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:

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.

Earlier, I described a new approach to teaching that is very old:  the flipped or inverted classroom.  Here I’ll start describing some of the nitty-gritty details, and some of the pitfalls.

The most important thing to remember:  A video lecture is NOT an ordinary lecture that has been videotaped.

A lot of folks have videotaped their lectures and put them online.  These videotaped lectures have their role, but in general, they don’t work for flipped courses for two important reasons:

• A lecture is 50+ minutes with a captive audience.
• A human visual field covers 120+ degrees.

What this means is that you can talk for a long time on a topic, and you can cover the board with lots of information.

In contrast, a person watching a video on the internet has an attention span of about 5 minutes.  (Be honest with yourself:  How often do you sit through the entirety of a 15 minute YouTube video, and how often do you fast forward until you get to the interesting bits?)  This means if you videotape a 50 minute lecture, most people will watch it for five minutes…and it is nearly impossible for someone to fast forward through the lecture to find the relevant parts.

There’s another problem:  that video also covers maybe 60 degrees of the person’s visual field.  This means you have considerably less space to present information.  When PowerPoint first came out, most users forgot about that, and loaded slides with hundreds of words and loads of information.  These days…most users still don’t understand that if a slide contains more than about 100 words, it will be unreadable.

What does this mean for math teachers?  I’ve adopted the ten minute rule:  If the material takes longer than 10 minutes to present, break it down into 9-minute segments, and include 1 minute at the beginning to review the material of the previous 10-minute segment.

So here’s the first step in flipping a course:  breaking the entire curriculum into 10-minute segments.  If you’re a gonzo mathematician like myself, you’ll do this all at once, and even for courses you have never taught like abstract algebra.  But if you’re a normal, sane person who wants to stay that way, you’ll take a more sensible approach.  I’ll talk about that in a future post.

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.

Prime and Composite Numbers

If you’re a mathematician or a student of mathematics, you know what prime and composite numbers are.  But if you’re someone who teaches mathematics, you’ll find the standard definition of prime and composite numbers are problematic.  In the following, we’ll take apart the standard definitions, and come up with a better one.

Here’s a common definition:  A prime number is a number that can only be divided by 1 and itself; all other numbers are composite.

At this point, we have to inject some fine print:

• Actually, any number can be divided by any (non-zero) number:  5 can be divided by 3, since $5 \div 3 = \frac{5}{3}$, so we have to specify that we are talking about whole number divisors.
• Oh, and 1 isn’t prime, even though its only whole number divisors are 1 and itself.
• Incidentally, 1 isn’t composite either.

Thus, we get the following standard definition:  Any number larger than 1 is prime if it has no whole divisors besides 1 and itself; the number is composite otherwise.

Now there’s nothing really wrong with the standard definition…if you’re a mathematician.  But in terms of doing anything with a prime number, it’s a good example of how our choice of definitions can make mathematics easier or harder.

There are two problem with the standard definition of prime.  First, to most non-mathematicians, divisors implies division, so to find if a number is prime, we see if any number divides it.  Thus if I want to find whether 5 is prime, I’ll check $5 \div 2$, $5 \div 3$, $5 \div 4$, and since none of them work, I’ll conclude 5 is prime.  But this means that in order to verify whether a number is prime or composite, we must divide…and division is the hardest of the elementary operations.   As a result, this definition discourages verification!

There’s a second problem:  it defines a prime number by a property it doesn’t have. Determining whether something has a property is relatively easy; determining whether something doesn’t have a property is harder, and tends to produce answers like “It just doesn’t have that property.”

What do I mean by that?  I’ve asked questions like “Prove 5 is prime” on assignments, and more often than not, I get answers like “5 has no divisors, so it’s prime.”  While this is true, it misses the point of a prove question:  A politician can claim something is true without giving any supporting evidence, but a sensible person is supposed to do better.   Mere claims are meaningless:  you’ve got to present the supporting evidence.

So here’s an alternative definition:  A number is composite if it can be expressed as a product of smaller numbers.  A number greater than 1 is prime if it cannot be expressed as a product of smaller numbers.

Note several features of this definition:

• We don’t have to specify we’re dealing with whole number products.  While it’s true that 5 can be expressed as a product of other numbers, none of these pairs involve smaller numbers.
• This also makes clear from the beginning that 1 is not composite, because it can’t be written as a product of smaller numbers.
• We avoid division and focus on the easier operation of multiplication.

What about “Prove 5 is prime?”  It’s still possible to respond with a simple assertion:  “No product of smaller numbers gives 5.”  However, because the focus of the definition is on the product, and not the property, it’s more likely to cause a respondent to recognize they must provide evidence.

