The importance of analogies in math and science

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“Good mathematicians see analogies. Great mathematicians see analogies between analogies.”
Stefan Banach

A recent Cognitive Daily post called “Why aren’t more women in science” (part 1) reviews some of the lit on sex differences in cognitive abilities. Dave Munger notes:

In the verbal portion of the [SAT] test, the male advantage is eliminated if the analogy portion of the test is eliminated; arguably this is more a test of mapping relationships than literacy.

The analogy portion was, of course, scrapped as of the spring 2005 SAT. [1] The boldfaced clause above shows why it matters more than the other Verbal portions: figuring out relationships between ideas matters, and reporting what some author said does not. Analogies are highly g-loaded, reading comprehension much less so. But aside from better detecting who the smarties are, analogies are more reflective of real-world math, science, and engineering. (And they matter in the humanities too [2].) If A got one more math question than B, but B got three more analogy questions than A, I’d bet on B doing better in math, even if an IQ test showed they had the same IQ.

What follows is mostly a diversion to show the importance of analogies in math, starting with high school material and moving to some college material. I hope you learn something new, but mostly the goal is to put it on the record, with examples, how important a person’s verbal analogy score is in predicting their success in math and science.

Example 1. A bouncy-ball is dropped from 2 feet, and after hitting the ground, bounces up only 1/2 as high as its previous maximum height. Pretend that it bounces forever like this. In the long run, how much distance does the ball travel?

We can make a table that shows how much distance the ball travels in a particular trip, either up or down, like so:

Trip 1, 2, 3, 4, 5, 6, 7, …
Dist. 2, 1, 1, 1/2, 1/2, 1/4, 1/4, …

This problem is introduced in a pre-calculus class during the unit on the sum of an infinite geometric series — infinite because it starts but never ends, and “geometric” meaning you multiply by the same number to get from one term to the next. The formula for such a sum is t1 / (1 – r), where t1 is the first term, and r is the constant that multiplies one term to get to the next. So if we only had these values, we’d be all set! Unfortunately, if we guess that r is 1/2, when we try to go from 1 to 1 — we don’t multiply by 1/2 anymore (or from 1/2 to 1/2). Damn. Plainly, the above series is not geometric, and at that point most students will opt to make better use of their time by yakking with friends on their cell phone.

Ah, but the students in the class who are good analogical thinkers will notice a geometric series hiding behind the series above — in fact, they’ll discover two of them. The terms of one are interlocking with the terms of the other, like two rows of teeth that complete a zipper. That analogy suggests a strategy: unzip the above series. Then we have two series that go:

2, 1, 1/2, 1/4, … and
1, 1/2, 1/4, …

Bingo! In each of these, you multiply by a constant (1/2) to get from one term to the next. And we know the first term of each, so we can plug in values for t1 and r in the sum formula. We get 2 / (1 – 1/2) = 4, and 1 / (1 – 1/2) = 2. So all together, the ball traveled 6 feet. That’s a neat analogy, but it only makes sense when there are two series meshed into one. We’d like to generalize to any number of series that dovetail into one — and no one makes zippers with more than two rows of teeth. So a better analogy might be the following:

Here there are two strands woven one around the other infinitely, with beads bearing numbers that face us, and there is a knot at the start where the strands fuse. Could we think up series with three or more geometric series hiding inside them? Sure, just as we could make a rope with three or more strands. And to make that series easy to solve, we would just unbraid the strands and work with the beads of each one separately. See note [3] for more uses of this braid analogy.

Example 2. Here are some (x,y) pairs associated with a function. What is the degree of this function? That is, does it look like x, x^2, x^3, etc.?
x = 1, 2, 3, 4, 5, 6…
y = 2, 14, 34, 62, 98, 142…

This problem also comes from high school math — or middle school, if you took algebra then. There, you were taught to look for the difference between consecutive terms, and maybe repeat this process, until you got a sequence of the same number. The number of runs you have to make is the degree of the function. So for the above, the differences are:
12, 20, 28, 36, 44

OK, not the same number, but take the difference again:
8, 8, 8, 8

Ta-da. We had to go through 2 runs, so it must be some function like x^2 (in fact, it is 4x^2 – 2). I guarantee you never knew why this worked when you learned it — and even after calculus or more advanced math, you may still have treated it as a mysterious trick. But there are analogies between discrete and continuous areas of math, and they are pervasive. If you took at least a semester of calculus, you know that if you take the 1st derivative of a function like 4x^2 – 2, you get something with the independent variable still in it — 8x. And sure enough, in our discrete case, the first differences are 8x plus a constant 4.

