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Memorization in math


Roadrunner
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As we are working through AOPS preA, I am wondering what should be memorized. DS naturally retained a lot of perfect squares and recognizes quickly some numbers (example- oh yes, 64 is 2 to the 6th power). He doesn't know his cubes as well, which slowed his computations on problems. All of those got me wondering if there are some things we need to memorize (like we memorize multiplication tables) to make computational aspect of math easier. If yes, what do we need to focus on? I think thus far he can recognize all perfect squares of up to 25.

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I am very glad our math teacher made us memorize the squares through 20x20 and the cubes through 10^3;  I greatly profited from that exercise.

It's really no big deal.

 

Other than that, I don't think I'd memorize anything. But squares and cubes are quite handy. I notice that many of my students have absolutely no number sense and can't identify those easily - which means their expressions won't simplify.

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No, but if you work through the book, some of these things will pop up again and again. DD has a good memory, so with some exposure, she's able to remember them. I think if you did systematically memorize them, like squares to 20 and cubes to 10, or whatever limit you set, it's just easier to see when you can simplify. It's no fun watching DD prime factorize, given her messy writing and those branches jutting out everywhere.

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I am very glad our math teacher made us memorize the squares through 20x20 and the cubes through 10^3; I greatly profited from that exercise.

It's really no big deal.

 

Other than that, I don't think I'd memorize anything. But squares and cubes are quite handy. I notice that many of my students have absolutely no number sense and can't identify those easily - which means their expressions won't simplify.

That's what I am noticing. He can solve challenging problems if he identifies big numbers as powers of something. He easily simplifies and moves on. If memorizing some of this simplifies his life, it's worth an effort.

 

I don't know if it's just chapter two of preA, but it seems recognizing various powers of two and three is handy as well. I am tempted to make a poster for the wall. :)

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Fwiw, Roadrunner, you have given me food for thought. Not long ago I started to give powers of 2 and 3 to ds8 to calculate as part of his much-loved oral bedtime math, along with squares and cubes, trying to take advantage of his current joy of working with numbers. I have been so busy worrying about the older ones that I hadn't considered any plan for the future, like next year (his brother did the prealgebra text in 4th though he probably won't quite be ready - he really hasn't had much afterschooling at all)... he was an engineer for Halloween, LOL.

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I would not directly work on the memorization, but I would make the chart and have it where you can see it regularly.  In this way, it is not going anywhere and just through exposure the will begin to memorize.  After chapter 2, the need goes way down.  However, the knowledge is not a bad thing to have.  Doing memory work with flashcards and such can really kill the joy of the math, though.  This is why I would use more of an immersion/osmosis approach.  He will use them over and over, so I would not worry about them somehow getting skipped if you are just immersing him in the math.  If he repeatedly forgets one, then fine (I have always wanted 4 x 8 to be 36.  I have to still consciously remember it is 32).  In general, though, I think this is one of those things you pick up through action.

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I suppose it depends on the child.  I can tell you perfect squares well past 20x20, but I never consciously memorized any of them.  Useful patterns will recur time and time again over the next 10 years.  Personally, if it's a choice on where to spend a little extra energy, I'd choose theory over noncritical computational aspects.  I'd let the discovery of patterns be a joy, not a requirement.

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I am very glad our math teacher made us memorize the squares through 20x20 and the cubes through 10^3;  I greatly profited from that exercise.

It's really no big deal.

 

Other than that, I don't think I'd memorize anything. But squares and cubes are quite handy. I notice that many of my students have absolutely no number sense and can't identify those easily - which means their expressions won't simplify.

 

I also had a teacher who required us to know that same list of squares & cubes, and I've benefited from the instant recognition countless times since (thank you, Mr Grimm!)

 

In general, I'm not a memorize-y person at all, and I hate to require stuff that might squish a kid's love of math. But this doesn't take that long to accomplish. Just start with having him make a small poster to hang on the wall near his work space. I bet that by the end of the year, he'll know them by heart. :)

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I'm not a fan of lots of memorization, but knowing your squares especially as well as cubes is very handy. I'm more of a person that says as you use it more, you'll probably learn it. I'd also decide based on the child and what they want to do.

 

... my son has now memorized the first 50 digits of pi which is way less useful, but comes in handy as he's on an ultimate Frisbee team named team pi.

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I suppose it depends on the child. I can tell you perfect squares well past 20x20, but I never consciously memorized any of them. Useful patterns will recur time and time again over the next 10 years. Personally, if it's a choice on where to spend a little extra energy, I'd choose theory over noncritical computational aspects. I'd let the discovery of patterns be a joy, not a requirement.

