Wednesday, May 26, 2010

What must be memorized?

I say I have a terrible memory, and that's why I'm in math instead of science (with its myriad names for bones, muscles, organs, chemicals, etc.). I say there's very little to memorize.

But other people disagree. I used to think it was only my students who did badly at math who had the 'wrong' idea that math was about memorizing. Math people will tell you it's not about memorizing, but many of them do think memory has a bigger place than I give it.

What do you think?

Here's what needs to be in your memory, eventually:
  • addition 'facts' (if add 1, double, and add 2 to an even are easy, then there are 25 harder facts)
  • multiplication 'facts' (2 is doubling, 4 is doubling twice, 5 is easy, 9 can be easy; only 9 facts left)
  • circumference and area of a circle (I know which is which by using dimension - circumference is 1D and has r, area is 2D and has r2. If I'm unsure of the formula, I can also check reasonableness of my answer by estimation.)
  • Pythagorean theorem (a2+b2=c2 for right triangles)
  • quadratic formula 
  • [edited to add ...] sine and cosine, tangent and reciprocal identities (the rest of trig pretty much follows, 6 facts)
That's a pretty short list. What would you add to it?

When I started teaching, I had a BA in math and still didn't have 7x8 memorized. I also didn't have the quadratic formula memorized. Whenever my topic of the day would include quadratic formula, I'd put it at the top of my notes. 7x8 I could figure out in 2 or 3 seconds.

To me, math is all about connections. When I think of slope, I think of a rise and a run (and a little right triangle under the slanty line, showing them). I know steeper lines want a bigger slope, so the rise has to be on top in the ratio that compares rise and run. To me, that's not memorizing.

So tell me what I'm forgetting.  ;^) What more do we have to memorize in math?


    1. Ooh, this is probably a bad question to ask someone like me (math circle instructor and math team coach).

      Unfortunately, many math competitions depend on time limitations, so success depends greatly on being fast. Usually that means memorizing theorems that could otherwise be derived. For example, it is possible to derive most of the trig identities from just a few of the simple ones.

      If you extend your question to include memorization of things other than straightforward theorems and facts, then general frameworks for solving problems, such as calculus and linear algebra can be included as necessary items for understanding many advanced topics in science and engineering.

    2. Some more simple stuff -- the area of a square. But maybe that is obvious. How about the area of a triangle?

      Inches in a foot and feet in a yard is good. Feet per meter? Centimeters per inch?

      I think it is good to know f=ma but then that is science, not math. ;-)

    3. @Hao: Yep, trig identities would be my downfall if I were in a competition. I've taught trig probably over 20 times, and still don't have those in my memory.

      But I don't think that would interfere with me being able to use the ideas. If I need them, I can either derive them or look them up.

      To me, understanding is very different from memory. I understand calculus and linear algebra, and can use those ideas to problem solve. For me, that sort of understanding is like learning to ride a bike - it will be with me forever.

    4. @Craig: Area of a square is, to me, pretty much part of the definition of area. Area of a triangle is something we figure out by putting two triangles together.

      It's unfortunate that inches in a foot needs to be memorized. That reminds me of how hard it's been for my son to learn the days of the week, and I imagine it'll take years before he has the months all down.

      Feet per meter? It's the image of a meter stick being just a tad longer than a yardstick that allows me to remember this. I think of it as 39 inches in a meter.

      Are these math? We need them to use our math. Pretty easy to look up...

    5. I totally agree with this post. I think in normal public schools, memorization is much overrated and I greatly struggled with times tables and hated biology and even geography for all the memorization. Regardless of that I was voted most intelligent - do you know why? I *understood* the concepts in math. The best math teacher I had was the one who figured out this about me and told me I didn't have to do any homework. I went on to get a 5 on the AP calculus test and a year of college credit for it. I cannot express how this rings so true with me - I am horrible at memorization, but all you need to do in math is understand the principle and you can always generate the formulas when you need them. I also aced some math trig competitions and such when I was in HS - 20+ years ago.

