7 ms·

How should mathematics be taught to non-mathematicians? (2012)

141 points by poindontcare 10y ago

ankurdhama 10y ago
Mathematics is just the extreme end of the two most fundamental concepts Abstraction and Generalisation. Start the intro to these concepts including showing how these concepts are useful and used by everyone in their day to day life in using natural language.
- semi-extrinsic 10y ago
  My wife has studied mathematics pedagogy, and one concept that really struck me from what she learned is compression. Put simply, you can't learn a new thing until you've compressed the old thing it builds on. If you have not compressed "addition" to the point that it requires little effort, you won't be able to learn "multiplication". Same holds for e.g. "derivatives" and "Taylor series", or "group theory" and "rings and fields". (I think this was from Piaget, or maybe Brissiaud.)
  - andars 10y ago
    I would guess Piaget, because it reminds of this quote from Papert (who drew heavily on Piaget): > Slowly I began to formulate what I still consider the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can't, anything can be painfully difficult.
- fiatjaf 10y ago
  That is the best way to not learn mathematics.
- pbhjpbhj 10y ago
  Could you expand on your second sentence, perhaps with an example or two and how you'd imagine they're be taught in class?
Animats 10y ago
Gower suggests story problems. Many, many story problems.[1] [1] https://pbs.twimg.com/media/BTwHsS5CAAAzBBu.jpg https://pbs.twimg.com/media/BTwHsS5CAAAzBBu.jpg
- simonbyrne 10y ago
  Yes, in the loose sense they involve words, but he points out the problem with standard word problems in an earlier post https://gowers.wordpress.com/2009/07/11/help-im-stuck-in-my-ivory-tower/ https://gowers.wordpress.com/2009/07/11/help-im-stuck-in-my-...
- laurieg 10y ago
  The problem with story problems is, as the article states, that they are never really presented in the open ended way they claim to be. You almost always teach a class a fixed operation, multiplication for example, and then give them a bunch of word problems where multiplication is thinly disguised. A much better exercise is to give an absurdly open ended exercise. "I'm at the supermarket, which checkout should I go to?" is one I have used in classes before. You can get a discussion going and generate a lot of interesting ideas, and almost every time I do it in a class someone says something I've not thought of. Once students have given you some good ideas you can massage it into a model and do some more 'proper maths' work. Of course, this takes a good teacher that can engage and steer the class.
  - axiomabsolute 10y ago
    So much this. Through a series of unexpected events, I ended up studying math in undergrad with no clue why or what I was going to do with it. My senior year I took a class called "Applied Modelling". The first project was a simple, one sentence question: "What would happen if the Greenland ice cap melted?". It reminded me a lot of Randal Munroe's "What If" blog on the XKCD site [1]. Easy to understand, open ended questions that encourage readers to learn a little about topics _outside_ of math to answer the question. The class gave me an appreciation for math that was lost during all those years of study before that, and it's basically my career now. [1]: http://what-if.xkcd.com/ http://what-if.xkcd.com/
Myrmornis 10y ago
Great article. Did anything come of the Gove-era math education policy changes he's referring to?
- keithpeter 10y ago
  In my opinion: Gove's main change in Maths was in the exam syllabus that is taken at age 16. The new syllabus will be examined for the first time this summer, most schools started teaching the material in 2015. It will take another 3 years or so to see the effect of that on the GCSE classes (basically age 14 to 16) and then another five years for earlier years to shift what they do. Change takes a bit of time when it is 700,000 children in each year group moving through 10 years of compulsory education. Politicians know this but the news cycle requires changes on top of changes...
  - pbhjpbhj 10y ago
    Which all means changes have to be made at a slower rate than the parliamentary cycle; which politicians won't heed. But then Gove hadn't a clue and seemingly doesn't care either. Each parliament IMO should get chance to make one change - presented to parliament with optional amendment suggested by third parties (unions, parent groups, students). Let them go the Lords to win the right to make a further proposal to parliament. That should at least slow them down enough to allow teachers to weave something useful out of the crap that they can enhance over a couple of years before the next half-wit comes along and arses it all up.
  - Myrmornis 10y ago
    Can you say a few words about the direction he took the new GCSE syllabus in?
    - keithpeter 10y ago
      Well, there is plenty of stuff on t'web about the current state of play [1] [2] [3]. My recollection is that Gove's original idea was to scrap the GCSE completely and replace it with a new 16+ qualification consisting of a single 3h exam for each subject. Maths represents a difficulty of course as the range of ability encountered at age 16 for all 700,000 children each year is huge. Maths and science subjects differentiate by topic, wheras Humanities subjects differentiate by response to a brief. As a concrete example you can ask a group of 30 students to write two sides about the best learning experience they have ever had. Everyone can leave the room feeling they gave it their best shot. Some will be excellent and creative with a good range of vocabulary and demonstrating some self-knowledge and analytical ability. Others will produce a description description, possibly with limited vocabulary, possibly with deficient skills in punctuation, grammar and spelling. Now ask the same group of 30 students to take a single Maths test. Some will finish in 5 minutes with full marks (it wasn't the right test for them) and some will take an hour and score close to zero (it wasn't accessible to them either). 'The Blob' (the education establishment in the UK, i.e. the people at the sharp end) managed to head this one off together with the QCA and the House of Commons' select committee on education lead by Graham Stuart - a conservative but with experience of work in education unlike Gove. The result was a re-vamped GCSE Maths in which the Foundation tier has some topics previously only found on the Higher tier such as trigonometry, surds, rules of indices with fractional indexes, simultaneous equations, quadratics: solving by formula, factorisation and substituting into to plot graph. Much of the more useful statistics has been removed (graphical presentation &c) and replaced with harder probability. Much more emphasis on technical algebra and difficult fractions/ratio questions. This lot is working its way down the school system now. If they stick to it the result might be OK in 5 years but my guess is they will get stick for the atrocious pass rates for the next couple of years (or fiddle them somehow) and then fudge it. [1] http://qualifications.pearson.com/en/qualifications/edexcel-gcses/mathematics-2015.news.html?article=%2Fcontent%2Fdemo%2Fen%2Fnews-policy%2Fqualifications%2Fedexcel-gcses%2Fmathematics%2FDownload-our-new-SAMs-and-2-new-sets-of-specimen-papers http://qualifications.pearson.com/en/qualifications/edexcel-... [2] https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-topic-changes/ https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-topic-c... [3] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=29&ved=0ahUKEwiwwOSNo87PAhXII8AKHcckC_E4FBAWCFEwCA&url=http%3A%2F%2Fsmartfuse.s3.amazonaws.com%2Fshirebrookacademy.org%2Fuploads%2F2014%2F05%2FNEW-for-2015-Higher-AQA-GCSE-SoW-010414.doc&usg=AFQjCNHVq9hT-i_mdzu32vNjFPhrYJMsNQ https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web...
wisty 10y ago
