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?
Been there, done that, didn't work. This is the problem with current students.
ReplyDeleteSay a class has 40 students. Are you going to include 40 different applications to "motivate" each one of them? Even if you try to do that, each students will only be "motivated" 1/40% of the time. Do you think there will be time left to cover all the material?
So, this will not work, unless a student is willing to spend 10 years as an undergrad because the teacher had to include one "motivation/application" for each student.
In reality, motivation comes from within. If you didn't like engines, etc, you would still be saying that diff eqns and linear algebra were hard. The problem is not the course, the problem is each student that has no self-discipline.
By the way, your justification is just why many other countries are beating the USA in math/science/etc (not in English, though, as you may have noticed).
Yes, you found the fault. I have not solved anything, but merely transformed one intractable problem into another intractable problem. Unless all 40 students are engaged by the same thing, this doesn't work and the essential problem remains:
ReplyDeleteMaths cannot be taught; it can only be learned.