But if you then take the derivative of the derivative, you get a constant — 8, the same 8 that appeared in our constant sequence after the 2nd run. A constant second difference in the discrete case is analogous to a constant second derivative in the continuous case. That also shows why you knew, back in high school, that you didn’t have a polynomial function like x or x^2 or x^3 when you saw something like this:
x = 1, 2, 3, 4, 5, 6…
y = 2, 4, 8, 16, 32, 64…

You can take differences of differences of differences of… and you’ll never get a constnant sequence for this function, which is 2^x. In first-semester calculus, you learned that e^x is its own derivative, so that if you keep taking the derivative over and over, you always get back e^x — the independent variable never goes away, so you never get a constant. This resilience to your effort to tease a constant derivative out of it is true of all exponential functions, which by analogy tells us that we’d never come up with a constant difference in the discrete case above.

Since there are a billion other discrete-continuous analogies, I’ll leave it there. I don’t think they’re that neat since it’s only like switching between a British and American accent, not like translating between Farsi and Chinese. On a closing note, the entire domain of represenation theory in algebra is based on finding good analogies: they attempt to better understand how some group works by casting the problem in terms of matrices and linear algebra, which are better understood. All of this shows how indispensable this way of thinking is to fields that many assume are primarily about visuospatial skills (though those are key too). Analogies are to all types of thinkers what SONAR and nets are to deep-sea fishermen regardless of which species they hunt.

[1] According to CollegeBoard’s 2007 national report of college-bound seniors, it does appear that within the past couple of years, the male mean for Verbal is only about two points above the female mean, shrinking from a difference of about 11 to 12 points that had persisted since about 1980. And at the high end, in 2007, 1.98 % of males and 1.84 % of females scored 750 – 800. Data from other years on the elite scorers are not contained in the 2007 report, and I’m not interested enough in this topic to pursue them. The point is that gutting the analogy portion seems to have served its purpose.

[2] When the retiring of the analogy questions was announced, an educator named Ted Sutton got an op-ed into the very liberal Boston Globe and made a guest appearance on the very liberal radio show On Point (which airs on NPR). He lamented the change, focusing on the centrality of analogies to the great philosophical and humanistic traditions. Older-style liberals like Sutton appear unaware that their social engineering cousins are the ones responsible for flushing great ideas down the drain, so that the gap between the sexes on a test might close.

At least there are still analogies on the GRE — despite a plan to re-vamp the test with the same gap-narrowing agenda in mind. And thank God for the Miller Analogies Test — not a single “how does the author most likely feel about X” question at all!

[3] The braid idea can also guide your intuition when you have a homework problem in a college-level course that says: “Prove that a countable union of countable sets is countable.” I provided a visual proof here (with a more detailed proof at the end), but I didn’t think of the braid analogy, which makes it even easier to picture. The argument is as I wrote before, but when you’re introducing yet another countable set into the union, it’s like adding a new strand to a rope. You look at the place where the n strands have shown themselves once — and before the first strand winds around the second time, you push it over and braid in your new strand. When they n strands have shown up twice, you push the first strand over before it winds around the third time, and there’s the second place where the new strand goes. And so on to infinity. The union of these strands is a rope whose beads are countable and, more importantly, ordered in a straightforward way.

More explicitly, we can think of the strands as equivalence classes and the rope as the space they fill out. We can imagine a rope that extends infinitely in either direction, like the even and odd integers woven together. We’ve already seen a rope with a knot but which continues to weave itself forever in one direction. A rope with knots at both ends is pretty boring — unless they were the same point, i.e. the rope circled back so that each strand fed back into itself, as with a sequence that’s cyclic (for instance: x, y, x^2, y^2, x^3, y^3, x, y, …).

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  1. What worked in the past seems little guide to what might work in the future. Perhaps removing analogies will enable a new form of intellect to dominate for awhile thereby empowering women. Who knows what works? The past is not a logical guide.

