## Mathematical Thinking vs. Mathematical Content

I had an epiphany recently regarding my teaching.  This past spring I had decided to include a unit on Rubik’s cube.  My goal was to introduce cyclic notation, commutators and conjugates as methods for devising algorithms or uncovering appealing patterns on the cube.  To do this I would need the students to be able to solve the cube.  I decided to give them quiz credit for being able to solve the cube in under 10 minutes using a sheet of notes.   Most of them used the solution on Rubiks.com which is very straightforward.   Here is the material for the epiphany.  I did this, gave quiz credit, because I needed them to be able to restore the cube as a matter of practical importance.  When playing around with the mathematics that I wanted to share with them you enviably lose track and end up with a scrambled cube.  I felt a little guilty giving them a solid grade for this accomplishment because they were just learning an online solution.

Months later I realized that when they were solving the cube they were, in effect, going through the same thought process that many students have in a mainstream math course.  You begin with a problem similar to a previous problem.  You have been taught methods of rearranging, modifying, massaging or what have you in order to reduce the problem to another problem or solve it altogether.   If I ask a student to differentiate a function such as $\sin(e^{x^2-4x})$ then they recall a number of different facts and algorithms.  The derivative of the sine function, the chain rule, the derivative of the exponential function and the power rule. They apply these in a certain order to arrive at a product of their manipulations.  Contrast this with solving the Rubik’s cube.

You start with a scrambled state, unlikely to be one you’ve seen before.  You probably begin by correctly positioning and orienting the four edge pieces of a particular side forming a ‘plus sign’ (usually called the  bottom or top cross depending on how you hold the cube).  From there you proceed to position and orient the corners of that side which restores that layer of the cube.  This continues.  At each step there are various possibilities and accompanying algorithms for those possibilities.  The student has to decide which algorithm to use to accomplish the desired result given the initial state.

This is certainly analogous to the type of thinking we have our students perform during drills and practice and so is at least as valid as that mode.  However there are some key differences, differences that I can use to my advantage.  When a student successfully solves Rubik’s cube lay persons are generally impressed and interested.  They tend to engage the student and ask questions.  They might even ask if the student can teach them. So here we have a situation where a student who has likely not felt competent in mathematics and has likely been viewed that way by others ending up in a potentially empowering position.  They have learned something of interest to other people.  Something that is deemed difficult (and frankly it does take time and patience to learn these algorithms) and desired.  It’s not often the case that when you work out a problem from traditional mathematics content that lay persons engage you with praise and interest.  “Goodness, did you just work out that integral!  That’s impressive, what trick did you use?  Could you show me?” It’s also not likely the case that after a rousing good semester with partial differential equations you went home to regal your folks with solutions of the heat equation (though I used to send my mother Mathematica graphs of the functions with a short explanation :).

For non-STEM students the reason to teach them mathematics can’t be mathematics content which seems to be entirely for the purpose of preparing a student for calculus.  It must be that we hope that the process of studying mathematical content will result in mathematical thinking that they will carry with them after they have forgotten the content specifics.

Let’s put this in context here, because it’s important.  I teach a college course MAT101.  By the time my students walk into my classroom they have completed 11-13 years of mathematics instruction and I have them for about 14 weeks.  Whatever grand ideas I can imagine I have to set it into this reality.  This is not their only class.  This is an attempt to satisfy a requirement in a less painful way and hopefully an easier way.  The numbering of my course is suggestive: 101.  There isn’t a MAT100 on campus.  They’re generally not walking in hoping to undo all the unsatisfying experiences of their past.  They’re hoping to survive, to be done with mathematics and to move on with what they really enjoy.  I don’t mean to be negative.  The students are very often wonderful to work with and this class is my favorite to teach.  I’d prefer it over any other class, however I have understand how they perceive the class (through surveys) and work with that as my starting point.

