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This question is primarily discussing maths education for adult learners, on courses teaching engineering mathematics at an undergraduate level. These students generally never set out specifically to learn mathematics, but need to in order to obtain necessary qualifications for their career.

I frequently find that students arrive in my classes with unfortunate preconceptions about mathematics. In particular "I was never good at maths", "Maths is too complex for me" or "I just don't understand maths". This notion that mathematics is beyond the ken of mere mortals seems to be a widely accepted cultural phenomenon (How would a movie show the audience that this scientist is a genius? By showing them working on a blackboard filled with impenetrable equations). I've had onlookers walk past my office, and remark how complex my whiteboard looks, when it's normally some simple calculus revision that I was going over with a student. I politely say that it's only complex to them because they haven't seen it before, but I always just get the "Well I was never good at maths" answer. I've even over-heard colleagues teaching more qualitative modules say these exact words to the students.

This preconception is often so pernicious that it is very damaging to their future prospects. Whenever such students are taught some particular method for dealing with a problem, (e.g. transforming an integrand into a different coordinate system) they see it as "this was a technique I never could have seen myself, and therefore this is too hard for me to learn", rather than "this is the technique that I need to familiarise myself with and learn when to apply". When they struggle with a problem, as everyone does from time to time, they see it as reinforcing that "they cannot do maths", rather than seeing it as something that can be overcome through practice and effort.

Although many academic subjects have this problem, I feel that maths suffers from this "mystique" issue more than most. Are there any teaching strategies that are good at trying to break down the "I just can't do maths" barrier?

EDIT: Some comments are raising the issue of "what if the student actually can't do the maths"? This is a complex secondary issue with a lot of history behind it, that I believe is beyond the scope of this question. I think it is manifestly true that there exist students who have the ability to learn the material, but mistakenly believe they cannot. This question is purely concerned with the best way of helping those students, and not on adjacent issues.

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    $\begingroup$ @DanielR.Collins: I accept that there definitely students who have diagnosable learning difficulties when it comes to mathematics. I have previously taught students identified as having dyscalculia and dyslexia. However, I think that understanding how to help those students would be best served in a different question. I am mainly looking for teaching strategies to apply to students who would not be diagnosed with any sort of learning difficulty, but never-the-less feel that any mathematics above simple arithmetic is beyond them. $\endgroup$ – MadScientist Jun 21 '18 at 13:55
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    $\begingroup$ @MadScientist It's not just people who have some diagnosable learning difficulty. People differ quantitatively in ability. It's not either-or, it's on a gradient. Daniel is in fact much more on point: people say that math is hard, not because of some stereotype of the science/math genius, but because it is. You're doing everyone a disservice if you don't recognize that people differ. Just because you don't find mathematics difficult, that does not mean it isn't for other people, and it's not just about some mindset. $\endgroup$ – Eff Jun 21 '18 at 15:37
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    $\begingroup$ @Eff You could replace every instance of "math" in your comment with "sociology" or "playing violin" and it would make just as much sense. Anyone who is basically within the spectrum of neurotypical and who is willing to make the effort is perfectly capable of learning mathematics, sociology, or violin---there is nothing particularly special about these three fields. Yet most people believe that mathematics is a lot harder than sociology or violin. Therein lies the question of "mindset." $\endgroup$ – Xander Henderson Jun 21 '18 at 16:54
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    $\begingroup$ The problem with people who are there for the credit hours is that mathematics above simple arithmetic is beyond them. You mention calculus; for that you need a full understanding of arithmetic, algebra, geometry, and trigonometry (that's why math has this mystique; if you weren't paying attention for your entire life, you're screwed now). How many of your students that passed all their pre-calcs with flying colors go on to say that calculus is too hard? These other people probably need 10y worth of one-on-one tutoring to bring them up to speed. $\endgroup$ – Mazura Jun 21 '18 at 21:23
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    $\begingroup$ Mandatory quote: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is" (Von Neumann) $\endgroup$ – Taladris Jun 22 '18 at 14:24

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This is indeed a challenge, especially for adults. Three suggestions, none of which is a panacea.

(1) Emphasize a growth mindset. Make it clear to them that learning math is a skill accessible to everyone, with effort. It is not only accessible to those with a mythical "math gene."

(2) Compare understanding the abstractions of math and its notations as learning a language, or learning to read music.

(3) Wherever possible, concretize the math, avoiding equations. Here is one example I use: Cutting out a triangle. See also Physical vs. Virtual manipulatives in the high school classroom. Exploit the many "Proofs without words" collections.