  2. Yeah, it’s worked for 3000 years, but that seems “little guide” to what’ll work tomorrow. The past doesn’t tell us what will work in the future — true, and no one argues that. But it does tell us what has worked well so far. And barring an imminent change in human nature, analogies will continue to be the most highly g-loaded in the near future, and so do better at detecting who the smarter students rae. 
    Your lazy and smug argument would be better supported by pointing to the major 20th C. change from analogical thinking and reasoning — just collecting a whole bunch of data and seeing what it says. There’s a lot of that going on in biology, as we see each time a team reports on which genes show signs of selection. 
    But note that they aren’t replacing analogies with this kind of work. The sentence completions remain, but the analogies have been replaced by a bloated reading comprehension portion. Unless the person plans to write book reports for the rest of their life, this section has the least predictive validity.

  3. Good post! 
    “Since there are a billion other discrete-continuous analogies, I’ll leave it there. I don’t think they’re that neat since it’s only like switching between a British and American accent, not like translating between Farsi and Chinese.” 
    Well I think they’re pretty interesting, since knowing these relations can throw a little more light on both discrete and continuous mathematics. Know of any books that give a lot of attention to this?

  4. Anagrams are also highly g loaded if I remember correctly. Females have a marked advantage in anagrams compared to men. Anagrams has not been included in the SAT nor will it ever be.

  5. I don’t know of any that specifically focus on the relationships, but I think whatever Amazon reviewers think is a good discrete math book would do it — usually you study that after having had some calculus (not that it’s necessary), so they will tie the two together. 
    If you’ve had a semester of linear algebra and ODEs, you could watch the video lectures for a course that Gilbert Strang teaches called Mathematical Methods for Engineers. It’s a fancy name for “applied math.” 
    He’s really good at pointing out the analogies between discrete and continuous. If you haven’t had that yet, I would do the courses 18.03 and 18.06 on that Open CourseWare site (click “Mathematics” at the top in the link above). They also have a full set of video lectures, and they use textbooks that are easy to get used. Even if you’ve already studied differential equations, you should watch some of 18.03 just because Arthur Mattuck is hilarious. Very “eccentric New York Jewish” mannerisms. Or maybe he’s from Philadelphia… somewhere around there.

  6. Actually, anagrams are among the least g-loaded! No better than reading comp. 
    So I should’ve posted a link to what’s more g-loaded than what: 
    Download the free “full article” PDF, and look at the diagram on PDF page 2. The tasks closer to the center are more highly g-loaded. Anagrams and reading comp are on the outer ring.

  7. Thanks. I’ve been picking up linear algebra this semester, and going to be brushing up on my calc next semester. Will probably check out the Strang lectures later.

  8. Strang also has put up an entire three-semester calc text online for free: 
    You can also download the solution manual to check your answers. It’s pretty good.

  9. Interesting that men do so much better with analogies. Maybe this has something to do with why men still dominate literature, even though women tend to do better with language in general. A big part of literary merit comes from powerful analogies, i.e. metaphors and similes.

  10. Bug agnostic, if we empower women and have more women in computing and engineering and science, there will be more interoperability, less evil, and the world will be a better place. That seems to be what Tim Berners-Lee believes. 
    You can’t argue with all the above, can you.

  11. “Bug agnostic, if we empower women and have more women in computing and engineering and science, there will be more interoperability, less evil, and the world will be a better place. That seems to be what Tim Berners-Lee believes.” 
    Sarcasm will get you nowhere, sir. Academics are in a field which is, at its very core, inegalitarian and dependent on inborn brains, and yet they have to pretend that any human-dropping scraped off the street ought to be better employed winning a Nobel Prize. Naturally these academics have become insane. 
    However, if one persists in declaring that humans have an “evolutionarily” different approach to life according to gender, then it follows that said gender will bring that evolutionary difference to whatever field of endeavor it inhabits in any significant percentage. Understandable resentment occurs when someone affirms that those differences are automatically “better.” Moms don’t like to be told that some Dads might be better child-raisers, no matter how much greater dad-participation they say they want. Nobody likes to be told they should be supplanted by their betters, whether the supplanters are really better or not. 
    Not to worry–evidentally Utopia will not occur due to female entry into science. At least not in the near future. Almost 50% of the medical doctors are female, and while I appreciate that, we don’t have Utopia. OTOH, medical science is in a less lofty and authoritarian position than it occupied 50 years ago. Good development, IMHO. Makes room for alternative approaches. 
    A woman who attended med school in the 1930s, related a cafeteria tale of some jocular male colleagues who put a cadaver penis in the her hotdog roll. She didn’t say how far the joke went, and nobody wanted to ask. Long time ago, but you get the idea, that the lady fielded a lot of missiles in her non-traditional life-choice trajectory. 
    Her son observed, “and women nowadays think they have it hard.” 
    They should not jiggle the SATs to make certain groups look better. As I said, academics are already insane, and they may soon have to be committed. Women were entering computer science more before they started to try and drum them in. The percentages have actually declined. 
    Ultimately the proof is in the pudding. Or the hot dog roll.

