About Mathematics Education

Paul B. Hinds

Much mathematics instruction is poor. Only 5% of 17 year olds are competent in basic high school mathematics (algebra and geometry). The mathematics scores on the SAT tests have fallen (yes they have risen recently but the reason is in question: was it due to renorming?). What should be an easy subject to learn has been made into a educational disaster.

Mathematics is an invention of humans and, as such, mirrors the human brain. It is human nature to order objects and count them, and all mathematics (even the most sophisticated calculus) is based on counting.

For background: keeping track of a herder’s sheep was one of the first utilizations of counting. A tally stick with a notch for each sheep allowed a herder to move his finger over the notches as each sheep was counted.

These earliest of mathematicians found that it is hard to determine the number of objects in a collection of more than four objects. For this reason and because people have hands with five fingers, the number five mark on the tally sticks was often made differently. Often it was shaped as a V notch, which eventually gave the Romans the symbol V for five. At ten, another unique mark was made.  Often this mark was an X.  From this, of course, the Roman ten became an X.

The natural divisions at five and ten and the fact that humans have ten fingers led to a number system based on ten. With few exceptions, all over the world most of the counting systems were based on ten, with a few based on five or twenty. One notable exception was found in the ancient civilizations between the Tigris and Euphrates rivers, Assyria, Babylon, etc., where they devised a system with a base of sixty. We still live today with remnants of this system: —60 minutes in an hour, 60 seconds in a minute, and 360 (a multiple of 60) degrees in a circle. It is interesting that the mathematicians who devised the metric system, putting a measurement system to the base of 10, did not change this over-3000-year tradition.

Initially, written numbers were useful only for recording; when computation systems were needed, the abacus was used. Objects were placed in fixed orders which usually represented ones, tens, hundreds, etc. and manipulated so as to add and subtract. It was not until the Hindus invented our present number system that written numbers became useful for computation. This new system, brought to Europe by the Arabs, is often called the Hindu-Arabic system. It is believed that in Europe the abacus and the Hindu-Arabic system were used side by side for a hundred years before the abacus was replaced.

A new abacus has now appeared in the form of a calculator. Many in the mathematics community consider this abacus to be the natural replacement for our present paper and pencil calculating system; and they are not content to wait one hundred years for the new system to prove itself as an educational means. But with the adoption of the calculator, they have decided that memorizing the addition and multiplication tables is not necessary. They have concluded, correctly, that calculators will do most of the computation in the future. But can calculators teach students mathematics?

This new abacus is different from the old ones which required the persons using them to manipulate the abacus in certain ways to obtain the correct answer. The manipulator had to know how the abacus operated because he or she controlled the parts. On the other, new hand, is the calculator which conceals its work from the operator and gives only the answer.  It is a box which information is put into and answers emerge. But, if you don’t understand how to solve a problem, you won’t know how to use the calculator. So at Basics+, we believe that a calculator should always be considered an accelerator for someone who can do their math themselves.  And for someone to learn to do math for themselves, they need to start at the basics.

A reluctance to demand that students memorize the addition and multiplication tables is the source of many problems students have in mathematics today. The weak foundation which results inhibits student progress. Time after time I have witnessed students struggling with fractions, not because of the difficulty of grasping the concepts of fractions, but rather from the distraction of having to count on the fingers or get a calculator to solve the simple arithmetic inherent in the fraction problems.

The Basics+ program is designed to build the foundation necessary for understanding mathematics. A person who is skilled at, and understands, mathematics knows the basic tables; and people who do not know the basic tables generally do not like and do poorly in mathematics. Knowledge of the basic tables and the ability to do computation are absolutely necessary for success in mathematics.

The Basics + Program Sequence

12 Algebra

11 Signed numbers

10 Powers & Roots

9 Ratios & Percents

8 Basic Algebra

7 Decimals

6 Fractions

3 Subtraction

5 Division

2
Addition

4
Multiplication

1 Counting

The accompanying diagram shows the structure of the Basics+ program and the sequence in which the basic topics of arithmetic are learned. The foundation is counting; and the pillars of the system are addition and multiplication.

We build this structure by requiring that students memorize tables.  Students do not move on until they have mastered the material. Each topic is divided into small parts with adequate drill. The system is kept simple; extraneous material is omitted. Students work at their own level.

Students achieve competence and independence in computation by following the Basics+ stair-step method of learning.  Each step builds logically on the last, and helps students efficiently move upwards in mathematics. When students start the Basics+ Mathematics Program in kindergarten, it is not unrealistic to have most students in algebra by sixth grade or before.

This program is used in classrooms, for tutoring, for home schooling, and as an individualized self-learning program. It allows each student to progress at his or her own pace; or, it can be a lock-step program with all students assigned the same specific lessons. It is also used as an adjunct to other programs for students who need structured remedial work. And lessons from the Basics+ program can be used individually by the students who need polishing on specific areas of study.

To get started, all the teacher needs to do is to determine the student’s level with the Basics+ Placement Test, evaluate the student’s results on the Basics+ Placement Chart, and move directly to the appropriate lesson book. Or, with a young child who is just beginning mathematics, the teacher just picks up the flashcards and first workbook, reads the special instructions and begins. It’s that easy. Simple-to-follow instructions are included each step of the way, so that the learning can begin immediately.

Our system gives students the math to think for themselves.  In a hundred years, perhaps, people will not need to think for themselves, having the luxury or curse of having machines do it all for them.  In the current world, however, students that can do their own thinking and who can understand the world around them are students ahead.

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