Added. (3') To quote user @Mars, "then bring the ideas back to the equations." In other words, use the concretization to help the students to grasp the concept, and then use that to walk toward the abstraction that generalizes the concrete, usually via symbolic equations. This responds to user @Džuris' concern.

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    $\begingroup$ I wonder if it would be better to concretize the math but then bring the ideas back to the equations. That way students will be more likely to overcome their fear of equations, which in the end are very powerful tools, and are invaluable in clarifying mathematical thinking. (Caveat: I'm not a math teacher, but a professor in another field, teaching content that can be enriched when students are unafraid of math.) $\endgroup$ – Mars Jun 21 '18 at 16:38
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    $\begingroup$ @Mars: I completely agree. $\endgroup$ – Joseph O'Rourke Jun 21 '18 at 19:48
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    $\begingroup$ The "Growth Mindset" materials are interesting, and something I will look at, although it may be difficult to apply to adults. As for concretising, it's something I use fairly often but with mixed results. We want the students to be fluent with the equations, but more importantly, we want students to be able to understand the underlying source of those equations so that they can apply these tools to different fields of problems. I've sometimes found that when concretising, students will focus on the example to the detriment of understanding the underlying method to apply to other problems. $\endgroup$ – MadScientist Jun 21 '18 at 20:52
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    $\begingroup$ @Džuris: I myself grasp the general from the particular: bottom-up, concrete examples to general theorems. $\endgroup$ – Joseph O'Rourke Jun 22 '18 at 20:47
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    $\begingroup$ @Džuris: We do not teach young children addition and subtraction via group theory. $\endgroup$ – Kevin Jun 24 '18 at 0:17
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I think it's something you have to tackle head on on the first day of class. Some materials I pull from are:

  • Jo Boaler's work on mathematical myths, such as: Some people are math people and some are not; struggle is a sign of weakness; math is about methods and procedures; you are not good at math unless you are quick at math; math is best learned alone.

Something else I learned from Jo Boaler: if you put people in an MRI and show them math problems, the same part of their brains lights up as if you had shown them pictures of snakes and spiders. So the myths aren't real, but the anxiety is.

  • Dana Ernst's “Setting the Stage” lesson plan, in which he has students discuss in groups what they think it means to learn, and what they want from their education. It might be different with nontraditional students, but I'm sure there are relevant themes.
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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – quid Jun 25 '18 at 15:41
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As a child, I was subjected to aptitude testing in various settings partly due to my very poor grades in mathematics. What these tests showed was had high aptitude with regard to understanding math concepts but that my skill level with computation was very poor. I think it was something like 30th percentile. I did manage to pass a fair number of college level mathematics courses but I struggled greatly. I hope that I can provide some insight that will assist you in your worthy goal.

Your mention of the whiteboard hits home for me. To this day, when I am faced with any mathematical formula of moderate complexity, I get a anxious feeling in my stomach. I can labor through these if I am familiar with the symbols but it takes a long time and I find it really unpleasant. However, a description of the same formula in text is generally quite easy for me to understand. The difference between the teachers whose classes I enjoyed and the ones I hated was largely related to whether they 'spoke' in symbols. Often a teacher would fill up an entire board with formulas while I was still trying to make sense of the first one.

What I find interesting is that in software development, there's a general agreement that one-letter variables or terse names are a bad practice. But in math a single letter or symbol is used to convey what are often extremely complex concepts or algorithms and practitioners seem fine with that. I think I understand why that is: time and space concerns. So one suggestion would be that when teachers are writing formulas they should describe what they mean out loud as they are written. just writing and saying things like 'and that gives us', 'and that leads to' is extremely frustrating and disheartening to someone who cannot easily read mathematical symbols. I'm not suggesting that you are doing this but I have had many teachers that do this, especially in higher education.

The other big thing that helps me a is using pictures. I took two statistics courses in undergrad and I don't recall being shown a histogram or having anyone even describe what one was. Just lots of nasty-looking formulas. I decided I hated statistics. Then I took a graduate course in statistics and everything was pictures and demonstrations. I loved it and continue to learn more on my own.

From my experiences, it seems that there is a general belief that skill in mathematics means the ability to execute calculations flawlessly and the ability to read and understand obscure symbols. If someone is not good at those two things, they will probably say something like: "I'm not good at math(s)." You are right in questioning this and I applaud you but you need to accept that what may seem like a simple matter of effort to you may be a life-long struggle to others. Make the mystical hieroglyphic symbology of mathematics secondary and the conceptual understanding primary and you may get better results.