    6. I think there are really 2 distinct questions here:

      1) What mathematical facts do you need to have ready, automated access to in order to do math? (In other words, what information do you need to have in memory?)

      2) What mathematical facts do you need to engage in an act of "committing to memory"? (In other words, what do you need to bother memorizing?)

      I think the answers to these two questions are really different, and also depend on what you are using math for.

      My personal answer to question 1, based on how I use math, would actually be very large. I can't even begin to list the number of different results and theorems I've needed to have immediate mental access to at one point or other in order to do something else.

      But almost none of these facts, now firmly ensconced in my memory for immediate access, got there via an "act of memorization." Most of them got there by repeatedly re-deriving the results until they were firmly lodged; and by looking for ways to understand what was going on so that I understood in a mentally efficient and meaningful way how all the parts fit together.

      An example is the residue theorem from complex analysis, which is a totally crazy-looking formula that I nonetheless have completely in the bank, not because I ever "tried to memorize" it but because I spent a lot of time thinking about why it was true and trying to understand it in different ways.

      An example of a formula I have not managed to lodge into memory but that would actually be pretty useful to me is the formula for the discriminant of a cubic polynomial in terms of the coefficients. (I know it's got a couple 4's and a 27 in it...) If I were in a situation where I needed it for a test that was coming soon, I might have to resort to trying to memorize it, which I would hate. Since I am not under any time pressure, the thing to do is clearly just to re-derive it a few more times (and keep paying attention to how the parts relate as I do so). If it doesn't stick in my mind after that, then it wasn't meant to be.

      What am I getting at?

      I feel like the answer to question (2) is all about what time constraints you are under. I do think it really helps in doing math to have ready mental access to what you know; but you don't need to engage in an act of memorization to build this up. On the other hand it takes a long time not to. So if you have all the time in the world, I say you don't even need to "memorize" the times tables. Just keep solving single-digit multiplication problems any way you know how, stay awake to the relationships going on (can't remember 6 x 9? know 6 x 10? how are they related?), and eventually, possibly very slowly, you will come to know them all automatically, and will have learned a whole lot else as well. But if you are in a rush, you have to give up this lovely way of learning them if you need the facts. The trade-off for speed is that memorization can be a pretty miserable experience, and at the end of it, the facts will definitely be the only thing you learned.

    7. Ben, Can you give me an example I might be able to relate to? I don't know that residue theorem, nor why I'd want the discriminant for a cubic.

      I teach courses up through calc II and differential equations. The trig identities that get used repeatedly in calc have lodged in my memory, as you suggest. (If not, I'd keep looking them up, since the derivations seem pretty ugly to me.)

      Am I being handicapped in math by my poor memory, or are we just framing it differently?

    8. need to memorize which is sine and which is cosine.

      the rest can, more or less, be derived. Except the quadratic formula (and a bunch of others) are worth knowing... even if you can live without them.


    9. Yep! I'm going to add that one. Tangent and the other 3 need to be memorized too, right?

    10. I actually haven't seen any reason why any of this "has" to be memorized. For the most part, since leaving high school I haven't had many "closed book" tests and even though I got a degree in electrical engineering with a math minor and not do computer networking I still have never once run into a situation where I wished I had memorized more. Why do teachers focus so much on what must be memorized?

      I can't remember half these equations people have mentioned, but I am quite sure if I needed them I could look them up and apply them quite easily. I do remember sohcahtoa... But most of college is open book tests, and all of real life and technical career has been for me.

      I am really looking forward to getting my kids excited about math (only 1 and 3 right now), but I see no reason "memorization" should play any role in that. For now we have fun with counting and playing games and I think mostly the games just keep getting more fun... (:

      I do agree that when you use something enough, then you tend to remember it and can apply it quicker by memory. That doesn't mean it needs to be memorized - when cooking I don't always look up the recipes anymore either.

