Mark Eichenlaub: How to develop scientific intuition

Link post

Re­cently on the CFAR alumni mailing list, some­one asked a ques­tion about how to de­velop sci­en­tific in­tu­ition. In re­sponse, Mark Eichen­laub posted an ex­cel­lent and ex­ten­sive an­swer, which was so good that I asked for per­mis­sion to re­post it in pub­lic. He gra­ciously gave per­mis­sion, so I’ve re­pro­duced his mes­sage be­low. (He oth­er­wise re­tains the rights to this, mean­ing that the stan­dard CC li­cense on my blog doesn’t ap­ply to this post.)

From: Mark Eichen­laub
Date: Tue, Oct 23, 2018 at 9:34 AM
Sub­ject: Re: [CFAR Alumni] Sugges­tions for de­vel­op­ing sci­en­tific intuition

Sorry for the length, I re­cently finished a PhD on this topic. (After I wrote the an­swer ker­spoon linked, I went to grad school to study the topic.) This is speci­fi­cally about solv­ing physics prob­lems but hope­fully speaks to in­tu­ition a bit more broadly in places.

I mostly think of in­tu­ition as the abil­ity to quickly co­or­di­nate a large num­ber of small heuris­tics. We know lots of small facts and pat­terns, and in­tu­ition is about match­ing the rele­vant ones onto the cur­rent situ­a­tion. The lit­tle heuris­tics are of­ten pretty lo­cal and small in scope.

For ex­am­ple, the other day I heard this physics prob­lem:

You set up a trough with wa­ter in it. You hang just barely less than half of the trough off the edge of a table, so that it bal­ances, but even a small force at the far end would make it tip over.

You put a boat in the trough at the end over the table. The trough re­mains bal­anced.

Then you slowly push the boat down to the other end of the trough, so that’s it’s in the part of the trough that hangs out from the table. What hap­pens? (I.E. does the trough tip over?)

The an­swer is (rot13) The trough does not tip; it re­mains bal­anced (as long as the move­ment of the boat is suffi­ciently slow so that ev­ery­thing re­mains in equil­ibrium).

I knew this “in­tu­itively”, by which I mean I got it within a sec­ond or so of un­der­stand­ing the ques­tion, and with­out putting in con­scious effort to think­ing about it. (I wasn’t cer­tain I was right un­til I had con­sciously thought it out, but I was rea­son­ably con­fi­dent within a sec­ond, and my in­tu­ition bore out.) I don’t think this was due to some sort of gen­eral in­tu­ition about prob­lem solv­ing, sci­ence, physics, me­chan­ics, or even float­ing. It felt like I could solve the prob­lem in­tu­itively speci­fi­cally be­cause I had seen suffi­ciently-similar things that led me to the spe­cific heuris­tic “a float­ing ob­ject spreads its weight out evenly over the bot­tom of the con­tainer it’s float­ing in.” Then I think of “hav­ing in­tu­ition” in physics as hav­ing maybe a thou­sand lit­tle rules like that and know­ing when to call on which one.

For this par­tic­u­lar heuris­tic, there is a clas­sic prob­lem ask­ing what hap­pens to the wa­ter level in a lake if you are in a boat with a rock, and you throw the rock into the wa­ter and it sinks to the bot­tom. One solu­tion to that prob­lem is that when the rock is on the bot­tom of the lake, it ex­erts more force on that part of the bot­tom of the lake than is ex­erted at other places. By con­trast, when the rock is still in the boat, the only thing touch­ing the bot­tom of the lake is wa­ter, and the wa­ter pres­sure is the same ev­ery­where, so the weight of the rock is dis­tributed evenly across the en­tire lake. The to­tal force on the bot­tom of the lake doesn’t change be­tween the two sce­nar­ios (be­cause grav­ity pulls on ev­ery­thing just as hard ei­ther way), when the rock is sit­ting on the bot­tom of the lake and the force on the bot­tom of the lake is higher un­der the rock, it must be lower ev­ery­where else to com­pen­sate. The pres­sure ev­ery­where else is $\rho g h$, so if that goes down, the level of the lake goes down. Con­clu­sion: when you throw the rock over­board, the level of the lake goes down a bit. When I thought about that prob­lem, I pre­sum­ably built the “weight dis­tributed evenly” heuris­tic. All I had to do was quickly ap­ply it to the trough prob­lem to solve that one as well.