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  • $\begingroup$ +1. I UV answers that don't use LaTex, and instead explain the concept using English... Meanwhile, on Creative Writing SE, a question is being asked: why mathematicians 'just can't even'. $\endgroup$ – Mazura Jun 21 '18 at 20:29
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    $\begingroup$ You should check out 3blue1brown's videos. Less formulas, more actual math. $\endgroup$ – heather Jun 21 '18 at 20:50
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    $\begingroup$ "using pictures": This is why I mentioned in my response "Proofs without words" collections. $\endgroup$ – Joseph O'Rourke Jun 22 '18 at 0:12
  • $\begingroup$ @JosephO'Rourke Yes, that is something that I think could be really helpful. I think though that the words aren't the primary issue, it's the focus on arcane symbols which I believe is the primary barrier that limits accessibility to mathematical knowledge. I think the question is about more advanced math but another issue is that there is way too much focus on computation without understanding in childhood education. Computer is no longer an occupation. $\endgroup$ – JimmyJames Jun 22 '18 at 14:34
  • $\begingroup$ This hits home. I bought a book "burn math class" and it's seriously one of the first books that allows me to glimpse into how math works. It starts by explaining, and re-iterating which each symbol means, and why the symbols are used, which eases one into the concept of the symbols, what it stands for. There are so many standard symbols in maths, and the only one I know for sure what it means is PI, the rest would mean looking it up. E=mc2 would be easier to read for me as "Energy equals Mass times Speed of light Squared" $\endgroup$ – Tschallacka Jun 25 '18 at 7:56
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I think that MadScientist described perfectly well the "I just can't do the math" barrier. It is something that most of us recognized in (a subset of) our students sooner or earlier.

The question is: what kind of teaching is effective in breaking this barrier? That's tough.

Why, first of all, the barrier is there? Because doing math is frustrating. In the end your exercise is right or wrong, nothing in between. And if it's wrong it's frustrating. And at times you do not even know how to start. Nothing of this happens if you're studying history or writing an essay on literature or discussing biology. This frustration often happens very early in your life. It is a common experience that many of the "I am not a math guy" will start you talking about bad experiences in primary schools. And, unfortunately, many math teachers at the primary school level are content with landing on you the label "not math guy". So that many (young) adults who are math blind have a long record of bad feelings associated to math: frustration, low self esteem. No one wants to face situation in which he feels like he is not "fit to". Funnily enough, often research mathematicians have more ease in recognizing this feelings than high school professor, since in their research work they have to deal on a daily basis with this kind of frustration...

How can one reverse this? On a very trivial level by associating math also with positive feelings, i.e. building a basis of successful experiences in math. Building situations in which you will feel proud by having accomplished a task (which is, btw, how research mathematicians are able to cope with periods of frustrations: balancing hem with successful moments).

Which is the first obstacle about this? While in sociology or history or literature there is clearly a relation between the effort you put in studying and the results you're obtaining (I'd say almost a linear relation), this is not the case with math. Often our students have no idea on how you study math. They'd like to make an effort but how can they do it? They sit in front of the exercise book: they fail the first, they fail the second, if they're brave enough they fail the third and at this point their level of frustration is high enough for the afternoon. Maybe they'd go for an ice-cream at this point :D.

I often compare studying math to learning to ski. I feel that really a small percentage of people is not capable of learning skiing at a decent level (say a full red slope without falling). But imagine this situation. Learning how to ski is compulsory and everyone has to go to a class where a teacher teaches you the principles of good skiing and then you're left alone with your skis on a slope. After you fell 10 times and hurt yourself a bit you start feeling frustrated. After the 100th fall you decide you're not the ski kind of guy.

  1. We should tell students that frustration is something they will experience in learning math, just like they will feel goofy on their first vacation on snow. There is nothing bad/wrong in this.

  2. We should tell students that mistakes are not something to blame but something to learn from: if most of your mistakes are because of sloppiness you have to work on your concentration skills, while if you make big conceptual ones they will signal you that there is something you've understood incorrectly.

  3. Exercise is important. You would not go skiing without some training.

  4. We should tell them how you do study a definition, a theorem, a whole theory. We all know what it means to give a definition. It is not so clear to our students. Definition= set of rules that identifies uniquely all objects that deserve that name. Have your students try to define what is a chair, or what is a car or even what is a student. Use this to show them how tricky a definition can be. Teach your students how you study the proof of a theorem, like identify at all points in a proof which hypothesis is used, check that all hypothesis are used at least once, signal eventually hidden hypothesis.

