Why Study Higher Mathematics?

Just about everyone will tell you that mathematics is important and should be taught in school.  All the arguments are over the type of mathematics that should be taught.  Algebra?  Statistics?  Probability?  Real world problems (“John has 137 erasers to distribute among 17 students…”)?  Math world problems (“How many ways can Ellen arrange six different books on a shelf…”)?

The problem is that you can’t answer the question of “What should we teach in school?” until we answer the question of “Why should we have an educated population?”  There are three reasons for getting an education:

  1. To help yourself.
  2. To help your community.
  3. To help your government.

These goals aren’t mutually exclusive:  If you help yourself, for example by getting a better job, then you help the government, because you pay more in taxes (and, because you have a better job, you are more likely to want to keep things the way they are…so you support the government).  Likewise, if you help your community, you’re helping yourself, because you’re making your own living environment better.  At the same time, helping the community might not be helping the government, and vice versa.

This is where things get interesting.  If school mathematics focuses on “real world problems” and “useful mathematics,” then it’s set up to help individuals and the government.  However, there’s no guarantee that this will help the community:  wealthy persons exist, even (and perhaps especially) under repressive regimes.

Here’s why higher mathematics is important.  First, a quick definition (my own):  Higher mathematics is any mathematics that focuses on proofs.  Every branch of mathematics can be taught and learned at a higher level:  3 + 2 = 5 is arithmetic and very basic; higher mathematics occurs when you ask (and answer) why 3 + 2 is 5.

I’ve talked elsewhere about why mathematicians do proofs. Despite my silver-tongued eloquence, not everyone is convinced:  students continue to ask “Why should we have to prove things that everyone knows?”  But consider this:  Throughout history…

  • Everyone knew that men were smarter than women.
  • Everyone knew the sun went around the Earth.
  • Everyone knew that slavery was acceptable.

Progress occurred when people began asking “Well sure, everyone knows these things…but are they really true?”

And this is one of the things that higher mathematics trains you to do:  Proof causes you to ask questions about what “everybody knows.”  Take that “3 + 2 = 5”.  What do we really mean by that?  There’s a few different answers, but one of them is “3 + 2 is the second number after 3.”  (You can go in a few different directions after this point, but that’s another story)

We can go further:  proof requires us to construct a logical argument, with each step based on the step before it.  We can’t make bold leaps, like “Since this happened one time, it must happen all the time.”  Instead, we have to establish a chain of causality, with each step carefully constructed.

We can go further still:  proof forces you to constantly question your own beliefs.  Many people accept 5 \times 3 = 3 \times 5 without question.  But if you’ve been taught to do proofs, there’s a little voice inside your head saying “Why do you believe that?”  If the answer is “Because someone told me it was true”, then you’re uncomfortable…and try to find evidence.

Now for the punchline:  You can slant history to make sure your country is always right, and other countries are always wrong.  Philosophy can be wrangled to serve the state:  China did this for two thousand years.  Even science can be forced into doctrinal correctness:  physics and chemistry did very well under the Soviets.  The arts can be browbeat into submission; literature can be stifled; engineering can be co-opted.

But questioning belief and demanding evidence is an integral part of higher mathematics.  You cannot remove these components from higher mathematics without destroying higher mathematics.

And if the state chooses not to teach higher mathematics?  Then the mathematics needed to solve new problems will not be developed.  Individuals and society will suffer…and the government will be replaced.    Thus I claim:

The development and continued existence of a free society depends on the teaching of higher mathematics.

Two Is the Oddest Number

One of the biggest problems facing anyone in a creative field is:  How do I create something new?  For obvious reasons, you can’t be taught how to be original.  But there are some ways you can make it easier for your creativity to emerge.  Here’s one strategy:  Two is the oddest number.

What does that mean?  One of my thesis advisors (and the original source of the saying) explained it like this.  To a mathematician, there are only three numbers:  zero, one, or infinity.  Either something doesn’t exist at all (zero); it’s unique (one); or it happens infinitely often (infinity).

For example, suppose you have a line and a point.  In Euclidean geometry, there is a unique straight line through the point that is parallel to the given line.  In spherical geometry, there are zero straight lines through the point that are parallel.  In hyperbolic geometry, there are an infinite number of straight lines through the point that are parallel.  Thus zero, one, or infinity.

Or consider primes.  There is a unique even prime:  2.  All other primes are odd.

So how does this help you create something new?  One way is to try and find a second example of something.  Here’s an example, that requires some background in permutation groups.  Here’s the short version, though if you really want to delve into the topic, take a look at my (in progress) videos on abstract algebra.

Suppose I have a set of distinct symbols a, b, c, d, \ldots.  A permutation occurs when I rearrange the symbols.  It’s best to think of the permutation as what happens when you do a “replace all” in a document.  A compact way to represent these permutations is cycle notation, where an expression like (abc) indicates you’re going to replace all as with bs, all bs with cs, and all cs with as.  Because this cycle has three elements, it’s called a 3-cycle.

