Category Archives: Math Communication

Every x has a story

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Recently I attended training on a platform our Math Department has chosen as a tool in their Algebra curriculum next year.  Part of the training involved, unsurprisingly, doing some math problems. I would choose x and y as my variables as I wrote equations, but I noticed none of the math teachers did. They would choose s, b, or h.   “Don’t you guys use x anymore?”, I asked. “No”, I was told, “math can and should tell a story”. The choice of the variable can help tell the story, and so, when asked to determine how many books a certain number of student read in so many hours, s as the variable for student continues the story, as does h for hours and b for books. I was delighted because there is nothing I like more than storytelling. And I agree, the story and the problem make more sense when the variable name has some kinship to reality.

Then I learned the x and y axis can be called the independent and dependent variables.  Oh, my! What stories one can tell when you consider things that way.  Now, I know music and math are intertwined, and I had a breakthrough when I was listening to Malcolm Gladwell’s podcast Revisionist History on country music http://revisionisthistory.com/episodes/16-the-king-of-tears.  Country music songwriters tell some of the best stories. These songs have a dependent and independent variable! The amount of bad luck and heartbreak are variables dependent on the actions of the independent singers and their choices.
bl+hb =(s*c)/2.  Thank you Math Department. Every x does have a story.

An Algebra Teacher’s Lament

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Here are two questions that were on my recent Algebra 2 with Trigonometry test:  A car dealership offers a 10% discount on a car.

a.  Write a function, f(x) to represent the price of a car after the discount.
b.  If the original price of the car is $20,000, what is the price of the car after the discount?

Could you answer these questions?  Is one easier than the other?

I pose these questions to you because I believe that these are questions that most people with high school math education should be able to answer.  In fact, these were on my test somewhat by accident.  These were lead-up questions to a much more advanced topic, composition of functions, and as such we were not in a unit on percentages.

As my title suggests, so many of my Junior students in this elective upper-level math class were unable to do this correctly.  In fact over half of the students got [a] wrong.  Of those who got [a] correct, only one student wrote an efficient function:  f(x) = .9x.  The other “correct” responses included:  f(x) = x-.1x and f(x) = x-(x/10).   While these are correct they don’t represent a clean, efficient solution.  It’s almost like a student who writes a sentence and you know what they mean but there is incorrect grammar.  It’s the same here.  I know what they meant and I appreciate their reasoning, but as Junior mathematics students I expect the more “grammatically correct” version.

Also interesting, 25% of those who got [a] wrong, were able to do [b] correctly.  Here are two examples:

What does this tell us?
These students can do the problem, but can’t connect it to math language.  Our students can do problems, but they can’t communicate as Mathematicians.

Finally, many of the students who go this wrong wrote something along the lines of f(x) = .1x and calculated the new price of the car to be $2,000.  Here’s an example:

It’s amazing to me how many of them didn’t realize their mistake after seeing that the new car, by their calculation, would be so inexpensive.  And this gives us the other important lesson:  that our students often forget to interpret their answer and apply number sense.  We would hope that there is an intuition as to how big 10% is, and a feeling that 10% off means 90% left and thus that the answer should be close to $20,000.

As sad as this grading experience makes me, I find it somewhat fascinating too.  And, it makes me excited for the changes we’re making in the Math Department, and in particular the way we’re planning to welcome our freshman into our new Math culture that emphasizes communication, understanding and number sense in addition to doing.  Too often, the focus of Math classes is just on the doing, and solving problems, this ability to communicate and make sense of our work gets lost.