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I perceive that among the general population the belief that mathematics is some sort of highly intellectual esoteric subject is still quite prevalent. After studying math proper I've discovered that even though the notation might look weird sometimes, the concepts that are dealt with are very natural and some of it can be observed in almost any real-world situation. Therefore mathematics is not esoteric, but is related to our surroundings (which everyone should be able to observe).

So why do people fear maths, even though they can see a lot of math around them all the time. It's just not notated, but the phenomena are there.

Also, surely it'd be better for mathematics itself, if it wasn't perceived to be esoteric, but accessible to everyone?

https://en.wikipedia.org/wiki/Mathematical_anxiety

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    $\begingroup$ @MichaelE2 I think that is to be expected. A teacher who sees and teaches the beauty of math, shows the ideas and the concepts rather than just the calculations will of course motivate a greater love for math than a teacher that simply states "this is the problem, here is the formula, put the numbers in and get the result out". Unfortunately, there are way to many teachers of the second kind; maybe because they themselves did learn it that way... $\endgroup$ – Dirk Nov 22 '17 at 8:59
  • $\begingroup$ Actually, you already state one of the reasons, the notation. It is hard for pupils to understand and get used to it, but at the same time the notation is being used to introduce further concepts. It is like learning Chinese in Chinese (from a perspective of a non-chinese person obviously). Pupils frequently fail to recognize the concepts because they are lost in notation... $\endgroup$ – Photon Nov 22 '17 at 18:13
  • $\begingroup$ @Photon Yea I once had this idea that if math didn't use greek symbols, then it might be more accessible to people. For starts at least. The notation can be anything, it's just a convention to use those greek symbols. And for real mathematicians it serves some sort of nostalgic value I believe, because it ties maths to the maths of the Greeks. But the notation might also make math look more weird than what it is. $\endgroup$ – mavavilj Nov 22 '17 at 18:17
  • $\begingroup$ @mavavilj: I'm not referring to Greek symbols in particular, Latin symbols can be confusing enough! $\endgroup$ – Photon Nov 22 '17 at 19:12
  • $\begingroup$ @mavavilj I agree that Greek can be a bit of a hurdle, but even variables are a problem. I'll often do entire computations with words. For example, if we want to show that $x \mapsto x^2$ is continuous at $a$ using an $\varepsilon$-$\delta$ argument, I might say (roughly): We want to find a (tolerance) such that the difference between $a^2$ and $(a\pm(\text{tolerance}))^2$ is less than some specified (error). That is, with a specified (error), we want to find (tolerance) such that $$ a-(\text{tolerance}) < x < a+(\text{tolerance}) \implies -(\text{error}) < a^2 - x^2 < \text{(error)} $$ $\endgroup$ – Xander Henderson Dec 15 '17 at 23:59
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Certainly not a final answer. But to me, maybe the most important aspect is the behavior of the teacher. The way he or she understands mathematics and the beliefs that are hold affect the learning of the students - a triviality...

But once you go on and read the classic by Stella Baruk (1971) Echec et maths, you get more than a glimpse on how math anxiety may develop. You may then appreciate, that besides being knowledgeable about math and teaching, why it is so important to hold a "Growth Mindset" belief and care about all your students as human beings. Within math teaching, Jo Boaler's work with YouCubed provides many examples that I find inspiring.

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I believe it is a combination of the following, among other things.

  1. The writings of Hung-Hsi Wu have plenty of examples of the gibberish and pseudo math that goes on in many math classrooms. This practically guarantees that only a tiny minority of students benefit from their "math" experiences in school.
  2. A huge share of elementary teachers, victims of pseudo math themselves, hate math. At least they hate what they believe math to be. As I've said elsewhere, an art hater becoming a children's art teacher is obviously stupid. But having math haters teach math to children is the norm. Negative attitudes towards math are passed on from teachers to students.
  3. Many passionate math educators constantly tell themselves to defeat their own curse of knowledge, they obsess over empathizing with students. And they fail. A lot. I include myself in this group. I look back at learning activities from a few years ago that I thought - no, I knew - were good... and now I feel like I'm one of those doctors who used to recommend cigarettes to their breathing-impaired patients. We educators overload our students all the time but students blame themselves when they're overloaded and get wrong answers.
  4. Traditional math assessment is extremely algorithmic, to the exclusion of logic and meaning, to the point where you can memorize the algorithms and earn an A in your decimal calculations test... while believing $7.30$ is greater than $7.3$. "Isn't 30 bigger than 3? Who cares? That's not on the test so it has nothing to do with math so I don't care and I don't need to know!" After 10 years of rote "learning", a student gets to a teacher who asks conceptual questions and the only way to even begin to understand those questions is to learn 10 years of concepts in time for the high-stakes test tomorrow.
  5. Math is extremely cumulative. In the early grades, it's obvious that one must learn to count to 10 and beyond before learning how to determine $8+6=\_\_\_\_$. You wouldn't really care what was on the test; you'd have the teacher teach the child how to count. But in high school, students, parents, and teachers complain angrily if I suggest the student should understand that $\frac{4}{3}>1$ before tackling $y=-\frac{4}{3}x-12$. "Tomorrow's test isn't on comparing pies vs numbers. It's on graphing!"