> Objection 5. You’d never find enough teachers who were capable of teaching a course like this. To do it well, you need to have a very sophisticated understanding of probability, statistics, game theory, physics, multivariable calculus, algorithms, etc. Objection 6: If it's hard to find teachers to teach it, maybe it's a little challenging for students (even though a good math expert might find it interesting). Just because a math expert thinks something is interesting doesn't mean low performing students will find it interesting. For a more HN friendly example - what bunch of high school students wouldn't want an IT class that taught compiler design, instead of stuffy old Excel? Even Python is more fun that spreadsheets, right? Certainly there are large swaths of high school math that can be cut, and replaced with more relevant stuff. But some care needs to be taken that it's actually teachable. The article does partly cover this though: > Thoroughly road test questions before letting them loose on the nation’s schoolchildren. In fact, that applies to the entire course: make sure one has something that definitely can work before encouraging too many schools to teach it.
- striking 10y ago
  > what bunch of high school students wouldn't want an IT class that taught compiler design The majority of high school students can hardly wrap their brains around the AP curriculum (probably for lack of time or effort, rather than ability). There are some that are honestly, actually interested in computer science and are thus capable of stuff like that... but they are low in number. What might be able to work is a fully-fledged web design course, using modern standards instead of boring stuff from ten years ago. With HTML, CSS, and finally JS (probably React, then Node). Maybe even SQL. With knowledge like that, you have more than enough of a base upon which to stand. You could likely even get a job. > Even Python is more fun than spreadsheets, right? Spreadsheets are easy computation for a wide audience, with a little learning curve. What can Python do, out of the box? What could you convince a high school kid to program with it that isn't a derivative of "10 PRINT HELLO WORLD; 20 GOTO 10;" ?
  - az0xff 10y ago
    A course designed around using a particular "modern" web technology stack will have to change too often (every time it doesn't become "modern" anymore) for it to be sustainable. Imagine that in 2016 you have a course centered around using what was modern in 2006. That would inevitably happen with a course like that. I'd rather teach programming from 0 to making a really basic 2D game (be it in C++ or Python or whatever language and whatever library). The results are eye-catching and the coding process is engaging, and there's no need for it to rely too much on how trendy the framework is in the current year.
Pica_soO 10y ago
Capturing the interest must always be first. So to capture the interest, one must use math applications like game or physic simulations that allready captured the audience, allow beginners to modify the laws and abstractions within and create a "im at home with this"-intuition to delete the "im just not cut out for math" bias. Questions would not come from the teacher, questions will come from the students on how to modify the application.
partycoder 10y ago
Applied math around areas of interest.
M_Grey 10y ago
When you consider how mathematics, an understanding of probability, and so on enhances your ability to see through bullshit in advertising, government, and to some extent religions (not faith, just religions); it's not hard to understand why so many people are not thrilled at the notion.
- frozenport 10y ago
  Much of advertising is emotional appeal.
  - M_Grey 10y ago
    Plenty of advertising, car sales, consumer financial products, lotteries, etc... rely on people not having an innate understanding of math, and especially probabilities. Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college!
    - wolfgke 10y ago
      > Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college! It's not obvious whether this is good or bad (too many cooks spoil the broth).
      - M_Grey 10y ago
        The cooks are in the kitchen with their votes already, how could it be worse to have them actually know what the hell they're doing there?
        wolfgke 10y ago
        Perhaps I am a little bit biased by the situation in Germany. In the past years there was an increasing number of non-voters in elections (typically people who are dissatisfied with the political system). Now in the last years a rather right-wing populist party (AfD) appeared that convinced many of these non-voters to vote for them. Compare this a little bit to the fictional situation as if Donald Trump would bring many people who did not vote in the last presidential elections back to the ballot boxes to vote for him.
maus42 10y ago
The title is missing year 2012. ("This entry was posted on June 8, 2012 at 4:53 pm") Does anyone know what happened regarding all this?
- keithpeter 10y ago
  Probably not a direct consequence but... The UK elected a Conservative government who decided to base all 16+ Maths for non-mathematicians on the revised GCSE Maths syllabus. Colleges are now coping with large numbers of students aged 16, 17 and 18 being required to take the GCSE exam again while studying vocationally based qualifications. I'm teaching maths to trainee hairdressers, trainee car mechanics and would-be fine artists. Pass rates are not high (we are starting with a selected sample after all and schools are pretty good at getting non-mathsy youngsters through). The statutory requirement ends at age 19 and so the majority of late teens will experience three more years of failure in a subject that they experienced failure in at school. That should guarantee another generation of the general public whose loathing of Maths is pretty marked. Taking a wider view, I think that we all tend to learn things in a situated way and I therefore have a lot of time for the 'Functional Skills' approach to Maths and English if done properly. This was the approach adopted in most vocational training courses prior to the Gove era. I also think that there is a place for a qualification based around probability, statistics, and critical thinking. I'd love to call it "How to spot bullshit when you see it". I'd make discussions of issues like genetic defects and screening, obesity and health education and so on a core component. It will never happen of course...
  - djaychela 10y ago
    Not sure if you're teaching in the above environment, but one of the schools I teach A-level at has a high number of maths-failing pupils, and as a result, many of those I teach are re-sitting maths in both their first and second years that I teach them. It's a typical case of Gove not thinking of the consequences of his (backwards, IMO) ideas - these people are definitely part of the generation you outline; it spreads so they not only think they are bad at maths (which admittedly is often true), but that they are incapable of any meaningful learning. Those who do take functional skills and pass at least feel they have made some progress - despite this being a level of maths which most people would feel was far below the age level of someone taking at aged 17. I know that anti-Gove rants are all over the place, but most of the people I know outside education don't have a clue about how poisonous his ideas have been; they don't withstand any serious scrutiny, but on the face of it may seem sane, so people who haven't thought about it will defend them. His replacement isn't far off his level, and I seriously fear for the future of education in the UK.
onetwo12 10y ago
The post is about suggesting more fermi estimates problems, see for example http://lesswrong.com/lw/h5e/fermi_estimates/ http://lesswrong.com/lw/h5e/fermi_estimates/ Fermi estimates require you to look for relations in the real world, construct a model of the situation and then iterate improving the model by adding more information. You don't use that process just to give an example of using a rule or solving an equation, instead you emphasize how the problem could be tackle mathematically. So there are more room for open questions, incentives for exploring and suggesting new approaches, in a more relaxed atmosphere creativity can flourish, perhaps maths is more than a single equation written in an old book.