  12. One point about the analogies: It seems like there are a couple different skills you need to use these: 
    a. You need to be able to hop between analogies in some problems. I recall reading something by Knuth along these lines–that computer science tended to involve hopping between many different levels of abstraction for the same problem.  
    b. You need to be able to keep track of where your analogies are likely to mislead you and where they’re likely to be helpful. 
    c. You need to be able to use your analogies to guide your intuition in investigating some idea. Say, you’ve got “differential cryptanalysis,” can we do second derivatives? (Yes.) How about integration. (Yes, in a funny way. But the relationships between derivatives and integration isn’t quite what you’d expect from calculus.)  
    d. When this becomes really powerful is when you can prove some kind of mapping from your own system to some really well-defined system of math. It’s like you can just import the brains of a long line of geniuses who’ve developed some other area of math.

  13. If we assume that women have pretty different mental styles on average, then it’s quite possible that more women entering a field will open up different approaches, which just make more sense for womens’ brains than mens’. I don’t know that we’ve seen any of that, and there’s no guarantee it will happen, but it would be consistent with the idea of different mental strengths and weaknesses on average.

  14. [Sorry Ryan, but you're outta here. One of the reasons our blog is better than most others is that we chase away the weasels and morons. This is a "safe space" where people won't be harassed by identity politics types. Conditioning on your initial remarks, the probability that you would've ever contributed anything of worth is negligible.] 
    If we assume that women have pretty different mental styles on average, then it’s quite possible that more women entering a field will open up different approaches, which just make more sense for womens’ brains than mens’.
    Here’s a whirlwind tour through some of the sex diffs in cognitive abilities: 
    Females do do better on some things, mostly having to do with perceptual speed and memory. But you can always look stuff up or ask someone who wrote that article / what its gist was. It’s the problem-solving, analogical reasoning, etc., tasks that males tend to do better at, and they’re more what success in math & science depends on. 
    But as we know, the diff in means isn’t a yawning chasm — it’s not like men think this way, while women think that way. It’s just that females are underrepresented at the high and low ends of a shared trait. So I doubt that women will bring a revolutionary way of thinking. That’s why I really dislike these gap-narrowing agendas: they make it more difficult for the truly able to showcase their talent, regardless of sex, race, class, etc. 
    When the Verbal section goes from analogies to mostly reading comp, a girl who can whip the analogies in her sleep could easily be outscored by a girl who’s had more practice reading and writing book reports but isn’t a very good thinker. May not be a huge diff since verbal skills correlate, but at an elite school, 50 or so points can make a person doubt you. As with most types of affirmative action, these measures are there mostly to benefit those who are somewhat smart but who will kill themselves if they don’t get into Stanford — mostly middle and upper-class kids who’ve regressed a lot toward the mean. 
    If you want to detect who the next Emma Noether is, it’s analogies rather than reading comp that’s going to give you a good hint. 
    Maybe this has something to do with why men still dominate literature, even though women tend to do better with language in general.
    I think so too. That’s why males especially dominate poetry, which is more abstract and figurative, compared to the novel.