  5. Give them a rule on what to do when they feel completely stuck. Like a behavioural psychologist would do with someone in distress.

  6. Teach them to work in groups in an effective manner. Learning from peers some times can put less pressure. It may be easier to admit you do not understand something to a friend than to a professor.

  7. Build confidence. They are math guys. They can learn this. It is not their fault if they've not been able to learn math correctly up to now, it is just that they've not found an effective way of working on it. They will do and succeed.

It is difficult. The obstacle is mainly psychological. As such each individual has to walk through his very specific path to overcome the barrier. If you teach someone on a 1-1 basis it is easier. If you teach a class of 20-30 you can do something. If you teach a class of 60 you'd be happy if you succeed in helping one of them.

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From my experience in professional training (teaching adults things like accounting, economics and statistics, and so unavoidably using some mathematics), it is not math, it is abstract thinking in general. This is the activity with which many people have a problem.

I strongly support the advice given in another answer: suggest to the adult students to treat mathematics as a foreign language, and one moreover written in a foreign alphabet. Add to this warning the complexity that this foreign alphabet includes also some familiar symbols, but with a different meaning and functionality that they have in the language they speak.

This psychologically mitigates the shock of looking at "unintelligible scribble" on the whiteboard because they anticipate it.

And remind them constantly about this "frame of mind" by saying things like "let's now translate in the language of mathematics the following real-world situation", etc. By being able to show them that two different specific situations that they fully understand, have the exact same translation in mathematics, is a concrete step in helping them understand the essence of abstraction.

Not everybody is good at learning a foreign language, but those who are, they have a good chance of understanding that there is nothing special about learning to understand mathematics, and go further. In fact one could argue that mathematics is an easier to learn language compared to human languages, because the rules of grammar and syntax of the former is much more rigid and without exceptions compared to the latter.

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As a mature student in engineering I definitely had a lot of math anxiety at the beginning of my studies. My professor mentioned to us on the first class, and regularly throughout the semester that anyone in the class can do well and that he is there to help us all get the marks we want, be that the 50% required to pass or 100% or anything in between. He also mentioned that when he was a student, he really struggled in his first semester math class and only scored in the 50s. He kept putting the effort in anyway and his marks increased each semester after that. It was very reassuring to have someone be so candid about earlier struggles and to know that you can come out on the other side.

Through a lot of hard work on my part and support from him I managed to get over 100% in both of my math classes last year and have been hired as a tutor by the school for the first year math class.

My recommendations for you, based on my professors attitude and methods as well as everything I did in class:

  1. Be open about your math journey. Was a topic difficult for you? How did you get past it?

  2. Share some of your more "unique" study strategies. For example, I am a tactile learner. I find it really helpful to write out the steps to a math problem on flashcards and then tape them to the floor. I then actually walk myself (think hopscotch) through each of the steps to solving the problem. This helps me focus on what I'm doing at that part of the problem and helps prevent me from skipping steps.

  3. Create a positive space in the classroom, where students can ask and answer questions without worrying about being made fun of or looked down on. My professor was really great at this, and would explain questions 6 or more times if necessary if several different people were getting the answer wrong.

  4. Encourage students to work together in groups and to access any tutoring services available on campus. My school has 3 hours of free (!) peer tutoring that each student is eligible for weekly for every semester they are enrolled. There are also special tutorials for first year math classes. The students who took advantage of these options do an average of 14% better in math classes than the students who don't, according to the school's own data.

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One of the things I find most helpful for teaching adults to math is math history. Math is known for is precision. Everything fits together with the precision of a fine Rolls Royce carburetor. (which, if I might say, are black magic and should be burned at the stake for their heresy against the laws of physics!) Frankly, it's terrifying to grapple with.

I like math history for showing how it was hard for others too. Math did not spring into existence, cleaved from its father's head with a gleaming axe. Real men and women struggled to get it there.