More importantly, students sometime have difficulties trying to prove “5 has no divisors.” I suspect it’s because they see “No divisors” and the immediate response is “But the divisors could be anything at all, so where can I start?”  In contrast, “No product of smaller numbers gives 5” automatically limits the scope of the problem:  products of smaller numbers.

I’ve started using the above definition for prime and composite numbers in all the classes I teach.  I hope you’ll start to use it as well.

Why Do Proofs

Proof is an essential part of mathematics, but it’s sometimes not clear why we prove things in mathematics.  By way of example, let’s consider a simple problem:  Multiplying two even numbers.

If you were a scientist, you would collect data. For example:  $4 \times 6 = 24$$2 \times 8 = 16$$10 \times 10 = 100$, and so on.  After a few hundred pieces of evidence, you would form an hypothesis:  The product of two even numbers is even.  You might then perform some more experiments to test the hypothesis:  $82 \times 14 = 1148$$14 \times 32 = 448$$11 \times 4 = 44$ (then sheepishly realize that $11$ isn’t even, so we’ll ignore that), $24 \times 4 = 95$ (better recheck that…yup, it’s supposed to be $96$), and so on.

But if you’re a mathematician, you’d next try to prove that the product of two even numbers is an even number.  The first thing to realize is this:  No one ever tries to prove anything they don’t already believe to be true.  If you asked me to prove the product of two even numbers is an odd number, I’d say “But that’s not true…why should I waste time trying to prove it?”  What this means is that we don’t prove things because we doubt their truth:  proof isn’t a way of obtaining truth, because in some sense, we already have it.

So why do we prove things?  Let’s consider what we’d have to do.

• First, what do we mean by “the product of two even numbers?”  We need to clarify what we mean by an even number:  “I know it when I see it” isn’t good enough.  We come to the conclusion:  An even number is any number that can be written as the product of 2 and some other whole number $m$.
• So now we multiply two even numbers:  $(2m) \times (2n) = 4mn$.
• But we need to write this as a product of 2 and some other number, so we have $4mn = 2 (2mn)$.

There’s our proof.

But stop and smell the roses.  Halfway through, we found the product of two even numbers was $4mn$.  This means that the product of two even numbers is in fact a multiple of 4.  If we go back to our data, we’ll see that it’s there…but it might not have been obvious at the time.

So what has this effort of proof given us?

• We had to dig down deep and elaborate on what we man by “even number.”  In general, proof requires us to define our terms in a meaningful and useful way.
• We had to use a little bit of algebra.  In general, proof requires us to review mathematics we’ve already learned.  (In fact, since most terms in mathematics have a standard definition, the first step also requires a review of the mathematics we’ve learned…or the mathematics we should have learned)
• We discovered something that we might have missed the first time around.

How To Get Good At Math (In Ten Minutes a Day)

Some years ago, a Certain Toy Corporation got into quite a bit of trouble for marketing a (girl) doll that spoke phrases.  In particular, one phrase:  “Math is hard.”

I’ve always argued that that phrase isn’t objectionable.  Math is hard.  So is throwing a three-point shot in basketball, doing a triple gainer, bowling a perfect game, and changing out a car engine.  One of my favorite episodes of The Bernie Mac Show was when Bernie Mac made this very point:  yes, math is hard…but you do hard things all the time, so why not do math?  (Season 1, Episode 16, Mac 101)

So how do you get better at basketball?  You practice, practice, practice.  The same is true in math, and teachers often tell students this.  And while that’s all true, and certainly good advice, it occurs to me there are two components to being “good at math.”  The first is being good at doing math.  Maybe you’ve just learned how to solve a quadratic equation, and so solving $x^{2} - 3x - 7 = 0$ takes a little effort.  But after you’ve solved a few hundred quadratic equations, it becomes second nature, and you can throw down the solution ($x = \frac{3 \pm \sqrt{37}}{2}$) without hesitation.

But the other component of being “good at math,” and ultimately what it means to be a mathematician, is being good at creating math.  This is far more difficult.  It’s the difference between doing 30 push-ups a day, and inventing a new calisthenic.

So how do you do that?  Let’s consider two of the greatest mathematicians ever:  Gauss and Euler.  They actually talked about what it took to be a great mathematician, and the short form is this:  Never solve a problem one time.

For example, in 1736, Euler proved a result of Fermat, namely that if $p$ is prime and $a < p$, then $a^{p - 1} - 1$ is divisible by  $p$.  Euler proved this, using an induction argument so obscure that it keeps being rediscovered by mathematicians, both great ones (Laplace and Cauchy) and obscure ones (me, actually…this may be the only time I’ve sat at the same table as Euler, Laplace, and Cauchy).