    11. I pretty much agree with you, Jason. 'Memorization' isn't a good way to do it, but eventually getting certain things into your brain ends up being very useful.

      Perhaps I should say more about why I brought this up. If you've been following my blog, you know I'm working on a book Playing With Math). In the current draft of the introduction I said, "There's very little to memorize." I know that will sound odd to many readers.

      People who aren't into math think a big part of it is memorization. Parents at my son's school (where I taught part-time for 2 years) worried about kids learning their multiplication facts, which was a low priority for me. My priority was increasing their comfort with the meanings of multiplication, through interesting problems.

      I'd say this is also an issue because of the way the 'standards' are framed. They talk about what kids should know, instead of the ways of thinking students should be comfortable with.

    12. Maybe an example of what Ben is talking about (for your students) could be the laws about how multiplication and addition operations both commute, and the distributive law, etc.

      I'm glad to see it's not on your list of things to memorize, but at the same time, you'd be unhappy if they didn't have instant access to that information. It's something they get by understanding (or practice, or whatever).

      It's hard for me to think of examples of the other kind, things you actually should just memorize. But I guess I can get behind wanting them to have some of this information in their heads, and maybe I'm coming around to memorization being a useful skill to have learned. Just not very much of it!

    13. It sounds like we're pretty much in agreement; tell me if I'm wrong. Here's what I think we're all saying:

      • The best way to learn any of it is to play with mathematical problems enough that the relationships make a home in your brain.

      • It's pretty important to have some of these in your brain to go further.

      • Some of them feel more like a memory task than others - 7x8, for instance. (While multiplying by ten isn't about memory at all, if you understand our base ten system.)

      • Commutative and distributive properties aren't obvious to children, and must be learned (hopefully through playful exposure), but once you 'get it', things like that will forever seem obvious.

    14. Yeah, it sounds like you and I are saying the same thing. I was tempted to just say "Nice post! I agree! Thanks" and leave it at that.

    15. JD2718's comment reminded me that all math's notational, graphical and linguistic conventions (not just sine and cosine but which one is the x axis and which the y; the fact that slope is defined as rise/run and not the reverse; etc.) need to be committed to memory. They're different from this other stuff we've mostly been talking about (theorems, etc.) because they're decided by common consent rather than being products of mathematical reasoning. So they can't, for example, be derived. I defer to the language teachers (esp. foreign language teachers) on how this stuff is best committed to memory. (Repeatedly encountering and using the terms in comprehensible contexts, right?)

      Sue, in answer to your question, pick any theorem you know in a heartbeat. E.g., the fundamental theorem of calculus:
      If f(x) is the derivative of F(x), then
      Integral from a to b of f(x) dx
      F(b) - F(a)
      All I am saying is that you probably need to have this theorem at the ready - so, you "have it memorized." But you may never have "memorized" it. Instead, you may have found ways to understand why it is so and these are what give you your easy access to it.

      All I was saying is that I think all math facts (as opposed to conventions, as above) can act like that. If you work in an area (math, teaching, etc.) where you need to use a particular fact, then you may need to have it committed to memory for ready access. But this doesn't mean you actually have to "memorize" it by quizzing yourself on it or whatever. The most effective and powerful way I've found to commit math facts to memory is to try to understand why they're true in as many ways as possible. It's a very slow process but the fact becomes permanently lodged and I usually learn a lot of surrounding information as well that helps me use it more effectively.

      In some ways I think having a "bad memory" could even be an asset! Because it prevents a person from getting ready access to math facts any way but the one I've described as the best. You're forced to learn something deeper than just the fact. This is just me speculating, but actually I believe I've seen this dynamic in students who have trouble memorizing things and consequently are forced to do a deeper sort of learning than their peers.

      Actually a close friend of mine describes this same experience: he couldn't learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can't remember 9x5, but I know 8x5 is half of 8x10, which is 80, so 8x5 must be 40, and 9x5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college, and is now a cancer researcher who deals with a lot of statistics.

