Why Is Math Hard?

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And why do we make your children suffer?

Math is synonymous with difficulty for most students. Even those who love math and do well at it have to work hard. It seems that there’s no easy route to learning math, though it can be enjoyable.

So why do we bother? Very few people use any math more difficult than percentages in work or daily life. Why learn years more than that? And why is it such hard work?

The difficulty comes from the enormous and often unrecognized requirements for doing mathematics. We can group those requirements into four major categories.

First, we have to learn the language of mathematics. Math has its own letters and symbols in addition to those in English: equal signs, pi, division signs, etc. Math has its own vocabulary, with many novel words. Where the words are the same as those in English, the definitions usually differ. Math has a unique grammar, that often spills off of lines with fractions and many unusual notations for calculation. This grammar doesn’t use nouns and verbs: it uses math’s own vocabulary categories. So our students need to learn to read, write, and speak fluently a new language vastly different from natural human languages such as English. Small wonder that task takes years!

Second, students have to master the unique concepts of mathematics. The best understandings of most mathematical concepts are very abstract, but most students need concrete, everyday examples to approach understanding those ideas. Still worse, often the concepts require that you understand earlier concepts. Some mathematics uses many layers of concepts.

Third, an important aspect of mathematics is computation. We learn many difficult methods (algorithms) for computation: addition, multiplication, division, etc. In addition, we learn quite a number of tools for computation, such as calculators, tables, spreadsheets and graphs. Techniques such as algebra and proofs are actually also methods of computation.

Fourth, the most difficult and important requirement of math is problem solving. Later, I’ll list Polya’s four principles: they outline the complexity of problem solving.

Given this horrible difficulty, why do we make your children suffer to learn it? Is it a conspiracy of sadistic math teachers to remain employed and pass on the misery? No.

There’s more than one answer. Perhaps the most powerful answer is that problem solving skills in math transfer: they can be applied to non-math problems that we often encounter. Skill in problem solving is valuable, and one good measure of useful intellect. The National Council of Supervisors of Mathematics claims “Learning to solve problems is the principal reason for studying mathematics.” Another important answer is that math is important for understanding science, technology, and some real-world situations. It can provide a common language for quantity, calculation, and many abstract concepts.

And finally, math is important for intellectual development: not only does it teach reasoning, but it provides necessary stimulation for brains. Our brains continue to develop into our late twenties, and just as exercise for muscles develops greater muscle capabilities, exercise for brains produces greater intellectual capabilities.

Polya's Principles

Polya’s First Principle: Understand the Problem This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve a problem simply because they don’t understand it fully, or even in part. Polya taught teachers to ask students questions such as:

  • Do you understand all the words used in stating the problem?
  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?

Polya’s Second Principle: Devise a plan Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

* Guess and check * Make an orderly list
* Eliminate possibilities * Use symmetry
* Consider special cases * Use direct reasoning
* Solve an equation * Look for a pattern
* Draw a picture * Solve a simpler problem
* Use a model * Work backward
* Use a formula * Be ingenious

Polya’s Third Principle: Carry out the plan This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled; this is how mathematics is done, even by professionals.

Polya’s Fourth Principle: Look back Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.

(Polya's principles are from http://www.math.wichita.edu/history/men/polya.html)

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