Mathematics in STEM fields: GATEKEEPER OR GATEWAY?

[fresh_42]

I found this interesting article

Some recent writers on mathematics education have been talking about mathematics as a field enjoying ’unearned privilege’ as a ‘gatekeeper’ in our society. The more I think about it, the less sense this makes.

https://blogs.ams.org/matheducation/2019/02/14/mathematics-gatekeeper-or-gateway
written by Mark Saul

So the question is: What is mathematics: A necessary evil or the universal language physics is written with?
If you tune in on a random tv quiz show and wait for a "mathematical" question, usually some very basic algebra, chances are that it will be accompanied by a comment "how bad someone was in 'math' at school" or a similar fishy statement. People seem to be proud of their ignorance when it comes to mathematics. Why is it so? Or is my observation a local one?

Moreover I like to claim, that 'math' at school is usually calculations, not mathematics! Shouldn't we start to teach mathematics instead? E.g. explain why the exponential function is crucial to integration, instead of partial fraction decompositions?

 

[gleem]

People seem to be proud of their ignorance when it comes to mathematics. Why is it so? Or is my observation a local one?

No, for I also feel there is this tendency to justify ones lack of skill (or effort) by forming a consensus of underachievers in "numbers there is safety". Algorithmic thinking should be easy since it is prescriptive and does not evolve much creativity. Follow the rules. Look how easily kids pick up video games with complex rules which they may not even been explicitly told. My feeling is that the problem is not lack of skill but lack of interest. I believe we all are more capable than what we admit to but just do not have the interest to invest the effort in developing those skills whose value we do not appreciate .

Moreover I like to claim, that 'math' at school is usually calculations, not mathematics! Shouldn't we start to teach mathematics instead?

I would tend to agree with you. Most math taught at an early age is at least aimed toward computation. The one exception is geometry. In the mid fifties when the US feared it was falling behind in science and math it instituted a "revolutionary " approach to teaching math that started with set theory and this had its origin I am led to believe from the works of the infamous Nicolas Bourbaki. This " new" math was not well received because of it non traditional approaches to teaching math.

We all have different aptitudes for various subjects but often do not attain the level of proficiency of which we are capable. So how do we inspire an interest in math so people can truly determine their capability.

 

[nuuskur]

Can't read the article at the moment, but from the outset: why is mathematics under 'fire', so to speak? Couldn't we argue along similar lines with any subject, not necessarely STEM, even.

 

[fresh_42]

Couldn't we argue along similar lines with any subject, not necessarily STEM, even.

I don't think so. I've never heard somebody brag about ignorance in geography, a language or even biology or physics. Also the gap between school and university appears to be a bigger one than in other non STEM fields. A kind of evidence are the relatively high drop out rates during the first year. Mathematics is essential in certain studies and students are regularly confronted with the fact, that what they thought is mathematics is actually just computation. At school, at least here, mathematics is a sequence of algorithms to solve certain problems. And at university, all of a sudden, algorithms are not a central element anymore; the language itself occupies this role now and algorithms are merely a matter for some exercises.

 

[Wrichik Basu]

Moreover I like to claim, that 'math' at school is usually calculations, not mathematics! Shouldn't we start to teach mathematics instead? E.g. explain why the exponential function is crucial to integration, instead of partial fraction decompositions?

School education system is flawed to a great extent. Not only in the case of maths, what you have mentioned extends to physics and chemistry as well. Almost all chemistry books explain spin of an electron as "a particle spinning on an axis". How many teachers would believe a student if the latter claims that spin is an intrinsic property of an electron and has its origin in QFT, and has no classical analogue? Molecular orbital theory is taught to students who have little knowledge in QM (quantum means $mvr = n \hbar$ for most high school students). As a consequence, most of them can't understand what it is.

In high school, we have learned how to find inverse of matrices using elementary row and column operations. I find that an absolutely useless thing, because if a simple function like $\text{inv(A)}$ in MATLAB can give me the inverse, why should I waste my time in doing all the guessing? Binary operations are taught without even mentioning that they have a greater application in group theory.

In fact, the examination system itself is flawed. Think for a minute. How can a few questions judge whether a student knows a topic, not only in maths, but in Science as well? Maybe the student couldn't do that very problem, but in general knows the topic(s).

 

[gleem]

Also the gap between school and university appears to be a bigger one than in other non STEM fields

At university non STEM subjects do not build upon previous foundational courses for successful completion like math and physics do at least in the early years but rely on common skills as reading and writing which are developed continually throughout school so non STEM students are in general more prepared. STEM students should be but I believe that STEM programs cast their nets too wide attracting students who do not appreciated the value of really learning a subject until it is too late.

