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### The Nuisance of Memorization in Mathematics EducationPrevents Children from Learning to Think

Millions of children in the world are required to memorize an enormous amount of arithmetical and algebraic information as they move through elementary and intermediate levels of their education. They struggle mightily to memorize addition, subtraction, multiplication and division facts. Examples of addition facts are 5+3 and 8+7. Examples of subtraction facts are 7-4 and 13-8. Examples of multiplication and division facts are 7X6 and 28/4 respectively. The mighty struggle continues unabatedly as they memorize the "steps" for the following abbreviated list of algorithms:

1. addition, subtraction, multiplication and division with whole numbers;
2. addition, subtraction multiplication and division with fractions and mixed numbers;
3. addition, subtraction, multiplication and division with decimals;
4. operations involving percents;
5. operations on signed numbers, monomials and polynomials;
6. solutions of different types of equations and inequalities;
7. various solutions of systems of equations; and
This list grows longer and longer as it covers higher levels of algebra, geometry and trigonometry.

Is memorization the natural way that human beings retain content? If it is not, then the enormous amount of memorization in traditional mathematics education constitutes a serious abuse of young learners.

For insight into the mental activity of the natural learner, consider how you (and all the rest of us) learned stories. Did you memorize them? When we say we have memorized some content, it is understood that an intensely concentrative and often stressful effort was the price paid for its achievement. Did you pay this price for learning stories? Clearly, you made no such effort; but you learned them. Even more amazingly, beyond no effort, you had no intention to learn them. What an awesome gift it is to learn and retain content without effort or intention!

If you recite a poem today and again a few days later, the words are the same. Was the effort to learn a poem the same as the effort to learn a story? As you recall, your learning of a poem required both intention and intensely concentrative effort. If, without reading, you tell a story today and again a few days later, are the words the same? They are not. However, on both occasions, the events and their sequence are the same. The events and their sequence constitute the structure of the story. The effortless and unintended learning of a story is powerful testimony to a natural "wiring" of virtually every human mind that inevitably assimilates the structure of a story (its connections and flow).

Let us take note that this is an article concerning mathematics education. Please also note that mathematics, of all the academic disciplines, is the one most concerned with structure for its own sake. Consequently, educators should consistently activate and cultivate young learners' universal aptitude for assimilation and retention of structure (as exemplified with stories) for the similar mastery of elementary through high school mathematics. Children also exemplify their aptitude for assimilation and retention of structure when they understand verbally expressed information. Incidentally, the conveyance of structure, from a speaker to a listener by means of language, occurs with amazing speed and efficiency.

How do we activate this aptitude for the mastery of "lower" addition facts such as 5+3 and lower subtraction facts such as 8-2? Look at the charts below.

We call this our "eight chart." As you can see, the sum of the two numbers on any horizontal level of the chart is eight. Our program presents an exciting game with the objective that after the numbers (below 8) have been erased from the chart, the children use the structure of the numbers on the left, counting backward from 7 to 4, and the structure of the numbers on the right, counting forward from 1 to 4, to recall the missing numbers on any level. This is the mental structure that, with relatively little practice, even kindergarten learners can use for rapidly and accurately responding to facts such as 5+3 and 8-2. By similar application of our ten-, nine-, seven-, six-, five-, four-, three-, and two-charts, kindergarten learners achieve mental mastery of all lower addition/subtraction facts without use of fingers.

With respect to the higher subtraction facts such as 13-8 and 12-4 for example, our strategies enable kindergarten/first grade learners to mentally transform them to 10-8+3 and 10-4+2, respectively, which, in turn, are transformed to 2+3 and 6+2. Hence, they conclude that 13-8 equals 5 and 12-4 equals 8. Eventually, recall of these facts from this kind of mental structure becomes (with some practice) as rapid as recall through memorization.