  15. *Emmy

  16. I was channel-surfing and came upon the audition of Beauty and the Geeks  
    So there is this pretty girl in the spotlight, she was asked if 
    right is your north, where is your west. 
    She pointed upwards, and the auditioning group were shown 
    straight face, like that deer in the headlight look, the correct answer is that she should point in front her. 
    FUnny thing, I got what she meant. Is it a female to  
    female thing? 
    She was seeing the analogy in another plane, which is parallel 
    to her body, not perpendicular. 
    Also if you notice, women can jump from topic to another 
    and still follow..each other, as if they didnt stop thinking  
    about it 20 minutes ago.  
    So bottomline, I think women are still smarter, in any plane. 
    Its just that their brain process information faster they get  
    lost between childrearing and female talk. 
    What if tge test were designed to measure a male brain, not measure  
    the female brain. 
    Do you think if we design the test, the male population can pass them? 

  17. If north is to your right, straight up is not west, it is east. It is nice that she could think in two planes, not so neat that she got the answer wrong in that plane. 
    Do you have any evidence that women are smarter, besides remembering old conversational threads? 
    There are lots of women who don’t have children. Where are their Nobels?

  18. That’s a shame. Pulling out the analogies would seem to reduce the value of a high score, anyway; I remember being pretty tickled with my verbal 800 back in 1996. (But not with my predictably sad math score.)  
    I’m told colleges rely much less on SAT scores than they did fifteen or twenty years ago. Seems that diluting the test would contribute to that, and you’d think the test officials would want to avoid becoming irrelevant!  
    Official mediocracy is here to stay, though– at least till people find new ways outside the official system to break out and “showcase their talents.”

  19. Anyway, what is the deal with this female-to-female communication nonsense? That sort of talk was always rampant in my undergrand English program, but hardly any of them could write a coherent sentence, let alone put together a compelling argument.  
    If that is the “new sort of intellect” that women are supposed to be cultivating in our universities, then I’d rather stay home and save my money. I can get empty-headed “feminist” prattling online for free, without the shackles of mean old patriarchal “standards” holding any of the discussion back.

  20. If north is to your right, straight up is not west, it is east.  
    Only if you rotate the person so that they are facing downwards. Rotate the person so that they are looking stright up and they are looking westwards.  
    Better still, imagine that a MAP is being rotated on a wall. Set it that so N points to the right. If you’re looking at the top-side of the map, W is straight up.

  21. oh shut up. lol. My bad, mistypo. i mean west.

  22. Anyway, what is the deal with this female-to-female communication nonsense?  
    Typical male – anything thats female-related is nonsense. 
    To follow darwin’s logic.. if a female has lesser strength than male, how did she survive?  
    Why didnt male species just turned into hermaphrodites? 
    The female made up for the lack of male strength by being smarter. 
    Do you think that those hermaphrodites are females who finally got rid of the males? 

  23. I AM female. ;) I just have a soft spot for old horrible patriarchal inventions like complete sentences and well-developed ideas with supporting paragraphs, and I’m still not especially fond of finding pointless, unrelated tangents in the middle of a supposedly upper-division paper on the environmentalism of Gary Snyder. 
    Some would call that internalized sexism. Those people can generally bite me.  
    Wouldn’t hermaphroditism reduce species biodiversity? Sexual reproduction obviously has something going for it, though I wouldn’t argue that the “something involved” is the idea that females are “smarter” than males. Last I heard, though maybe this is not correct, men average 5 IQ points above women. Nothing huge, but probably not totally inconsequential, either.  
    As far as I know, human hermaphrodites are not capable of self-reproduction. One or the other sets of organs are incomplete and non-functional.  
    More a strange birth defect than a feminist revolution.

  24. human hermaphrodites are not capable of self-reproduction 
    NOT yet. Were still evolving.  
    Wouldn’t hermaphroditism reduce species biodiversity? Then the hermaphrodites can impregnate not itself but another one. 
    men average 5 IQ points above women Like I said, what if we design the test according to our liking, something they cant pass.  

  25. But what selective pressures exist for hermaphroditism to come about? I can’t see any. On balance, men seem to be much more of a social benefit than a liability.  
    I’d say that the few female-dominated societies that exist are doing rather poorly. Our inner cities are practically matriarchal, since the men head off to jail and aren’t exactly “Dad of the Year” material. They’re squalid and deteriorating, not feminist paradises flourishing under the wise rule of elder Gramma Jones.  
    Yes, if living in the patriarchy means continuing to enjoy the benefits of traditionally divided labor, then I will happily continue my so-called sexual enslavement.  
    Adding children to the equation changes EVERYTHING.  
    Anyway, what has this got to do with the SAT? Oh yeah, I’m in favor of retaining traditional measures of academic knowlege, not throwing a random gamble at the intuitive wisdom of women under the age of 21.

