A Mathematician's Lament

I recently came across an essay titled "A Mathematician's Lament" by Paul Lockhart. If you'd like to read it for yourself, here's the link:

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

In the opening, Lockhart wonders what it would be like if music were taught the same way as math. What if, throughout all of grade school and undergraduate college, students were to take classes like like AP Music Theory, AP Musical Notation, and Introduction the Musical Transpositions. Of course, all of this happens without once having the students play music or, God forbid, compose pieces of their own--those are reserved for later in the undergraduate curriculum and graduate school.

What a nightmare! After all, the joy, fun, and beauty of music come from actually being able to experience it! Lockhart uses this example is to introduce the main point of his essay: that this is precisely the state of math education today.

In grade school, students memorize multiplication symbols. In middle school, they learn to mechanically manipulate symbols using a fixed set of rules and shortcuts that come from some magical faraway land called "Algebra." In high school, the same rules are extended and students are taught a new set of rules and shortcuts, this time from a place called "Calculus."

One might argue that math is fundamentally rooted in logic and arithmetic, so we should naturally reinforce "fundamental" concepts in order to better prepare students for more advanced topics. But take, for example, the visual arts. Before allowing students to draw, should we could teach them the benefits of using water colors versus oil pastels or the differences between classical and impressionist styles of painting? After all, if students are to become advanced painters, its only natural that they start by learning all about the techniques which so many artists have dedicated their entire lifetimes to develop. Instead, we do the unthinkable by giving them a paintbrush and letting them paint a self-portrait however they want! Yet, students seem to enjoy art much more than math. No one even has to defend the value of art education by bringing up points like how useful it is or how much more money you can make if you list still-life painting as a skill on your resume. They enjoy it because it's, well, enjoyable!

Imagine if math could be taught like that! Imagine if we could present students with a blank canvas, a very small set of rules to follow, and see where they take themselves just by mixing and matching their preexisting ideas in order to form new ones. Art and music give students something that is almost nonexistent in a math class: freedom. It is that freedom that allows students to develop their own appreciation for the things they do rather than accepting someone else's argument about why it's important or necessary.

I can't say that I have a way to reform the entire grade school math curriculum. But I do think that classes which spend more time talking about the ideas behind math can do so much more for students then classes which ask them to regurgitate a handful of definitions and theorems on a test. What happens if I add all the numbers from 1 to 10? 1 to 11? 1 to 12? Could there be some kind of pattern here? Give a student a chance to think about these questions, come up with answers on their own, and you might discover that even an elementary student is capable of inventing one of the most useful summation formulas in the history of math (spoiler: this problem was solved by Karl Friedrich Gauss in the late 1700s while he was in elementary school!).

"Just a Theory"

I don't like taking sides in debates about religion, but when it comes to whether or not we should teach both evolution and creationism in biology classrooms, there's a certain phrase that gets thrown around which I think is being used very incorrectly.

A major section of the debate revolves around those on the side of creationism mentioning that evolution is simply a theory while those on the side of evolution retort that it has been established as a fact. Both of these arguments are founded upon a misunderstanding of science and the scientific method: what it means for science to be “right” or “wrong.” A hypothesis is an initial guess which a scientist has taken in an attempt to explain some observed phenomena. Many mistake a theory to be the same as a hypothesis, when instead a theory is a hypothesis whose predictions have matched experimental observations and continue to do so. In this sense, a theory is not necessarily an established fact but it does hold the condition that it has yet to be proven false. The only way a theory can be proven absolutely true is if has withstood, does withstand, and will continue to withstand all experimental attempts to disprove it. Since this is clearly impossible as it would take an infinite amount of time to confirm, a “good” theory is simply one which has survived a great number of attempts to disprove it.

At the same time, EVERY theory carries with it a scope of accuracy, essentially stating that the theory may not be all-encompassing but that there do exist particular scenarios in which the theory remains valid. Like I talked about in an earlier post, Isaac Newton’s theory of gravity is an example of such a case. Not long after Newton published his theory of universal gravitation, it was demonstrated via extremely accurate measurements that the orbit of Mercury deviates slightly from that predicted by his law. Physicists searched for disturbances which might have led to the deviation, such as large clusters of asteroids or clouds of dust which might have been distorting Mercury’s orbit. Ultimately, nothing was ever found which was substantial enough to account for the discrepancy, and the anomalous orbit of Mercury was one of the first confirming pieces of evidence which allowed Einstein’s theory of General Relativity to replace Newton’s theory as the most accurate description of gravity because it was able to give the correct orbit. Even so, Newton’s theory is still taught to students at both the high school and university levels because its description of gravity is still sufficient in all but the most extreme cases.

Waves of Gravity!

