

Recently I was asked to briefly explain the difference between math and arithmetic.  Here’s what I came up with:


Arithmetic can answer the question:

“What is 3 plus 2?”

Arithmetic cannot answer the question:

“What are 3 apples plus 2 oranges?”

Math, on the other hand, can take a higher order look at the problem and answer:

“3 apples plus 2 oranges are 5 pieces of fruit.”

Math is about relationships and patterns.

Math is arithmetic plus context.

#WeHateMath The Minimum Wage in Context

I rarely watch television news because my yelling at the TV frightens the cats. But my wife likes it so this morning while I was putting on my socks I saw our local business correspondent report on a recent vote to raise the minimum wage in Switzerland to the equivalent of $25.00 per hour. He noted with a grin that this measure was defeated, with 76% of the voters rejecting it. This was all offered ‘as is’ with no context and they went on to the next story.

What does this have to do with math, you ask?

Economics is math plus people.

Math is arithmetic with context. (If you add 2 and 3 you get 5 but what do you get when you add 2 apples and 3 oranges?)

Without context, this story was meaningless and by reporting it, they had stolen moments of my life that I could never get back.

I needed to add context to this story. I thought about what was missing.

How about:

What is the current minimum wage in Switzerland? – It turns out that Switzerland doesn’t have a mandatory minimum wage. Your pay is set either by negotiating with your employer yourself or through a representative, ie a union. (According to the OECD, union membership represented 17% of the work force as of 2010. By comparison, the US is 11%.)

How meaningful is $25/hour to the Swiss worker? – For that, we need to compare the cost of living in Switzerland versus the United States. But that’s a complex metric, so let’s pare it down to a few, everyday measurements (NOTE: I’m comparing Denver, where I live, to Zurich):

A tank of gas – Gas is $7.99/gallon in Switzerland. The average gas tank size is 18 gallons. It would cost you $143.82 to fill up your car in Switzerland. The average gas price in Denver (as of 05/19) is $3.44 so a tank of gas would run you $61.92.

A gallon of milk – $6.45 in Switzerland, $3.69 in the United States.

Average Monthly Salary (after tax) – $6652.07 in Zurich, $3466.28 in Denver.

It turns out that Swiss workers are among the highest paid in the world and 90% of them already earn more than the proposed minimum wage of $25/hour AND they work an average of 35 hours per week. In addition, the unemployment rate in Switzerland is a paltry 3.2%.

Now I have some context. It would appear that the Swiss are doing pretty well for themselves (at least, most of them) and now it’s not that surprising that this proposal was rejected.

#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.


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.

#WeHateMath Calculators Considered Harmful

“For this invention will produce forgetfulness in the minds of those who learn to use it, because they will not practice their memory. Their trust in writing, produced by external characters which are no part of themselves, will discourage the use of their own memory within them. You have invented an elixir not of memory, but of reminding; and you offer your pupils the appearance of wisdom, not true wisdom, for they will read many things without instruction and will therefore seem to know many things, when they are for the most part ignorant and hard to get along with, since they are not wise, but only appear wise.”(Phaedrus 274c-275b)


The above quote is Socrates complaining about the invention of the written word. He claimed that allowing students to write down facts instead of memorizing them would weaken their minds.

I’m reminded of this when I see some of the reactions to suggestions that we embrace the use of computer technology in the way we teach math. The most noted advocate for this is Conrad Wolfram, who described his philosophy in a 2010 TED Talk.

My first introduction to technology-assisted math was in my high school chemistry class, where we learned how to use a slide rule. (Yes, I’m thatold.) We didn’t have them in math class (or calculators for that matter) so all calculations was done by hand. I didn’t get my first calculator until I went off to college. As this technology got cheaper and more readily available, it began to filter down towards K-12, where I recall a lot of controversy over allowing their use in class. It was felt at the time that it would make students unable to do arithmetic by hand and therefore become dependent entirely on machines to do it for them.

Back in 1957, Isaac Asimov wrote a short story “The Feeling of Power”, about just that kind of future and how society had to rediscover the technique of doing arithmetic by hand. The comments at this page about this story are…..okay, I’m trying to think of how to say this….well, let’s just say there’s a certain ‘you kids get off of my lawn’ quality to them.

Please don’t misunderstand. I absolutely believe that grade school kids should learn how to do arithmetic by hand, to start. But I don’t believe that it’s as simple as we either do all of our math by hand or become completely dependent on machines.

First of all, I have no problem doing arithmetic using a machine. Arithmetic is a very mechanical activity, which is one of the reasons why teaching it involves so much memorization. It isn’t something that comes naturally to us humans. However, we should be comfortable with arithmetic so that we can visualize what our answer should look like and to make sure the problem was entered correctly.  (I prefer using a text editor to writing by hand. This doesn’t make me illiterate.)

Math, on the other hand, involves intuition, creativity, imagination and logical thinking. Machines can make the arithmetic part of it easier but you still need to understand the problem well enough to explain it to the machine. We teachers don’t have to fear the use of calculators or computers in a math class if we use them intelligently. Where these machines can be used to our advantage is to reduce student anxiety about the mechanical parts of the problem so they can focus on the part requiring human-based thinking.

In my experience, students hate math because they fear arithmetic. They are so scarred by their grade school arithmetic classes where the slightest error in a long chain of arithmetic would ripple down and cause them to get the problem wrong (and fail the test) that they don’t want to be anywhere near any class that reminds them of that.

I encourage the use of calculators (and spreadsheets and Wolfram Alpha) in my college math class. They take away fear and give us more time to actually talk about math.

#WeHateMath Two Teachers

We recently celebrated Teacher Appreciation Week so I’ve been thinking about the teachers in my life who have influenced me. Since this is a blog about math, I decided to mention two of my math teachers who have made a lasting impression on me.

(Unfortunately that impression didn’t extend to their names, since I’m terrible with names and it was about forty years ago. However, they know who they are.)

Middle School Algebra – My teacher was a tall, lean, severe-looking man whose horn-rim glasses and dark suits gave him the air of a church deacon. He had a deep, mildly monotonic voice, with an odd rhythm that forced you to pay closer attention. He addressed all of us as “Mr. X” or “Miss Y” and had a way of looking at you that felt like he could see inside of your skull. He gave off an air of mild disdain that for some reason pushed us to work harder to prove ourselves.

High School Geometry – My geometry teacher was a former Marine D.I., with a gymnast’s build and a wicked Van Dyke. He seemed to radiate energy and his eyes took in everything and missed nothing. His enthusiasm for his subject was contagious and he spent the majority of the class going through proofs. This might sound dull, but he presented them as if he was letting us in on this Great Secret and before long we were excited to get to the proofs and solving the puzzle. The most important thing I took away from this class was that I could assert that something was true but that was no better than simple faith. If the only evidence of a truth is belief, what happens when the believer goes away?

But if I can assert a truth through a rigorously logical proof, then this is not just true because I say so. The Universe says so. Not only that, but my proven truth will always be true and will have always been true. It doesn’t matter what happens to me.

Despite their different styles, both teachers had something in common. They treated their students as individuals and were able to express their passion for the subject matter in a way that energized us to learn.

With each new class I teach, I try to live up to the standards set for me by these two men.