Friday, August 3, 2018

The Mathematical Method

What/Why/How is Math?
I have the chance this year to teach a semester-long math elective for juniors at my school. I have pretty much full curricular freedom. The course is titled "Discrete Math," so I intend to focus on the topics of Modular Arithmetic, Number Bases, and possibly Graph Theory. I am excited at the opportunity, and am super glad I had a chance to see Joey Kelly teach two semester-long Discrete Math courses during my teaching residency.

As I sit here this summer and try to figure out what I want to teach, I have found myself slipping down the big philosophical rabbit hole of "Why?" I think that many math teachers believe that our objective is to not only teach specific math content (e.g., polynomial multiplication, estimating fractions, converting numbers to binary). We are also trying to teach students about the bigger general practices and skills that guide mathematicians. Understanding the specific content standard that the powers of i cycle every 4 powers is only long-term helpful for a kinda-narrow band of people. But teaching students to notice patterns, explore and discover truth, state and argue their position based on evidence...those are about as transferable as skills get. I believe that we are trying to teach students to think and work like mathematicians, because the way that mathematicians think and work is powerful and useful. The more math that I experience as a learner, the more I develop an understanding of what it means to think and work like a mathematician. And I have been looking for a schema that allows me to understand what that actually means.

The Common Core Standards for Mathematical Practice capture some of this, by removing mathematical work from specific content standards. I have tried to think about how to value these practices in my math teaching, alongside specific content standards. But these math practices feel disjointed, and they still haven't helped me to develop a unified "Big Picture" of "What/How" mathematicians do. But I think I have an idea of how I can begin think about this, and I want to run it by you folks.

Introducing the Mathematical Method
Scientists have a Scientific Method that reflects how they "do science." What if we had a Mathematical Method that reflects how mathematicians "do math"? The goal of developing and understanding this Mathematical Method would be to offer a vision of what mathematical work can look like, across math content. I have created a first draft of what I think this method could look like. Below, I ask some questions about this method and offer some initial responses. I would love to hear your questions and thoughts about this idea.