But Euler didn’t stop with one proof.  About every ten years,  he came up with a new way to prove the theorem.  His re-examination of the problem led him to discover the $\varphi$-function (where $\varphi(n)$ is the number of numbers less than $n$ which are relatively prime to $n$) and generalized it to what is now called the Euler-Fermat Theorem:  For any number $N$ and any number $a$ relatively prime to $N$, the least value $x$ for which $a^{x} -1$ is divisible by $N$ is a divisor of $\varphi(N)$.

Incidentally, this result is the basis of modern computer security (the RSA algorithm).

What about Gauss?  In 1799, Gauss proved the Fundamental Theorem of Algebra,namely that a $n$th degree polynomial with real coefficients has $n$ real and/or complex roots.

And then, over the course of his career, Gauss proved the Fundamental Theorem three more times, each time extending the result and developing new mathematics.

What’s the practical application?  Let’s consider something really basic:  multiplication of two numbers, say $47 \times 153$.  We all know how to do this:  we were taught how to do this computation in school.  We can practice the multiplication algorithm by trying different products:  $47 \times 153$ today, $23 \times 17$ tomorrow, $153 \times 301$ the day after, and so on.  If you do this, you will develop your skills at applying the standard algorithm.

On the other hand, suppose that instead of doing new products the way you were taught, what if you tried to find the same product using a completely different method?  You know what the answer is supposed to be, so you’ll have a good way to check if your method works.

How might that work?  In this case, $47 \times 153$ is the sum of forty-seven $153$s.  So you could add $153 + 153 + 153 + \ldots + 153$.  That’s one method of multiplying; one nice feature of it is that it’s something a first grader can do.  (Granted, you’d probably have them do something easier, like $5 \times 4 = 4 + 4 + 4 + 4 + 4$, but the important thing is that they don’t have to know multiplication to be able to solve the problem “Find $5 \times 4$“)

Obviously, you don’t want to spend the next half hour adding forty-seven $153$s together…but progress comes when someone asks “Can we find a better approach?”  So you start thinking about how to improve the efficiency of your sums.   Maybe tomorrow, you realize $153 = 100 + 50 + 3$, so adding together forty-seven $153$s is the same as adding together forty-seven $100$s, forty-seven $50$s, and forty-seven $3$s.

And even that gets a little tricky, so the day after, you come up with a new insight that allows you to make the addition even more efficient.

Order of Operations

So if you’re on social media, you will sooner or later get exposed to one of those math questions tagged with “What’s the answer:  9 out of 10 people get this one wrong!!!!!”

Many of these deal with mathematical problems:  $8 \div 4 \times 2 - 1$, for example.  Before talking about this, here’s an important insight:

Order of operations is like driving on the left side of the road:  it’s a convention, decided by society, and there’s no right or wrong as long as everyone agrees on it.

In fact, the existence of a standard order of operations is because there was no general agreement, so you couldn’t be sure how to answer a question like $8 \div 4 \times 2 - 1$.  It was only during the twentieth century that order of operations became standardized.

So what’s the standard?  The basic rule of order of operations is this:  All operations are to be performed from left to right unless

It’s the unless that makes it complicated:

• Expressions inside grouping symbols (parentheses usually, but there are implied grouping symbols; I’ll return to this topic later) go first.
• Multiplication and division are equiprecedent:  they’re dealt with, left to right, at the same time.
• Addition and subtraction are equiprecedent:  they’re also dealt with, left to right, at the same time.

Those who remember PEMDAS may think I’ve omitted something.  But read on…

Let’s take a look at that $8 \div 4 \times 2 - 1$.  We have a division and a multiplication, which are equiprecedent, so we proceed from left to right:

• We find $8 \div 4 = 2$,
• Next we find $2 \times 2 = 4$,
• Finally we find $4 - 1 = 3$.

Now many of us learned PEMDAS:  parentheses, exponents, multiplication and division, addition and subtraction.  One problem with this nifty little mnemonic is that it makes it appear that multiplication should be done before division, and addition before subtraction.  But again, multiplication and division are equiprecedent:  they’re done at the same time, left to right; likewise addition and subtraction.

What about exponents?  This one’s a little tricky, but it comes down to this:  Exponentiation isn’t an operation:  it’s shorthand.  In particular, when you write $2^{3}$, you are not actually performing an arithmetic operation; you’re using mathematical shorthand:  $2^{3} = 2 \times 2 \times 2$.

So consider:  $3 \times 2^{3} \div 4$.  You don’t do exponents first.  You do recognize that $2^{3} = 2 \times 2 \times 2$, so this problem is really $3 \times 2 \times 2 \times 2 \div 4$.  Now you can do the multiplication and division from left to right to get the final answer:  $6$.