All of this reinforces the belief that math is esoteric, that only a tiny share of people can understand it, that there is no beauty behind it, that math is just fearful rules to memorize.

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  • $\begingroup$ Per 3, thanks for mentioning it. I actually often encounter a lot of blind spot which is a form of insensitivity. Very bright people who think everyone should go to Harvard are not that aware of what it is like in other person's shoes. And actually not that bright, if being shrewd and avoiding intellectual blind spots is part of being bright. $\endgroup$ – guest Nov 22 '17 at 20:50
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  1. We are not computers and the human mind did not evolve to do math problems. We are much better at looking for the snake in the grass or recognizing our children's faces in a crowd than we are at solving for X (let alone proving epsilon delta). Our minds are actually stronger at speaking and hearing than they are at reading and writing. But even reading and writing is much more suitable to our minds than math. That doesn't mean we can't do it at all. Obviously we can. But it is not as natural. That's just life. We are not as agile in the trees as monkeys or squirrels either, even though we can do some tricks with enough training.

  2. For a lot of people, lack of skill/understanding drives this fear. I think this is logical and understandable. (A lot of people have fear of physical tasks that they are not good at also.)

  3. I don't think that nomenclature and pedagogy always helps either. I get more out of a PDE book that starts with units I know than if it is a blizzard of Fraktur and Greek symbols (and not alpha, talking "squiggle" and "other squiggle"). Or consider quantum mechanics. Lowe's graduate QC text makes the good argument in preface that it is better to teach the basics of the concept first before hitting the students with bra and ket. It is just hard to learn too many new things at once. You can get a lot of basic benefits from learning the basics of Schroedingers Equation for the hydrogen atom first in normal PDE land before getting hit with bra and ket and postulates. And already knowing some of the basics, makes it easier to have some ground to learn these new concepts on). Sometimes how the topics are taught or described in texts is poor as well (lack of motivation, not enough progression, not enough "small wins" or drill.)


I would put the issue as more on 1 and 2 (seems like same thing after I wrote it) than on 3. Even if you have wonderful teachers and texts, it does not make all the problems go away. And it is a long, long, loooong fallacy of educational reform that if we just changed this (no THIS) than everyone would be an A student. But sometimes the "harden up and handle your Rudin" attitude does do the student a disservice.

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    $\begingroup$ I study maths, stats and computer science at uni and I've noticed that, yes, intially there's what you say by 1., 2. and 3. But I've noticed that once one reaches a sort of "break-in" period for math, then further (and previous) mathematics becomes much easier, because one can perceive that further maths is just extension of the previous. The break-in period is when one becomes accustomed with the language and logic in order to understand the reasonability and simplicity of the notations as well as may come to appreciate things such as mathematical beauty, elegancy in proofs, ... $\endgroup$ – mavavilj Nov 21 '17 at 22:02
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    $\begingroup$ ... One also forms an overview on what different subareas of math are about and how they relate to each other. $\endgroup$ – mavavilj Nov 21 '17 at 22:04
  • $\begingroup$ Your question asked about issues in the general population but your later comment reflects a tiny subset of the population. Yes, I do agree that things can become better over time. But the key is progressive training. It's not that much different than weightlifting and getting stronger or learning gymnastics. And that some level of mastery will lower the fear level even of higher levels of master than you have attained. Oh...and not everyone will be able to achieve the same physical levels or want to (even though progressive training will help anyone). $\endgroup$ – guest Nov 21 '17 at 22:14
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First, we need to stop insisting that "math is all around us!" (usually said with Carl Sagan-like starry-eyed wonder ;-) (jk)

As a STEM major, I had a natural interest in things that math talked about. However, I was not quite so enthralled with Chaucer, The Stamp Act, Art, or most of the humanities (I've gradually come to enjoy them too).