dmitripopov 10y ago
There's no need to invent popular mathematics. You just need a teacher that can show a student how beautiful mathematics is. It's an art. A way of thinking and creating.
ivan_ah 10y ago
Wow, that's a nice collection of problems. They are exactly the opposite of the dry and artificial "word problems" students are used to. The discussion at the bottom of the blog post is also very interesting. The socratic approach is very good to "break the ice" and introduce the application, but I wonder how scalable this approach is. Does the teacher need to be very knowledgeable/entertaining to pull this off? BTW, I'm working on a new project, which is essentially "math lessons by email" that will walk readers through the math material from the NO BULLSHIT guide to MATH & PHYSICS. Anyone interested in learning or reviewing basic math (expression, equations, functions, algebra, geometry) should signup: https://confirmsubscription.com/h/t/4C2D9C45B88734F3 https://confirmsubscription.com/h/t/4C2D9C45B88734F3 (it's free)
- TheOtherHobbes 10y ago
  They are, but I think non-math people will look at just about all of them and think "I have no idea where to start." Worse, I think they don't teach generalisable skills. That's probably the core problem with all school-level math and science teaching. You learn a vocabulary of basic symbols and some rules for manipulating them, but you don't learn math skills - in the sense of understanding the real world well enough to make the leap from symbols and abstractions to useful life skills. The point of math teaching shouldn't be to know how to solve problems like these, but to learn how/when you can use math to answer your own questions for yourself. There's also a deeper level where you can teach the process of abstraction as an end in itself. I suspect that may be too far for most people - although I haven't completely convinced myself that's true yet.
blahblah3 10y ago
The problems he suggests overall seem too difficult for the average non-mathematically inclined student. And they also require quite a skilled teacher to teach. I'm not sure stuff beyond "algebra 1" needs to be taught to everyone in high school. Even the concept of using "x" to stand for an unknown is very difficult for some to grasp. Instead, schools should make sure all students can properly understand how to use addition, subtraction, multiplication, and division, with applications to things like personal finance. In my experience, even many college graduates have trouble understanding when to multiply, divide, etc...
- ghaff 10y ago
  At the risk of being sarcastic, the suggestion seems to be that mathematics is best taught through stereotypical management consulting interview questions. Or the apocryphal (?) Google interview questions like how many ping pong balls can fit on a bus. ADDED: I also suspect that the average high school student lacks the world knowledge to come up with meaningful guestimates for the inputs to many of those questions. A lot of the high school mathematics that I learned such as geometric proofs and trig are not all that useful. And it seems as if things that would be more generally useful like probability and stats are not that broadly taught--and are often taught in a very theoretical way when they are.
  - blahblah3 10y ago
    Unfortunately, probability and stats are not easy to teach, and even many professional scientists / researchers have major confusions about the subjects. Common sense actually provides a decent enough guide for most people (i.e a baseball player with a high batting average is more likely to hit the ball). Euclidean geometry as taught in school does seem rather archaic and out of place though. Some people say it's an introduction to "proofs/rigorous thinking", but it seems to me that that purpose could be better served with a first order logic class.
    - inimino 10y ago
      First-order logic is much more abstract. I think a major benefit of geometry is that it introduces visual thinking and is very grounded and real because you can see and draw the proofs. This foundation of visual/spatial intuition seems to be very useful in higher math, as a counterpart to the exclusively symbolic manipulation of algebra or first-order logic.
andars 10y ago
If anyone cares to share, I'd love to hear some opinions about at what point someone switches from "non-mathematician" status to "mathematician" status.
- Koshkin 10y ago
  One becomes a mathematician, even if only temporarily, if one becomes interested in an investigation into purely mathematical matters (e.g. in finding a rigorous proof of a statement) rather than in using mathematics as a computational tool in one's (original) area of interest, such as physics or biology. Even when one happens to invent a new mathematical method that works, one does not, in general, instantly become a mathematician - unless, of course, they lose the focus and turn their attention to making the method just discovered more efficient or better substantiated from the purely logical standpoint.
- contravariant 10y ago
  I think you switch to a non-mathematician as soon as you proclaim your hate for mathematics. I don't like seeing 'mathematics' as some kind of specialist activity that only a specific group of people is capable of.
- yummyfajitas 10y ago
  When you mentally translate theories and discussions from words to equations rather than equations to words.
- inimino 10y ago
  In the context given in the article it sounds like non-mathematicians are students who choose to stop maths at age sixteen, presumably for the rest of their lives. This reflects the observation that children are born mathematicians and explorers and remain so until something kills their curiosity (often school is named as the culprit).
- tsomctl 10y ago
  I'd say for me, it was when I properly learned how to write proofs. It just formalized everything I knew, and forced me to think about it logically.
losteverything 10y ago
Your question made me think. Could be innate. My part time job forces me to make 10-20 micro decisions an hour. Most involve minimizing negatives and max positives. And knowing what to ignore. Yet my co-workers are quite unable to even know how to get 10% from a cash register total. Other managers lack a "math approach" imo to want to get sales numbers or staff assignments. For example, what should you do if you have 80 hours of work and only 70 hours of workers? Some fail because they can't even frame the task that way. Could be vocation only. Get paid by using math, you are one. I'm still thinking
Koshkin 10y ago
Teaching mathematics has two aspects, or goals, that are largely not related to one another. One is teaching mathematical methods as a computational tool, as used in a certain area of expertise (such as, for instance, electrical engineering). The other one is teaching how to do mathematics, i.e. how to generalize facts, find efficient proofs and algorithms, etc.
- godelski 10y ago
  I've always thought one of the most important goals of learning math was learning how to think logically. This may be coming from a physics perspective where the language is used because it is (arguable)the most accurate tool we have to analyze the world around us. The average person doesn't need efficient proofs and algorithms. But they can use generalized facts.
  - Koshkin 10y ago
    I think that the idea that studying math is necessary to train one's logical abilities is a misconception. Learning any of the sort of non-trivial activities, such as cooking, which requires a high degree of awareness and logical thinking, would have the same side effect.
    - godelski 10y ago
      Sorry if it came off like that. I don't think it is necessary, but it sure is a helpful tool. So if we're going to teach people basic math skills we might as well focus more on this aspect so that they can use those basic skills and know why knowing how to find the angle of a triangle can be useful.
imgabe 10y ago
