Math Talk # 02 -- What to Memorize

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What in basic school mathematics do you feel is "Need to Memorize" and why?

What content, facts, patterns or relationships has your combined insight, experience and education led you to realize is truly useful for students to know as they continue through the continuum of mathematics.

Memorized in this context simply means reliably committed to a memory that is easily and reliably accessed and activated when needed. I don't want to get super technical about what it means for something to be "memorized" in this context. Use your best judgement.

It can be anything from the scope/sequence of school mathematics. You may quantify your response, such as "Before Grade_, kids benefit from having memorized..." or "To thrive in ____ topic, kids need to have..."

One last thing, I know that this question invites a list-style response, but please take the time to explain why something made the list or what it is about it that leads you to feel that it should be memorized. Your justification could provide useful insight for someone else who reads this topic down the line.

🙂

Edited by mom2bee
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Well, I am only teaching through elementary right now, and probably once my kids get to middle/high school, I will be switching to something probably online based.  So, take that with a grain of salt.

Having multiplication facts memorized by like 6th ish is vital, IMO.  Not so much for concept understanding, just mostly for speed.  Without having basic multiplication facts memorized, it slows down a LOT of more complicated processes.  I struggled getting DD11 to learn math facts and it makes long division and fractions take so much longer than it has to because she often has to think out the multiplication.

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I don't think it's a matter of 'before a specific grade' but an issue of'before you move to certain topics.  Like, I realized that learning to tell time on a traditional clock is much easier if kids can count by 5s.  I loved the way that Singapore math had students regrouping with 10s all the time - my kids learned arithmetic much faster because there are only a handful of facts to learn with numbers smaller than 10.  We used blocks to physically make the groups (13-6, so take 6 from the group of 10, then add the 4 to the leftover 3 to get 7).  I think that for a long time they quickly visualized it, and for all I know they still do.

Equivalent fractions probably shouldn't be taught until kids are fluent with multiplication and division facts.  I have spent nightmarish sessions trying to help the kids that I volunteer with find equivalent fractions but how do you do 5/6 = x/30 if you can't figure out that 6 goes into 30 5 x and then multiply 5x5?  Cross multiplying can be taught as an algorithm, but they still have to be able to multiply and divide, and the numbers are bigger.  Likewise, long division prior to having a solid grasp of multiplication facts is awful.  I haven't had problems with my own kids because the program that we used (Singapore) is well-ordered, even if a little different from the old US sequence that I had as a kid.  The kids that I volunteer with seem to get a more scattershot approach, or maybe they just move on before mastering the first thing - it's hard to tell from worksheets what the overall plan is.

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I am going to relate this to real life needs because that is how I think in terms of math and memorization. There are a lot of things that are good to memorize before high school math, but I personally think math application is worth more in elementary grades. Once using math is mastered, I will move on to other things like order of op.

So in elementary I think:

multi. div. facts to 12 (maube just 10)

Common baking fraction equivalencies (1/4 is clean so I need 2 to make 1/2 kind of thing)

Ft in yard for sewing, inches in foot, cm in meter (we use a lot of metric so for us these are needed)

boiling and freezing points for understanding weather, baking

For us memorizing Roman Numerals was important as we do a lot of historical site seeing.

Money- how many dimes in a dollar, what each coin is worth

We also started this year to learn the Greek and Latin roots for numbers. This is a linguistic area, but applies to math. So learning tri means 3 helps in a lot of areas in geo. but also outside of math too. I also think it helps make it easy to see how numbers relate- when I get 1 kilogram of beef my son can now understand it better.

Right now that is what I am working on with my ds, 8, as far as memorization. There may be more, that is just what is on top of my mind. But to me these are all because they relate to regular use of math, so I want them automatic.

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This topic has been on my mind a lot lately as my kids are moving into higher math ... On the one hand, I'm often wondering whether they need to commit a new formula or equation to memory, or whether it's better that they just really understand the underlying concepts. On the other hand, I'm seeing how certain concepts, facts, and formulas come up again and again at every level of math.

To answer your question, then, I think that it is really helpful for kids to have these facts and concepts really nailed down as they progress through the levels (many of these won't be relevant until the upper middle and high school stages):

• Addition, subtraction, multiplication, and division facts through 12 x 12 (we got lazy, stopped at the 10s, and neglected to memorize multiplication facts through the 12s – I regret it)
• Squares of 1–15 (they really come up a lot and it's so satisfying to be able to quickly recognize 169 as a perfect square)
• Formulas for calculating the circumference of a circle and areas of 2D shapes (circle, square, triangle) – perimeter is easy to figure out without a formula, but circumference and area aren't always as intuitive
• # of degrees in a triangle and in a circle
• How to determine mean, median, mode for a data set
• Order of operations (once this is learned, it's used so often that it becomes intuitive, but important anyway)
• Quadratic equation – so annoying to memorize, but so helpful to have on hand whenever you're presented with solving a challenging quadratic
• Rules for exponents: x^a • x ^b = x^(a+b) and (x^a)^b = x^(a•b) -- these come up often enough that kids really need to know them but infrequently enough that they always seem to mix them up. Maybe it's not worth trying to memorize because they do get addled in kids' minds (or at least my kids'), but whenever we encounter them, I ask them to figure out for themselves which rule applies by substituting 2 for a and 3 for b, then running the experiment to see which is correct. [So, if you're multiplying two x's to two different powers, and you don't remember what to do with the powers (add or multiply), test it out in the simple case: x^2 • x ^3 = (x • x) • (x • x • x) = x^5 = x^(3 + 2) -- then you'll see that you add the powers.]
• Pythagorean theorem! It's absolutely everywhere in Geometry, Algebra 2, and Trigonometry
• Trig functions (sin, cos, tan) are easily confusable – we memorized SOH CAH TOA. They should also memorize the csc, sec, and cot
• Values of sin, cos, and tan for 30-60-90 and 45-45-90 triangles. This seems kind of random, but they come up all throughout Algebra 2, Trig, Precalculus, so it's really handy to have them memorized by the end of Geometry.

One last thing – this isn't a 'memorization' item, per se, but the one concept that I absolutely would make sure your kid understands forward and backward is fractions – add, subtract, multiply, divide, find the reciprocal, simplify, find common denominators, everything fractions!! Once you get into advanced math, you can always use your calculator for long division and multiplying decimals and adding and subtracting huge numbers, but your calculator can't really help you with fractions if you don't understand how they work and how to work with them.

Edited by RebeccaMary
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Assuming that you are interested in ensuring that your student is reasonably numerate by the end of high school, I would want to ensure mastery of and fluency with essentially everything that is taught in K-8.

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