    16. By the way, my own memory of the trig identities is all a product of the same type of process: seeking as rich as possible an understanding of why they're true. Here are some examples. My ways-of-understanding may do something for you, or they may not. My point is just, this is how I remember the formulas.

      sin^2 t + cos^2 t = 1

      This is really the pythagorean theorem. (For a triangle with one vertex at the origin, one on the unit circle, and one on the x axis below the last one.) That's how I remember it.

      tan^2 t + 1 = sec^2 t

      This is the above identity divided by cos^2 t; alternatively, it's the pythagorean theorem for a triangle with one vertex at the origin, one at (1,0), and an angle of t at the origin.

      sin (x+y) = sin x cos y + cos x sin y


      cos (x+y) = cos x cos y - sin x sin y

      I understand these two as parts of the same whole. It's a little more elaborate than the last ones, but it works for me:

      First, I believe deeply that
      e^iz = cos z + i sin z
      (Why I believe this is a whole other thing. I used to believe it based on power series, but I got a whole new level of satisfaction when I read the "moving particle argument" in Tristan Needham's awesome book Visual Complex Analysis.)

      But I also believe that
      e^(x+y) = (e^x)(e^y)
      Which also means that
      e^[i(x+y)] = (e^ix)(e^iy)
      Putting this together with the previous thing means that
      cos(x+y) + i sin(x+y)
      = (cosx + isinx)(cosy + isiny)
      And now the sum formulas for both sine and cosine fall out from the nature of complex multiplication.

      Actually, in my brain it's more holistic than this. It's just:
      How do complex numbers multiply?
      (c+si)(C+Si) = (cC-sS)+(cS+sC)i
      And now the right side gives the sum formulas for sine and cosine if I think of c=cos x, C=cos y, s=sin x, and S=sin y.

    17. >all math's notational, graphical and linguistic conventions (not just sine and cosine but which one is the x axis and which the y; the fact that slope is defined as rise/run and not the reverse; etc.) need to be committed to memory.

      I agree that there's a big difference between things that can be proved, and these conventions. But if we think about which way makes more (or the most) sense, we can often see why the convention was chosen.

      I guess there'd be nothing terribly wrong with swapping x and y axes. I'd have to think about that one more. But slope makes sense with change in y on top - a steeper hill gets a bigger slope.

      Order of operations is a convention, but I tell students that it allows polynomials to have no parentheses, and that's my best guess as to why the conventions were set up the way they are.

      Saying that makes me embarrassed I don't know the history, though. Ok, I've goggled it: Dr. Math has researched it and agrees with me.

    18. The fundamental Theorem is a funny example. I would say that I had it all wrong until I started teaching calculus. I don't remember for sure, but I'd bet I used to think that the definition of the definite integral was antiderivative, evaluated at left and right endpoints.

      I know that the first time I tried to teach it, I read the text, thought I understood, got to class, sat on my stool, and ... stuttered. After a minute of so of that sort of agony, I said, "Well, I'm sorry I can't explain why, but here's how we're going to do the problems." (I know. It's a terrible thing. Someone call the bad teacher police!)

      I found a project that walks students through it. I don't know how much it helps them, but I sure understand it better now. ;^)

      I hate the order we teach indefinite and definite integration in with most texts. Students are led to make the same mistake I did. If the indefinite integral symbol means the antiderivative, why doesn't the definite integral symbol also?

    19. Understanding that it's an arbitrary choice where to put the (also arbitrary) labels on the axes (knowing you could label the vertical one 'x' and then it would be the x-axis) is way, way more important than knowing that conventionally the vertical one is the 'y' axis.

      And even if that seems like an easy idea, I think it's actually hard. Something my calculus students (back when I had calculus students) would have trouble with.

    20. I'm lusting after that book, Ben. (Visual Complex Analysis) I put it on my wishlist.

      I first saw the e and sin, cosine relationship recently. I don't quite 'get it' yet, but it's very cool, and I'll get there soon enough.

