How should mathematics be taught to non-mathematicians? (2012)(gowers.wordpress.com) |
How should mathematics be taught to non-mathematicians? (2012)(gowers.wordpress.com) |
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.
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.
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.
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.
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.
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.
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.
Sort of: http://paulgraham.com/essay.html
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.
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;" ?
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.
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)
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.
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.
(I think this was from Piaget, or maybe Brissiaud.)
> 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.
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)...
The average person doesn't need efficient proofs and algorithms. But they can use generalized facts.
* 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.
Can yiu tell my why, please? I'm not necessarily disagreeing, but I'm curious as to whst you're alluding to.
Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college!
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.
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.
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
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...
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.
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.)
Does anyone know what happened regarding all this?
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...
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.
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...
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.
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.
Actually, this is an endless source of discussion among instructors, not only maths. Should classes be driven by applications in order to motivate students?
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.
Then teach mathematician-to-mathematician.
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.
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.
https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
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).
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...
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...
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.)
[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.
It's not obvious whether this is good or bad (too many cooks spoil the broth).
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.
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.
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.
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.
