#WeHateMath Storytelling in Math Education

Recently my math class was studying basic probability and I had one student who was really having a hard time wrapping her head around the topic. One evening I got an email from them, stating that they didn’t understand one of the practice problems:

Given that P(E) = 1/4 find the odds in favor of E.

I replied:

P(E) means the probability of E happening. 1/4 means that there are four possible outcomes and that E (the one we want) is only one of them.

We figure our odds for something by comparing the number of possible things that we want (in this case just one) versus the number of possible things we don’t (in this case the other three).

So the odds are 1 to 3 that our event E will happen because out of four total, only one of them is E.

I hope this helps.

They responded that it didn’t help and that they had read the chapter multiple times and just weren’t getting it.

I gave this some more thought and decided on a different tactic:

Let me give it one more shot:

We’re having a dog race — four beagles: Evelyn, Sam, Henry and Charlie.

All four dogs are evenly matched – same age, physical condition, everything – so each of them is equally likely to win.

However, Evelyn is your favorite and you’d like to know the probability that she’ll win the race.  There is a total of four dogs so she has one chance in four to win.  In math-speak, this is known as :

P(E) = 1/4

What about the odds? Well, It’s Evelyn against the other three dogs so we say that the odds for Evelyn winning are 1 to 3 or 1:3

Once I had expressed it in the form of a story, it clicked. They replied that they now understood what the problem was about and had a better sense of the material.

 

In my experience, the mere mention of ‘story problems’ (sometimes known as ‘word problems’) is enough to send my students into fits. (Google ‘story problems suck’. This is not an isolated issue.)

But stories are the primary way we humans communicate with each other. I use stories in my classes all the time. Sometimes they’re personal anecdotes (usually of the ‘See what I did there? Don’t do that’ variety), historical references or just something I made up like the example above.

I think that the issue with traditional story problems is that they’re not very interactive. Most seem to be little more than a written statement of the math problem we’re trying to solve.

So maybe we need better stories.

One promising project is Oppia. This is a Google project described as a “Tool for creating interactive educational content”. In other words, you can create a story that guides your student through a topic, periodically asking them to apply what they’ve learned to solve a problem within the context of the story.

You can set up your own copy of Oppia on your own computer to test it, contribute to or browse through the lessons at the official hosted server or just use their test server to see how it works. I played with one of the tutorials and it was very engaging. So head on over there and give it a try, let me know what you think.

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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 Blog Review – Math Babe

I was directed to Math Babe by a link from Fawn Nguyen’s blog Finding Ways to Nguyen Students Over and it’s definitely a different kind of math blog than I’m used to reading. For one thing, blogmistress Cathy O’Neill was a former data scientist who co-authored the text Doing Data Science and before that she was a quant on Wall Street. (“Quants” or quantitative analysts are much sought after in the finance sector.)

Dr. O’Neill is currently the director of The Lede Program at the Columbia School of Journalism and she brings her unique perspective to her MathBabe blog. Primary topics of discussion include data science, statistics, finance, journalism and data modeling. Her writing style is friendly but not oversimplified. Her innate sense of humor always shines through, particularly with her regular feature Ask Aunt Pythia where she takes questions from readers in a mock advice column format (Pythia was the priestess at the Oracle of Delphi).

The questions aren’t restricted to finance wonkery, however. For example, this one from “Breasts of Oppression”:

Dear Aunt Pythia,

I am a female physics PhD student. A colleague once said to me that “If women want to be respected, they should not show cleavage.” What do you think?

Aunt Pythia’s answer?

Dear Breasts,

I’ve always thought quite the reverse. Namely, if men had boobs, they’d be showing them off all the time.

MathBabe has earned pride of place in my RSS feed so head on over there and tell her I sent you. (She has no idea who I am but, as I’ve mentioned, I’m trying to start a thing.)

#WeHateMath Book Review: Debt – The First 5000 Years

“Surely one has to pay one’s debts.”

The reason it’s so powerful is that it’s not actually an economic statement: it’s a moral statement. After all, isn’t paying one’s debts what morality is supposed to be all about? Giving people what is due them. Accepting one’s responsibilities. Fulfilling one’s obligations to others, just as one would expect them to fulfill their obligations to you. What could be a more obvious example of shirking one’s responsibilities than reneging on a promise, or refusing to pay a debt?”

Excerpt From: David Graeber. “Debt.” iBooks. https://itun.es/us/3z6RA.l

I was first put on to this book when I heard an interview with the author on the Majority Report podcast. David Graeber is an anthropologist who teaches at the London School of Economics and has a unique perspective on the intersection between economics, politics, religion, war and human nature. As the quote above suggests, this book was born out of a desire to answer a deceptively simple question: How does morality enter into what should be an objective business transaction?

At over 500 pages (including copious footnotes), this is not what you might call a breezy summer read. However, it’s very engaging and Graeber writes in a style that is friendly while still maintaining an academic rigor. He does an excellent job explaining how the very concept of ‘debt’ has hooked itself into some primordial part of our collective souls.

You’ll never look at your mortgage statement in the same way again.

 

References

Graeber, D. (2011). Debt: The first 5000 years. New York: Melville Publishing.

#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.

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.