Slides from the math/art presentation

Here are the slides from my section of the presentation.

For the other slides, and the activity, please see Kelsea's and Adam's blogs.

Mathematical understanding and multiple representations

 
In my experience, a good math problem always prompts me to bring out pencil and paper and make a quick diagram, table, or notes. Externalizing my internal representations of these ideas helps me to clarify my thoughts, see new connections, and catch errors with my ways of thinking, so I agree from personal experience that "internalization without externalization is non-holistic and incomplete" (Pape and Tchoshanov, 2001). The idea of representations (both internal and external) as symbols that help reduce the cognitive load when thinking about complicated concepts also made a lot of sense to me, and helped explain exactly why these representations are useful in solving problems.

The article often mentioned graphs as representations for functions, but there is another common reperesentation for functions that was omitted: the input-output machine. In high school I preferred the graph representation, and never really understood the point of the input-output machine, but now I realize that it has some advantages: some functions, for example, are very difficult or even impossible to graph accurately, so an input-output machine is a better framework for understanding such a function. Also, the machine makes more explicit (although often not as immediately obvious) the relationship between the dependent and independent variables. I think it's important, therefore, to teach both representations, and have students explore the advantages and disadvantages of each, so that their internal representations of functions can be as complete as possible.

Fictional letters from future students

Dear Mr. Kuepfer, I wanted to thank you for the way you taught me enjoy math. Your classes were full of exciting discoveries, as you guided me into seeing connections that I had never seen before, and taught me how to solve problems in ways I'd never have expected. I remember that I struggled with math before your class, and found it boring, but your explanations were always crystal clear, and the activities and problems that we worked on were always fresh and interesting. I loved how you would let us explore a difficult problem in groups for a while, giving just enough hints to nudge us towards a solution - that was when I felt I learned the most. I also think those group activities just gave the class a good feel - everyone was included and having a good time, and everyone learned a lot. I hope you're continuing to inspire your students like you inspired me!

Sincerely,
A. Student

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Dear Mr. Kuepfer, I was going over my old yearbook the other day, and I thought I should write my old math teacher. I guess I have some complaints I wanted to air, but never did until now. To be honest, your class confused the heck out of me, and it wasn't until I took a few other math classes that I realized that I wasn't actually bad at math, you just made it too confusing. For the sake of future students, please try to be clearer in your explanations, and stop putting completely new kinds of problems on your tests - it's too much to ask of high school students under the stress of a test. Also, your delivery is boring, and sometimes hard to understand - if you could try to speak more dynamically and clearly that would be very helpful.

Sincerely,
B. Student

My math teachers: the best and worst

 One teacher that I give a lot of credit for my interest in and appreciation of math was my teacher in grades 6 and 7. He was clearly very passionate about math and about students getting a solid understanding, and he was always challenging us to be creative in our math. One of his favourite activities was to write four digits on the board, and challenge us to create expressions with them that were equal to the numbers 1, 2, 3, ... as high as we could. Activities like this were very approachable for every student, but challenging for everyone as well. That philosophy informed all of his teaching: he was patient and clear with students who were struggling, but also gave challenging extensions for students who were ready for them.

My worst experience with a math instructor was in my Calculus 2 course in university. The professor emphasized difficult calculations over conceptual understanding - one assignment was simply a list of 24 very complicated integrals to solve. He also introduced more advanced concepts like the Laplace and Fourier transforms without giving much explanation or motivation for them, then tested us on them on the midterm. Since his lectures were not engaging and consisted of him reading off of slides, I absorbed very little and felt like I understood nothing in the course.

Sept 21 Exit Slip

One group wrote that it doesn't make sense to teach relational understanding early in life, making the connection to how people learn languages. I think the language comparison actually makes an argument for teaching relational understanding from an early age - very young children learn how to express ideas verbally much sooner than they learn to use proper grammar, spelling, and syntax. I think something like spelling is more akin to instrumental math, and while it's important to learn when young, it never replaces the goal of being able to communicate (relational understanding. In math, I think young children should build their conceptual understanding at the same time as, or before, they learn instrumental procedures for calculation.

The Locker Problem

 I had seen this problem a long time ago, and while I didn't remember the exact solution, I had some vague memories that definitely helped. Here are my (slightly idealized) steps for solving it:

My first thought on seeing the problem is that numbers are way too large to solve the problem manually, so there must be some sort of pattern. To look for this pattern, we'll shrink the problem down a little: take the same situation, but say there are only 10 lockers and 10 students. We can draw the following table:

The numbers in squares represent the lockers, the numbers down the left side represent the students, and we put a dot in a cell of the table if the corresponding student opens or closes the corresponding locker. For example, the first student visits every locker, while the third student only visits every third, so we put a dot in those places. The numbers along the bottom indicate how many students opened or closed each locker.

Now, which lockers are open at the end? Well, since they start open, a locker will be open at the end if it is visited by an even number of students, and will be closed at the end if it is visited by an odd number of students. Therefore, in this small example, lockers 1, 4, and 9 will be closed at the end, and all the others will be open.

These numbers suggest a pattern: 1, 4, and 9 are the first three square numbers. Let's take a look back at the table to try to understand why this might be. One thing that we can see is that a student visits all the lockers that are multiples of their number. For example, Student 3 vists lockers 3, 6, and 9. Looking down each column, then, we can see that th n'th locker is visited by each student whose number is a divisor of n.

Now, divisors of n come in pairs whose product is n. For example, the divisors of 6 are 1,2,3,6, where 1x6 = 6 and 2x3=6. Since the divisors come in pairs, it makes sense that most numbers have an even number of divisors. What about square numbers, for example 9? Well, the divisors of 9 are 1,3, and 9, where 1x9 = 9 and 3x3 = 9. The divisor 3 forms a pair with itself, so 9 has an odd number of divisors. In general, a square number will have some pairs of divisors, and its square root, which is paired with itself, so all square numbers will have an odd number of divisors.

This is the key realization that allows us to solve the big locker problem: if a locker's number is a perfect square, it has an odd number of divisors, so an odd number of students will visit it, and therefore it will be closed at the end. If a locker's number is not a perfect square, it has an even number of divisors, so an even number of students will visit it, and therefore it will be open at the end.

On Instrumental and Relational Understanding

I would like to add one more argument in favour of teaching relational mathematics over instrumental mathematics: instrumental mathematics justifies the modern student's common complaint that they need not learn the subject when they have access to a computer. If they are being taught only to follow strict procedures to arrive at an answer, then they are correct in believing that a computer can do this much more quickly and reliably. To stay relevant, we need to teach students more than what can be done by Wolfram Alpha - we need to be teaching relational understanding.

It worries me, though, that my own experience of math in school, over thirty years after this article was published, was still more focused on the instrumental than the relational (and I consider myself lucky to have had some excellent math teachers). I was motivated enough to seek out relational understanding on my own, but in retrospect it's not surprising that so many students complained about not being able to understand math when they were not being taught to understand it so much as to do it. As far as I have seen, the math concepts Skemp suggests as ways to promote relational learning - sets, mappings, equivalence - have not been introduced into our curricula. Maybe this is because of another of his points - math curricula are already overstuffed, and we need to trim away some of the content if we are to teach any content effectively. Since this problem hasn't been solved in the last 30 years, it's certainly very difficult, and teachers and curriculum designers will need to work closely together to make any progress.

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