Science is easy, but what about Maths?

For many years I have wrestled with the problem of how to get better at Maths, and how to help others do so as well. It isn't easy! I tried all kinds of ways of explaining Maths to people in plain-English. Eventually I became convinced that teaching Maths was actually not possible: Someone either "gets" it or they don't, and nothing you can say or do seems to make any difference.

While that will certainly be the case most of the time, I have recently modified this theory slightly as a result of several chance conversations. One was with a Maths teacher at a local university, the other a student in Social Work who, to my great surprise, happens to love Maths.

In my own life Maths was not a strong suit, initially. I struggled in primary school with the tasks of adding and multiplying. Long division in particular is what I imagined Hell must be like for someone like me who wants to see the Big Picture, and who only worries about details that haven't been worked out before. Dear old Dad finally caved in and bought me a Calculator in about 1975. LED digits. 9-V battery good for about an hour. Fixed decimal point which made the answers off by powers of 10. It was a God-send! A Miracle!

(I now own dozens of calculators, and none of them are completely adequate in all ways. The best calculator for numerical analysis is actually Matlab, a very good and very expensive piece of software. Perhaps my obsession with calculators will be the subject a future post, if enough Readers vote for it!)

University Maths didn't make life any easier for me. I spent an entire year (and a painful one) studying Differential Equations and Linear Algebra. I got C's in both subjects. How is that possible?

I won't blame the teachers, although I won't thank them either. Aside from their impenetrable foreign accents, they presented themselves with an air of utter boredom which I imagine they thought was "professionalism" or even "cool." As if to say, "Whereas you are all morons, I find maths so incredibly easy I can do it in my sleep. I will now demonstrate this fact by pretending to be in a coma while I lecture."

No, students will either learn because of their teachers or in spite of them. I did neither. Why?

The university Maths teacher I mentioned recently gave his theory of how to teach the subject:

"Strip it of all applications and meanings, and deal only with pure notation first. That will get right to the heart of any conceptual difficulty the student may have, without the distraction of trying to interpret 'word problems'."

I immediately recognized the fallacy of this, and realized what the key to learning Maths must be.

My hypothesis was confirmed when I had a chance conversation a few days later with a student in Social Work. I asked whether she found it frustrating to have to take Maths courses which detract from her core interest in getting out there and doing something to make a difference. She said, "No! quite the opposite. I really like Maths, and have always found it easy."

"What do you like about it?"

"I really enjoy order and logic. I like it when things make sense and fit together. It gives me a feeling of satisfaction and control when I can solve a problem and get The Right Answer. Life makes sense when things work out correctly."

Why will some people enjoy Maths and learn it easily, even in spite of poor teaching? Why will otherwise capable students hate Maths and struggle endlessly with it? And most importantly, how do we make Maths easier to learn and to teach?

The human unconscious mind, in order to save us from Information Overload, filters out 99.99...% of all incoming data (sounds, images, stimuli, information), allowing into our conscious awareness only that which it deems relevant. The decision is made based on an individual's unconscious values, beliefs, fears, and desires. Further, memory is also activated in the same way, and we remember only things that unconsciously are important to us or that we care about emotionally. We already have "bins" or structure in the brain for retaining such information. Information far outside our experience is harder to classify and link to previous experience, with the result that there will be few neural connections created to constitute a memory of it.

Therefore in order for a student to have an activated memory and be open to information, it must be information that has emotional meaning to the student, and which relates to something the student already knows well. In the case of that Social Work student and most other "born mathematicians," the emotional meaning of Maths is built-in: the love of order, the satisfaction of being able to solve puzzles, and the sense of "all's right" when they get the one right answer.

Most other students, however, care about different things. Whether it's sports, music, art, books, friends, cars, fashion, money, animals, or Physics, there is always a way to make Maths relevant and something to which a student can and will attach emotional importance. Additionally, the teacher can generate the emotion in the classroom through enthusiasm, a personal story, and showing caring for the students individually. In other words, exactly the way a very good presenter or salesman "sells" any message.

If we want our kids to do better in Maths, (and it is definitely the one subject essential for success in virtually any field), then we should look at changing the way Maths teachers are trained. Or better yet, recruit teachers from the ranks of Salesmen! You don't have to be an expert in N-Dimensional Topology just to teach first-year Algebra, after all. You only need to be an effective communicator and understand the principles of Influence.

How did I eventually get on top of Maths? A decade after my Physics degree, I entered a Master's Degree program for Engineering. I just couldn't get enough of engines, spacecraft, cars, motorcycles, electronics, and gadgets generally. I loved fixing things, building things, and inventing things. When I took a most fascinating course in Control Systems Theory, for example, I needed to know both Differential Equations AND Linear Algebra. Suddenly these were no longer boring, difficult millstones around my neck, but exciting and useful tools that I couldn't get enough of. I was even teaching the other students the finer points of how to use them.

Stripped of all application and meaning, these subjects made no sense to me and I could not produce the excitement and discipline necessary to gain competence. But in the context of something I cared a lot about and had high interest in, they made perfect sense. They are now subjects I am very comfortable with. My engineering professors were absolutely dumbfounded that I had previously earned C's in those subjects.

But is it really such a mystery?

A 14 year-old Discusses Relativity and Science

The most recent Making Sense of Science Newsletter (available at http://www.wallingup.com/newsletters.php) sparked an online discussion with a Year 8 student. From it, I learned that there is not necessarily an age barrier to understanding advanced science topics, and that these challenging topics can supply the interest levels prerequisite to student engagement in the topic.

It also highlights the importance of integrating science with the humanities. This student had been reading the classic Ender's Game science fiction series by Orson Scott Card for his middle school English class (we both give it ***** five stars out of five).

