Memorizing In Math Is A Necessary Evil
Hi there. In this post, I talk about how memorization is mathematics is a necessary evil. Anyone who has a mathematics/engineering/statistics (or related) degree has had to learn and memorize concepts on their way of earning their degree.
Some people like memorizing and are comfortable with it, some are neutral with it and some just hate memorizing as they prefer understanding more. (I am in the neutral camp where I do not like pure memorization but if I have to do it in math then I'll do it.)
In this post, I separate it into four topics. The first section talks about how math is like a language. High competency in languages require a lot of practice and attention to detail. The second section is an overview how there is a lot to know and memorize in mathematics due to a wide variety of topics (and sometimes too much algebra homework). Section three talks about paying attention to details on top of memorizing stuff. The fourth section shares some tips and tricks for math memorization.
Topics
- Math Is Like A Language
- There Is A Lot To Know & Memorize In Mathematics
- Paying Attention To Details When Memorizing In Math
- Some Memorizing Tips & Tricks In Math
Math Is Like A Language
Learning a new language outside of your own native tongue language is not easy. It is generally encouraged for young kids to learn languages as soon as possible as they are in the early learning phases. This gives young students a lot of time for practicing by the time they hit their teenage social years and work ready years. With mathematics, it is not much different to language learning. Okay, the one key exception is where the mathematics gets theoretical and abstract to the point where seeing a number is rare.
Mathematics has its many symbols in the forms of numbers, shapes, lines, equations, algebra, ABC letters and Greek letters. As students get older and learn more mathematics, they start to see more and more symbols related to mathematics (until they stop learning math). There are times where you have to memorize key concepts or formulas for solving a problem. Not knowing something could cost you an entire question that could be 5 to 10% on a test.
(Side note: A lot of non-French people are learning French as a second/third language in Toronto. Entering a French-immersion school has great benefits. Once you leave French-immersion, the student cannot re-enter.)
There Is A Lot To Know & Memorize
As math is like a language, there is so much to know and remember. It is a mentally demanding subject by nature. (Generally, many students score lower grades in math related subjects versus non-math related subjects) On top of knowing your numbers, letters, symbols and shapes, you have to know how certain things works, how to solve problems and how to communicate in mathematics.
A big reason why there is a need to know a lot and memorize things in mathematics is because there are so many different types of problems which requires math memorization. These problems include:
- What score do I need on the final exam so I can get at least an eighty percent final grade? (Uses average concept)
- Why is a rectangle not a square?
- Determine the mean and median hospital waiting times.
- What is 77 multiplied by 10?
- What is the quadratic formula?
- Explain the confusion matrix from machine learning.
Paying Attention To Details When Memorizing In Math
This section refers more to algebra, formulas with subscripts, theorems and proofs when it comes to details in math memorization.
Most memorization tasks are not too difficult. Examples include facts such as Toronto is a city in Canada, Michael Jordan was a famous basketball player, one foot is 12 inches, vinegar is an acid, spinach is a vegetable and so on. When it comes to memorization in mathematics, it can get confusing for kids and adults (at higher level mathematics).
When it comes to mathematics definitions, the definitions tend to be more precise. Some people don't like it as it is a bit nitpicky (and/or they don't think writing is that important in math but it is). Some people prefer this precision as it makes things clear and people do not confused about certain math terms.
Example One
A half means one out of two equal pieces. Note that the word equal is used as it is important that the pieces are equal. Saying one out of two may not be enough for some picky math people.
Example Two
The famous quadratic formula is a formula that can be applied to any quadratic function of the form y = ax^2 + bx + c. The solutions to this formula are either real numbers or complex numbers. (Formula rendered in QuickLaTeX.com)
For people new to this formula, it looks scary as it kinda looks like some sort of alphabet soup with a random line. If you look at piece by piece, you have a negative b then a plus and minus sign, the square root of b^2 - 4ac and then a denominator of 2a.
Over time, the math student will eventually memorize this formula easily until they stop learning/using math.
Example Three - Absrtact Example - Monotone Convergence Theorem
I used to study some Real Analysis but now I don't use it nor remember everything from it. I do know some bits and pieces from this Monotone Convergence Theorem.
Source: http://mathonline.wikidot.com/the-monotone-convergence-theorem
There is another version of this Monotone Convergence Theorem but for Lebesgue Integrals. I've seen it once (I think) through a Measure Theory course. Don't ask me about it as I'm not really a pure math specialist nor do I remember the details.
Some Memorizing Tips & Tricks In Math
Even though there are a lot of math concepts and formulas to memorize, there are a few tricks and memory aids here and there. I list a couple here.
Even Numbers
Even numbers (above 9) end with digits of 0, 2, 4, 6, and 8 in the ones place.
Multiplying & Dividing By Tens
Multiplying by tens is quite easy once you recognize that you add zeroes (from moving the decimal place to the right). Examples include 7 x 10 = 70, 88 x 10 = 880, 100 x 100 = 10 000, 7.8 x 10 = 78 and so on.
Dividing by tens is pretty much the reverse of multiplying by tens. Instead of adding zeroes, zeroes are removed when dividing by ten or multiples of 10 (moving decimal place to the left). Examples include 90 ÷ 10 = 9, 120 ÷ 10 = 12, 11 ÷ 10 = 1.1, 1000 ÷ 1000 = 1, 888 ÷ 10 = 88.8 and so on.
Prefixes
Math prefixes are nice when it comes to naming shapes. Triangle contains the prefix tri- for three, pentagon contains the prefix penta-, octagon contains the prefix octa- and more.
Trigonometric Ratios - SOH CAH TOA
In high school mathematics, trigonometric ratios are used to find missing angles and missing side lengths from right angle triangles. The sine, cosine and tangent functions can be equated to their respective ratios depending on the angle positioning relative to the opposite side of the angle, the adjacent side to the angle and the hypotenuse side.
The memory aid for memorizing the trigonometric ratios is called SOH CAH TOA. See DuckDuckGo Image Search Result Photo below.
Calculus Derivatives - Product Rule And Quotient Rule
In introductory calculus, computing the derivative of a product of two different function types requires a formula and concept called the product rule. For computing the quotient of two function different function types requires the quotient rule.
Instead of using the full function notation such as f(x) and g(x) it is more easier to memorize the product and quotient rule using f and g. Shorthand derivatives notation represented by f' and g'. (Screenshot from http://tutorial.math.lamar.edu/classes/calcI/productquotientrule.aspx)
