Why Do Some People Struggle with Math?

[dkotschessaa]

If math were just the grade-school procedures for performing the basic arithmetic operations, it would be one of the most tedious subjects around.

But there's another, very different mathematics, and that's what excites mathematicians: Math as the study of pattern. Pretty well the earliest-studied patterns were geometry, number, and music.

Which is why I wish we would teach number theory and graph theory or some related topics from the start. Just turn the whole thing into a game. Forget about relating it to the "real world." The ability to do arithmetic is actually not a useful real world skill. The ability to recognize the underlying patterns behind things (like arithmetic operations) and to reason is a very useful ability in many aspects of life.

-Dave K

 

[fresh_42]

I was a musician (classical and jazz guitarist) for years before I went back to school for math. So yeah, possibly there is a certain thing about abstraction here.

Many mathematicians are musicians, as well. An interesting contribution can be found in the books of G. Mazzola:

 

[UsableThought]

Just turn the whole thing into a game. Forget about relating it to the "real world." The ability to do arithmetic is actually not a useful real world skill. The ability to recognize the underlying patterns behind things (like arithmetic operations) and to reason is a very useful ability in many aspects of life.

This is exactly the position taken by the guy who wrote A Mathematician's Lament, an essay and later a book by Paul Lockhart, a math teacher at St. Ann's School, a K-12 school in Brooklyn Heights, NY. I assume most here know of at least the essay, given that it was publicized back in 2008 by Keith Devlin in his MMA column. The essay is here as a PDF and the book is available on Amazon and elsewhere. Book & essay attack traditional teaching along the lines given by @dkotschessaa, and suggest play as a far more suitable approach, e.g. treating math more like art class than history or science. He also has a more recent book Measurement, which is a self-teaching guide for adolescents or older; the reader is invited to "play" with math via geometric patterns such as symmetry, rotation, etc. I started that book but found it a bit daunting, plus it doesn't meet my current goals (re-learning high school algebra) so I put it aside.

In response to a couple of comments saying that learning mere calculation (as we are supposed to in grade school & secondary school) will never be anything but tedious - I disagree; I think it depends on your circumstances and attitude. If you are re-teaching yourself high school algebra, as I am for example, you can go at your own pace; and you can concentrate on those things you find interesting. This is similar to what I've read about the concept of "flow" as espoused by psychologist Mihaly Csikszentmihalyi: almost activity can support an enjoyable state of flow so long as the person doing it able to take charge of how they do it, can set their own goals & receive immediate feedback, and engage in it as if it were a game.

I will add that in my case, my ability to guide my own learning is probably much greater as an adult than it was when I was very young; and although I no longer have the full measure of wonder that everyone misses from childhood, I can understand certain difficult subjects better now than I could then. Also, I have been heavily influenced by a MOOC I took early on, Keith Devlin's Introduction to Mathematical Thinking, that taught predicate logic and simple proofs; quite a few of the proofs involved number theory. So now when I do my simple algebra problems in Gelfand and Shen, or Brown et al, I often go beyond the problem as stated and do a proof; and also look for interesting patterns. It's actually good that I've been so bad at math most of my life, because now even high school algebra is rich territory for me!

 

[Hendrik Boom]

If you're relearning algebra, look for patterns in the formulas.

 

[UsableThought]

If you're relearning algebra, look for patterns in the formulas.

Someone said that already . . . oops, it was me:

So now when I do my simple algebra problems in Gelfand and Shen, or Brown et al, I often go beyond the problem as stated and do a proof; and also look for interesting patterns.

 

[dkotschessaa]

This is exactly the position taken by the guy who wrote A Mathematician's Lament, an essay and later a book by Paul Lockhart, a math teacher at St. Ann's School, a K-12 school in Brooklyn Heights, NY. I assume most here know of at least the essay, given that it was publicized back in 2008 by Keith Devlin in his MMA column. The essay is here as a PDF and the book is available on Amazon and elsewhere. Book & essay attack traditional teaching along the lines given by @dkotschessaa, and suggest play as a far more suitable approach, e.g. treating math more like art class than history or science. He also has a more recent book Measurement, which is a self-teaching guide for adolescents or older; the reader is invited to "play" with math via geometric patterns such as symmetry, rotation, etc. I started that book but found it a bit daunting, plus it doesn't meet my current goals (re-learning high school algebra) so I put it aside.