  • Have a student that's having trouble with negative numbers? So did the ancient Greeks. The Greek mathematician Diophantus even argued that an equation equivalent to "4x+20 = 4" was absurd. The Chinese were the first to grapple with the concept, and they had the advantage that Chinese philosophy was amenable to the concept. So yes, it's okay to struggle with them! (taken from wikipedia, source)
  • A student has trouble with irrational numbers? Remember Hippasus. As the story goes, the Greeks were so certain that every number was rational that when he proved that irrational numbers must exist, he was literally thrown overboard. Okay, so maybe if you follow my link, the connections between Hippasus and drowning is a little less extreme. But it does demonstrate that these numbers are infuriating enough that such a drowning story could grow and persist!
  • Having trouble with limits? Zeno had trouble with them too. His famous paradox argued that we could not possibly move anywhere, despite the obvious observational evidence that we do move. His argument dealt with infinite sums. This was not satisfactorily resolved until the invention of limits, which provided a concrete set of tools to manipulate infinite sums in ways which aligned with our observed reality.
  • Admit that we sometimes just make things up, and see if it sticks. This is a very coarse way of phrasing what Michael Stevens does in his VSauce video: How To Count Past Infinity at around the 12 minute mark.

You can also look at the attempts made to grapple with these problems, and see where they fell apart. It's not that math is only for brilliant people, it's that it's really good at letting a thousand people try, and moving forward on the best path any one of them came up with. I, for one, find it fascinating to look at the attempts of Whitehead and Russel to create a self-consistent proof system, only to have Godel demonstrate that the specific task they were trying to undertake could only possibly fail, with the twist of the knife in the form that the system they created would provide the proof that it must fail. (from my understanding, Whitehead later left mathematics behind and got himself well known for his philosophy)

Now about those carburetors...

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  • $\begingroup$ Last semester when I told the Hippasus story in class, the whole room was yawning and looking at the clock. $\endgroup$ – Daniel R. Collins Jun 23 '18 at 18:31
  • $\begingroup$ @DanielR.Collins That's a shame. Did they think irrational numbers were hard, or just not worth their time? $\endgroup$ – Cort Ammon Jun 23 '18 at 19:02
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    $\begingroup$ Cort, I suspect Daniel's students had correctly guessed that the stuff about Hippasus would not be on the test. $\endgroup$ – Dave L Renfro Jun 23 '18 at 19:58
  • $\begingroup$ Case study: I had a student come after failing the final two weeks ago, very upset, claiming he "absolutely knew everything on the test", and it must have been misgraded (score on final ~40%). When I checked the final I saw that this question (identifying which of 5 numbers was irrational), the first on the test, was wrong (among most everything else). I mentioned that to him as an example and he protested, "But that's English, it's not math". $\endgroup$ – Daniel R. Collins Jun 23 '18 at 22:02
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Not an educator but couldn't resist replying.

I always struggled (and still struggle) with math.

A couple of years ago, randomly browsing YouTube, I came across this home made video asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon.

My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.

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    $\begingroup$ Any anecdote like this needs a caveat: The distance to the moon is something that you're essentially interested in, and that may be a minority of people/students. For people who aren't interested in that particular application, the problem becomes worse; now there's a bunch of abstract formula and trigonometry, plus "wasting my time with something about the moon that I'll never have to deal with in real life". $\endgroup$ – Daniel R. Collins Jun 22 '18 at 1:53
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    $\begingroup$ ... I've even had a student in a basic algebra class, who apparently ran their own business, say that computing percentages or tax rates was not useful to know, because he could just hire someone to do that instead. I'd say that most instructors have the experience that, for any application they add to the class, most students dislike it. $\endgroup$ – Daniel R. Collins Jun 22 '18 at 1:54
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    $\begingroup$ @Daniel R. Collins: I experienced this problem a lot, namely thinking students will find something really neat and they don't. Sometimes you get lucky and it goes over well --- in my case, one that worked a lot was folding in half a piece of paper 42 times will get you to the moon --- but a lot of times you don't get lucky. Of course, when you aren't lucky, then you can later turn that seemingly neat idea into a math discussion post! $\endgroup$ – Dave L Renfro Jun 23 '18 at 20:06
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I have experienced this problem, from both sides.

Words won't break the barrier. Action and examples might.

I'd suggest approaching the problem by answering two questions for yourself, then devising a way to approach the issue that implicitly answers those questions: "Why are you teaching this?" "When would I use this?" "Only mathematicians do that stuff, and I'm not a mathematician."

As a child first learning functions, I remember asking my teachers to explain what they were and why. The best answers I ever got were "just because" and "it's a black box that does whatever 'f' is to 'x' to produce 'y'". I was taught to stop asking those questions; math is just to be trusted and memorized, not understood (I doubt I'm alone). But there are good answers to both what and why. An example: the Sum() function in Excel. It takes in a bunch of numbers, and returns a new number. f(x)=Sum(x)=y.

Maybe not everyone appreciates parallels to programming as much as I do, but the point is that there is a tangible reason why a topic is being taught, but so often educators focus on the mechanics absent the motivation, (or the educators don't really understand what they're teaching well enough to actually explain it).

