Beginnings of an Automathography

(This post is part of the coursework for Justin Lanier’s smOOC. Read about it here and check us out here!)

I picked this story because I feel it shows the beginning of a lot of my strengths and weaknesses when it comes to my relationship with mathematics. Here goes.

My first experience with extended mathematical problem solving was in the fourth grade. My math teacher was Mrs. Adair. I was lucky enough to be put in her class with the fifth-graders for math time. On top of a little bit of nightly homework, we had P.O.W.s; I guess that means Problems of the Week, but P.O.W. is what stuck in my memory. I remember that they were always assigned on Tuesday and due on the following Tuesday, and each assignment had one problem, and five steps, which I can still vaguely remember, and here they are:

1. What is the problem? Write it in your own words.

The problem was: how many ways are there to get $1.21 using only half-dollars, quarters, dimes, nickels and pennies? There may have been some story I don’t remember, but it was definitely $1.21.

2. How are you going to solve the problem? Are you going to draw a picture, make a list?

I was a big fan of brute forcing problems, so I was going to make a list. But how? It seemed like there were probably at least 10 or 20 ways, so writing them down might be rough. I complained to my dad, since most P.O.W.s I could knock out in 30 minutes or an hour. He could have told me to suck it up, or to go away, but instead it turned into an Excel lesson. He showed me how to label the columns “Half-dollars,” “Quarters,” “Dimes,” “Nickels,” “Pennies,” and how to enter the numbers using the keypad at the right side of the keyboard. I started at it, starting with as many half-dollars as possible and working my way towards 121 pennies. He guessed there would probably be a few dozen.

3. Show your work. Be neat so that someone else can understand your work.

It turns out that this is not a good problem if you want your kid to give the one family computer back anytime soon. Several hundred rows later, each row representing a way of making one dollar and twenty-one cents, I proudly print the list.

4. What is the answer?

I take it as a good sign that I have NO IDEA what the answer was, other than that it was over 100 and less than 2000 (I think).

5. Did your plan work? What could you have done differently?

I thought my plan worked great! Good job me! (Now it seems mind-numbing and slow just imagining typing in all those numbers, but I was happy that I had an exhaustive method, a feat in itself.)

I would estimate that I spent around three or four hours on this problem (and my dad spent an hour with me). When the next Tuesday rolled around, I felt pretty pleased with myself, since there had been some talk about this being the worst P.O.W. ever. There were lots of different answers; no one had the same answer that I did, but most people had made a list and gotten numbers lower than mine; I had already checked mine over for double-counting, so I knew they had left something out and started feeling even more smug.

One girl, Sophie, had a number bigger than mine, but we ran out of time and all but the two of us were clearly tired of the problem. As recess began, she came up to me. I remember her making fun of my list, that she had a shorter way. She had taken all the ways to get a dollar, all the ways to get 21 cents, and multiplied them! This whole story sticks out in my mind because of this idea, even though it’s wrong. I told her that her way was wrong, that she must have double-counted, but I couldn’t explain why and refused to entertain the idea she might be right.

It turned out later that I was the closest, having left out only a few cases, but this story still comes to mind when I think about two things:

1. My own strengths and shortcomings as a mathematician. Sometimes I try to make up for in speed and brute force what I lack in elegance. I can come up with a plan, but it may not be the best plan and I can stick to it too long instead of hearing others out.

2. What does it mean for a mathematical idea to be a good idea? Does it have to be correct? I used to think that correctness was a necessary but not sufficient condition for a mathematical idea to be good, but now perhaps it’s neither?

8 comments
  1. suevanhattum said:

    Nice!

  2. Jeff d. said:

    This reminds me of the scenario in class when you (think you’ve) set up a task perfectly, practically forcing students to find a more efficient way, and yet you have that one group who is more than happy to try to use the brute force…they end up making a table with 1200 entries or hitting “=” on their calculator 500 times. I never stop them because they are, in their own way, establishing that need for efficiency and yet they still know what’s happening conceptually. Fun stuff.

  3. Yeah, I’d be interested to know what the solution my teacher (or whoever she got it from) had in mind, since careful enumeration is the only way I can think of that would be accessible to a fifth-grader.

  4. Love this story, both for how it played out, what you learned, and how the different relationships (yours and Sophie’s, I guess) played out. How did your teacher debrief it with your class? Whatever happened to Sophie? (does she teach math now?) I feel like I come at math problems with a brute force approach as well and am trying to get better at being strategic. Perhaps we should solve this in our sMOOC? And I feel like I am getting further and further from correctness being necessary but sufficient for a mathematically good idea. Some of the students I worked with this year had amazing thoughts, even if they didn’t get the correct answer (and desperately needed to hear that they were being smart, mathematically). Thanks for sharing!

  5. Thanks!

    I vaguely remember a sense of class-wide disappointment, since none of us got it right, and the problem was way too big for the allotted class time, so it was tabled. As for Sophie, no idea.

  6. What a wonderful first anecdote as you plumb your mathematical past!
    I’m curious to find out how some of the details of your story play out in future “episodes.” Did your acceleration in math classes continue, and how did that go for you? Will your experiences with math and with computers continue to intertwine? How will your math relationships with others—teachers, family, peers—evolve over the years? I guess I should stay tuned!
    I’m also struck by and appreciate the range of emotions that’s packed into this one anecdote—feeling in turn fortunate, excited, pleased, smug, and something like frustration and stubbornness in your conversation with Sophie.
    Great start!

  7. Mary Olsen said:

    I can relate! I often did my math problems in a longer more difficult way. I sometimes still do. I remember the first time teaching proportions to my 7th grade class a light bulb went off in my head saying wow….I could have solved soooo many problems much easier if I just understood proportions when I was a young student. I still find myself doing things a more difficult way at times and love it when my students show me another (easier) way to solve the problem we are working on. Probably why I always emphasize there is often more than one way to solve a problem.

  8. Cristina said:

    David, what you call “brute force” I’m calling “discipline, self-confidence, ingenuity, and a darn good amount of sticktoitiveness.” I love that you had that kind of fortitude in the fourth grade- to spend whatever time it took to come to an answer, even when the process was not the most aesthetic, nor the correct answer was achieved, but you were making discoveries of your own, and I think that turned out to be the most valuable lesson, after all.

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