In response to a couple of comments saying that learning mere calculation (as we are supposed to in grade school & secondary school) will never be anything but tedious - I disagree; I think it depends on your circumstances and attitude. If you are re-teaching yourself high school algebra, as I am for example, you can go at your own pace; and you can concentrate on those things you find interesting. This is similar to what I've read about the concept of "flow" as espoused by psychologist Mihaly Csikszentmihalyi: almost activity can support an enjoyable state of flow so long as the person doing it able to take charge of how they do it, can set their own goals & receive immediate feedback, and engage in it as if it were a game.

I admit I kind of keep wanting to go back to some of those "mental math" books and learn all manner of tricks for arithmetic. But mainly my goal is a kind of brain training, and secondly I think the tricks use some neat "number theoretical" types of ideas to get the answer. I don't think arithmetic is that useful as a skill anymore but it would be fun.

I will add that in my case, my ability to guide my own learning is probably much greater as an adult than it was when I was very young; and although I no longer have the full measure of wonder that everyone misses from childhood, I can understand certain difficult subjects better now than I could then. Also, I have been heavily influenced by a MOOC I took early on, Keith Devlin's Introduction to Mathematical Thinking, that taught predicate logic and simple proofs; quite a few of the proofs involved number theory. So now when I do my simple algebra problems in Gelfand and Shen, or Brown et al, I often go beyond the problem as stated and do a proof; and also look for interesting patterns. It's actually good that I've been so bad at math most of my life, because now even high school algebra is rich territory for me!

Oh, I think algebra (high school level) is beautiful stuff. It is really ones first exposure to mathematical thinking since it requires some kind of manipulation of objects, and there is more than one way to get from one place to the other. There's also all manner of "tricks." Even last year as a master's student I watched a fellow TA teach an algebra class and learned new ways to factor.

-Dave K

 

[RogueOne]

In, fact, I can say from personal experience, that "most" things do.

We had words/names for people that couldn't pay attention, before we labeled it ADHD/ADD. Silly-Nilly, Wiggle-Wort, daydreamer, wandering mind, undisciplined, stupid, mule-headed, stubborn, dim...and many, many more. I was on the receiving end of them, more than I could ever hope to count.

It was the teachers that knew how to reach through the ADD fog I lived in, and shine some light on the subject at hand that I remember and appreciate the most. Believe it or not, my high school Physics Teacher, who gave me a D+, will always be my favorite. That man just knew how to reach me, and while I could not remember formulas to do well on his tests, I learned an awful lot from him. His lessons on vectors improved my pool game, too.

You know, I do appreciate my highs school physics teacher. My experience with him is very similar to your experience with your physics teacher. I did well in the class, but that teacher was one who gave me several paradigm shifts in the way that I looked at math and science. Physics teachers and teachers with engineering backgrounds were always able to teach me very well. Thank god for those guys.

 

[UsableThought]

I don't think arithmetic is that useful as a skill anymore but it would be fun.

Speaking of arithmetic: One of the first things I discovered upon starting my review of algebra was that my ability to subtract when "borrowing" was required - especially with 3-digit or longer numbers - was terrible. I found this out because Gelfand and Shen's little book Algebra (which I recommend as a fun romp) begins by reviewing certain basics, including long division (with problems involving some neat patterns of repeating digits), as well as both subtraction and division in binary. All three of these required subtraction; and I was making too many mistakes to be sure of my answers, especially when it came to binary.

Whatever rote procedure I had learned as a kid in grade school, it apparently hadn't been needed for 40-plus years of adulthood; and so I no longer could remember it well enough to use it. Addition I still remembered, mostly because every few months I would find myself writing up a deposit slip at the bank for multiple checks; but subtraction of 3-digit or bigger numbers? Apparently not required to be a typical functioning U.S. citizen.

So I set out to relearn subtraction. This was quite interesting, because as an adult working on my own time, versus a little kid crammed into in a littler desk with a tyrannical cheek-pinching teacher to deal with (that's another story), I could make relearning a game. The single best source I found was Wikipedia's article "Subtraction by hand"; this was where I learned that my reliance on "borrowing" was a clue that I had been taught "the American method” of subtraction, as opposed to other methods such as “Austrian," "left to right," "partial differences," or partitioning & other non-vertical approaches. I got to try some of these alternative methods; the one that seemed most like a neat mental trick was partitioning, which is taught extensively in a little book meant for British parents, Maths for Mums & Dads.

What was most interesting of all was to see for myself that an understanding of "borrowing" depends on a good understanding of the radix or base you're working in. So that right there is a sort of "number theoretical" type of idea that I'm guessing I was never taught in grade school.

