Elements of Learning and Enjoying Mathematics

The Dawn

I would be ignorant as the dawn
That has looked down
On that old queen measuring a town
With the pin of a brooch,
Or on the withered men that saw
From their pedantic Babylon
The careless planets in their courses,
The stars fade out where the moon comes.
And took their tablets and did sums;
I would be ignorant as the dawn
That merely stood, rocking the glittering coach
Above the cloudy shoulders of the horses;
I would be -- for no knowledge is worth a straw --
Ignorant and wanton as the dawn.

William Butler Yeats
June 13, 1865 - January 28th, 1939

The wonderful book Euclid’s Window by Leonard Mlodinow references Yeats poem in his assessment of the Babylonian approach to learning and using mathematics.

“Yeats refers to Babylonian indifference to knowledge in this poem. This was a trait, which in mathematics, held them back from greatness. Pre-Greek humanity noticed many clever formulae, tricks of calculation and engineering, but like our political leaders, they sometimes accomplished amazing feats with astonishingly little comprehension of what they were doing. Nor did they care. They were builders, working in the dark, groping, feeling their way, erecting a structure here, laying down stepping stones there, achieving purpose without ever achieving understanding.”

I tutor high school student in mathematics and more and more I am forming the opinion the current method of teaching mathematics has adopted the Babylonian approach. Let me give some examples to support my claim. First of all I see students being given a multitude of problems on the same process. Requiring them to calculate, solve, etc. the same type of problems over and over again. A certain amount of repetition is needed and one does need to give them the opportunity to work variations, but I see too much overkill. The real disappointment is I rarely see the more abstract, applied or thought provoking problems given to them. These are problems that will really determine their level of understanding versus being able to just follow a defined process, which is easily forgotten when not associated with understanding. I also tutor for SAT preparation. Everyone has his or her opinion about the value of SAT test. One thing the mathematics portion of the test does attempt to test is the understanding of certain mathematics principles. The level of mathematics on the test is not high, but it does require understanding and a problem solving process to get a score above 600.

I will illustrate further with some comments my students have made. “We finished factoring today with the completion of our test. The teacher told us not to worry about factoring again until the end of grade testing”. A student was working a process to determine the best fit of a line. She had the process down and was able to do it perfectly. I asked her if she knew why they were doing this, where it could be used, etc. She said they had not been told. After explaining the benefit, usefulness and power she thought it was very interesting and wanted to know more. While tutoring another student on matrix multiplication I asked if they had done any proofs on this topic. She said, “Oh no, our teacher promised no more proofs after we finished geometry”.

I know every student will not be a mathematician and many students have some poor skills in mathematics. Neither is a valid reason to not give them the opportunity to enjoy mathematics. No wonder most dislike it and cannot wait until they are done. The students I see having a high degree of math anxiety are just worried about doing a process, not interested in understanding. They feel they cannot and just want to pass the test. Once I can get them past this feeling and approach, they actually surprise themselves and say this is fun.

A problem based approach to teaching and learning mathematics can be a more effective method. I am currently developing a better explanation and some lesson plans to illustrate. Take a look at this diagram which illustrates what I believe are the key elements of learning mathematics. One analogy is to thing of it as a project. This is another pet peeve of mine about the teaching of mathematics. How many times are students required to do a project/paper in a mathematics class? Very few I would venture to guess. A project offers the advantage of developing a clear objective, research, building up an understanding of supporting information, going deeper as necessary, connecting what you have learned to other topics or subjects, and finally being able to explain your project to someone else. Nothing like having to explain to someone else to see if you really know what you have learned.

• Strong mathematics problem solving and reasoning ability.
• Firm grounding in essential mathematical concepts and skills including computation and estimation.
• Connections within mathematics and other disciplines.
• Ability to use appropriate tools including technology to solve mathematics problems.
• Ability to communicate understanding of mathematics effectively.
• Positive attitude and beliefs about mathematics

Let’s not make our student’s “Ignorant and wanton as the dawn”. The Greek’s flourished because of their interest and approach to learning. Give our students the same benefit.