The Scientific Method
The Mathematical Method

Questions I Have
  • Are there any other teachers, scholars, or researchers that have done work on this kind of "Mathematical Method"?
    • These are the only two places where my research has found a discussion of a math equivalent to the Scientific Method. Both of these two articles claim that the mathematical goal of absolute proof makes the work fundamentally different from science. They don't discuss the ways in which the similarities can/should be described.
  • If math requires absolute "proof" in a way that science typically can't--does that affect the process?
    • I claim that in both math and science, you take the same big step of proof writing, where you develop an intuition as two why your conjecture/hypothesis SHOULD be true. We believe that humans are driving climate change, because it makes sense that many of the things that humans do directly affect the climate. But can we "prove" this in the same way that we can prove that the Pythagorean Theorem?
  • What are some consequences of the fact that this Mathematical Method seems similar to the Scientific Method?
    • Our science teacher friends have been teaching and advancing the Scientific Method for a long time now, and from a very early age. If we can present a parallel structure between the two methods, then we can take advantage of the fact that students already kind of understand this Mathematical Method (insofar as it resembles the Scientific Method).
    • If the methods are so similar, this can make the distinctions between the two methods more obvious.
    • If both science and mathematics seem to have a similar process, that seems to suggest that there is a broader structure of "problem solving." 
  • How is this Mathematical Method different from the Scientific Method?
    • That penultimate step of the Mathematical Method, "Develop a Conclusive Complete Proof," doesn't apply in many (any?) other fields.
    • The language of the processes is different, even if they are describing largely the same concepts (hypothesis vs. conjecture vs. thesis). More philosophically, "prove" means different things in math, science, and other fields.
  • Should we teach to this bigger "Problem Solving Process"?
    • Should we just teach a generalized "Problem-Solving Process" that can transfer between problems in math, science, and other fields? Drawing connections between big ideas within and across fields of knowledge feels useful, and allows a student to better package and organize their knowledge. This feels like part of a bigger question of what it means to "know" something?
    • What does this generalized "Problem Solving Process" have to do with "Critical Thinking?"
    • There are meaningful differences in the different problem-solving processes in math, science, and other contents. If they teach this process in science classes, in addition to teaching "general problem solving," should we do the same in math class?
  • What about content areas outside of STEM? Is there a "Historical Method?" An "Artistic Method"? A "Language Arts Method?" Do they have a similar "process"?
    • I feel like any content area in which students "solve problems" likely has some kind of general problem-solving process. I definitely don't know enough about these other content areas to begin to answer this question!
    • From my own experience in my high school English Language Arts classes, it feels like there are parts of ELA that aren't problem-centered. In particular, it seems like in parts of ELA the goal is to create or communicate something. (Personally, I always loved writing short fiction.) But there are problems posed in this process too, right? How can I make the reader "feel" what I want them to feel? How can I communicate exactly what I mean to communicate? Is there another articulated "process" that writers and artists use in this situation? This also makes me wonder: what are the things we do in math that aren't problem-centered? Which of these "processes" fit in "Critical Thinking"? Which don't?
  • How do the Math Practices fit in?
    • I think that mathematicians use the math practices throughout the Mathematical Method, some more at certain times. For example, during the Exploration phase, a mathematician will do a lot of MP8 where they "look for and express regularity in repeated reasoning." This is how they might develop their initial suspicions that something mathematical is going on that they might not yet understand. And at the end, when in the phase where they Communicate Results, mathematicians will do a lot of MP3: "Construct viable arguments and critique the reasoning of others."
    • I think science teachers have a similar set of Science Practices as a part of their NGSS standards. A lot of them feel similar to the Math Practices. For example, #2 "develop and use models" connects to MP4 "Model with mathematics." How do science teachers combine the Scientific Method with their own Science Practices?
An Example of the Mathematical Method
Maybe an example would help me to think through Mathematical Method. I will use a common Algebra II problem, based on the powers of the imaginary number i.
  1. Develop a Question: What is i^357? (Or more generally, any power of i.)
  2. Explore: use your present understanding of i to calculate some easy powers of i. What is i^2? What is i^5? What is i^6? What is i^9? 
  3. Make a Conjecture: "All odd powers of i are i and all even powers of i are -1."
  4. Test Conjecture (with strategic exploration):
    1. Strategically Explore: Test a bunch of odd powers of i. The 3rd, 5th, 7th. Maybe you remember to check the 1st power. You get -i, i, -i, i respectively, and realize that your conjecture is wrong because it doesn't account for this cycling between i and -i. You realize this same problem comes up with the even powers of i.
    2. Revise Conjecture: Odd powers cycle between i and -i, even powers cycle between -1 and 1. And you go through all four before you see a power again.
    3. Strategically Explore: Test a bunch of powers of i. This time maybe you make a table, or a diagram of some kind, and work more systematically through all the integer powers of i starting at 1.
  5. Confirm Conjecture: Your strategic exploration has shown you that you see your repeating cycle of i, -1, -i, 1. You also notice that the 1 always shows up when the power is a multiple of 4, i when the power is one after a multiple of 4.
  6. Develop Intuition As To Why Conjecture SHOULD Be True: Depending on how you have come to understand what i is...
    1. (Algebraically) i = sqrt(-1): you know that when you square i you get -1. And if you square that again, you get 1. Squaring a square is the same as taking something to the 4th power. So it's like every group of 4 i's you multiply get together, and turn into the multiplicatively useless 1. And you take away all the groups of 4, till you have 0, 1, 2, or 3 left. So every power of i is really just one of those 4 situations. (This becomes easy to articulate when you have modular arithmetic in your toolbox.)
    2. (Geometrically) i <==> quarter-rotation. Every 4 quarter-rotations you have gone back to where you started. So it's like you didn't even do anything. So you can subtract 4's from your exponent until you get to 0, 1, 2, or 3.
  7. Develop A Conclusive Complete Proof (here, I chose a more algebraic proof)
    1. For any natural number n, n=4s+r for some r in the set {0,1,2,3}. (Depending on your level of rigor, you may need to prove this as a lemma, or just cite an existing proof.)
    2. i^(n)=i^(4s+r), given
    3. i^(4s+r)=(i^4s)*(i^r) =((i^4)^s)*(i^r) by exponent laws
    4. (i^4)=i*i*i*i= -1*-1 = 1 by definition of i and some hand-wavily used axioms (definition of exponents, association, definition of additive inverses)
    5. ((i^4)^s)=1^s, from above
    6. 1^s=1 by definition of 1 as multiplicative identity
    7. Therefore i^n=((i^4)^s)*(i^r)=i^r, where r=0, 1, 2, or 3.
      1. If r=0, then i^n=i^0=1
      2. If r=1, then i^n=i^1=i
      3. If r=2, then i^n=i^2=-1
      4. If r=3, then i^n=i^3=-i
    8. Specifically, 357=4*89+1 ==> i^357=i^1=i. Q.E.D.
      1. Since I have written the rigorous proof I wanted, I may want to go back and work out some language so that it is clear that my proof results here are actually saying the same thing as my revised conjecture).
  8. Communicate Results: publish my proof in my high school journal of mathematics. Win Fields Medal, have it stolen immediately, and so on...
I think that the "experimental phase" of the mathematical method is super under-experienced in math, and we push kids to "believe" pattens that we tell them to believe (like the patterns in the powers of i, or the volumes of prisms, or that tangent is the quotient of sine and cosine). The degree to which students are forced to innovate their own explorations, develop their own conjectures, and convince themselves and others of their findings, is the degree to which they understand and believe their own findings! And this is also the degree to which students own and live by this "problem-solving process." One design problem that math teachers face is how to make sure we pose problems that allow students to go through this process. We also have to decide when we want students to go through this process, and when we don't.

What does the #MTBoS think about this schema for thinking about what mathematicians "do"? Does this feel useful to you all? Given my own developmental space (a few weeks from starting Year 2), this feels like something I'm going to keep in the back of my head throughout this year, and then maybe next summer I'll think about what it would mean to teach to this Mathematical Method.

In general, this feels like a tremendous opportunity for math teachers to collaborate with their teacher friends outside of the math world. Where is the #STBoS? I should definitely talk to science teachers and see how they teach and support the Scientific Method. Does this mean that math class can/should be filled with labs in the same way that science classes are (ideally) filled with labs where you are actually "doing science"? What do our teacher-scientists have to say? What about the #ELATBoS? How does this include understanding history? Art? Literature?

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