 

[dkotschessaa]

Speaking of arithmetic: One of the first things I discovered upon starting my review of algebra was that my ability to subtract when "borrowing" was required - especially with 3-digit or longer numbers - was terrible. I found this out because Gelfand and Shen's little book Algebra (which I recommend as a fun romp) begins by reviewing certain basics, including long division (with problems involving some neat patterns of repeating digits), as well as both subtraction and division in binary. All three of these required subtraction; and I was making too many mistakes to be sure of my answers, especially when it came to binary.

Whatever rote procedure I had learned as a kid in grade school, it apparently hadn't been needed for 40-plus years of adulthood; and so I no longer could remember it well enough to use it. Addition I still remembered, mostly because every few months I would find myself writing up a deposit slip at the bank for multiple checks; but subtraction of 3-digit or bigger numbers? Apparently not required to be a typical functioning U.S. citizen.

So I set out to relearn subtraction. This was quite interesting, because as an adult working on my own time, versus a little kid crammed into in a littler desk with a tyrannical cheek-pinching teacher to deal with (that's another story), I could make relearning a game. The single best source I found was Wikipedia's article "Subtraction by hand"; this was where I learned that my reliance on "borrowing" was a clue that I had been taught "the American method” of subtraction, as opposed to other methods such as “Austrian," "left to right," "partial differences," or partitioning & other non-vertical approaches. I got to try some of these alternative methods; the one that seemed most like a neat mental trick was partitioning, which is taught extensively in a little book meant for British parents, Maths for Mums & Dads.

What was most interesting of all was to see for myself that an understanding of "borrowing" depends on a good understanding of the radix or base you're working in. So that right there is a sort of "number theoretical" type of idea that I'm guessing I was never taught in grade school.

Yes, I've noticed similar things related to arithmetic too, when regarding it as a kind of number theory or something abstracted up a bit higher than just following some prescribed steps. Some only work in special cases. Which I guess that is where all those mental math tricks come from, since they tend to be for certain situations.

 

[PetSounds]

When subtracting, I always think in terms of addition. For example, when evaluating 91 - 42, I never think, "Take 42 away from 91 to get 49." Instead, I think, "What do I need to add to 42 to get to 91? 49." It's interesting to discover all the shortcuts others use. Reminds me of a Feynman anecdote describing how, when people count mentally, some "hear" the numbers and others "see" the numbers.

 

[Logical Dog]

this is normal as your teachers are probably not very good, i wouldn't worry about it as maths is a very strange subject indeed, just learn as well as you can and get some philosophical outlook on it (you may search for books or internet). Usually pure math courses in school tend to be the easiest, physics and science are always much tougher generally(and in university too), but it depends on every person.

Learn every definition, theorem and try to see and understand the proof of the theorems. and procedures. for example factorization rules stem from axioms of real number such as the distributive of multiplication over addition etc. Your teacher in high school may not be able to provide you these, look for them by yourself and remember them and life gets much easier later on. Also don't believe you cannot learn math or anything because of "ADHD"...question everything (who told you you have this? is it a proper diagnosis? which even then is not saying much, when we are young we tend to have a lot of energy anyway) , read more philosophy and stay calm, set small goals and don't overstudy maths. Perhaps it could be that you are extremely bored in normal school too, and need something more challenging.

 

[chardyoss]

You should try to see math as a game. You will have a goal by resolving problems and increasing your level. Maybe your motivation can come like that.

 

[Hendrik Boom]

"As a game" is the only way I've found to approach math. Otherwise the easier stuff gets too boring to hold my attention and the harder stuff too frustrating. But as a whole set of challenges there are always a few I can figure out. Finding those is a good route to learning.

In a video game, the challenges are often graduated, with tougher and tougher bosses. Math textbooks should be the same, but often aren't. One technique I've found useful is, when a chapter gets too hard, skip to the next! You won't get far into the next, but you'll get a glimpse into where the story is going. My wife tokd me her technique studying linear algebra was to read an entire chapter whether it made sense or not, and then reread it the next day. The next day it made more sense.