      I figure out tan^2+1 = sec^2 the same way you do, almost. I start with the Pythagorean theorem, and dividing by x,y, or r gets me each of the 3 Pythagorean identities.

    21. I just read Shawn Cornally's post on Mandelbrot fractals.I'm so amazed by what he's doing there, I didn't notice the mistake that Blaise Pascal pointed out. Shawn was saying series when he meant sequence.

      I used to do that. A lot. I had to have those two words, with their definitions (or an example) on my notes while i was teaching, so I wouldn't mess up. We spend so much more time on series, and have so many named sorts of series, that I finally got it into my head.

      Definite memory task. Anything where the words used don't separate out easily. I never could remember which were 'whole numbers' and which were 'natural numbers', but I didn't care about those terms as much...

    22. I used to say similar things about memorization until I realized that I could take a problem I solved a decade or two ago and not only tell you the problem but also the solution and the process that I went through in getting there.

      I see that as "remembering" more than "memorization" but in either case it's a huge fraction of my problem-solving approach: relate what I'm seeing to something that I saw before, pick out some pieces that will still work, and make a bunch of progress without having to do too much new thinking.

      I suppose what I'm trying to say here is that I don't know how to draw that line between remembering and memorization, nor how to teach kids to be better rememberers other than giving them memorable problems to struggle with. At the same time I think that remembering is crucial.

    23. I love the distinction you're making, Josh. I tell my students that the kind of learning we want is like learning to ride a bike. If you don't do it for years, you can still get back on and ride. That kind of remembering.

      I forget how to do problems I've done before, and then in the process of trying to solve them, it comes back vaguely.

    24. Hello from another mathmom! I just found your blog and I LOVE it. When you said that you have trouble with 7x8 I had to laugh, that one is my problem fact as well. My 8 year old and I compare which facts we "like" and which ones we don't, and they are different.

      I think the last comment sums up how I feel about memorization as well---it is a bit like knowing your friends names. If you get comfortable with them, and pay attention and get to know them, it's not something you have to memorize, but something you remember.

    25. Hi MathMom, Thanks for coming by!

      Yep, if only we knew all the numbers as well as we know our friends, it would all be easy, wouldn't it? From the biography (The Man Who Knew Infinity, by Kanigel), it sounds like that's how it was for Ramanujan.

    26. I have to disagree- strongly. Memorization is an important took for kids to have in their math toolbox. I'm a poor memorizer (is that a word?) and always relied on word association to remember things. Still to this day I remember 7x8 is 56 because I was 7 turning 8 and my favorite channel was 56. It was the channel with all the cartoons.

      I'm not saying students need to memorize EVERYTHING, but their times tables? DEFINITELY! In my experience as an algebra teacher, the kids who have their times tables memorized have a MUCH easier time with math, especially with factoring, which is a big topic in algebra and that carries right through to calculus. Try asking a kid "What two numbers multiply to -56 and that add to -1?" and watch the kids who don't have their times tables memorized. It's painful.

      As far as formulas go, refer to the poster as far as I'm concerned. When would a kid not be able to look the quadratic formula up on the internet. Answer: never. Just memorize the times tables to 12x12, it'll make your math life a lot easier as time goes on.

    27. But ZSR, who are you disagreeing with? I think everyone here wants kids to have their multiplication 'facts' in their memories. I think the biggest question we have is how to help them get there.

      My main point was that people who aren't comfortable with math think that math is all about memory, but really the number of memory tasks is pretty small. The more we help kids see the beauty of it, the more willing they may be to put in that effort to get things in their memory.

      I love factoring myself, but there was an interesting blog post a while back (of course I can't find it now) showing how rare factorable quadratics are among all possible quadratics. Interesting...

    28. I'm currently a fan of rote memorisation as the core with understanding and maths games to support the understanding but...
      I'm starting to think it has a lot to do with the type of learner you are; my preschooler memorises numbers because he sees his older brother memorising digits of pi for fun. His big sister on the other hand needs to understand the ideas before memorising.
      This is why I advocate on my blog, parents getting involved in their child's maths learning. Who knows a child better than a parent?