I wholeheartedly agree with this and vote no on memorization. My kids never practiced rote memorization for any math topic, including multiplication tables. They learned through use.

 

(I also let them learn whatever concept interested them and did not follow any particular order, especially for my most math talented son. This approach might not work with every child.)

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Just out of curiosity, what are some other things that people have found useful on their mathematical lives when it comes to memorization? Not necessarily active memorization. It could be just from use. It would be interesting to see what various memory work would be helpful to continue even as Ds gets older. So often I think memory work gets dropped by the logic stage and that isn't necessarily helpful in the long run.

 

For me the quadratic formula and derivation using binomials & Pascal's triangle for expansion of binomials. Those were the first major ones that came to mind.

 

Anyone else?

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Here are some from prealgebra/algebra 1 that *I* think are important to have committed to memory.

 

factoring - difference of squares, perfect squares, difference/sum of cubes

lines - y = mx + b, y - y1 = m(x - x1), m = (y2-y1)/(x2-x1) = rise/run, parallel lines have the same slope, perpendicular lines have negative reciprocal

distance formula - d = sqrt ((y2-y1)^2 + (x2 - x1)^2)

pythagorean theorem -- a^2 + b^2 = c^2, where c is the hypotenuse of a right triangle

areas and perimeters of triangles, circles, rectangles

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apologies if i have told this story too often.  i am reminded that when my younger son started a new prestigious private school in 2nd grade, I made him learn his multiplication tables up to 12 x 12 before showing up the first day.  when he came home all smiles i asked how it had gone and he said fine.  i asked if his multiplication tables were up to snuff, and he began laughing uncontrollably.  when he calmed down he said in this class they did one multiplication table per month, and the first month was the zeroes times table.  it was a really good school in lots of ways, but somewhat math phobic, and i thought about how much tuition i was spending.

 

 

i have noticed a lot of mathy kids enjoy memorizing things like a lot of digits of pi, so if they enjoy it, let them.  I agree too it can be useful to know several squares and cubes.  (I myself probably know squares up to 17^2.)  this includes the expansions of things like (a+ b )^3, (a+ b )^4, and the factorization of (a^3-b^3), ((a^3+b^3), (a^4-b^4).  these latter are actually more important.  i see kiana beat me to these.

 

amazingly, in the age of dependence on calculators i have had students in calculus who did not know the cube root of 8, nor how to multiply 2 digit numbers even using pencil and paper.  a propos of nada, i happen to know the random 14 - symbol alpha numerical password for my wifi connection, which always amazes guests, but i find it useful when they ask what it is, instead of turning the modem over and squinting.

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intelligent memorization is always useful.. i used to memorize the formula 1^2+ 2^2 + 3^2+....+n^2 = (1/6)(n)(n+1)(2n+1), at least i think that's it, to use in my calculus classes to integrate the area under the graph y = x^2.

 

then i realized that the only part i needed for that was the first term: (n^3/3 +......).  then i realized that all the formulas are like that:

 

i.e.  1^3+2^3+3^3+.....+n^3 = (n^4/4 +.......),  and 1^4 +2^4 +3^4+....+n^4 = (n^5/5 +......), and that you never need the part where the dots are..... 

 

Very few calculus textbooks point this out, most just bash out the result using the full formula, showing to me their lack of appreciation of what matters.

 

 

always try to find the understanding that makes the memorization easier.  if someone has trouble with (A+ B )^2 = A^2 + 2AB + B^2, show her/him the picture of a square with sides A+B, and how it decomposes into an A square and a B square and two AB rectangles.  that's how Euclid did it in the Elements.  No one forgets it after that.

 

 

 

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Honestly, factorials and sequences pop up more frequently than many squares or cubes.  In probability, knowing 6! off the top of your head can be useful -- sometimes.  Recognizing the binomial expansion 10 levels deep is actually helpful, believe it or not.  But, I still don't think it helps day-to-day.  I didn't need to know such facts to get either of my degrees (math / physics), but I DID need to recognize numbers as composites, be it squares, factorials, cubes, complex, or whatever.

 

I would never fault anyone for memorizing squares or cubes, but a well-placed poster would be as good as anything.

 

For composites, number theory is a real treat.  Quickly: what's the greatest prime factor of 15! + 16!?  It's easy to do in your head, if you recognize the composite nature of numbers...  ;)

 

 

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always try to find the understanding that makes the memorization easier. if someone has trouble with (A+ B )^2 = A^2 + 2AB + B^2, show her/him the picture of a square with sides A+B, and how it decomposes into an A square and a B square and two AB rectangles. that's how Euclid did it in the Elements. No one forgets it after that.

