Some Mysteries of Arithmetic Explained:
Have you noticed that information presented in school and in textbooks is perceived by students as much too sacred to be touched? For example, how many students would have done this? Using a book borrowed from the library, a young teenager studied trigonometry on his own because he wanted to be a scientist some day. But the names of trig functions printed in the textbook did not make sense to him. So he created his own names! It worked beautifully until he earned a doctorate in physics and began working with other physicists who did not know what he was talking about. Whoa! Time to go back and memorize the textbook labelswhich he did in a flash. That young man was Dr. Richard P. Feynman who won a Nobel prize in physics.
Even testing to assess learning implies that students must duplicate information in their brain without editing. The more exact the replication by the students memory, the higher the grade. Should we encourage students to doodle with information instead of merely duplicating information in memory to pass a test?
A closer look at doodling
Only when one advances to a masters or doctorate degree, do the rules change from duplicating information to the discovery of information. All the way up to graduate school, the model is that of monks in a monastery carefully copying sacred scripture exactly as is. Then suddenly, one is in graduate school where the rules change. Students are now like people in a think tank attempting to decipher mysteries of the atom or speculating about the future in technology. By that time it is too late for most students. Their stunning success in school has been a by-product of by heart learning which is signaled by cliches such as cramming for a test.
Some fun examples from arithmetic
Here is a fun example from arithmetic usually too sacred to tinker with. Mathematics is, after all, a field of absolute truths that have been proven for hundreds of years or has it?
Show me an exciting example
Like everyone else, I always believed that there was only one algebra and that was the algebra I learned in school. Laurie Buxton in a revealing book, Mathematics for Everyone, convinced me otherwise with the statement that there are many algebras, one of which by the Irish mathematician, Roland Hamilton, was used by Albert Einstein to predict the location of Mercury. Not only that, but Buxton suggested that anyone is capable of creating a new algebra by selecting some symbols and a few rules. The test is whether the new algebra is self-consistent, meaning there are no internal contradictions. More about this in a moment.
My attempt to create a new arithmetic that will explain some mysteries
I confess to you that I am not sure what the difference is between arithmetic and algebra. It looks as if one uses numbers and the other uses letters of the alphabet. But, that isnt right either since arithmetic and algebra both use numbers and letters, but arithmetic does seem to use mostly numbers and algebra uses mostly letters.
Doodling with the arithmetic we learned in school
I discovered that the standard arithmetic we learned in school has some severe limitations that are hidden until one begins to doodle with arithmetic. Here are some examples:
Multiplication is nothing more than repeated addition
With utmost confidence, teachers present to their students this premise: Multiplication is repeated addition. If this premise is true, I believe it only holds for whole positive numbers. It certainly does not explain negative numbers or fractions. Let me show you how I arrive at this conclusion.
Lets start with positive numbers
An alternative model that explains the mystery
The schools interpretation of multiplication is that it is simply repeated addition, but I have demonstrated that this may not be true for negative numbers. Here is a new interpretation of multiplication which will explain positive numbers, negative numbers and, as a bonus, it will also explain fractions.
New interpretation of addition and multiplication
Addition means to copy the first number, copy the second number and combine like this:
2 + 2 = 4
Multiplication is not the same as addition because the first number is not a real number, but an instruction to copy a certain number of times the second number which is real and then add. For example,
Application to positive numbers
Application to negative numbers
Whether you multiply, for instance, - 2 times + 2 or reverse the order and multiply + 2 times - 2 the product is the same. In this case, - 4.
Lets kick it up a notch and add a little complexity
Division of whole positive numbers
If we start with a fraction such as and add an increment such asoror, the result is a value larger than . Intuitively, that makes sense. Adding something to something results in something bigger or larger than the we started with. Thats the nature of addition.
So, since multiplication is repeated addition, if we multiply the fractions above, the result should be an increase in valuesomething bigger or larger. Lets test to see whether this is true.
If we multiply the initial fraction of , with some increment, contrary to expectation, the result is a decrease in value.
Lets solve the mystery with an alternate interpretation of multiplication
With my new copy rule, it makes sense that when one fraction is multiplied by a second fraction, there is a decrease in value.
Dividing fractions using standard school arithmetic:
There is a satisfactory explanation for each question which requires more detail than we have space for in this article. The curious reader can find the answer in my book: Brainswitching: Learning on the right side of the brain. See Chapter 9: Use brainswitching to learn the second most difficult subject in school.
Now, I would like to explain the division of fractions using the new copy rule for multiplication.
Application of the new rule to division with fractions
I selected simple examples so that the underlying pattern is transparent.
While algorithms that we all learned by heart in school enable us automatically and efficiently to apply arithmetic to solve for addition, subtraction, multiplication, and division, the explanation for why the procedures work is a mystery. For whole positive numbers I have demonstrated that multiplication is simply repeated addition. However, the premise is false for negative numbers and fractions.
The solution to the mystery is a new copy rule for multiplication which is internally consistentmeaning there are no contradictions. I do not believe that the new copy rule is efficient for actual computation, but the new rule does solve some mysteries in arithmetic by explaining the inner structure of arithmetic with positive numbers, negative numbers and fractions.
Inner structure is a term I borrowed from the Gestalt psychologists who believed that understanding only comes from insight which is suddenly seeing a cause-effect relationship. Seeing a cause-effect pattern is the key to one-trial learning which they advocate rather than memorization by many repetitious trials as in by heart learning.
We now believe that one-trial (or first-trial) learning happens in the right brain while multiple-trials to learn by heart takes place in the left brain. The reason: The right brain is looking for a cause-effect pattern to explain something. The left brain is looking for flawsreasons to filter or block incoming information from long-term memory. Why retain something that is not true? Why retain a lie?
Any information is perceived by the left brain as a potential threat to the stability and security of the individual. Stick with the tried and the true. Better to be safe than sorry. The blocking mechanism of the left brain is to erase information so that many trials are necessary until the left brain fatigues and concludes, I give up! If you insist, I will store the information in long-term retention, even though it is against my better judgment.
The ape experiments
A famous example of insight that produces one-trial learning is the ape experiments by the Gestalt psychologists. A hungry ape is in a cage with several boxes placed at random on the floor along with a stick.Hanging from the ceiling are bananas high enough to be out of reach.
The ape tries to leap up and grab the fruit many times but is unsuccessful. He sits on a box and seems to be puzzled. Finally, he picks up the stick and tries a number of times to knock the bananas down, but that strategy does not work.
He sits on a box again and looks perplexed. Then, suddenly, he stands up and places several boxes on top of each other, climbs up and retrieves the bananas. In an instant flash of recognition, he seems to see a cause-effect connection between boxes on top of each other and access to the food.
Wrap it up: Benefits for kids in math classes
My hypothesis is this: If parents and teachers prepare youngsters with my interpretation of why multiplication works, it may be easier for the brain to assimilate nitty gritty procedures in arithmetic.
With my interpretation, they have a chance to see cause- effect relationships for procedures that now do not make sense. They have a chance for insightthe marvelous Aha, I get it! response rather than settling for the instructor asserting: Just do it because Im telling you it works! Dont ask me why! Just memorize it!
Part of this article is excerpted from my new book, The Weird and Wonderful World of Mathematical Mysteries: Conversations with famous scientists and mathematicians.
My conclusion is that the numbers 1 and 2 behave differently compared with the numbers to follow such as 3, 4 and 5. How come?
Recommended follow-up reading
Asher, James J. (2005). A Simplified Guide to Statistics for Non-mathematicians:
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