I cringe when people suggest that "I can't do math," just like I cringe when people say "I just can't write well." Neither are things beyond the means of people who want to learn those things, but those people carry baggage of past failures, and people are afraid of failing, so they don't try. They also think that they've been fine thus far without those skills, so why bother now. I think a practical approach can help here. Math isn't magical, and so much of math is practical, especially for the types of courses I understand you to be describing (eg, an engineering student learning boundary value problems vs a physics student studying abstract algebra).

As an aside, if people are scared off by the Greek, then translate the Greek. If capital sigma is scary, then write it as Sum(X for 23 steps). If delta is scary in a derivative, then remove the shorthand: the change in x for each unit of time. In math, we use symbols for clarity and expressiveness and convention. If the "shorthand" impedes learning, then remove it, and ease it back in.

Unfortunately, math is hard to learn because it's harder to teach, not because it's hard.

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While I'm not a maths teacher, I tutored some GCSE students in afterschool sessions a few years ago and I found the easiest way was to show them some other math concept they understood like long division and explain that while they understood it and knew how simple it was, somebody who had never seen that concept before would find it "too complex" the same way they do about things that are ultimately quite simple.

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My experience tutoring algebra 1 students in college led me to believe the best way to get past this issue is to do 1-on-1 sessions with them. I'm a programmer, so diagnosing how a system is malfunctioning is one of my strengths. I would look at the questions they asked, and diagnose what was actually going on in their minds to confuse them, then address those issues on a case-by-case basis.

Emphasize how important it is to write every step down, so that you can help them learn to diagnose their own mistakes once they gain more confidence.

Using this method I was able to get several second- and third-timers pass the class. They expressed how amazed they were that it wasn't nearly as hard as they initially thought.

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  • $\begingroup$ I agree that students in this situation likely need 1-on-1 tutoring to succeed. But for a standard classroom teacher, that's simply not an option. $\endgroup$ – Daniel R. Collins Jun 26 '18 at 13:32
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Just to add something to the already very good existing answers:

When you solve an exercise-like problem in course in front of the students, start with a very intuitive, but wrong approach. Explain your thoughts and why you’re doing what you’re doing, then when it can be seen that the approach fails, explain what went wrong and what to take from it. Then come up with a better approach.

I think doing this could be fruitful, because very often I think that this unbelievably common conception comes from the fact that in class, the professor seems to solve everything instantly and everything is immediately clear to him, and so when they go about solving a problem, they expect the same to happen for them, and when it doesn’t, they are disappointed and give up. They don’t even try. Showing that it is normal, even essential, to just start, and when something is wrong to just restart with a more adapted approach, could led to the students trying more.

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  • $\begingroup$ Disagree. This approach might work (or be of interest) to stronger students...who can tolerate the complexity, confusion. But think this is a bad approach with weak students, struggling to get the material. Pick simple examples and only progressively get more complex. Perhaps occasional remarks about "watch out not to make this mistake" are useful. But not deliberate false paths. This will be confusing for them and waste their time (remember they are struggling already). $\endgroup$ – guest Jun 26 '18 at 8:54
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    $\begingroup$ @guest I see your point and it’s true; I think I implicitly had in mind proper math students (and not say engineering students who have to do maths). However I do think that honestly admitting and showing to the students how difficult some things may be can be beneficial and even motivating if done the right way. Because it shows that there isn’t a dichotomy between those who understand maths and those who don’t, but that everyone has the same kind of struggles, but possibly on different levels. $\endgroup$ – Redundant Aunt Jun 26 '18 at 12:02
  • $\begingroup$ Done the right way, I agree. Even things like making a class joke about "watch out for forgetting the +C". But I wouldn't go down deliberate long wrong paths. You will get (rightfully) killed on the course evals if you do so. $\endgroup$ – guest Jun 27 '18 at 2:46
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Math is a practical, physical skill. Like tackling a football player, doing a backflip, tying a tie, riding a bike, etc. Teach it as such. Practical...repetitive, etc. End, end, end of story. These are NOT natural processes like running, talking, loving or recognizing faces.

Humans are not computers. They are better at certain things than other things. They have a lot of flexibility...so they can perform abstract tasks. But it is not natural. So approach it with that in mind. And DON'T assume that all the kids are budding Ph.D. students destined to teach community college.

P.s. Sure there is more to math than that. But 95%+ of students will benefit from that. For the 5% who need more or suffer from that, they will claw your arm off for something different.

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