He had this in his preA chapter 2 problems. He used distributive property to solve it. Off to explore visuals. :)

 

 

Ooooooooo :)

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interestingly, my mother used to cut my sandwiches into the same visual representation of a+b squared that mathwonk is mentioning, while telling me that this was ab, this was a^2, etc.

I will most definitely share this story with my son the next time he tells me to stop being such a math nerd! He needs to realize it could always be worse 😄

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I will most definitely share this story with my son the next time he tells me to stop being such a math nerd! He needs to realize it could always be worse 😄

 

Haha!

 

We also had posters, both hand-made and purchased, on the wall with fibonacci numbers, the golden ratio, pascal's triangle, 800 digits of pi, all the multiplication tables, and probably more that I've forgotten. Whenever we baked cookies, she'd come up with some reason we needed to make 1.5 times the recipe or something and quiz us on fraction multiplication. This is what happens when an engineer homeschools, you see.

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is it 17?  (took me a few minutes of thought.)  hint:  factor it.

 

 

quickly:   why is the smallest prime factor of  16! + 1, at least 17?

 

Yes: 15! + 16! = 15! + 16 (15!) = 15! (1 + 16) = 15!(17)

 

Yours isn't quick computation -- it requires a little theory.

 

Assume 16! + 1 had a prime factor a, 1 < a < 17.  Then, by definition, it also has a factor b such that ab = (16! + 1).

Note that 16! also contains every factor larger than 1 and less than 17, including both a and b independently (edit: challenge to the reader - why must they both be included independently?  Be careful!)

This means (16! + 1) - 16! = ab - abc for some c >= 1, or 1 = ab (1 - c), where c >=1, an impossibility.

Therefore, neither a nor b exist, and the smallest possible prime factor of 16! + 1 is 17.

 

The shortcut trick is to just remember it, but you should be able to reproduce some form of proof for that one.  :)

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The distributive property.    Many students memorize a(b+c) = (ab) + (ac), and they are fine if they are presented with a(b+c)..but ask them to take ab+ac and find a(b+c) and the sweat starts pouring. Important to know and understand, not just memorize.

 

Furthermore, they *really* struggle when doing something like (a+b+c)(x+y) = (a+b+c)x + (a+b+c)y. 

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@Mike: Nice argument!   it can also be done "by inspection" as follows: the remainder of (16)!+1 upon division by any integer from 1 to 16, is visibly 1 (why?).  so the smallest prime factor is at least 17.

 

(hint if wanted by anyone:  "the 3 term principle"  says: if n divides a and a+b, then n also divides b.)

 

 

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@Mike: Nice argument!   it can also be done "by inspection" as follows: the remainder of (16)!+1 upon division by any integer from 1 to 16, is visibly 1 (why?).  so the smallest prime factor is at least 17.

 

(hint if wanted by anyone:  "the 3 term principle"  says: if n divides a and a+b, then n also divides b.)

 

Nice with the remainder.  I always shy away from visual inspection arguments because I get myself in trouble, but this one is completely sound.

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Haha!

 

We also had posters, both hand-made and purchased, on the wall with fibonacci numbers, the golden ratio, pascal's triangle, 800 digits of pi, all the multiplication tables, and probably more that I've forgotten. Whenever we baked cookies, she'd come up with some reason we needed to make 1.5 times the recipe or something and quiz us on fraction multiplication. This is what happens when an engineer homeschools, you see.

Where does one buy math nerdy posters? I think my son would like some.

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  • 3 weeks later...

My high school math teacher would tell us, "Memorization is a mathematical malpractice."  I take that to heart.  To wit, today my dd asked me to remind her if 3-1 equals 1/3.  Instead of just telling her, I went through the AoPS rationale for 3-1.  

Start with 33 = 27

32=9

31=3

30=1

3-1 = ?? what is the pattern?

 

This also reminds me of the snarky comments made by smart kids in my high school math class, along the lines of, "I forgot the formula on the exam, so I derived it."   :glare:

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Things I have found worth memorizing are measurements. I grew up with metric and I live in the US which uses imperial and i still sometimes check a chart when I am cooking. I have cookbooks in metric but all my cooking utensils are in imperial.

And it is super helpful to know how to convert Fahrenheit to Celsius when talking to my mum in Australia.

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