Thursday, May 23, 2019

Computers in Math, Part 1: Authentic vs. Inauthentic

Over break I had a brief conversation with Melynee Naegele (@MNmMath, blog: mNm Math) about the appropriateness of having computer-based standardized assessments. In particular, they raised the issue of how restrictive the medium of computers can be. It got me thinking about what a coherent philosophy about computer-use in math class can be. I started to write this blog by listing what kind of math computers should/shouldn't be asked to do. But then I realized that there is a more important guiding philosophy that I needed to think through and articulate before going into that. So I'll introduce it with an analogy:
  • "You wouldn't chop down an oak tree with a butter knife--you wouldn't butter toast with a chainsaw."
    • I mean, you could. It would suck. And if it was all you had, I guess you'd have to make do. But it would suck.
    • This is because chopping down trees is an inauthentic application of the butter knife tool. It would be an authentic application of the chainsaw tool--it's what it was made for, it's what it's good at.
  • Big Idea: Part of learning math is learning when to use what math tool.
    • What is a math tool? Here, when I say tool, I mean more or less a physical or technical devise, like spreadsheets, rulers, calculators, lasers, string, Latex, MATLAB, graphing calculators, etc. I don't mean intellectual tools like, "Try a simpler problem," or "try a different representation."
    • What do many people think the main math tools are? When a lot of people think of mathematicians, they think of someone standing at a blackboard, with chalk, just scribbling all over the place. In my math studies, this is common, in one form or another. Personally, most of my own math education has happened in pencil, on thousands of sheets of blank white printer paper.
    • How have computers-based tools been used in math?
      • The Traveling Salesman Problem, a classic graph theory problem, was ultimately solved with computers, because the nature and volume of calculations and algorithms were enormous.
      • At least two of my calculus professors were disappointed in us when we weren't proficient in MATLAB, a computing platform with powerful applications in math and science, especially with partial differential equations, which have tremendous applications in the physical sciences. MATLAB can really allow you to see and experience what happens when you change your model, conditions, and representations.
      • Almost ALL of my college math professors were incredibly bummed when we weren't able to type up our problem sets in Latex, the premier language for typing math. This is because it can be way easier to read typed out math than handwritten math.
      • Most of the common fields of applied math (finance, physical sciences, etc.) would be almost impossible without spreadsheets. Ask anyone who knows me: I'm the spreadsheets guy. The capacity to manipulate, analyze, and represent huge amounts of data is immeasurably better than what humans can do alone.
      • Modern cryptography literally exists because of computers, and our need to publicly transmit private information.
    • What if I use a tool inauthentically? You waste time, energy, and it sucks, generally.
      • What if I asked you to use a spreadsheet to find the square root of 80? You open a spreadsheet, figure out what function to use, what the syntax is, click some cells, and boom: 8.9ish. If I only asked you to do computations with spreadsheets, you'd probably think I was ridiculous for not using a calculator this whole time, and you'd have NO idea of the utility of spreadsheets. Why is that bad?
      • Because what if I THEN asked you to find the how many numbers less than 10,000 weren't divisible by a square number (square-free)? My gut tells me there's a reasonable way to do this with combinatorics, but when I first experienced this math problem, my instinct was guided by some slick spreadsheet functions, which revealed some very interesting stuff! (No spoilers!)
      • If all I knew about spreadsheets was that they could do computations, then I would be missing ALL the wonderful things that they can do, and feel forced to use less-good tools to do the job. Or worse, I might come to not feel like spreadsheets are good for anything!
  • The difference between math education and professional mathematics
    • There is definitely a difference. In math education we have to deal with things like grades, lots of standardized assessments, a criminal lack of funding and resources, and children, parents, and teachers/administrators who may not themselves be experienced mathematicians. In general, math education is a different field of mathematics, with different needs and capabilities. So the use of tools will necessarily be different.
      • So this raises a new dimension of authenticity. What are authentic math tools in the field of math education, that might not otherwise be authentic for professional mathematics. For example, one of the biggest applications of computers in math education is assessment. There aren't very many super meaningful "math assessments" that professional mathematicians engage in. I guess there's the Putnam?
    • But this difference doesn't mean that the math that children experience has to be super different from professional mathematics. The goal of math education is to teach young people how to think and do like mathematicians, because it is a useful and interesting way to think and do things. So ideally, math education would teach students how mathematicians work by asking them to do mathematical work, and have all the mathematical experiences that shape the way they think and do math, and things in general.
    • So as a teacher, when deciding what tools to teach your kids to do what math with, the degree to which you teach your kids to use the tool authentically, is the degree to which they will develop a REAL understanding of the tool, the math they did with it, and what it means decide how and when to use tools in math.
  • Conclusion: Computers don't belong in math any more or less than patty paper, calculators, pencils, chalkboards, string, and lasers. They are all tools of math, with the power to be authentically used (to great benefit), or inauthentically used (to great detriment).