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How should mathematics be taught to non-mathematicians? (2012) (gowers.wordpress.com)
141 points by poindontcare on Oct 8, 2016 | hide | past | web | favorite | 86 comments

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.

Yes, all the time. I've heard it for physics, I've heard it for CS and I've heard it for programming, all with good reason: the way you teach somebody deeply invested in your field is different than the way you teach somebody deeply invested in a different field, learning yours as a supplement. Or maybe a better way of putting it is that the way you teach somebody interested in a field in and of itself is different than the way you teach somebody who has other interests and motivations, but still needs to learn.

And hey, maybe before college, almost everyone is a non-mathematician. There's the small majority of students who'd love learning group theory because it's fun and beautiful, and then there's everyone else who need practical applications and real-world examples. But the way you teach them is ultimately just like the way you'd teach non-mathematicians in college or further on in life, when there's a clearer delineation.

This situation actually started a pretty nasty divide at my university. Engineering and Math are in separate organizational units which gives each a degree of independence. The politics worked out that most of the operational budget of the Math department came from teaching students from other disciplines. Over the years Engineering was dissatisfied that after going through the prerequisite Math courses their students had plenty of theoretical knowledge but couldn't actually do the Math -- even those that took the applied variants. So after Engineering came into more money they hired engineers to teach "Calculus/LinAlg/DiffEq For Engineers" courses that focused on application and topics that more directly applied to their other curriculum.

It was a success overall for the Engineers but Math wasn't happy. The engineers working on advanced coursework needed higher level math courses that were only available in Math. Not only were they constantly failing which angered Engineering but the professors teaching them had to devote more time to the Engineering students which took away from the math students and angered Math.

The sort of uneasy truce that they eventually came to was Engineering students take the regular Math courses and their Physics/Engineering curriculum supplements what they're taught in Math. This annoys double major Math/Eng students because there's far too much repeated material in their curriculum but it's the best they could do.

Absolutely agree with this comment. If you get beyond the title, what Gowers is talking about specifically is "If everyone were compelled to take 2 more years of mathematics, what should it be?" It is very different from what we would teach undergraduates getting engineering degrees; it would also be very different from what we would teach math majors. If we were to replace "mathematics" by, say, "programming," it would also not be surprising that one might come up with a list of topics differing from what we would want an aspiring computer scientist to know.

I did not find anything really surprising in the first part of Gowers's essay, but I thought the list of questions was great, independent of whether or not it is really appropriate for a required course.

When I was in school I taught a physics lab to non-physicists, and this was a conversation professors and I had. Granted this was at university level so most people had an idea about whether they (thought they) were a math person or not. I went in with a completely different philosophy to that class than with physicists. Because here's the thing, a physicist will care about different things than the non-physicist.

What I focused on was less of the physics and mathematics and more on making sure they gained an intuition about how nature works and how to think critically. These people didn't need the same toolbox as the ones trying to get a degree in physics or engineering. I could care less if these students knew the equations for Newton's laws, but I did care if they had an intuition. I didn't care if they had Bernoulli's principle memorized, but I did care if they could analyze an experiment.

The difference is that these people needed different skills in their lives. You must know it is popular to say things like "School taught me the Pythagorean theorem but not how to do my taxes." These people aren't realizing that math is giving them a toolbox that can help them do their taxes and other things. I teach my young nephews math whenever I visit them. They don't care about it in school but they like what I teach them because I make it fun and challenging. You can get a lot of topics covered and a lot of ideas and principles conveyed if you aren't worried about them being able to do every case. An example of this being that the basic principles of calculus can be used in your daily life and could benefit everyone (thinking about things like rates of change, tangents, series, limits, and squeeze theorem), but they won't need to be able to take the derivative of a function unless they are going into a job that requires that.

So I guess what I'm getting at is that there is a lot of benefit from math to the average person without actually requiring them to be fluent in the language. Think about this like being a moderate speaker in a second language but not being able to write or hold a deep conversation. They have a lot of advantages over someone who might know more about the written language and grammar, but don't know many words. I think the same argument of learning basic phrases in a second language would apply here as well. Most people don't need to be fluent, but we are teaching them like they are.

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.

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.

I agree with "we should teach mathematics to non-mathematicians in the same way we do to mathematicians". But I do not agree with teaching mathematicians the same way as we do to non-maths. I personally was taught like non-math. And it was a pain. It was boring. Every year I finished reading through math textbooks in a month, it was fun. But then the rest of the year it was a total boredom. I hated math classes for this. Literature classes was much more fun, than maths, despite I didn't like all those stupid discussions about stupid books of stupid authors.

> How should writing be taught to non-writers?

Sort of: http://paulgraham.com/essay.html

People keep asking this about programming all the time. There are many movements that aim to teach everyone (kids or adults) to program.

> 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.

> 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;" ?

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.

I love games as an introduction to programming. But then you have to teach kids how to do collisions (or physics), you have to teach them how to keep track of multiple sprites that behave the same but are in different places at the same time (I'm talking about classes, yes), and it's harder to point to a "real world" usage of game programming... which makes it harder to get your course approved.

I know, teaching today's web standards means they'll be out of date within the next ten years. But I believe the improvement of the web is asymptotic and will slowly come to a halt in the coming years... that and I would never forgive myself for not refreshing a course when it's too old to be applied to the real world, as a pragmatist. Finally, I don't think Vue will beat React too quickly.

(I'm not an educator, but I've spoken with a few on some of these topics.)

I went to a private school, where the only CS class was game programming; and it was offered as an art elective [0]. The class itself was very much a programming class, requiring only a couple of supplemental lunch time classes to be prepared for the AP exam [1].

[0] According to the teacher, the class was originally planned as offering a CS credit; but was changed to art when they realized that a CS credit is not a graduation requirement, while art credits were.

