Today's topic is in response to a suggestion, as promised, which may have been left on Facebook. The question is, "I have many able calculus kids who score a 4 or 5 each year on the AP test (to go along with their 4 or 5's in AP chemistry) and the vast majority of them end up studying the humanities, with a ridicluous proportion choosing psychology - something that most will never make a living from - and relatively few choosing math/science/tech. Meanwhile, in huge proportions, the Chinese, Indian and Russian kids are studying the hard sciences. So, why, in your humble opinion, do our American kids duck the hard sciences at a rate that is not seen in other countries?"

As someone who has bachelor degrees in both math and philosophy, yet still cannot spell bachelor, I think I have an informed perspective upon this question. First off, although in might in part be the perception that the humanities are easier than math/science/tech degrees, I doubt this is a significant factor. I imagine this perception exists even in the communities which choose math/science/tech degrees, and I do not believe it is true. Considering my "fondness" for essays, I think my homework for the philosophy degree was, overall, more stressful than the math degree.

One thing I do think is a factor is the wonderful freedom of the US higher education program. From what I understand, in many places in Asia and Europe, one is channeled into a course of study at the university, rather than choosing one freely from a myriad of options. Thus, subjects that society places greater value upon are emphasized in schools and end up with the greatest number of students corralled in that direction. Whereas, in the US, upon arriving at university you are presented with a plethora of options and permitted to switch even rather late in your academic career, which can lead to Philosophy majors being declared in the second half of Junior year.

Another issue is something like an American sense of entitlement, or optimism, compared with caution or pragmatism exhibited by citizens in less privileged nations. To wit, you say yourself that a humanity degree is less likely to lead directly into a career, and to someone from a culture of economic caution, that might be a major deterrent. In comparison, an American who assumes that things will turn out alright in the end might feel more empowered to follow an interest for interest's sake.

I must admit that, as a cautious person by nature, this last point did play a major role in my decision to pursue graduate studies in mathematics rather than philosophy. Programs seem easier to enter, and the job market is definitely kinder, to mathematics students than those who study philosophy. However, the idea has been growing in me to attempt to switch, now that I am in a math program or after I complete it, as I feel teaching philosophy would be a more fulfilling career personally.

This is the best answer I can give to the question at the moment. As I mentioned when it was asked, I would like to hear your thoughts on the issue, as mine are based primarily upon my own experiences as a student, while you have the wider perspective of an educator who has watched many students head off to college. As always, I heartily welcome the answers, reactions, or further thoughts of all my readers. If the editing is rough today, I beg forgiveness, as my trip to Oregon has left me exhausted at the moment.

## 9 comments:

OK, so here is my opinion: I think for most kids, math/science/tech is much more difficult than the humanities and that is why kids duck those subjects, because they do not want to work their butts off. And, yes, I read what you said about math and philosophy, but, Kenny, you ARE a math prodigy, and so your experience is somewhat skewed by your high intellectual gifts.

I think my experience is much more typical, as I am like most people and possess only average intelligence. I strolled through my English major, and I do mean stroll, Kenny, working hard in only one upper level course on Chaucer. But in math, oh my, it kicked my butt again and again, and I can't even tell you how many hours I put in studying.

I also think back to my fellow students in both majors. Every single math major that I knew at that time could have walked into an upper level English class and passed, maybe not turned the world upside down, but passed, nonetheless.

And my English major friends? Well, they would be hard pressed to pass a math class that they were qualified to take. I mean, of course, you wouldn't toss them into an abstract algebra course and expect them to pass, but you would be shocked at how many could not pass a 100 level college algebra course, let alone trig with radians, and as for calculus, forget it. I know this, because I was essentially the English department's math tutor my last two years and business was indeed brisk. And yet, all of these types of questions are only viewed through our respective filters, and this one is mine.

I do like your critique on the choices that are available to people and how kids in other countries are tracked, and that makes a huge difference in their career choices. For us, it's almost like staring at this huge buffet table and not being able to choose all that readily.

So, thanks for the answer; I asked you specifically, because like me, you have majored in both the math and a humanity and thus have been in both worlds. Thanks

On a related topic, my office-mate yesterday was lamenting teaching Math 060 at a community college to students who could be kindly described as having their priorities elsewhere. Algebra, or whatever topics they get to in that class (order of operations, maybe?) are not particularly difficult, but he could see a divide growing between the students who cared and got it, and those who didn't care and didn't get it.

The funny thing is, as soon as he asks them questions about money -- if so-and-so owes you this much per week and it's been six and a half months, how much do they owe you? -- they wake up and find their abilities. Practical applications connect.

But as Mathematicians we want to teach an abstract understanding of the material. Mathematics doesn't just apply to cash transactions, though for most of these people, pursuing programs with broadly spurious names like "Hospitality Management," that may be the vast bulk of its application.

It is particularly hard to strike a balance in a shortened summer course between applications and theory, and from our perspective, applications are an extension of the theory, so they can be excised from the lectures, right? Or is that

exactlywhat's not working?@Frank: You could certainly be right about the difficulty, but for some reason I have a strangely resilient optimism for people. One factor with the difficulty is that a math course is not a stand alone examination of a subject, each draws on precise tools one has gained in previous courses. On the other hand, one could walk into an upper level English (or psychology as I have more experience with) course and understand most of what is going on, since the analytical tools used are not so specialized, so prerequisites are not as critical.