    29. Wow, this is a fascinating discussion. I am on a quest to find better ways to teach my kids math, and this was on the path. I am homeschooling and my kids are 9 and 6.

      I really resonate with this idea of learning math by thinking deeply about the concepts, exploration, and discovery. I am looking for some examples of how I can coach my kids along this path. Thanks for the post!

    30. A also am reading this blog primarily to figure out how to "teach" my kids math - or more importantly show them how fun and useful it is. I started reading an interesting book "Developing Math Talent". My kids are only 1 and 3 so maybe I'm a bit ahead of the game, but the main thing I take away from my own math schooling through college and this book is the best way to learn is to be interested and have fun.

      I mean really - if you aren't interested in calculus but love geometry, then maybe you will love to study non-euclidean geometry. If.when you find the calc useful you will probably learn what you need and still could make great mathematical discoveries if so inclined.

      My point is it is better to be interested, fascinated and love the tools than memorizing any specific thing. As far as the times tables - I think it is ridiculous to suggest anything about them needs memorizing. Sure it helps if you want to shove math down your students throats and they will probably find it distasteful and resent it and vow to take as few math classes as possible - that happened to me. When did I start to love math - geometry I think. The best teacher I ever had saw that I was a natural at it (h.s. calculus) but found the homework boring so she gave me an exemption from doing it. I found it so interesting without that pressure I aced the AP exams to get a full year of college credit. I can only think how much I would have loved it if I had the 9 years prior back from the time the multiplication tables were shoved down my throat and I thought math was boring.

      Needless to say - I will not do that to my own kids who thankfully will be home schooled. Math will be dad's subject and time for math games when I get home from work and weekends. For now it is mostly games and counting, but who says it ever has to be much more than that? The games and counting just gets a lot more complicated. (:

    31. Hi Christen and Jason, and Welcome!

      I'm working on a book, which I talked about here. I think you'll both find it helpful.

    32. The ONLY piece of math I advocate memorizing is multiplication tables. My students who know them have an easy time in math, and my students who don't have a difficult time in math.

      Are multiplication tables the "gateway math"? I don't think so. If I had to guess, I'd guess that knowing one's times tables is a simple confidence builder that gives the student a feeling that he is a "math" student. Other kids around him will begin to see him that way, too, when he's able to reel off his times tables.

      At the end of the day, "We are who people think we are."

      btw I've been making animated math videos for my blog. I'm not all bidness.

    33. I agree that not knowing them sure hurts. In my beginning algebra class at community college, I've handed flash cards (privately, in my office) to anyone having trouble with their times tables.

      And I made a quicky quiz on that which I give the first day of class. It's one of my mastery tests that they have to get at least 85% on to pass the course. (I've got one foot in the SBG sea.)

      I think this year I'll have a handout with more suggestions, so they can pick flash cards or ... (When I've pulled that together, I'll post it here.)

      I like your style on those videos, but I have a few problems with the explanations. I'll comment at your blog. Thanks for pointing the videos out!

    34. Here's a great post on this topic, at Math Mojo.

    35. Sue,
      I'm with you. There's not a lot that must be memorized. There may be a lot to remember, but rote memory is a lousy way to go.

      I do love memorization, though, but that is a personal choice. Since I learned mnemonics, I've learned to love to way the mind makes new pathways. I spend a lot of time memorizing things for fun. I know that sounds screwy, but it really can be addictive fun.

      Here's my point, though: Finding ways to understand and remember is easier to do than spending your life regretting what you couldn't learn. If you bite the bullet and "dig in" to math for awhile, It becomes fun, so you don't have to suffer for a lifetime.

      Thanks for your great post. I'm amazed at the amount and intensity of your comments. Great work, team!

      Brian (a.k.a. Professor Homunculus at )


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