25 Nov 2009
Why Some Math Formulas Shall be Memorized
Frank Ho, Amanda Ho
Canada certified math teachers at Ho Math and Chess
Shall math formulas be memorized by students? Many times we hear the answer is either YES or NO and both these answers are absolutely wrong.
If the answer is “yes” then it seems to imply that all formulas learned from grade 1 to grade 12 shall be remembered, this is a wrong way of teaching math. In contrast if the answer is “no” then it means students are not required to remember anything and in this case what is the consequence if students do not bother to remember any formulas? My answer is that it really depends on what formulas. Some shall be remembered and some can be just to be familiar with but not necessarily remembered. For example the formula of sin (x+y) shall be familiar but not necessarily memorized since they do not appear very often and can easily get confused with other similar formulas such as cos (x+y).
Many times, we hear that some experts demonstrate on how great it is to use fingers or other ways to get answers for times table, but as I pointed out in my other article it is embarrassing to stand in front of cashier to use fingers to calculate a simple transaction when grown up. The other problem is these students seem to have more difficulty in calculating the factors of a number since the “fingers” only teach them one way not the reverse way of finding out what times what is the product? The most apparent impact is these students who do not have times table memorized by heart will lose number “instinct” in doing any number theory problems; problems like to find GCF, LCM, factors, primes, factorization, divisibility etc., Their thinking is slow and perhaps will lose their confidence when seeing other kids can get answers much faster. The other problem is they can not do division well since it is basically a reverse calculation of multiplication. So from this point on, their ability in math is affected when learning decimal, fractions, percentage, ratio etc. They are not mental alert and have not developed any instinct of number sense or have weakness in finding number pattern.
What are the formulas shall be remembered by students? The typical answers will be some of these (not meant to be exhaustive):
1. Pythagoras property, if you do not remember then you can forget about doing well in any math contest.
2. 30-60-90 or 45-45-90 degrees ratio.
3. Rules of 2 triangles are congruent.
4. Rules of multiplying 2 fractions or dividing 2 fractions.
5. How to get answers quickly for 1234.567 divides by 0.001.
6. All the factoring formulas like (x+y) squared, (x+y)(x-y) etc. Perhaps some math teachers did not memorize themselves when they were students so they did not emphasize the importance of remembering these formulas when teaching later in their life, so their students have troubles in working on completing the square or rational equations. They can not see any clues on how to work out problems. By the time, I have my grade 9 students finish practicing these factors problems, they have done over 2000 questions. These students have no problem later in grade 10, 11, or 12 to continue to work on factors related problems when compared to other students who only had done 2 pages of questions (standard assignments in textbook).
7. Trigonometry sin, cos, and tan. How can a student prove identity when they do not even know 1 can be converted to . The tragedy is some teachers did not tell students other than give a “cheat” sheet but students still could not do proof. I always have to demonstrate on why I know how to prove and why they did not know how to prove the obvious difference is I “see” things and they don’t. The reason for they not seeing things is they have no instinct in knowing since they have no information (formulas) stored in their brain.
8. Squares from 11 to 15.
I list a few examples above to only demonstrate how important for a student to remember some formulas so they can work out problems easier and faster too. It shows from my teaching experience if they do not bother to remember any formulas then even given a cheat sheet with all formulas on it they still cannot work out the solutions.
Amanda Ho, Frank Ho