So far, we have looked at strategies for mental mastery of the various addition/subtraction facts that activate, for mathematics, our universal gift for learning, retaining and recalling stories from structure rather than memorization. Let us now look at structures for learning, retaining and recalling some arithmetical algorithms rather than relying on memorization of meaningless "steps." With respect to the long division 2,347÷4, for example the contextual structure within which second graders function, is as follows:

1. Find the maximum thousands of 4 they can subtract from 2,347 (zero thousands of 4);
2. Find the maximum hundreds of 4 they can subtract from 2,347 (5 hundreds of 4);
3. Find the maximum tens of 4 they can subtract from the remaining 347 (8 tens of 4 or 80 fours);
4. Find the maximum ones of 4 they can subtract from the remaining 27 (6 ones of 4 or 6 fours);

They conclude that they have subtracted 586 fours from 2,347 with a remainder of 3.

Our approach to the teaching of all algorithms of arithmetic and algebra similarly activates and cultivates children's universal gift for learning, retaining and recalling stories through structure. We totally abolish meaningless memorization (rote learning) from mathematics education.

There is a horrible consequence of the traditional requirement to memorize one's way through arithmetic consistently without meaning. Consider the thousands of times, day after day, week after week, month after month, year after year, from first grade through seventh grade and beyond, when you were required to recall arithmetical or algebraic information from memory only. How does one's mind function for recalling meaninglessly memorized information? For example, let us suppose some children memorize the higher subtraction facts for a test and they would be under vigilant surveillance to prevent the use of fingers. On the day of the test, a member of the class, looks at the first problem, 15-9, looks up pensively, waits a little bit, the answer "pops" into his mind and he eagerly writes it down. He now looks at the second problem, 13-8, looks up pensively and waits. This time, however, the answer does not pop into his mind. It is taking too long and he is beginning to worry. He knows that sometimes the answer is there and sometimes it is not. Memory has often disappointed him and this time it feels like it is about to fail him again. Sadly, in spite of the contortions on his face, as he desperately attempts to squeeze the correct answer into his mind, he makes a blind guess and writes down the wrong answer.

Now let us return to the point of observing Johnny as he waits for the answer to pop into his mind. What is his mind doing? Absolutely nothing! Even as he desperately squeezes his brow, his mind is completely blank: a tabula rasa. He is not thinking. Can there be thinking without thoughts? Now multiply this unhappy occasion for Johnny by the thousands of other similar occasions through elementary, intermediate and high school (day after day, week after week, month after month, year after year) when, as a consequence of his attempt to recall information from memory, his mind is completely blank. Is there a more effective way to condition the minds of young learners so they never make the personal discovery of cognitively processing mental structures for the recall of mathematical information? Is there a better way to prevent impressionable young learners from becoming thinkers and, consequently, good problem solvers? The damage that prevents young learners from becoming good problem solvers is a result of the non-thinking approach to arithmetical facts and algorithms: recalling through memory.

In this article, you have already seen how Johnny's mind could function if the answer to 13-8 did not pop into his mind. He could resort, calmly and confidently, to using thoughts for cognitively processing a structure as his means of recall. Consequently, he would transform 13-8 to 2+3 and write 5 as the answer. Consistently using structures for the recall of arithmetical and algebraic information (day after day, week after week, month after month, year after year) activates, cultivates and nurtures our universal capacity to learn contextually, which was the means whereby we taught ourselves our native language and retained stories. This thinking (rather than non-thinking) approach to arithmetical facts and algorithms (basic arithmetic) from kindergarten through fourth grade constitutes a critical prerequisite for conditioning children to become good problem solvers. Condition them to think consistently through the basics, using strategies that work, and they cannot prevent themselves from thinking when solving problems.

The total elimination of memorization from mathematics education allows impressionable young learners to rely on context (structure) as their means of recalling mathematical and algebraic information. This is precisely the nurturing necessary for the healthy development of the "embryonic mathematician" in all children: the universal acumen for making connections to decipher meanings and retain content.