So, I sympathize with students who are not interested in a STEM career -- unlike, say, history, writing/literature, or even science, mathematics is quite removed from daily life. And it's not just because most people will never have to solve the quadratic equation. It's that its supposed "knock-on" benefits appear oversold to most non-STEM folks. Here's a possible view from their perspective:

Take history -- I'll never write a history article or scour the archives, yet, knowing history helps put everyday life into perspective. History directly affects our lives and we constantly talk about it in politics and our own lives. Relevant!

Now, take English/writing -- I'm not going to be an author or journalist, but I need not tell anyone here that being able to write clearly and coherently is probably one of the biggest determinants of success in your career. From cover letters, to reports, to presentations, if you cannot express yourself, you're limiting yourself. Relevant!

No, onto our sister discipline -- science. I'm not going to be a scientist or engineer (well, OK...I did become one, but that's beyond the point!) but science is about tangible things and is responsible for our technical wealth. Not to mention global warming, stem cells, cancer treatment, space exploration, and others are things many people are interested in talking about or having an opinion on. Relevant!

Now, on to our collective favorite -- math. While the above subjects give knowledge that is quite useful outside the academic discipline, math gives us points, lines, planes, quadratic formulas, and calculus. I've found that "everyday examples" of these ideas are quite contrived, as most people will never need to write a formula down or solve anything but the simplest equation.

The problem is the other fields are geared towards being useful to the general citizen while math classes appear to be attempting to groom the next generation of STEM majors. I'd say, leave that to college. To people who aren't interested in solving equations or proving theorems, math class is just one puzzle after another, with no relevance (and, guess what, they're right!).

Now, I'm not being down on math. I think there is hope, but it's not by making non-STEM-focused students love the quadratic equation, trigonometry, or integrals by making up examples of cannonballs falling, bridge building, or leaky buckets. It's by copying from the playbook of History, English, and, to a lesser degree, science. We need to focus on areas where complex mathematical concepts creep up in actual daily life. In my experience, this area is statistics.

Statistics pops up everywhere in life, and even if you're not going to be a statistician, the quantitative reasoning skills (and knowing the logical traps) required to understand statistical arguments is extremely useful in everyday life. Politics, sports, theology (Bayesian arguments for/against god .. Ok, hot button issue, but still ;-), medicine, gambling...these are things ordinary people actually do and actually could be helped by some numeracy.

Until the curriculum is changed, I think we'll be stuck with contrived examples from algebra, trig, and calculus. Also, I agree that we need some basic arithmetical numeracy like finding areas/volumes, and understanding arithmetic operations. However, I think that Algebra/Trig/Calculus sequence is really best left for students who will be doing STEM.

I'd really, really, like the general ed math to focus more on asking quantitative questions and being skeptical of numbers. I think that number-phobia is manifested by people either trusting or mis-trusting an argument simply because it has numbers. They've never been given the confidence to "challenge the teacher or the numbers", which is what you need for correct quantitative reasoning.

Imagine students leaving after spending four years honing their ability to deconstruct a quantitative argument in a newspaper, to question a politicians statements, to be able to appreciate clinical research, to incorporate uncertainty and risk properly in their lives...that would be something!

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Another reason, I think, is how mathematics is typically graded. Do 20 problems, get red Xs through 7 of them, you earn a D.

Every time a student exposes some misunderstanding, they are punished. This is backwards: students should be rewarded for showing that they do not understand something, because this gives them an opportunity to grow. Since exposing misunderstanding is punished, students hide (even from themselves!) what they do not know. This leads to a death spiral of mathematical knowledge.

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some months ago, on the Flemish radio, somebody mentioned another problem:

It's cool to be bad at mathematics

And face it, it's true: when people are coming on television and they are asked about their skills or knowledge, when they need to admit being bad at history, geography, languages or other basic knowledge, people openly feel bad about it.
But when admitting being bad at mathematics, people feel proud about it! In a television show, I remember having seen once following dialog between a locally famous person and the television host:

  • Famous person: "I was actually not that bad at mathematics"
  • TV host: "Really???" (as he meant "Aren't you ashamed for that???")

When popular persons (singers, artists) would start showing another attitude towards mathematics (not saying they were bad at it, or in case they were, showing real honest shame for it, or, when they were good at it, showing real pride for it), that would change the whole mindset about mathematics, the mentality around it might change and side-effects like math anxiety might be reduced.

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