And if some­one else also had a back­ground in physics but didn’t find the trough prob­lem easy, it’s prob­a­bly be­cause they sim­ply hadn’t hap­pened to think about the boat prob­lem, or some other similar prob­lems, in the right way, and hadn’t come away with the heuris­tic about the weight of float­ing things be­ing spread out evenly.

To me, this pic­ture of in­tu­ition as small heuris­tics doesn’t look good for the idea of de­vel­op­ing pow­er­ful in­tu­ition. The “weight gets spread out by float­ing” heuris­tic is not likely to trans­fer to much else. I’ve used it for two physics prob­lems about float­ing things and, as far as I know, noth­ing else.

You can prob­a­bly think of lots of similar heuris­tics. For ex­am­ple, “con­ser­va­tion of ex­pected ev­i­dence“. You might catch a mis­take in some­one’s rea­son­ing, or an er­ror in a long prob­a­bil­ity calcu­la­tion you made, if you hap­pen to no­tice that the ar­gu­ment or calcu­la­tion vi­o­lates con­ser­va­tion of ex­pected ev­i­dence. The nice thing about this is that it can hap­pen al­most au­to­mat­i­cally. You don’t have to stop af­ter ev­ery calcu­la­tion or ar­gu­ment and think, “does this break con­ser­va­tion of ex­pected ev­i­dence?”. In­stead, you wind up learn­ing some sorts of trig­gers that you as­so­ci­ate with the prin­ci­ple that prime it in your mind, and then, if it be­comes rele­vant to the ar­gu­ment, you no­tice that and cite the prin­ci­ple.

In this pic­ture, build­ing in­tu­ition is about learn­ing a large num­ber of these heuris­tics, along with their trig­gers.

How­ever, while the in­di­vi­d­ual small heuris­tics are of­ten the eas­iest things to point to in an in­tu­itive solu­tion to a prob­lem, I do think there are more gen­eral, and there­fore more trans­ferrable parts of in­tu­ition as well. I imag­ine that the para­graph I wrote ex­plain­ing the solu­tion to the boat prob­lem will be largely in­com­pre­hen­si­ble to some­one who hasn’t stud­ied physics. That’s par­tially be­cause it uses con­cepts they won’t have a rigor­ous un­der­stand­ing of (e.g. pres­sure), that it tac­itly uses small heuris­tics it didn’t ex­plain (e.g. that the rea­son the pres­sure is the same along the bot­tom of the lake is that if it weren’t, there would be hori­zon­tal forces that push the wa­ter around un­til the pres­sure did equal­ize in this way), par­tially that it made sim­plifi­ca­tions that it didn’t state and it might not be clear are jus­tified (e.g. that the bot­tom of the lake is flat). More im­por­tantly, it re­lies on a gen­eral frame­work of New­to­nian me­chan­ics. For ex­am­ple, there are a num­ber of tacit ap­pli­ca­tions of New­ton’s laws in the ar­gu­ment. For ex­am­ple, I stated that the to­tal force on the bot­tom of the lake is the same whether the rock is rest­ing on the bot­tom or float­ing in the boat “be­cause grav­ity pulls on ev­ery­thing just as hard ei­ther way”, but these aren’t di­rectly con­nected con­cepts. Grav­ity pulls the sys­tem (boat + wa­ter + rock) down just as hard no mat­ter where the rock is. That sys­tem is not ac­cel­er­at­ing, so by New­ton’s sec­ond law, the bot­tom of the lake pushes up on that sys­tem just as hard in each sce­nario. And by New­ton’s third law, the sys­tem pushes down on the bot­tom of the lake just as hard in each sce­nario. So un­der­stand­ing the ar­gu­ment in­volves some fairly gen­eral heuris­tics such as “ap­ply New­ton’s sec­ond law to an ob­ject in equil­ibrium to show that two forces on it have equal mag­ni­tude” – a heuris­tic I’ve used hun­dreds of times, and “de­cide what ob­jects to define as part of a sys­tem fluidly as you go through a prob­lem” (in this case, switch­ing from think­ing about the rock as a sys­tem to think­ing about rock+boat+wa­ter as a sin­gle sys­tem) – a skill I’ve used hun­dreds to thou­sands of times across all of physics. (My job is to teach high school­ers to be re­ally good at solv­ing prob­lems like this, so I spend way more time on it than most peo­ple, so ap­ply­ing a heuris­tic spe­cific to solv­ing in­tro­duc­tory physics prob­lems in a thou­sand in­de­pen­dent in­stances is re­al­is­tic for me.)

