More Creative Mathematics

I believe that in the teaching of mathematics, there is far too much use of pre-fabricated problems and not enough on creativity in setting up a problem. Figuring something out using numbers or geometry involves setting up the problem and then solving it. Students, in my opinion, learn much more of how to solve a problem that has already been set up for them than how to observe the real world around them and apply the math to it by setting up the problem.

In figuring out the things that I have presented on these series of blogs, I find that real-life mathematical problems are often relatively simple but require considerable creativity to set up effectively. I wish the teaching of math was more like art, such as painting or drawing. A proficient mathematician is not one who can solve any pre-prepared problem that is presented to him or her but one who can observe the real world and effectively apply mathematics to it by setting up the problem to be solved.

Calculators make the setting up of the problem even more important relative to the solving of it. Calculators can solve problems quickly but only after it has been correctly set up. No matter how proficient one may be with a calculator, it is useless unless it's user can observe the real world and select the mathematics to apply to it. There may be several ways that a real-world calculation can be accomplished.

I am sure that at least half of the mathematics one learns in school will never be used in real life, no matter what occupation the student ends up in. Remember that imaginary number, i, supposedly the square root of -1, which cannot actually exist, from algebra class? I am still mystified as to what that is supposed to be actually used for. The same for factoring polynomials, although it is a useful mental exercise.

In the calculations that I have done for these series of blogs, the spatial branches of mathematics such as geometry and trigonometry have been by far the most useful. I have rarely used algebra and never calculus. I do find fractions to be very valuable since this is often how the real world operates.

In making real-world decisions, it is often a "sense" of numbers or geometry that is required, rather than an actual calculation. I was never brilliant in math class but later found that I had a knack for creating my own mathematics and applying it to the real world and universe.

One day, I was figuring something out and wanted a quick way to add up all the numbers up to a certain number, 1+2+3... There was the factorial function on a calculator (!) to multiply all the numbers up to a given number but I had never heard of a similar function for addition. However, after a few minutes of trial, I noticed that if you divide a number in half, add one half (.5) to it and multiply it by the original number, it will provide the answer. This means that the numbers from 1 to 10 should add up to 55 and they do.

Another time, just as a mental exercise, I wondered what the odds would be of winning a game in which the odds of winning were one in three and we played the game three times. I knew that the answer would have to be more than one in three but less than 100%. I had never heard of this in class but I realized that the only practical answer would be 1/3 + (2/3 x 1/3) + (1/3 x 1/3). In other words, 6/9 or 2/3.

Mathematics is the underlying patterns of how reality works. The objective of math class should be more like art class, to see how the system operates and then create your own. There should be more emphasis on setting up problems in the real world as opposed to solving pre-prepared problems.