[1] This was not out of the norm for my school. The only class that was designed for the AP was calculus.

When I taught myself basic as a child I did so by building my own single-user dungeon. In my first c class in high school the project was to create a playable terminal-based hangman game. Neither of these engaging projects required me to learn anything about physics or sprites. Though I did build an ascii-animated hanging sequence for those who lost at hangman.

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 (it's free)

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.

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.

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.

I'm thinking of an analogy: Musicians. The definition covers all levels of involvement and skill, but every "musician" pursues some amount of musical activity and learning, voluntarily.

Nobody has a choice about math in K-12 school, though there are kids who will choose the more advanced classes on their own. For instance my daughter proactively convinced her high school to let her skip a grade in math.

In college, you can choose to major in math, or in a math intensive subject like CS or physics. Within those disciplines, you can choose the more mathematical specialties. In the work world, you can volunteer for assignments that involve math, or get a reputation for being willing to solve hard math problems. That's me.

I don't see it as a "switch" because my interest in math was evident (so I'm told) before I could even talk.

I think you need to define your terms because I see at least a couple different classes of answers.

Speaking personally, a lot of people would probably say that I'm "good at math" in the sense that I got through a fairly rigorous engineering program and that I've never had issues with the quantitative side of business or other such pursuits. On the other hand, I've never personally considered myself a "mathematician" in the sense that anything approaching upper-level university math was something that came remotely naturally.

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.

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).

When you mentally translate theories and discussions from words to equations rather than equations to words.

I thought Terrence Tao had an interesting viewpoint on this:


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.

The post is about suggesting more fermi estimates problems, see for example 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.

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.

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.)

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.

Could you expand on your second sentence, perhaps with an example or two and how you'd imagine they're be taught in class?

That is the best way to not learn mathematics.

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)...

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.

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.

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.

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.

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.

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.

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.

Why exclude faith? For PC sake?

Not many people change their foundational epistemic stance in response to mathematical arguments.

It would help if fewer people had foundational epistemic stances. Fallibilism is much more realistic than foundationalism, as I understand the terms.

Faith isn't a logical process, from what I can tell, but it's also not a supposedly coherent system as most religions claim to be. Religions offer predictions (past and present), and supposedly consistent statements about morality and life, which are demonstrably inconsistent.


Can yiu tell my why, please? I'm not necessarily disagreeing, but I'm curious as to whst you're alluding to.

Much of advertising is emotional appeal.

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!

> 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).

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?

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.

>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.

Well, maybe the Left should try actually appealing to the interests of the working class instead of the SDP going quietly along with the austerity measures.

>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.

That's not fiction.

That's an interesting perspective, and a good point.

Gower suggests story problems. Many, many story problems.[1]

[1] https://pbs.twimg.com/media/BTwHsS5CAAAzBBu.jpg

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.

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/

My university's Engineering Science department runs a competition each year for high school students along the same lines -- mathematical modelling of an open-ended question.

I also found it very valuable; it was one of the factors that pushed me over the line into studying STEM at university (I was a better English and economics student in school).

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-...

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

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

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...

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.

Can you say a few words about the direction he took the new GCSE syllabus in?

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-...

[2] 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...

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.

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.

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.)

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?

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...

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.

Charlie Chaplin's film Modern Times has a section where the tramp is working on an assembly line and parts are coming down the conveyor. He has at it and manages to get 10 yards ahead of the conveyor, and has a rest. Just as he is relaxing, the conveyor belt catches up and he has to start again.

In my opinion a lot of the Coalition policies (housing benefit changes, nursing/police/teaching pension changes, reduction in the number of nurse/teaching training places) were like Charlie's conveyor belt. They got a couple or three years of savings but now the belt has caught up. It is almost as if they did not expect to form a second government.

Many colleges are using the legacy syllabus this year. So when the majority of those come back again next year, we will be catching up with the conveyor because of the topics previously on the Higher tier added to the Foundation. I especially like it when a youngster 'taught' (i.e. coached in a smattering of topics) Higher tier can tell me some half remembered facts about the sine rule but has to use her fingers to work out seven sixes...

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...

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.

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.

>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).

I'm not sure how much I agree.

Sure, the math and the principles involved in designing scientific studies etc. can get pretty complicated. But there are a number of fairly basic ideas that could be usefully taught. And I'd argue that many people don't have a great common sense view of stats and probability. Sure, they have some idea of what batting average means--though there are lots of interesting sabermetric discussions to be had around baseball measurements--but there are also many well-known and consistent biases that many people have. For example, around ideas like streaks.

I've argued before and continue to believe that a semester long intro-level course on stats and probability that didn't get overly wrapped up in a lot of complex equations would be more useful at the high school level than some of the ways that time is used today.

How is learning facts about the space we all live in is "out of place"? Geometry continues to be extremely useful. In fact, in its generalized forms it is one of the most important parts of the modern mathematical thought. If anything, for a mathematically inclined student learning geometry, I imagine, would be much more both instructive and fun, than some "first order logic".

well I can only speak to my own experience. personally I really enjoyed geometry but can't say the same for most students

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.

As an anecdote, I actually had a fair bit of trouble with geometry in high school even though I did very well throughout high school in math/science generally and went on to major in engineering in college.

I'm not so sure about the visual thinking part but wrt symbolic representations at least you're probably right as I've never felt a particular connection to higher level math and theoretical physics.

By the way, what I mean by "geometry" is basically reading Euclid and working out the proofs with a straightedge and compass. What I see in the high school geometry homework I've come across is something else altogether.

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?

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.

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/

[1] http://www.inference.phy.cam.ac.uk/sanjoy/benezet/

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

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

Then teach mathematician-to-mathematician.

Applied math around areas of interest.

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