## Thursday, August 25, 2011

### Fixing Math

An OpEd piece in the New York Times this morning made a very interesting proposition on how to teach high school math.  From the article:
Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.
As people who use math as a tool to do science, I wanted to get your thoughts on this idea.  May I propose two questions:
1. One of the benefits of learning math is an ability to think abstractly rather than simply act as a calculator.  Does teaching math in an applied context hurt the student's ability to think abstractly?
2. In your experience, do high school math teachers have the technical background to effectively teach those courses?

1. I think this is a great point.  Too many kids (me included way back when) sat in algebra class thinking that this stuff was okay, but we'd never use it again.  But if we could combine math and science (physics, chemistry, engineering) and teach kids how to use the formulas in real world settings, it becomes an "Aha!" moment.

2. It's an interesting proposal.  I am not completely against it as I think people will use things like finance more in their life then geometry and trigonometry.  However:

1.  You are right, it is hard to teach logic and abstract thinking better then a "pure" math course.  Geometry for example is one of the few high school classes where making a proof starting with axioms is done and every step of logic is scrutinized.

However, I think you could make the argument that logic and abstract thinking aren't as important as we think they are.  (And I'm not being sarcastic.)  For tens of thousands of years mankind has made incredible progress without learning formal logic.

What would be awesome is if they replaced geometry with group theory.  Then they would really learn to be rigorous with abstract thinking! :)

2.  No.  Or I should say maybe.  Maybe if the course was watered down.  But you have to remember: many of these teachers have never taken such courses themselves, (for example, you can get a math degree without taking a math of finance class).  WHo is going to teach the teachers?

Although, if it is watered down enough I guess any teacher should be able to learn it quickly.

I still think the best thought I got out of this is how great life would be if we replaced geometry with group theory.  I mean, if you are going to learn math that doesn't require a ton of prerequisites you might as well have some fun. :)

3. Weren’t we taught arithmetic, then word problems in arithmetic – which were really applied math, then algebra was introduced demonstrating how the word problems can be solved through formulas; and, then same or similar formulas applied in various fields?  Why do we need to change this?

We already have young men sitting at computer screen using joysticks and buttons blowing up thing 10,000 miles away!  Darn it, this is practical, applied, and the best part is you never have to see a dead body!  No need to conceptualize anything.

Thanks Nick, for posting it, and more so for raising two serious questions.

4. This surprised me: "However, I think you could make the argument that logic and abstract thinking aren't as important as we think they are.  (And I'm not being sarcastic.)  For tens of thousands of years mankind has made incredible progress without learning formal logic."

Joseph, I would really like to see that argument.  I think you are wrong for two reasons (neither of which is the fact that I am teaching an intro logic course this semester!): (1) the issue isn't whether mankind must have formal logic in order to make progress, it's about whether having training in formal logic -- or abstract, mathematical thinking -- makes progress easier, more efficient, better, etc. (I think it does); and (2) one of the largest advances that mankind -- as opposed to any individual person -- has ever made was the development of digital computers, which were a direct product of hard work in formal logic.

But as I said, I found your remark very surprising, and I would really like to hear your side of the argument.

5. Replying to the explicit questions:

(1) Whether it hurts a student's ability to think abstractly depends a lot on how it is taught.  If you have a math finance course that is taught like -- (a) here are some numbers, (b) here is a formula, (c) compute -- then yeah, the students are only going to learn how to be computers.  But if the course goes like -- (a) here is a problem, (b) here is a problem, (c) what is common to those problems and could you describe and solve a more generic version of the problem? -- then no, I don't think it hurts abstract thinking.

(2) I don't have a good sense of what the proposed courses actually look like, so I'm not sure whether high school teachers have the relevant ability.  I want to say that the answer is yes, and that the really hard work is done by people farther up the food chain who are writing the textbooks.

I wonder, though, like the Ancient1, how is this really all that different from what we do now?  I mean, I learned some applied math in my chemistry and physics classes in high school, didn't you guys?  And math classes had application problems -- often having to do with money -- didn't they for you?

Moreover, I wonder why we couldn't have both pure and applied math classes.

6. Jonathan,

I must say a few things.