Do other disciplines ask similar questions? How should physics be taught to non-physicists? How should writing be taught to non-writers? How should car maintenance be taught to non-mechanics? I guess my point is, why should we teach mathematics any differently to "non-mathematicians" than we do to "mathematicians"? I mean, at the point when you're first teaching someone, how do you even know if they're a "non-mathematician" or a "mathematician"? After all, they haven't learned enough yet to know if they'd want to continue in that field of study.
- blahblah3 10y ago
  Maybe this is an unpopular view, but I think there is something unnatural about mathematics for most people. It tends to be much more abstract than other subjects. I say this as someone who has always enjoyed math and majored in it. And it does make sense to have "math for mathematicians" or "physics for physicists" classes. You see this a lot in college (often intro classes are divided into different "levels"). This allows some students to accelerate and others to still learn relevant materials but at a more appropriate pace.
- spion 10y ago
  > How should writing be taught to non-writers? Sort of: http://paulgraham.com/essay.html http://paulgraham.com/essay.html
- zengid 10y ago
  Usually mathematics is taught in a progression, and layers are built upon more basic layers. A 'non-mathematician' is someone who does not consider it their life's goal to solidify their expertise in mathematics, so they may choose other disciplines to obsess over, or no disciplines at all. Either way, a teacher of a class for 'non-mathematicians' who wants to explore depth in a certain area of mathematics cannot assume that this population of 'non-maths' will understand jargon or any concepts that aren't specifically explained within the scope of the class. Thus they must proceed axiomatically, being sure to avoid using concepts that haven't previously been defined. Conversely, a class for "mathematicians" would be one with some background that can be used to begin from, depending on the average grade-level or prior education of the students. So in a nutshell to teach deep math concepts to 'non-maths', one has to start from the beginning and define each concept in order, ignoring wider scope for the sake of reaching the goal. 'Math' students can start at a middle-depth, depending on their background.
chx 10y ago
There's a book by a Hungarian mathematician for this exact purpose. Péter Rózsa: Playing with infinity. That's the Hungarian name order of her name. https://books.google.ca/books?id=pj5G-3boMBwC&redir_esc=y https://books.google.ca/books?id=pj5G-3boMBwC&redir_esc=y
kazinator 10y ago
Bootstrap: teach enough math to turn non-mathematicians into mathematicians. Then teach mathematician-to-mathematician.
yodsanklai 10y ago
These problems are examples of questions that can be tackled with the help of mathematical tools. They can show students some applications of mathematics, but IMHO they don't really teach mathematics. Maths have to do with statements and proofs about abstract objects. I'd rather learn about euclidean geometry than trying to figure out the number of piano tuner in manhattan (I'm sure most students will be bored either way). Actually, this is an endless source of discussion among instructors, not only maths. Should classes be driven by applications in order to motivate students?
- danarmak 10y ago
  Do applications motivate students? Personally I would expect examples like these to de-motivate, since they make clear that the subject does not, in fact, apply to the students' idea of real life. If you teach abstract mathematics and make vague promises they will be really useful, some students may believe you. But if you try to demonstrate how they're useful, and the best examples you can come up with are estimating the number of piano tuners in Chicago or the number of air molecules in the room, things students know they won't ever need to do in their lives, they should become less interested.
baby 10y ago
I think music is the best example. There are two ways of learning music: * learning solfege * learning how to play directly Most great school of musics will teach you solfege first, and you will have to go through hardcore solfege classes while you start learning how to play an instrument. Some teacher will do the same, or some family will make their kid do the same. Now I can tell you this is not fun, but this is the way to become a great musician. You need the theory, you need to know how to read that stuff like you're reading English. But this is not fun, and I've known many in my youth who gave up learning an instrument because of this. Now if you would teach every kid to play an instrument first, and have fun with it, and actually producing music with their fingers/mouth then... they would maybe enjoy it enough to get interested into taking solfege classes and music theory later on.
- jimmydddd 10y ago
  I agree that music is a good analogy. For example, in the US there is a guy who has "shortcut" piano courses. His point is that yes, if you aspire to be a concert pianist, then you should spend years learning scales, reading music, and playing music from classical composers. But if you just want to play a few Billy Joel or Beatles songs, you can skip all that and cut to the chase. You won't have that great "foundation" needed to be a master player, but most people don't want that in the first place.
kkylin 10y ago
I really like the collection of problems, but I'm not sure it is easy to teach this kind of problem solving. I, for one, would love to try someday. (I'm a mathematician teaching at a large research university in the US, and most of the courses I get to teach are not anything like this.) I'm also not sure this can completely replace the more "traditional" way we teach math, which is not to say I don't think it has problems (there are lots). If I may make an imperfect team-sport analogy, traditional classroom teaching of mathematics is all drilling and very little scrimmage / play. These problems are sort of on the other extreme. If we are to (1) equip students with intellectual tools that they can use, and (2) convey, to at least a fraction of the students, the sense of beauty and joy that attracted many of us to mathematics in the first place, we would need a balance between the two. I get the impression that this is something like what Gowers is actually advocating, but I haven't had a chance to read all his blog posts on this topic to find out (will have to do that later)...
danarmak 10y ago
Fermi estimates require at least some knowledge of the quantities you're using as inputs. Are the students expected to know these, and tested on that, or are they allowed to state any values they like for the inputs? Do the teachers and the test-writers think only the process is important, or the results as well? For example: > How many molecules from Socrates’s last breath are in the room? We can estimate the number of air molecules in the room by using Avogadro's number (6.022e23), the rough chemical composition of the air (roughly 80% nitrogen, element 7, and 21% oxygen, element 8), and the room's volume (which can be hard to estimate by eye unless it's a very small room). We also need the total mass of the atmosphere; we can estimate it from the Earth's surface are and atmospheric pressure, but we need to convert 1 atm to kilos per square meter. Would an average student remember that conversion rate? I certainly don't. (Turns out that 1 atm ~ 10,330 kg/m^2). That's quite a few constants to remember from physics and chemistry. (I'm aware that Fermi estimates are only one of many kinds of example questions in the post.)
sn9 10y ago
Sanjoy Mahajan [0], who is the author of perhaps the two best books on Fermi estimation (freely available online, too!), has linked on his website a really fascinating account of an experiment to reform mathematical K-12 education from 70+ years ago [1]. [0] http://web.mit.edu/sanjoy/www/ http://web.mit.edu/sanjoy/www/ [1] http://www.inference.phy.cam.ac.uk/sanjoy/benezet/ http://www.inference.phy.cam.ac.uk/sanjoy/benezet/