I'd like to share the ensuing discussion with you.

Student: Aliens travelling to earth may have spent 1000's of years on a ship but the theory of reletivity would mean that they would not feel the ravages of time.

John: You are completely correct that aliens in a spacecraft moving near the speed of light would experience less time elapsed than on either their home world or destination planet. To them it might seem only a few years or even weeks. But to get going that fast requires such absurd quantities of energy it seems incredible that they would chose to do so.

So the aliens could survive the trip if they wanted to bad enough and had virtually unlimited energy to waste. But it would still take many hundreds of years of earth time for them to get here. They would have left on their journey at a time long before there were any radio signals to indicate that someone interesting lives here. We have only been sending out radio signals strong enough to be picked up in space for about the last 60 or 70 years, meaning that aliens living 70 light years away, if they are listening, would only now be aware of us. We can't expect to hear back from them for another 70 years, and we certainly can't expect them to drop by for many hundreds of years at least.

Student: Faster-than-light travel might be possible someday. Look at mobile phones! A few years ago people would have said they were impossible, too.

John: One thing to notice is that we did not exactly "discover" mobile phones, we invented them. The natural laws that allow them to operate were discovered more than a hundred years before.

But the computer technology that makes them work was invented step by step over the last 40 years. We did not discover any natural law that said "computers can do this" or "computers cannot do that," so we just kept trying new things.

There is a huge difference between technology and nature. Nature is the tree, and technology is the decorations. Technology must always follow Nature's rules, but Nature does not have to obey or allow technology. We do not discover technology, we invent it. Nature can only be discovered, and never re-invented to suit us. Technology only works if it is allowed by Nature.

Student: Just because we've never seen something doesn't mean it doesn't exist.

John: You are right about that. Don't make that mistake! If a guy says, "Dogs do not exist," because he has never seen one, he will be totally unprepared the first time a dog shows up and bites him on the bum.

We are not saying "there is no faster than light travel" merely because we have never seen it. We say there is no superluminous travel because we have discovered a natural law which says, "all speeds are less than the speed of light." Many experiments have proven that this law is true. They also prove that the opposite of this law is false. All experiments attempting to disprove this law have failed, too. It seems Nature is trying to tell us something, and the message is loud and clear.

We are not exactly in a room with the light turned off, speculating on what might be or might not be in it. We are also not like the guy who has never seen a dog before, jumping to a false conclusion. The light has been turned on. We have discovered that the faster you go the heavier you get until at the speed of light, you weigh infinity. It also takes infinity energy for matter to go that fast. By analogy, we have discovered dogs, and found that all dogs bark and poop. (and have sharp teeth). There is no need to speculate on the existence of dogs, or on the speed of light, because we are in possession of the facts.

Student: You cannot be certain that for example every element in existence has been discovered yet. New things might be discovered at any time.

John: I am particularly impressed with the way you think. Good job. And good on your teachers who cultivated your thinking skills. Your question is an important one: how can we be certain of what hasn't been discovered?

I'd like you to examine, if you would, all the whole numbers between 1 and 118. Are there any whole numbers missing? Are there any that we have not "discovered" yet? Are there any whole numbers in that range that are not known and still need to be invented?

If not, then there are also no new elements waiting to be discovered or invented either. We know of all elements as surely as we know about all the whole numbers from 1 to a million and beyond. Every element in the universe is a whole number of individual protons in a nucleus, with the same number of electrons hovering around it. To work out what possible elements might exist, simply write down all possible whole numbers. There aren't any missing.

This means that often it is specific knowledge of what does exist and how it is put together that enables us to make absolutely certain statements about what else might exist, and also things that cannot exist.

The only way to know the difference between what might yet be discovered and what will definitely never be discovered is to get as much understanding of the laws of Nature as you can. Things that those laws allow are possible at least in theory; things that the laws disallow are never possible no matter how hard we try or how much technology we get.

Student: Yes, but new laws are discovered which prove that the old laws were wrong.

John: Now who told you that? I'm sorry to say this is the first statement that is completely false. This has never happened, and people who say so are simply mistaken.

Newton's laws of physics replaced Aristotle's "laws". But those so-called laws were actually wrong to begin with, as any simple experiment could show. They were not really laws, but actually philosophical beliefs that were never tested in practice before they were written down. Galileo proved them to be wrong in his experiments. Finally, Newton worked out the simple laws of motion, forces, speed, mass and distance.

Einstein's laws of Relativity are sometimes said to prove Newton wrong, but the truth is that Relativity proves Newton was exactly right for all possible earthly speeds we encounter in everyday life. It is only speeds above 20 or 30 million meters per second that more information is needed.

Quantum mechanics is said to prove Newton was wrong, too. This deals with objects smaller than atoms, and it says they don't behave according to Newton's laws. But here too, Newton is proved right. Quantum mechanics tells us that large numbers of particles in a bunch together behave exactly as Newton would predict. Things even as tiny as a grain of sand obey Newton's laws perfectly, for all practical purposes. (How many atoms are in a grain of sand? See an older post in this blog.)

So a law that is proven experimentally does not stop being true, even when more information about totally different situations is uncovered.

Student: So the best way to invent stuff is to learn exactly about the laws of science so you can take advantage of them?

John: Good for you! Now you're on the fast-track! People who don't do what you suggest get bogged down forever basically re-proving the laws of Nature that are already known, and never invent anything useful. Do what Isaac Newton said, and Stand on the Shoulders of Giants to see farther than you would on your own legs. And Happy Newton's Birthday, 25 December!

Watch out for this kid. He's only 14 now, but he's going to make a difference.