
How should mathematics be taught to non-mathematicians? (2012) - poindontcare
https://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/
======
imgabe
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.

~~~
tikhonj
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.

~~~
Spivak
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.

------
wisty
> 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
> 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
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.

~~~
striking
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.)

~~~
gizmo686
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.

------
ivan_ah
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
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.

------
andars
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
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.

------
onetwo12
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.

------
ankurdhama
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
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
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.

------
kkylin
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)...

------
Koshkin
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
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
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
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.

------
baby
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
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.

------
M_Grey
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.

~~~
rimantas
Why exclude faith? For PC sake?

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

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

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

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

~~~
laurieg
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
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/)

~~~
nxtrafalgar
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).

------
losteverything
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

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

~~~
keithpeter
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...

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

~~~
keithpeter
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-...](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)

[2] [https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-
topic-c...](https://bettermaths.aqa.org.uk/2014/06/28/gcse-maths-topic-
changes/)

[3]
[https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web...](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)

------
dmitripopov
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.

------
Pica_soO
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.

------
danarmak
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.)

------
maus42
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
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
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.

~~~
keithpeter
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...

------
blahblah3
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
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
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.

~~~
Koshkin
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".

~~~
blahblah3
well I can only speak to my own experience. personally I really enjoyed
geometry but can't say the same for most students

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yodsanklai
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
_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.

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sn9
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/)

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chx
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)

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

Then teach mathematician-to-mathematician.

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partycoder
Applied math around areas of interest.