141 points by poindontcare 10y ago

Mathematics is just the extreme end of the two most fundamental concepts Abstraction and Generalisation. Start the intro to these concepts including showing how these concepts are useful and used by everyone in their day to day life in using natural language.

semi-extrinsic 10y ago
My wife has studied mathematics pedagogy, and one concept that really struck me from what she learned is compression. Put simply, you can't learn a new thing until you've compressed the old thing it builds on. If you have not compressed "addition" to the point that it requires little effort, you won't be able to learn "multiplication". Same holds for e.g. "derivatives" and "Taylor series", or "group theory" and "rings and fields". (I think this was from Piaget, or maybe Brissiaud.)
- andars 10y ago
  I would guess Piaget, because it reminds of this quote from Papert (who drew heavily on Piaget): > Slowly I began to formulate what I still consider the fundamental fact about learning: Anything is easy if you can assimilate it to your collection of models. If you can't, anything can be painfully difficult.
fiatjaf 10y ago
That is the best way to not learn mathematics.
pbhjpbhj 10y ago
Could you expand on your second sentence, perhaps with an example or two and how you'd imagine they're be taught in class?

Animats 10y ago

Gower suggests story problems. Many, many story problems.[1] [1] https://pbs.twimg.com/media/BTwHsS5CAAAzBBu.jpg https://pbs.twimg.com/media/BTwHsS5CAAAzBBu.jpg

simonbyrne 10y ago
Yes, in the loose sense they involve words, but he points out the problem with standard word problems in an earlier post https://gowers.wordpress.com/2009/07/11/help-im-stuck-in-my-ivory-tower/ https://gowers.wordpress.com/2009/07/11/help-im-stuck-in-my-...
laurieg 10y ago
The problem with story problems is, as the article states, that they are never really presented in the open ended way they claim to be. You almost always teach a class a fixed operation, multiplication for example, and then give them a bunch of word problems where multiplication is thinly disguised. A much better exercise is to give an absurdly open ended exercise. "I'm at the supermarket, which checkout should I go to?" is one I have used in classes before. You can get a discussion going and generate a lot of interesting ideas, and almost every time I do it in a class someone says something I've not thought of. Once students have given you some good ideas you can massage it into a model and do some more 'proper maths' work. Of course, this takes a good teacher that can engage and steer the class.
- axiomabsolute 10y ago
  So much this. Through a series of unexpected events, I ended up studying math in undergrad with no clue why or what I was going to do with it. My senior year I took a class called "Applied Modelling". The first project was a simple, one sentence question: "What would happen if the Greenland ice cap melted?". It reminded me a lot of Randal Munroe's "What If" blog on the XKCD site [1]. Easy to understand, open ended questions that encourage readers to learn a little about topics _outside_ of math to answer the question. The class gave me an appreciation for math that was lost during all those years of study before that, and it's basically my career now. [1]: http://what-if.xkcd.com/ http://what-if.xkcd.com/

Myrmornis 10y ago

Great article. Did anything come of the Gove-era math education policy changes he's referring to?

keithpeter 10y ago
In my opinion: Gove's main change in Maths was in the exam syllabus that is taken at age 16. The new syllabus will be examined for the first time this summer, most schools started teaching the material in 2015. It will take another 3 years or so to see the effect of that on the GCSE classes (basically age 14 to 16) and then another five years for earlier years to shift what they do. Change takes a bit of time when it is 700,000 children in each year group moving through 10 years of compulsory education. Politicians know this but the news cycle requires changes on top of changes...
- pbhjpbhj 10y ago
  Which all means changes have to be made at a slower rate than the parliamentary cycle; which politicians won't heed. But then Gove hadn't a clue and seemingly doesn't care either. Each parliament IMO should get chance to make one change - presented to parliament with optional amendment suggested by third parties (unions, parent groups, students). Let them go the Lords to win the right to make a further proposal to parliament. That should at least slow them down enough to allow teachers to weave something useful out of the crap that they can enhance over a couple of years before the next half-wit comes along and arses it all up.
- Myrmornis 10y ago
  Can you say a few words about the direction he took the new GCSE syllabus in?
  - keithpeter 10y ago
    Well, there is plenty of stuff on t'web about the current state of play [1] [2] [3]. My recollection is that Gove's original idea was to scrap the GCSE completely and replace it with a new 16+ qualification consisting of a single 3h exam for each subject. Maths represents a difficulty of course as the range of ability encountered at age 16 for all 700,000 children each year is huge. Maths and science subjects differentiate by topic, wheras Humanities subjects differentiate by response to a brief. As a concrete example you can ask a group of 30 students to write two sides about the best learning experience they have ever had. Everyone can leave the room feeling they gave it their best shot. Some will be excellent and creative with a good range of vocabulary and demonstrating some self-knowledge and analytical ability. Others will produce a description description, possibly with limited vocabulary, possibly with deficient skills in punctuation, grammar and spelling. Now ask the same group of 30 students to take a single Maths test. Some will finish in 5 minutes with full marks (it wasn't the right test for them) and some will take an hour and score close to zero (it wasn't accessible to them either). 'The Blob' (the education establishment in the UK, i.e. the people at the sharp end) managed to head this one off together with the QCA and the House of Commons' select committee on education lead by Graham Stuart - a conservative but with experience of work in education unlike Gove. The result was a re-vamped GCSE Maths in which the Foundation tier has some topics previously only found on the Higher tier such as trigonometry, surds, rules of indices with fractional indexes, simultaneous equations, quadratics: solving by formula, factorisation and substituting into to plot graph. Much of the more useful statistics has been removed (graphical presentation &c) and replaced with harder probability. Much more emphasis on technical algebra and difficult fractions/ratio questions. This lot is working its way down the school system now. If they stick to it the result might be OK in 5 years but my guess is they will get stick for the atrocious pass rates for the next couple of years (or fiddle them somehow) and then fudge it. [1] http://qualifications.pearson.com/en/qualifications/edexcel-gcses/mathematics-2015.news.html?article=%2Fcontent%2Fdemo%2Fen%2Fnews-policy%2Fqualifications%2Fedexcel-gcses%2Fmathematics%2FDownload-our-new-SAMs-and-2-new-sets-of-specimen-papers http://qualifications.pearson.com/en/qualifications/edexcel-... [2] https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-topic-changes/ https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-topic-c... [3] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=29&ved=0ahUKEwiwwOSNo87PAhXII8AKHcckC_E4FBAWCFEwCA&url=http%3A%2F%2Fsmartfuse.s3.amazonaws.com%2Fshirebrookacademy.org%2Fuploads%2F2014%2F05%2FNEW-for-2015-Higher-AQA-GCSE-SoW-010414.doc&usg=AFQjCNHVq9hT-i_mdzu32vNjFPhrYJMsNQ https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web...