Then there may be more meta-level skills and heuris­tics that you de­velop in solv­ing prob­lems. Th­ese could be things like valu­ing non-calcu­la­tion solu­tions, or be­liev­ing that per­se­ver­ing on a tough prob­lem is worth­while. It’s also im­por­tant that in­tu­ition isn’t just about hav­ing lots of lit­tle heuris­tics. It’s about or­ga­niz­ing them and call­ing the right one up at the right time. You’ll have to ask your­self the right sorts of ques­tions to prompt your­self to find the right heuris­tics, and that’s prob­a­bly a pretty gen­eral skill.

There is a fair amount of re­search on try­ing to un­der­stand what all these lit­tle heuris­tics are and how to de­velop them, but I’m mostly fa­mil­iar with the re­search in physics.

In the Quora an­swer ker­spoon linked, I cited Ge­orge Lakoff, and I still that he’s a good source for un­der­stand­ing how we go about tak­ing prim­i­tive sorts of con­cepts (e.g. “up” and “down”) and us­ing and adapt­ing them, via par­tial metaphor, to un­der­stand­ing more ab­stract things. For a spe­cific ex­am­ple that’s well-ar­gued, see:

Wittmann, Michael C., and Ka­t­rina E. Black. “Math­e­mat­i­cal ac­tions as pro­ce­du­ral re­sources: An ex­am­ple from the sep­a­ra­tion of vari­ables.” Phys­i­cal Re­view Spe­cial Topics-Physics Ed­u­ca­tion Re­search 11.2 (2015): 020114.

They ar­gue that stu­dents un­der­stand the ar­ith­metic ac­tion “sep­a­ra­tion of vari­ables” via anal­ogy to their phys­i­cal un­der­stand­ing of tak­ing things and phys­i­cally mov­ing them around. How­ever, I think Wittman and Black’s work is in­com­plete. For ex­am­ple, it doesn’t ex­plain why stu­dents us­ing the mo­tion anal­ogy for sep­a­ra­tion of vari­ables do it cor­rectly – they could just as well use mo­tion to en­code alge­braically-in­valid rules. Also, they don’t ex­plain how the anal­ogy de­vel­ops. They just cat­a­log that it ex­ists.

A foun­da­tional work in try­ing to un­der­stand the com­po­nents of phys­i­cal in­tu­ition is:

DiSessa, An­drea A. “Toward an episte­mol­ogy of physics.” Cog­ni­tion and in­struc­tion 10.2-3 (1993): 105-225.

This work es­tab­lishes “phe­nomenolog­i­cal prim­i­tives”; lit­tle core heuris­tics such as “near is more”, which are tem­plates for phys­i­cal rea­son­ing. Draw­ing from these tem­plates, we might con­clude that the nearer you are to a speaker, the louder the sound, or that the nearer you are to the sun, the hot­ter it will be (and there­fore that sum­mer is hot be­cause the Earth is nearer the sun – a false but com­mon and rea­son­able be­lief).

That’s a long and some­what-ob­scure pa­per. I re­ally like his stu­dent’s work

Sherin, Bruce L. “How stu­dents un­der­stand physics equa­tions.” Cog­ni­tion and in­struc­tion 19.4 (2001): 479-541.

Like Disessa, Sherin builds his own frame­work for what in­tu­ition is. His scope is more limited though, fo­cus­ing solely on build­ing and in­ter­pret­ing cer­tain types of equa­tions in a man­ner that com­bines “in­tu­itive” phys­i­cal ideas and math­e­mat­i­cal tem­plates. He spells this out in de­tail more in the pa­per, and it’s in­cred­ibly clear and well-ar­gued. Prob­a­bly my fa­vorite pa­per in the field.

A more gen­eral refer­ence that’s much more ac­cessible than Disessa and more gen­eral an overview of cog­ni­tion in physics than Sherin is
“How Should We Think About How Our Stu­dents Think” by my ad­vi­sor, Joe Redish http://​​me­dia.physics.har­​​video/​​?id=COLLOQ_REDISH_093013 (video) https://​​​​abs/​​1308.3911 (pa­per).

