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Appendix E
How to Study Math and Why

Previous: D. What to do in School and Why

A  Before You Stop

In school before you stop studying mathematics, please do the following.

1. Look at the chapters Implication Rules and Chains of Reason. These two chapters may help you to use and understand precisely rules, instructions, patterns, definitions and recipes in every subject and every area of skill or specialization including mathematics.
2. Read the first chapters on algebra. The description in them of the three key skills for algebra and the algebraic examples will, I hope, help you step from arithmetic to algebraic way of writing and thinking plus a little beyond. These chapters try to explain and describe with everyday words, how (algebraic) shorthand notation is used to describe and do mathematics after arithmetic. There is more to mathematics than just doing arithmetic well.

B  Why Study

Mathematics courses are preparation for business calculations, for handling your money sensibly and for courses in sciences, engineering and technology. You should view mathematics as an opportunity to strengthen your thinking skills.

In mathematics courses you should not only meet calculations to do but also the chains and threads of reason and persuasion which justify them and links them together. Understanding and following the rules and patterns of mathematics, practices and nurtures an ability to think and reason well. Mathematics provides a neutral territory for the practice of rule and pattern-based reason and logic. The opinions and views you meet in daily life say and care little about what mathematical conclusions should be.

If you find yourself in a course which gives formulas and numbers to use in them but does not expect you to use algebra, you are wasting your time. Your time would be better spent studying algebra, and then taking a more advanced course that respects your intelligence. Similarly in college, if you find a course which gives you formulas and numbers to use in them and also talks at length about rates of change without expecting you to understand calculus,⁶ then a calculus course would be of better use of your time.

⁶ Calculus in the first instance provides formulas for the slopes of (nonlinear) curves and for the rates of changes of numbers or quantities.

C  When to Study

Look at the description of courses you will take in and outside mathematics. From their description, see what mathematics course(s) you are expected to take with them (co-requisites) or before them (prerequisites). Methods taught in a co-requisite mathematics course are too often covered after they are required in another course. So take a mathematics course before there is any possibility the methods in it will be needed in another course. Then you may master the methods before they are required and not after.

D  More Advice

If you follow the advice and the cautions below, you should have a mathematical foundation for any subject requiring calculation. In mathematics do the following.

1. Master arithmetic. Also master weights and measures. After you have mastered the rules of arithmetic, learn to use a calculator.
2. Master the algebraic way of writing and thinking. Also master the use of rules and patterns to arrive at conclusions. Mathematics after arithmetic builds on our abilities to talk about numbers and quantities, to describe calculations and to change the way calculations are done. Mathematics after arithmetic also depends on our ability to follow and understand rules and patterns. See the first chapters on reason and algebra in this book.
3. After algebra, take trigonometry and geometry.
4. Learn about money matters. Take a course on money calculations, preferably after a course in basic algebra. Most of us handle money for credit or investment. In your last year of studies before starting work, take a course on the mathematics and arithmetic of personal finances. This course should include balancing of budgets, description of typical household expenses for individuals or families in rented or mortgaged properties, problems involving simple and compound interest, and mortgage/pension calculations. Traces of such calculations appear in elementary and high school mathematics, but they are forgotten years before you need them. A course like the one described should be offered in schools and colleges for students in any art or science. Ask for one to be given, if it is not already offered. All this is practical mathematics. It should be more widely known.
5. If you go to college, take a year or two of the mathematics subject called calculus, a year of probability and statistics, and a year of matrix computations (or linear algebra). Calculus courses usually have trigonometry and algebra as prerequisites. Calculations in many trades, including business, engineering, computer technology, physics and health science, require calculus.

The problems you find easy to solve should be done to restore and build your confidence and to reassure yourself that you have understood what they require. But after you have done a few such problems, you should look at the ones which appear harder. The problems which appear to be too hard should be noted and remembered. You can return to them later by yourself or with help from another. What is hard for you to solve may be easy for another, and vice-versa.

E  Cautions

When I taught a remedial algebra course, one of my students was a high school gym teacher. One of his past assignments was to teach algebra.⁷  

⁷This should not occur, but in many school systems it does. When it does, it shows a lack of respect to students and a lack of purpose for education. It also suggests circumstances beyond the control of students and teachers.

In some schools due to circumstance beyond their immediate control, some instructors are required to explain ideas outside their own specialties. When or if you meet such an instructor, be polite and do not become a troublemaker. If a teacher sees you as a threat or troublemaker, you may suffer. When you meet a misplaced instructor, politely and diplomatically try to transfer to another class in the same subject or read the course textbook yourself and get a tutor.

F  More Keys to Better Learning

1. How you find a solution to a problem is not important provided you understand fully the solution. (Some teachers may disagree.)
2. If you have to copy solutions blindly then you will not understand ideas well enough to pass tests and the final examination.
3. You should ask another to check that your written responses or solutions are both understandable and well-written. Mistakes brought to your attention in any manner improve your understanding. If such checking improves your ability to avoid mistakes in the future, then such checking should I believe be encouraged. Again, some teachers may disagree.
4. Students who know and identify in their solution those steps which are doubtful deserve more respect than students who don't. Knowing exactly where one is sure and where one is not is the sign of an alert mind.
5. Correct answers obtained accidentally, for instance by canceling errors in a solution should not be given full marks. Errors in a solution show that the subject is not carefully mastered.
6. Learning is better done in a cooperative atmosphere where students help each other to understand instead of a competitive one, where the success of one student is at the expense of others. (But you can not always choose your environment.)
7. Seeing two or more approaches to a subject can be better than one. What appears hard in one approach may appear easier in another.

G  Calculators and Computers

Calculators lessen the need for us to do arithmetic but, in using them, mistakes can be made. Here you need to know in advance what kind of answer a computation will yield. If you think you have made a mistake in entering numbers or instructions, you need to reenter them again. If a different result appears from before, at least one of your efforts, the original or the check, will be in error. (Logic Question: What can you say for sure if the results agree?) Suggestion: remember or learn how to do arithmetic by hand and how to estimate the expected size of results for addition, subtraction, multiplication and division.

Computer programs can perform arithmetic and algebraic or symbolic operations. They can also draw graphs and solve some equations rapidly. These programs do not provide solutions to all possible problems. For the solutions they can provide, you have to understand the statement of the initial problem. Beyond this, a computer (or another student) cannot understand the chains of reasoning for you. Understanding is an personal affair. No computer and no other person can do this for you. But if you know what to expect from a calculation, calculators and computer programs can help you check your expectations and explore mathematical ideas. Here you can learn from your mistakes. In some cases, computer software can tutor you. They can tell what to expect in various circumstances. Today there are computer programs and on-line computer books which may help you master mathematics and other subjects. More are appearing everyday. I know of them, but I have no experience with them.

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