wisty 10y ago

> Objection 5. You’d never find enough teachers who were capable of teaching a course like this. To do it well, you need to have a very sophisticated understanding of probability, statistics, game theory, physics, multivariable calculus, algorithms, etc. Objection 6: If it's hard to find teachers to teach it, maybe it's a little challenging for students (even though a good math expert might find it interesting). Just because a math expert thinks something is interesting doesn't mean low performing students will find it interesting. For a more HN friendly example - what bunch of high school students wouldn't want an IT class that taught compiler design, instead of stuffy old Excel? Even Python is more fun that spreadsheets, right? Certainly there are large swaths of high school math that can be cut, and replaced with more relevant stuff. But some care needs to be taken that it's actually teachable. The article does partly cover this though: > Thoroughly road test questions before letting them loose on the nation’s schoolchildren. In fact, that applies to the entire course: make sure one has something that definitely can work before encouraging too many schools to teach it.

striking 10y ago
> what bunch of high school students wouldn't want an IT class that taught compiler design The majority of high school students can hardly wrap their brains around the AP curriculum (probably for lack of time or effort, rather than ability). There are some that are honestly, actually interested in computer science and are thus capable of stuff like that... but they are low in number. What might be able to work is a fully-fledged web design course, using modern standards instead of boring stuff from ten years ago. With HTML, CSS, and finally JS (probably React, then Node). Maybe even SQL. With knowledge like that, you have more than enough of a base upon which to stand. You could likely even get a job. > Even Python is more fun than spreadsheets, right? Spreadsheets are easy computation for a wide audience, with a little learning curve. What can Python do, out of the box? What could you convince a high school kid to program with it that isn't a derivative of "10 PRINT HELLO WORLD; 20 GOTO 10;" ?
- az0xff 10y ago
  A course designed around using a particular "modern" web technology stack will have to change too often (every time it doesn't become "modern" anymore) for it to be sustainable. Imagine that in 2016 you have a course centered around using what was modern in 2006. That would inevitably happen with a course like that. I'd rather teach programming from 0 to making a really basic 2D game (be it in C++ or Python or whatever language and whatever library). The results are eye-catching and the coding process is engaging, and there's no need for it to rely too much on how trendy the framework is in the current year.

Pica_soO 10y ago

Capturing the interest must always be first. So to capture the interest, one must use math applications like game or physic simulations that allready captured the audience, allow beginners to modify the laws and abstractions within and create a "im at home with this"-intuition to delete the "im just not cut out for math" bias. Questions would not come from the teacher, questions will come from the students on how to modify the application.

partycoder 10y ago

Applied math around areas of interest.

M_Grey 10y ago

When you consider how mathematics, an understanding of probability, and so on enhances your ability to see through bullshit in advertising, government, and to some extent religions (not faith, just religions); it's not hard to understand why so many people are not thrilled at the notion.

frozenport 10y ago
Much of advertising is emotional appeal.
- M_Grey 10y ago
  Plenty of advertising, car sales, consumer financial products, lotteries, etc... rely on people not having an innate understanding of math, and especially probabilities. Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college!
  - wolfgke 10y ago
    > Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college! It's not obvious whether this is good or bad (too many cooks spoil the broth).
    - M_Grey 10y ago
      The cooks are in the kitchen with their votes already, how could it be worse to have them actually know what the hell they're doing there?
      - wolfgke 10y ago
        Perhaps I am a little bit biased by the situation in Germany. In the past years there was an increasing number of non-voters in elections (typically people who are dissatisfied with the political system). Now in the last years a rather right-wing populist party (AfD) appeared that convinced many of these non-voters to vote for them. Compare this a little bit to the fictional situation as if Donald Trump would bring many people who did not vote in the last presidential elections back to the ballot boxes to vote for him.

maus42 10y ago

The title is missing year 2012. ("This entry was posted on June 8, 2012 at 4:53 pm") Does anyone know what happened regarding all this?

keithpeter 10y ago
Probably not a direct consequence but... The UK elected a Conservative government who decided to base all 16+ Maths for non-mathematicians on the revised GCSE Maths syllabus. Colleges are now coping with large numbers of students aged 16, 17 and 18 being required to take the GCSE exam again while studying vocationally based qualifications. I'm teaching maths to trainee hairdressers, trainee car mechanics and would-be fine artists. Pass rates are not high (we are starting with a selected sample after all and schools are pretty good at getting non-mathsy youngsters through). The statutory requirement ends at age 19 and so the majority of late teens will experience three more years of failure in a subject that they experienced failure in at school. That should guarantee another generation of the general public whose loathing of Maths is pretty marked. Taking a wider view, I think that we all tend to learn things in a situated way and I therefore have a lot of time for the 'Functional Skills' approach to Maths and English if done properly. This was the approach adopted in most vocational training courses prior to the Gove era. I also think that there is a place for a qualification based around probability, statistics, and critical thinking. I'd love to call it "How to spot bullshit when you see it". I'd make discussions of issues like genetic defects and screening, obesity and health education and so on a core component. It will never happen of course...
- djaychela 10y ago
  Not sure if you're teaching in the above environment, but one of the schools I teach A-level at has a high number of maths-failing pupils, and as a result, many of those I teach are re-sitting maths in both their first and second years that I teach them. It's a typical case of Gove not thinking of the consequences of his (backwards, IMO) ideas - these people are definitely part of the generation you outline; it spreads so they not only think they are bad at maths (which admittedly is often true), but that they are incapable of any meaningful learning. Those who do take functional skills and pass at least feel they have made some progress - despite this being a level of maths which most people would feel was far below the age level of someone taking at aged 17. I know that anti-Gove rants are all over the place, but most of the people I know outside education don't have a clue about how poisonous his ideas have been; they don't withstand any serious scrutiny, but on the face of it may seem sane, so people who haven't thought about it will defend them. His replacement isn't far off his level, and I seriously fear for the future of education in the UK.

onetwo12 10y ago

The post is about suggesting more fermi estimates problems, see for example http://lesswrong.com/lw/h5e/fermi_estimates/ http://lesswrong.com/lw/h5e/fermi_estimates/ Fermi estimates require you to look for relations in the real world, construct a model of the situation and then iterate improving the model by adding more information. You don't use that process just to give an example of using a rule or solving an equation, instead you emphasize how the problem could be tackle mathematically. So there are more room for open questions, incentives for exploring and suggesting new approaches, in a more relaxed atmosphere creativity can flourish, perhaps maths is more than a single equation written in an old book.