@Max: I was actually having a related conversation with my sister. Personally, I believe that children should learn calculations after getting an abstract algebra flavored introduction to the number system. Perhaps real applications can motivate why we need math, but to understand math you need some familiarity with algebraic concepts. Otherwise you are just memorizing rules, rather than understanding what is happening, which is antithetical to what mathematics is.

For example, "subtraction" and "division" make a lot more sense as special cases of addition and multiplication, using inverses. It avoids the issue of non-commutativity, easily explains why "division" by zero is verboten, and gives negative numbers and fractions a natural motivation.

I like these examples ("subtraction" and "division") so let's hang out with them for a little longer. In teaching subtraction, it's important to connect it to addition, certainly. An understanding of the properties of addition will motivate an understanding of the properties of subtraction.

But, if you give a student, through practice with word problems (for instance), the ability to generate tangible examples that test how subtraction works, I think they will be more likely to be able to recall fine details in a test or real-world situation.

What about associativity? Is it:

(1-3)-4=1-(3-4),

or

(1-3)-4=1-(3+4),

and why? Can a student really remember the algebraic motivation better than they can: I have $1, I pay $3 (so now I owe $2), and I pay $4 again (so now I owe $6)?

If the numbers were replaced by letters, and you asked if

(x-y)-z = x-(y-z),

I think you would trick a fair number of students who hadn't learned to give themselves examples with which to think critically.

However, your last question wouldn't even be an issue if you asked does:

(x+(-y))+(-z)= x+(-(y+(-z))

Well, as long as they are ok with distribution, which we can now emphasize because commutativity and associativity are no longer issues.

I remember once hearing or reading that Math was a game with rules and no objectives and that Philosophy was a game with objectives and no rules. I will freely admit that I have taken the minimum number of writing/english courses (well, I've FINISHED the minimum number, I dropped a couple that had horrible 15 page projects or wanted me to do something for every class) and so have little expierence with the soft sciences.

I would say that there is a different mindset that goes with the hard sciences than with the other subjects. In science I know that silane is not advisable to let freely into the air, and that I cannot divide by 0. However I cannot remember a single rule that isn't broken in English.

I'd guess that this helps explain the difference. I like knowing what the rules are and playing with them. I can operate very well and creatively in that environment, and more importantly, I enjoy it. I also know that there are those that always question the rules and don't like having them, and they may feel less satisfaction playing in a game with rules.

I would also agree that there is much more choice here, and I believe that is a great thing, however I think that having so few people in the hard sciences is a bad thing. Personally, I think that this is largely due to how we teach younger students, as both you and Max seem to think. However I don't think that the issue stems from how you teach the material (theory first vs. using examples to develop the theory, I think both are useful) but rather how the teachers act with the students. I remember when I was helping in a MTH020 class that the instructor said that Math was the only subject that it was socially acceptable (did I use the correct word, and did I spell it correctly?) to be bad in. I think he is right. When students have a hard time with addition, subtraction, multiplication, and division they are told that "It's OK, Math is hard." I'm sorry, it isn't difficult. I understand that this can't/shouln't be said, but telling a young child that it's OK Math is hard gives the OK to give up right there. And when you've lost them before you get past arithmetic how are you supposed to get them through calculus? And without some decent level of Math, how are you going to perform in any of the hard sciences?

That's my take.

That last post was me.

In regards to math specifically, I think you have a great point. One of my colleagues pointed out that when you tell people you do math, often the response is that they hate math. I mean, how many doctors, when revealing what they do, are greeted by, "I hate biology"?

To be quite honest (I know I'm a bit late on the conversation here, but it's an important one) I think a great deal of it has to do with the way educators are trained/accepted into education programs. This is my fourth year as an employee of our infinitely wise public school system. Would you believe that in one of my "teacher preparation" math courses one of my fellow students could not define the difference between a trapezoid and a triangle? I have worked with colleagues who came to me for help completing work simplifying fractions, creating graphs based on simple functions, and long division. How can they adequately teach students to do these things when they don't have a firm grasp of the how or why? Now, one might assume that there is a minimal level of knowledge required to teach these concepts, right? Well, there is, but clearly it's a very low standard. The last real math class I took was in high school, as I imagine is the case with most of my professional colleagues.

One of the great travesties in early education is that elementary educators are not required to specialize in a subject area like they are in secondary programs. As much as I consider myself a fairly well-rounded person I would not claim to be the most highly qualified math, science, language arts, music, and art teacher my students could have. Yet we continue to assume that elementary school subjects are unspecialized enough that anyone could teach them. ("People who can't do teach.") Now couple that with higher standards for all students and you end up with fifth grade science teachers who never took chemistry but are expected to teach their students how to balance chemical equations.

There is a fundamental flaw in our educational system, which is probably based on the attitudes held by our society at large. Math is hard. Elementary school is easy. Elementary teachers aren't that important. The real learning happens in high school. I ask you, if that's the case, what foundation are children working with when they enter middle and high school?

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