1.  Sorry, I wrote it slightly different then I was thinking about it now that I have reread it.  I didn't mean logic and abstract aren't important only that training humans formally in these areas may not be as important as we may think it is. (See my # 3 below)

2.  There is no doubt modern logic and mathematics are some of the greatest accomplishments of the human species.  If in any way I implied they are not I apologize.

3.  My argument is simple and probably not that logically sound :) : I think we often view things we specialize in as having greater importance then they really do.  For example, I have talked to like a dozen economics grad students who swear it is a shame that people aren't required to take economics classes because they are convinced that economics more than anything shapes society and many of the world's economic problems may be partially do to billions of people making economics decisions and voting on laws that affect the economy with no formal economics training.

Well, I don't know if you have taken an economics course.  I haven't, and I am guessing the world isn't as bad off with billions of people like myself not taking economics classes as some economics grad students make it out to be.

And by the principle of mediocrity if econ people can overhype the importance of formal economics training, I am sure that people with degrees in mathematics, like myself, might overhype the importance of teaching formal logic and mathematics.

7. Thanks for your thoughts.  In response:

1)  I like the idea that "applied math" courses could be taught in a way that effectively teaches abstract thinking.  I feel like a lot of what we call abstract thinking in math is really just intimidating notation and vocabulary.  What we really want is to show how a formal system like algebra can be applied to many different problems in a variety of fields.

2)  My feeling is that high school teachers are not well equipped to teach these courses, but that is generally also true of many standard math classes as well. My high school calculus teacher hadn't done calculus in at least 15 years when she taught our class and was effectively learning the material one week ahead of us.  There were several times where student caught details that she hadn't understood and we had to explain them to her.  I can still remember my buddy effectively teaching the class L'Hopital's rule.

Within reason, I believe that good teachers will be successful teaching just about any ciriculum and that no curriculum can turn a bad teacher into a good one.  I'm not sure how to get better math teachers, but I think that's really the key.

8. I agree that we are all prone to make our pet discipline out to be more important than it really is.  Following up on that line, I think it would be crippling to ask everyone to have the same level of education in every field.  There is just too much for one person to learn!

On the other hand, I do think that we all need basic training in -- or at least, exposure to -- core areas.  And while I would probably agree that economics is not as vitally important as economists (sometimes) make it out to be, it is a core area in contemporary society.

You personally are not in need of formal econ training in part because you have *other* formal, abstract, mathematical training.  So, when you see an economic argument, you can make sense of it.  And that is, I think, a good argument for everyone having abstract, formal training: the more abstract, the more transportable the training.  Anyway, you might have gained from having an econ class, all the same -- I certainly did!  (At least, in my case, opportunity cost, substitute goods, elasticity of demand, and so forth were new concepts when I took my econ class, and I'm not sure I would have learned those ideas anywhere else.)

9. Jonathan,

The more I think about it the more I want to take back what I said just a little.  Logic and abstract thinking are skills that I do believe should be understood on the basic level and I bet having a formal class goes a long way.

For example, I think basic reading skills are essential in today's society and I would hesitate to say with confidence that society would be literate without formal classes on reading.   So given that, and given I think the human's ability to  think logically and rationally is as helpful to humans as the ability to read, I think it is probably fairly helpful to make sure everyone is taught basic logic skills in a formal setting.

And I will do one better: At the college level most universities I believe require at least two writing courses and one math course.  I think you should also have two math courses like writing except I think one of the two math courses should mostly be a "how to prove things" course that could be supplemented by taking a  logic 101 course taught by philosophy department.  The other math course I believe should be more like an applied math course.

So I wish colleges required one applied math course and one "introduction to proof" type math course which the student may or may not want to take the equivalent thing through the philosophy department.

10. Fantastic!  I think we are now in agreement.  :)

11. Regarding pedagogy I think the issue really hinges upon how it is done.  I'm all for applied math but the history of doing this in elementary and junior high textbooks isn't exactly exemplary. It typically ends up far, far worse than if they'd just stuck to a more "traditional" approach.  Think of the so called "New Math" movement in the 70's and 80's which was such a disaster.  Yes, I know New Math was more a move to basics rather than application - but I think it highlights that the issue isn't the theory of how these things develop and that the real issue is how to deal with teachers (who often don't understand or like math) and students with limited resources and often little home support.