How should mathematics be taught to non-mathematicians? (2012) • Hacker News Board

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

#31

Post by nxtrafalgar » Sat, Oct 08, 2016, 1:50 PM UTC

axiomabsolute wrote: post #24

laurieg wrote: post #5

Earlier quoted context omitted.

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…

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 qu…

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).

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

#32

Post by blahblah3 » Sat, Oct 08, 2016, 1:55 PM UTC

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...

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

#33

Post by ghaff » Sat, Oct 08, 2016, 2:05 PM UTC

blahblah3 wrote: post #32

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 a…

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.

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

#34

Post by andars » Sat, Oct 08, 2016, 2:09 PM UTC

semi-extrinsic wrote: post #22

ankurdhama wrote: post #2

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 "rin…

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.

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

#35

Post by andars » Sat, Oct 08, 2016, 2:24 PM UTC

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.

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

#36

Post by blahblah3 » Sat, Oct 08, 2016, 2:30 PM UTC

ghaff wrote: post #33

blahblah3 wrote: post #32

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 a…

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.…

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.

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

#37

Post by az0xff » Sat, Oct 08, 2016, 2:35 PM UTC

striking wrote: post #30

wisty wrote: post #7

> 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…

> 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 f…

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.

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

#38

Post by Koshkin » Sat, Oct 08, 2016, 2:51 PM UTC

andars wrote: post #35

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.

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

#39

Post by losteverything » Sat, Oct 08, 2016, 2:51 PM UTC

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

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

#40

Post by inimino » Sat, Oct 08, 2016, 2:52 PM UTC

blahblah3 wrote: post #36

ghaff wrote: post #33

Earlier quoted context omitted.

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.…

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…

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.