We can also have 2-cycles:  (ab).  These occur so often that we have a special name for them:  they are transpositions.  

We can juxtapose two cycles (of any length) and form a composition.  For example, (ac)(ad).  For somewhat technical reasons, we read these from right to left.  Thus first we replace all as with $latex $d$s, and all ds with latex $a$s; then we replace all as with cs and cs with as.  The net effect is that all as have been replaced with ds; all ds with cs (because the first cycle replaced them with a a, and the second cycle replaced the a with a c); and all cs with as.  So (ac)(ad) is the same as (adc).

In a similar manner, we can write (ac)(bc) as (acb).

At this point, the mathematician says “Hmmm…two times we’ve managed to replace a composition of two transpositions with a 3-cycle.  But two is an odd number…maybe we can always replace a composition of two transpositions with a 3-cycle?”

So now we try it on (ab)(cd).  And we find…the first component of a rather important theorem in the study of permutations.

Welcome to the 21st Century!

Once upon a time, a teacher could assign homework questions to students, and the students could be relied upon to do the work on their own.

Then writing was invented.

A little later, homework solutions manuals became popular (and Cliff notes, for those in the humanities).  Still later, you have the internet where, if you can’t find the answer to a question, the answer doesn’t exist.

This can be a source of dismay to some teachers:  How can they ask a question and prevent their students from simply looking up the answer?  There’s an easy answer to this:  You can’t.

You have to accept the reality that students will look up answers…and adjust your expectations accordingly.  For example, here’s a question that you could look up online:  How many 3-digit numbers increase their value when the digits are reversed?

As worded, the answer “37” is correct (or not…I’m not going to give it away).  All that’s necessary is for a student to enter the question into Google or post it onto a message board, and someone will happily supply the answer.  This makes it a bad question in the 21st century.

What would make this same question a better question?  One added instruction:  “Explain your reasoning process.”  One of three things will happen:

  • A student will work it out, and be able to explain their reasoning process.
  • A student will go online and get the answer.  If they’re honest, their explanation is “I had no clue, so I asked someone to tell me.”  Otherwise, they might try to come up with a reasoning process…that is completely nonsensical.
  • A student will go online and get the answer and the respondent will explain how they got it.  In order to explain the reasoning process, the student would have to sift through the explanation.  This means that, even though they didn’t originate the thought process, answering the question still has learning value.

What about the student who simply cuts-and-pastes someone else’s reasoning process? One way to limit this is to have the student explain it to you, and ask them to elaborate on each step.  If they didn’t properly assimilate the reasoning process, they won’t be able to explain it to you.

Of course, this relies on having the time to teach.  This is the greatest threat to effective teaching:  overloading the curriculum so material can’t be covered deeply; and overcrowding the classroom so that students can’t get the individual attention they need.

Fortunately, technology takes with one hand, but gives with the other.  One way to make it easier for teachers to devote more time to developing student understanding is the inverted or flipped classroom.

Flip Your Class

Over the years, I’ve done quite a few educational videos for math courses, from Math for Elementary Education, through Calculus and Linear Algebra, and Cryptography.  Originally, I’d intended them to be resources for my students, but unknowingly I had started down the path towards the flipped (or inverted) classroom.

What’s a flipped class?  It’s a brand new thing in education…that’s very, very old.  Back in the Good Old Days (which never existed, but that’s another story), students would seek out a teacher.  The teacher would say “Read these books, and we’ll discuss them next time.”  This had several advantages:

  • The student got into the habit of learning on his or her own,
  • The student got more personal interaction with the teacher.

So what are most math classes like nowadays?  The teacher spends an hour or so lecturing on a topic, then sends the students away to do homework.  The lecture is impersonal:  as a student, you’re one of many; as a teacher, you give the same lecture whether there’s one person or a thousand.

But let’s follow the student after the lecture.  Every student has experienced the following situation:  leaving a lecture, thinking that the lecture was brilliant, since everything made sense.  They go home, ready and raring to tackle the homework, and then…discover that they missed something critical, and can’t do half or more of the problems.  They can try to see the teacher during office hours, or ask questions during class, but even if they can, by that time a week has passed by and the class is on the next topic.

The flipped class alters the timing:  Students get the lecture at home, usually through some web video; then they go to class to work problems.  This works best when students take advantage of several key features of online videos, namely:

  • They can be played anytime, anywhere:  students can learn at their own pace.
  • They can be paused and rewound:  students can learn at their own pace.
  • They can be replayed:  students can review at their own pace.

When the students go to class, they work problems.  This works best when students take advantage of several key features of the classroom, namely:

  • The instructor is there to offer guidance and clarification,
  • Other students are there to discuss potential solutions and to cross-check work.

You’ll notice that the flipped classroom model is exactly the “Read these books and we’ll discuss them in class” model of education.  If you enjoy the teaching process, a flipped class is one of the best ways to experience it.

But be aware…there are pitfalls along the way.  More about this later…