The ac­tual pro­cess of build­ing new heuris­tics is also stud­ied, but over all I don’t think we know all that much. See my friend Ben’s paper

Dreyfus, Ben­jamin W., Ayush Gupta, and Ed­ward F. Redish. “Ap­ply­ing con­cep­tual blend­ing to model co­or­di­nated use of mul­ti­ple on­tolog­i­cal metaphors.” In­ter­na­tional Jour­nal of Science Ed­u­ca­tion 37.5-6 (2015): 812-838.

for an ex­am­ple of the­ory-build­ing around how we cre­ate new in­tu­itions. He calls on a frame­work from cog­ni­tive sci­ence called “con­cep­tual blend­ing” that is rather for­mal, but I think pretty en­ter­tain­ing to read.

A rele­vant search terms in the ed­u­ca­tion liter­a­ture:

“con­cep­tual change”

but I find a lot of this liter­a­ture to be hard-to-fol­low and not always a pro­duc­tive use of time to read.

On the ap­plied side, I think the state of the art in ev­i­dence-backed ap­proaches to build­ing in­tu­ition, at least in physics, is mod­el­ing in­struc­tion. I’m not sure what the best in­tro­duc­tion to mod­el­ing in­struc­tion is. They have a web­site that seems okay. Eric Brewe writes on it and he’s usu­ally very good. The ba­sic idea is to have stu­dents col­lab­o­ra­tively par­ti­ci­pate in the build­ing of the the­o­ries of physics they’re us­ing (in a spe­cific way, with guidance and di­rec­tion from a trained in­struc­tor), which gets them to think about the “whys” in­volved with a par­tic­u­lar the­ory or model in a way they usu­ally wouldn’t.

I have writ­ten some about why I think things like check­ing the ex­treme cases of a for­mula are pow­er­ful in­tu­ition-build­ing tools. A preprint is available here: https://​​​​pdf/​​1804.01639.pdf

How­ever, I think it’s dan­ger­ous to have rules like “always check the di­men­sions of your an­swer”, “always check the ex­treme cases of a for­mula”, or even “always check that the num­bers come out rea­son­able.” The rea­son is that hav­ing these things as pro­ce­dures tends to en­courage stu­dents to fol­low them by rote. A large part of the cog­ni­tive work in­volved isn’t in check­ing the ex­treme cases or the di­men­sions, but in re­al­iz­ing that in this par­tic­u­lar situ­a­tion, that would be a good thing to do. If you’re do­ing it only be­cause an ex­ter­nal prompt is tel­ling you to, you aren’t build­ing the ap­pro­pri­ate meta-level habits. See https://​​www.tand­fon­​​doi/​​abs/​​10.1080/​​09500693.2017.1308037 for an ex­am­ple of this effect.

See pa­pers on “metarep­re­sen­ta­tion” by Disessa and/​or Sherin for an­other ex­am­ple of gen­er­al­iz­able skills re­lated to in­tu­ition and prob­lem solv­ing.

Un­for­tu­nately, I don’t think writ­ing books well or writ­ing courses of in­di­vi­d­ual study is some­thing we know much about. I don’t know any­one who has a sig­nifi­cant grant for that; the most I’ve ever seen on it is a poster here or there at a con­fer­ence. Gen­er­ally, grants are awarded for im­prov­ing high school and col­lege courses, or for pro­fes­sional de­vel­op­ment pro­grams, sup­port­ing de­part­ment or in­sti­tu­tion level changes at schools, etc. So adults who just want to learn on their own are not re­ally served much by the re­search on the area. If you’re an adult who wants to self-study the­o­ret­i­cal physics with an eye to­wards in­tu­ition, I recom­mend Leonard Susskind’s se­ries of courses “The The­o­ret­i­cal Min­i­mum” (the first three courses ex­ist as books, the rest only as video lec­tures). He ap­proaches math­e­mat­i­cal top­ics with what I find an in­tu­itive ap­proach in most cases. Of course the Feyn­man lec­tures on physics are also very good.

I’ll be build­ing an in­tro­duc­tion to physics course at Art of Prob­lem Solv­ing, start­ing work some­time this win­ter. It might be available in the spring, al­though stu­dents will mostly be mid­dle and high school stu­dents (but any­one is wel­come to take our courses). I cur­rently teach an ad­vanced physics prob­lem-solv­ing course at AoPS called “Physic­sWOOT”. I try to sup­port in­tu­ition-build­ing prac­tices there, but the main aim is in train­ing these many small heuris­tics which stu­dents need to solve con­test prob­lems.

There should be some­thing like mod­el­ing in­struc­tion for adult in­de­pen­dent learn­ers, but I don’t know of it.

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