dmitripopov 10y ago

There's no need to invent popular mathematics. You just need a teacher that can show a student how beautiful mathematics is. It's an art. A way of thinking and creating.

ivan_ah 10y ago

Wow, that's a nice collection of problems. They are exactly the opposite of the dry and artificial "word problems" students are used to. The discussion at the bottom of the blog post is also very interesting. The socratic approach is very good to "break the ice" and introduce the application, but I wonder how scalable this approach is. Does the teacher need to be very knowledgeable/entertaining to pull this off? BTW, I'm working on a new project, which is essentially "math lessons by email" that will walk readers through the math material from the NO BULLSHIT guide to MATH & PHYSICS. Anyone interested in learning or reviewing basic math (expression, equations, functions, algebra, geometry) should signup: https://confirmsubscription.com/h/t/4C2D9C45B88734F3 https://confirmsubscription.com/h/t/4C2D9C45B88734F3 (it's free)

TheOtherHobbes 10y ago
They are, but I think non-math people will look at just about all of them and think "I have no idea where to start." Worse, I think they don't teach generalisable skills. That's probably the core problem with all school-level math and science teaching. You learn a vocabulary of basic symbols and some rules for manipulating them, but you don't learn math skills - in the sense of understanding the real world well enough to make the leap from symbols and abstractions to useful life skills. The point of math teaching shouldn't be to know how to solve problems like these, but to learn how/when you can use math to answer your own questions for yourself. There's also a deeper level where you can teach the process of abstraction as an end in itself. I suspect that may be too far for most people - although I haven't completely convinced myself that's true yet.

blahblah3 10y ago

The problems he suggests overall seem too difficult for the average non-mathematically inclined student. And they also require quite a skilled teacher to teach. I'm not sure stuff beyond "algebra 1" needs to be taught to everyone in high school. Even the concept of using "x" to stand for an unknown is very difficult for some to grasp. Instead, schools should make sure all students can properly understand how to use addition, subtraction, multiplication, and division, with applications to things like personal finance. In my experience, even many college graduates have trouble understanding when to multiply, divide, etc...

ghaff 10y ago
At the risk of being sarcastic, the suggestion seems to be that mathematics is best taught through stereotypical management consulting interview questions. Or the apocryphal (?) Google interview questions like how many ping pong balls can fit on a bus. ADDED: I also suspect that the average high school student lacks the world knowledge to come up with meaningful guestimates for the inputs to many of those questions. A lot of the high school mathematics that I learned such as geometric proofs and trig are not all that useful. And it seems as if things that would be more generally useful like probability and stats are not that broadly taught--and are often taught in a very theoretical way when they are.
- blahblah3 10y ago
  Unfortunately, probability and stats are not easy to teach, and even many professional scientists / researchers have major confusions about the subjects. Common sense actually provides a decent enough guide for most people (i.e a baseball player with a high batting average is more likely to hit the ball). Euclidean geometry as taught in school does seem rather archaic and out of place though. Some people say it's an introduction to "proofs/rigorous thinking", but it seems to me that that purpose could be better served with a first order logic class.
  - inimino 10y ago
    First-order logic is much more abstract. I think a major benefit of geometry is that it introduces visual thinking and is very grounded and real because you can see and draw the proofs. This foundation of visual/spatial intuition seems to be very useful in higher math, as a counterpart to the exclusively symbolic manipulation of algebra or first-order logic.

andars 10y ago

If anyone cares to share, I'd love to hear some opinions about at what point someone switches from "non-mathematician" status to "mathematician" status.

Koshkin 10y ago
One becomes a mathematician, even if only temporarily, if one becomes interested in an investigation into purely mathematical matters (e.g. in finding a rigorous proof of a statement) rather than in using mathematics as a computational tool in one's (original) area of interest, such as physics or biology. Even when one happens to invent a new mathematical method that works, one does not, in general, instantly become a mathematician - unless, of course, they lose the focus and turn their attention to making the method just discovered more efficient or better substantiated from the purely logical standpoint.
contravariant 10y ago
I think you switch to a non-mathematician as soon as you proclaim your hate for mathematics. I don't like seeing 'mathematics' as some kind of specialist activity that only a specific group of people is capable of.
yummyfajitas 10y ago
When you mentally translate theories and discussions from words to equations rather than equations to words.
inimino 10y ago
In the context given in the article it sounds like non-mathematicians are students who choose to stop maths at age sixteen, presumably for the rest of their lives. This reflects the observation that children are born mathematicians and explorers and remain so until something kills their curiosity (often school is named as the culprit).
tsomctl 10y ago
I'd say for me, it was when I properly learned how to write proofs. It just formalized everything I knew, and forced me to think about it logically.

losteverything 10y ago

Your question made me think. Could be innate. My part time job forces me to make 10-20 micro decisions an hour. Most involve minimizing negatives and max positives. And knowing what to ignore. Yet my co-workers are quite unable to even know how to get 10% from a cash register total. Other managers lack a "math approach" imo to want to get sales numbers or staff assignments. For example, what should you do if you have 80 hours of work and only 70 hours of workers? Some fail because they can't even frame the task that way. Could be vocation only. Get paid by using math, you are one. I'm still thinking

Koshkin 10y ago

Teaching mathematics has two aspects, or goals, that are largely not related to one another. One is teaching mathematical methods as a computational tool, as used in a certain area of expertise (such as, for instance, electrical engineering). The other one is teaching how to do mathematics, i.e. how to generalize facts, find efficient proofs and algorithms, etc.

godelski 10y ago
I've always thought one of the most important goals of learning math was learning how to think logically. This may be coming from a physics perspective where the language is used because it is (arguable)the most accurate tool we have to analyze the world around us. The average person doesn't need efficient proofs and algorithms. But they can use generalized facts.
- Koshkin 10y ago
  I think that the idea that studying math is necessary to train one's logical abilities is a misconception. Learning any of the sort of non-trivial activities, such as cooking, which requires a high degree of awareness and logical thinking, would have the same side effect.
  - godelski 10y ago
    Sorry if it came off like that. I don't think it is necessary, but it sure is a helpful tool. So if we're going to teach people basic math skills we might as well focus more on this aspect so that they can use those basic skills and know why knowing how to find the angle of a triangle can be useful.