In 1916, just one year after Albert Einstein published his already-revolutionary theory of general relativity, Einstein once again revolutionized our understanding of the world by showing that his general theory of relativity implied the existence of wave solutions. In other words , if general relativity implies that space and time can be stretched like the surface of a trampoline, Einstein showed that it was possible for someone to send ripples vet the surface of that trampoline by jumping on the surface! The problem was that these ripples were incredibly weak, so detecting them would necessarily be a very difficult task. Yesterday, almost exactly one century after Einstein made this famous prediction, physicists amounted that they have in fact detected (and therefore confirmed the existence of) gravitational waves.

It's very fortunate that this happened to be announced almost right after my last post about the nature of scientific progress, because it means we can keep talking about it. One important thing that I think everyone should keep in mind is that no scientific theory can ever be proven to be 100% correct. After all, the only thing that can say that a scientific theory has yet to be proven incorrect is experimental confirmation of a scientific prediction. That means there is no way to show that a scoentific theory is absolutely correct, only that it had withstood the trial of experimentation up until the present day.

Nevertheless, is that not how we learn everything, I have no guarantee that the sun will rise tomorrow, but it has risen every day of my life up until today, and that alone is why I'm confident that it will rise again tomorrow. Science is simply a generalization of that line of reasoning.

The nice thing about physics is that it is all formulated mathematically. If I am to accept a theory of physics, I must be prepared to accept everything which is implied by that theory. For example, the notion that we might be able to send waves through a gravitational field. Thus, while I can't say that Einsteins theory of general relativity is definitely correct, what I can say is that, for the last 100 years, it has been able to withstand each an every attempt to disprove it's correctness. That, to me, is as sign that it must at least be a step in the right direction.

The Nature of Scientific Progress

Without having had a lesson, I don’t exactly have much to “reflect” on, but I definitely have things that I want to say about the future! The falling object experiment, which I plan to do for my first lesson, is one of my favorites. It’s not as exciting as making an explosion with some chemicals or having the kids build something that they can take home to show their parents, but to me it’s one of the most important experiments ever performed.

For centuries, people debated with each other about whether a heavy object or a light object would hit the ground if both were to be dropped from the same height. Aristotle reasoned that since heavier objects feel a stronger downward pull, they should hit the ground first. Others argued that lighter since objects are easier to move, the downward force they feel has an easier time pulling them to the ground despite being weaker, allowing lighter objects to win the race.

The issue was laid to rest when Galileo (the same one who famously defended the then-controversial heliocentric model of the solar system) decided to see for himself what the result would be by dropping to objects of differing weights. The answer he found was (C), none of the above! Any two objects, regardless of weight, will hit the ground at the same time when dropped from the same height! Although Galileo obtained this result through experiment, it wouldn’t be until the time of Isaac Newton that scientists began to understand why.

To put it simply, both arguments were in fact correct, just not individually—the correct explanation requires the effects of both arguments to be taken into account at the same time. It is through Isaac Newton’s 2^nd Law of Motion and his Law of Universal Gravitation that we can understand how to incorporate both arguments. Through the Law of Universal Gravitation, Newton stated mathematically something we already know: if object A has twice the mass of object B, A must then feel twice the gravitational force that B feels (i.e., A is twice as heavy). His 2^nd Law says that if A is twice as massive as B, then it is in fact twice as hard to push A as it is to push B (i.e., starting from rest, it is twice as hard to push A until it is moving at some speed as it would be to push B to the same speed). We can see both of the original arguments captured in these 2 statements. The incredible insight here is that the two different effects turn out to exactly balance each other out! Object A might be twice as heavy as B and is therefore pulled twice as hard by gravity, but it is also twice as hard to move A as it is to move B!

I think this simple experiment and the history behind it captures much of the philosophy of science. Regardless of how nice an argument sounds or who made it, it is still doomed if it fails to account for experimental results.

It also speaks about how science moves forward. Neither of the two arguments was wrong, but each failed to see the bigger picture. Scientific progress is ultimately about making our picture of the world ever so slightly bigger. Thus, the goal of science is not to find replacements for our current theories, but rather to expand upon them and make them more complete.

Today, Newton’s theory of gravity is known to be inconsistent with experiment when dealing with environments that have extreme gravitational fields. Our current understanding of gravity comes from Einstein’s General Theory of Relativity. Einstein’s theory is in complete agreement with Newton’s here on earth, and this was in fact used as an important test of the validity of General Relativity. What this test tells us is that Einstein’s theory explains at least as much as Newton’s. When it was confirmed that Einstein’s theory can also account for additional scenarios in which Newton’s fails, physicists knew this was the right theory to expand the horizon of our understanding of gravity. In non-extreme conditions, Newton's law of gravitation is a fine approximation of Einstein's (and also much simpler)—in fact, it was all we needed to go to the moon!