  26. oh, good gourd, Ryan. I think everyone understands the XY system by now.  
    That doesn’t say a durn thing about women being “dominant” to men. The default setting, maybe, but not “dominant.” After all, being born female is the safer choice, from a reproductive standpoint. It just ain’t that hard to find someone to knock you up. ;)

  27. Do bees take the SAT? :p

  28. Sigh, Ryan demonstrates his/her ignorance once again by proffering Bees and Ants. They are haplo-diplod, which changes things significantly. 
    In any event, it is interesting how easily people get upset when we start discussing the difference in variance between males and females …

  29. Oh Ryan,  
    If you notice, we are just having fun, considering the topic. Its like here we go again, male vs. female. 
    I wouldnt argue with you using scientific terms because I dont use that in my daily life,  
    I can put something up cut and pasted and borrowed from somewhere else, argue with the same arguments you’ve already heard from somewhere. Im just throwing ideas crappy it may sound. 
    I am going back and forth to my system design and this blog, because Im bored. You need to lighten up with your “stupid-female namecalling”.  
    When was the last time you got laid?  

  30. I’m not Ryan, but NOBODY here, male or female, is calling women in general stupid. I don’t see why this is a gender battle thing; most people would be pretty disgusted by the probable rationale behind making the SATs easier if they were aware of it.  
    And I got laid this morning. Thanks, though.

  31. Our inner cities are practically matriarchal, since the men head off to jail and aren’t exactly “Dad of the Year” material. They’re squalid and deteriorating, not feminist paradises flourishing under the wise rule of elder Gramma Jones.  
    Youre talking about single-parent household compared to a two-parent household. Thats just poverty.

  32. [Ryan: get lost. We are assuredly not open-minded toward morons and weasels. I don't know which you are. Identity politics people cannot help but make every issue about their cause, which utterly derails discussion -- as intended. Stop causing a public nuisance.]

  33. May:To follow darwin’s logic.. if a female has lesser strength than male, how did she survive? 
    Why didnt male species just turned into hermaphrodites? 
    The female made up for the lack of male strength by being smarter.Physical strength requires vital resources that could have gone into something else – men being stronger than women doesn’t make them more fit universally, there’s a specific tradeoff involved. 
    The differences imply that men and women inhabit subtly different niches, not that women are inferior to men.

  34. Wow. I thought this was a site about genes???

  35. Ryan has a dynamic ip address, probably using the att dsl and now hes stalking and harassing us. 
    Man, how desperate are you for attention?

  36. darwin’s logic 
    Males and females are the same species. Are you asking why we are not all YY? That sort of thing results in birth defects. So why aren’t we all YX? Every male carries both a Y and an X with equal probability of giving one or the other to any given offspring and every female carries two X chromosomes. It would be possible for a mutation to arise that kills off embryos of a certain gender to select in favor of the other, but an explanation why selection balances can be found here. YOu are of course right that there will usually be selection against being “fat, drunk and stupid” (it’s no way to go through life) among both genders, but to assume sexual dimorphism resulted in higher intelligence among females is a leap you haven’t supported sufficiently. The general consensus seems to be that males and females have the same average IQ. Brain volume among males is on average higher, but the composition of gray vs white matter is also different and this could be the reason for the higher IQ/volume ratio for females. 
    purple & floral pattern, this site discusses more than just genes, and this does involve genes (though they aren’t the end-all be-all). 
    I didn’t see any of Ryan’s posts other than what one person seemed to be quoting, but my general experience is that those who get censored here deserve it fully. I give thumbs-up to the moderators.

  37. IMO, “censored” is the wrong term. “Thrown out” is more appropriate. It’s not like you were publishing a newspaper and the police shut you down, it’s like you were trying to take over a bar to spout your half-baked ideas, and the bouncer invited you to find another place to be.

  38. That sort of thing results in birth defects. 
    I think we’re all product of birth defects, like were in a big trial and error process.