I already have a course which assumes little prerequisite knowledge (I don’t want the past to continue to convince them of what they can’t do).  Now I have another goal: choose content that has or could have, lay audience interest.  That is, choose content that if performed in a public area is more likely to illicit comments or questions, Rubik’s cube being the easiest example.  In fact I have a number of properties I’ve been thinking about when looking at content in addition to public interest.  For example, it’s clear when you have solved Rubik’s cube.  It’s not often clear (for those that struggle with the skills) when an equation is solved correctly.  We as teachers attempt to give our students methods of checking their work and these are important but these almost assume the same skills, or in some cases, deeper understanding, in order to carry out.  The ability to creatively experiment or engage in some sort of ‘play’ or fiddling with the possibility of discovery would be another important property.

Another property would be some cultural connection, though this is tied in with public interest.  Consider string figures (which I’ll eventually write about here) of which cat’s cradle is a two person version.  There are a great deal of string figures from around the world in many  cultures and this activity has many mathematical aspects, is visually appealing and is the sort of thing you’d like to share with others.  It, like the cube, also invite different levels of problem formalization which is a neat idea I came across somewhere on the internet.  Certainly one of the things we do as mathematicians (scientists as well) is to formalize a problem so we can understand it better and break it down.  Develop a language to talk about it, to communicate ideas.  Build a model to work out various aspects of the problem, etc.  For the cube and string figures, building a language, a short hand seems very necessary.  Writing out prose would be prohibitive (check out the online version of Jayne’s classic book on string figures: http://www.stringfigures.info/cfj/) or at least cumbersome.  Breaking down the process of string figures in order to experiment with different algorithms and the resulting figures is a wonderful way to connect mathematical thinking with tactile manipulation and culture studies of pre-literate societies.  (check out James Murphy’s excellent book on this topic, he taught string figures at a high school in NY for many years).

Beyond those two examples: origami and other folding math (Fold and cut theorem for example), Blackjack and card counting, paper topology (lot’s of room here for predict and check activities), compass and straightedge tessellations (particularly Islamic geometry art).

The properties mentioned above also have the potential for another benefit.  They might be able to save me from myself.  It’s very easy to become excited about a topic in mathematics and to want to share it with your students.  It’s hard to tell what will fly and with whom it’ll fly.  For several semesters I followed a unit on Fermi problems with a unit on ridiculously  huge numbers (exponential growth, factorials, Knuth arrow notation etc) and would include such stories as Borge’s Library of Babel.  While some students really enjoyed this unit, it mostly fell flat.  The mathematical content seemed too similar to traditional mathematics and the skills needed to really enjoy the content, while basic, were frankly out of reach (whether because of self confidence or inability I’m not sure).  Likewise I sometimes would follow that unit with an introduction of infinity.  Again, some students would be floored by these ideas, but for most not so much.  However, the abovementioned unit on Fermi problems generally went quite well, perhaps because of the perception of no single  right answer?

The mentioned activities certainly are used as vehicles to introduce and work on what would  be considered traditional mathematical ideas, but they’re pretty well disguised.  Students find themselves challenged and engaged and it’s that second bit that I constantly strive for.  Watching my students glow with pride having solved the cube, watching them help each other, watching them find patterns and share them, watching them fiddle with their cubes instead of their phones, watching them play blackjack and have wonderful conversations about card counting and odds these have been some really great experiences to have.

The hope is that by focusing on novel and engaging content, content that they want to share and content that others want to hear about, that my students will feel a sense of confidence or enjoyment in an activity they didn’t think could happen.  It takes objects of mathematical thought and makes them relevant to their lives.  Mind you, these kids know that mathematics is important and they understand that their phones and what not rely on mathematics, but they don’t want to build phones or power grids.  However, some of them may want to tessellate, solve puzzles, fold paper or play games.   If you’re the first to introduce them to this activity and if you can get them thinking mathematically about it, that connect may well persist and blossom.

Many ideas presented here today have been bubbling in my mind for the past two months.  I hope to develop them more this fall and will try to blog as the 101 course unfolds.