imgabe 10y ago

Do other disciplines ask similar questions? How should physics be taught to non-physicists? How should writing be taught to non-writers? How should car maintenance be taught to non-mechanics? I guess my point is, why should we teach mathematics any differently to "non-mathematicians" than we do to "mathematicians"? I mean, at the point when you're first teaching someone, how do you even know if they're a "non-mathematician" or a "mathematician"? After all, they haven't learned enough yet to know if they'd want to continue in that field of study.

blahblah3 10y ago
Maybe this is an unpopular view, but I think there is something unnatural about mathematics for most people. It tends to be much more abstract than other subjects. I say this as someone who has always enjoyed math and majored in it. And it does make sense to have "math for mathematicians" or "physics for physicists" classes. You see this a lot in college (often intro classes are divided into different "levels"). This allows some students to accelerate and others to still learn relevant materials but at a more appropriate pace.
spion 10y ago
> How should writing be taught to non-writers? Sort of: http://paulgraham.com/essay.html http://paulgraham.com/essay.html
zengid 10y ago
Usually mathematics is taught in a progression, and layers are built upon more basic layers. A 'non-mathematician' is someone who does not consider it their life's goal to solidify their expertise in mathematics, so they may choose other disciplines to obsess over, or no disciplines at all. Either way, a teacher of a class for 'non-mathematicians' who wants to explore depth in a certain area of mathematics cannot assume that this population of 'non-maths' will understand jargon or any concepts that aren't specifically explained within the scope of the class. Thus they must proceed axiomatically, being sure to avoid using concepts that haven't previously been defined. Conversely, a class for "mathematicians" would be one with some background that can be used to begin from, depending on the average grade-level or prior education of the students. So in a nutshell to teach deep math concepts to 'non-maths', one has to start from the beginning and define each concept in order, ignoring wider scope for the sake of reaching the goal. 'Math' students can start at a middle-depth, depending on their background.

chx 10y ago

There's a book by a Hungarian mathematician for this exact purpose. Péter Rózsa: Playing with infinity. That's the Hungarian name order of her name. https://books.google.ca/books?id=pj5G-3boMBwC&redir_esc=y https://books.google.ca/books?id=pj5G-3boMBwC&redir_esc=y

kazinator 10y ago

Bootstrap: teach enough math to turn non-mathematicians into mathematicians. Then teach mathematician-to-mathematician.

yodsanklai 10y ago

These problems are examples of questions that can be tackled with the help of mathematical tools. They can show students some applications of mathematics, but IMHO they don't really teach mathematics. Maths have to do with statements and proofs about abstract objects. I'd rather learn about euclidean geometry than trying to figure out the number of piano tuner in manhattan (I'm sure most students will be bored either way). Actually, this is an endless source of discussion among instructors, not only maths. Should classes be driven by applications in order to motivate students?

danarmak 10y ago
Do applications motivate students? Personally I would expect examples like these to de-motivate, since they make clear that the subject does not, in fact, apply to the students' idea of real life. If you teach abstract mathematics and make vague promises they will be really useful, some students may believe you. But if you try to demonstrate how they're useful, and the best examples you can come up with are estimating the number of piano tuners in Chicago or the number of air molecules in the room, things students know they won't ever need to do in their lives, they should become less interested.

baby 10y ago

I think music is the best example. There are two ways of learning music: * learning solfege * learning how to play directly Most great school of musics will teach you solfege first, and you will have to go through hardcore solfege classes while you start learning how to play an instrument. Some teacher will do the same, or some family will make their kid do the same. Now I can tell you this is not fun, but this is the way to become a great musician. You need the theory, you need to know how to read that stuff like you're reading English. But this is not fun, and I've known many in my youth who gave up learning an instrument because of this. Now if you would teach every kid to play an instrument first, and have fun with it, and actually producing music with their fingers/mouth then... they would maybe enjoy it enough to get interested into taking solfege classes and music theory later on.

jimmydddd 10y ago
I agree that music is a good analogy. For example, in the US there is a guy who has "shortcut" piano courses. His point is that yes, if you aspire to be a concert pianist, then you should spend years learning scales, reading music, and playing music from classical composers. But if you just want to play a few Billy Joel or Beatles songs, you can skip all that and cut to the chase. You won't have that great "foundation" needed to be a master player, but most people don't want that in the first place.

kkylin 10y ago

I really like the collection of problems, but I'm not sure it is easy to teach this kind of problem solving. I, for one, would love to try someday. (I'm a mathematician teaching at a large research university in the US, and most of the courses I get to teach are not anything like this.) I'm also not sure this can completely replace the more "traditional" way we teach math, which is not to say I don't think it has problems (there are lots). If I may make an imperfect team-sport analogy, traditional classroom teaching of mathematics is all drilling and very little scrimmage / play. These problems are sort of on the other extreme. If we are to (1) equip students with intellectual tools that they can use, and (2) convey, to at least a fraction of the students, the sense of beauty and joy that attracted many of us to mathematics in the first place, we would need a balance between the two. I get the impression that this is something like what Gowers is actually advocating, but I haven't had a chance to read all his blog posts on this topic to find out (will have to do that later)...

danarmak 10y ago

Fermi estimates require at least some knowledge of the quantities you're using as inputs. Are the students expected to know these, and tested on that, or are they allowed to state any values they like for the inputs? Do the teachers and the test-writers think only the process is important, or the results as well? For example: > How many molecules from Socrates’s last breath are in the room? We can estimate the number of air molecules in the room by using Avogadro's number (6.022e23), the rough chemical composition of the air (roughly 80% nitrogen, element 7, and 21% oxygen, element 8), and the room's volume (which can be hard to estimate by eye unless it's a very small room). We also need the total mass of the atmosphere; we can estimate it from the Earth's surface are and atmospheric pressure, but we need to convert 1 atm to kilos per square meter. Would an average student remember that conversion rate? I certainly don't. (Turns out that 1 atm ~ 10,330 kg/m^2). That's quite a few constants to remember from physics and chemistry. (I'm aware that Fermi estimates are only one of many kinds of example questions in the post.)

sn9 10y ago

Sanjoy Mahajan [0], who is the author of perhaps the two best books on Fermi estimation (freely available online, too!), has linked on his website a really fascinating account of an experiment to reform mathematical K-12 education from 70+ years ago [1]. [0] http://web.mit.edu/sanjoy/www/ http://web.mit.edu/sanjoy/www/ [1] http://www.inference.phy.cam.ac.uk/sanjoy/benezet/ http://www.inference.phy.cam.ac.uk/sanjoy/benezet/