 

[StoneTemplePython]

In the mid fifties when the US feared it was falling behind in science and math it instituted a "revolutionary " approach to teaching math that started with set theory and this had its origin I am led to believe from the works of the infamous Nicolas Bourbaki. This " new" math was not well received because of it non traditional approaches to teaching math.

From a serendipitous conversation with a (west) German fellow recently, they did something akin to this around 1970s in his part of West Germany. The schools pulled the plug on the experiment after a couple years.

Among other problems when done there and when done in the US: parents and teachers don't understand the stuff. Another possible problem: most students aren't mature enough to appreciate it.

 

[StoneTemplePython]

In high school, we have learned how to find inverse of matrices using elementary row and column operations. I find that an absolutely useless thing, because if a simple function like $\text{inv(A)}$ in MATLAB can give me the inverse, why should I waste my time in doing all the guessing? Binary operations are taught without even mentioning that they have a greater application in group theory.

Part of the issue is that clever applications of algorithms and calculations can be very useful in proofs. Elementary row operations / applying elementary matrices can be used to prove that determinants multiply. There are other ways to prove this very important result, but they involve a lot more machinery.

You mention groups-- consider the task of proving that upper triangular matrices form a subgroup-- this was a problem in the homework forums in July of last year. Among other things it required proving that the inverse of an upper triangular matrix is upper triangular.

I like simple and obvious stuff -- which to me said write out Cayley Hamilton and the result is almost immediate. (And even more basically it's implied by a dimension/linear dependence argument if you don't know Cayley Hamilton.) All the other suggested approaches seemed in the weeds of calculations to me... but I didn't get any likes.

That said, once you've done a few of the numeric matrix inversions by hand, the benefits decrease rapidly, and you should (hopefully) move onto something more interesting and push the future computations down to matlab numpy .
- - - - -
long story short: knowing some of the computational routines
(i) let's you know how to do the problem by hand
(ii) give you a flavor of what is going on under the hood when the computer solves the problem
(iii) can be useful in proving important results

 

[mathman]

I am a mathematician. However, arithmetic and spelling were my two worst subjects in elementary school. I suspect people who are proud of being poor in math are just talking about arithmetic.

 

[fresh_42]

Sure. But there are two things which disturb me. Firstly, this isn't even math, and secondly, it is as ridiculous as if someone would be proud of a horrible orthography. Why is the former usus and the latter shameful?

I once knew a mathematician who used to say: "I'm a mathematician, I cannot calculate." This was the other side: stretching that calculations have nothing to do with mathematics and he opposed this common equalization. He went as far, as he refused to multiply strict upper triangular matrices. But he had no problem multiplying nilpotent ideals!

I am challenging the way "mathematics" is taught at schools, namely as arithmetic. Why don't we teach mathematics instead and call the other thing "calculation"?

 

[mfb]

A kind of evidence are the relatively high drop out rates during the first year.

Even outside of mathematics: Physics has a somewhat high drop out rate in the first year - because of the mathematics.

Raw drop out rates can be misleading, however. I don't know about mathematics, but many young people enroll in physics just for student benefits in Germany. Physics because they are sure they don't want to study that, so dropping out after a year has no consequences on whatever they do afterwards. This could apply to mathematics, too.

 

[Wrichik Basu]

That said, once you've done a few of the numeric matrix inversions by hand, the benefits decrease rapidly, and you should (hopefully) move onto something more interesting and push the future computations down to matlab numpy .
- - - - -
long story short: knowing some of the computational routines
(i) let's you know how to do the problem by hand
(ii) give you a flavor of what is going on under the hood when the computer solves the problem
(iii) can be useful in proving important results

Actually that's what I meant to say. Learning once and remembering the process is a good option, but doing it throughout an year for one examination at the end is not a good option. For that matter, children could be taught the use of calculators, and they could do 1+2 using it, but that wouldn't be a good option.

 

[Svein]

“Anyone who cannot cope with mathematics is not fully human. At best, he is a tolerable subhuman who has learned to wear his shoes, bathe, and not make messes in the house.”

― Robert Heinlein

 

[lpetrich]

Galileo Galilei - Wikiquote:

Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth.

That is still true today. Mathematics is essential, not only in the physical sciences, but also in the biological sciences, and even in the social sciences. Yet many people find math to be almost impossibly difficult, and they often seem to run away from it.

So it's important to teach that math is more than doing calculations. One must start simple, of course, and one can do that with simple algebra and geometry. I think that algebra is critical, because one needs the notions of variables and functions in computer programming.