 

[theb2]

Take this for what it’s worth. Your gut will tell you if this is contributing to ur problem. I grew up in a house hold with video games and television. The tv is always on, even when I go home to visit my mom she still keeps that machine running lol. Long story short I struggled really bad with focusing on math, i was just so bored and felt like the material was pointless. This is calc3 I’m learning at the time. Anyways, I noticed I was listening to music a lot while studying and on breaks I’d surf social media. It was my way of relieving the anxiety but the problem was it made the math harder to focus in the medium to long term. Short term it worked like a charm. This is becoming a tangent my bad. So what I decided to do was take away all low effort dopamine release. It took a while to fully get rid of my social media and not listen to music or watch any dumb YouTube videos lol but now my ability I concentrate is a lot higher and I don’t have that bored feeling while studying. If anything I get it when I’m not doing anything. I dono, I think technology of today is very addictive. That it makes math or learning very boring in comparison. Seems to work for me. :)

 

[gibberingmouther]

this is an interesting thread that a lot of people seemed to have had something to add to. what i took from it was that peoples' learning styles differ in interesting ways.

i've known some very intelligent seeming people who claimed to have test anxiety - that their mind would go blank or something along those lines. I'm an example of the opposite as i am and have always been an adrenaline junkie. the anxiety i would get when i had to perform would make me do better rather than worse. having test anxiety doesn't make you stupid, i guess it's just something you have to learn to live with.

anyway as far as being a kid learning about math, i agree with what several other posters said. if you can learn about the connections of different parts of math to each other, and see how they can be applied to real world science or technology, that could really increase your motivation. that worked for me, definitely. on the other hand, you could treat it as a game or project to become good at something. in college, i'll say, being good at math helps with the opposite sex ... something to work towards!

 

[waternohitter]

I hate geometry but I love algebra. I hate chemistry but I love physics.

 

[Hendrik Boom]

I can sympathies with you regarding your viewpoint on calculus :-)

Unfortunately calculus carries a lot of weight in exams and that affected me in a negative way. I have always been fascinated by calculus though. Differentiation is easy, but integration is like a puzzle. I am slowly getting back at it.

Exactly right. Symbolic integration *is* a puzzle. Lots of puzzles. Treat them like that, see how many you can figure out. The ones posed in textbooks usually have solutions. The ones in real life can sometimes only be solved by numerical approximation.

 

[Drakkith]

I enjoy geometry and abstract scientific problems, but struggle in areas such as basic equations and terms. Everything just seems like it flies over my head, and despite people praising me as a good thinker and philosopher, I feel deep envy for people who understand math and people who can be in high-level classes, because I find physics fascinating (even though it takes mathematical prowess). I always panic when confronted with math homework, and my mind goes blank.
I feel like I need motivation to do better.

I think part of the problem is that you and most other people are not viewing math as a coherent whole, but as bits and pieces that you don't realize add together to make mathematics a truly beautiful subject. Imagine if instead of teaching math in school, we taught golf. What you're doing now is being taught how to use different clubs without really being taught how the whole game works. Instead of playing the game, you're just driving golf balls over and over in different ways. Some of you are good with certain clubs, others are good with different clubs. A small number may be good with all of the clubs. But you're still just doing the same thing over and over again, with no end in sight. Of course you'd be frustrated and confused with it! When are you ever going to drive golf balls in real life?! What's the point of this anyways?!

Unfortunately, unlike golf, you can't 'play the game' in math without mastering the basics. It's is literally impossible. If you don't know how to swing a golf club particularly well, so what? You just hit the ball a few more times until it goes in the hole. But in math if you don't know how to work with fractions, solve equations, or use some other basic concept, you're done. That's it. It's not possible to continue and eventually be successful in a higher level math course without understanding the basics.

Now, I'm sure that sounds pretty negative. But I don't want to discourage you or make you feel worse. Just like golf, you must practice to be good at math. The reason that math is so terribly difficult for most people is because they don't practice it. It would be like being forced to take play golf as a test every few weeks when all you do to practice is to go hit a few balls around every day in a half-hearted manner. You'd be terrible! That's not how you get good at golf!

I can tell you from personal experience that doing even 15-30 minutes of extra math every day, beyond your homework, can do wonders for you. I've been a math tutor before and the main reason that students do better after being tutored isn't because the tutor is some genius who can explain things in a way that you just 'get' it. It's because the student put in extra work. That's it. That's the main reason in my opinion. There are certainly other things that help, such as being able to have questions answered in a one-on-one manner, but in my opinion the main reason is that the student is putting in extra time to develop their math skills.

Finally, don't expect to be good at math and don't feel bad if you're struggling to understand something. If you look at any sports star, I can almost guarantee you that they had something that they had to work extra hard at to master. Some were even turned down because they were doing poorly and they had to go back and work extra hard to get good enough to eventually make it. The same is true for people with almost any skill set.

Remember, you don't have to be a genius at math to be successful. Hard work and persistence will get you far in life.

 

[symbolipoint]

I hate geometry but I love algebra. I hate chemistry but I love physics.

Each pair is a pair of two related fields. Algebra helps with Geometry and also Geometry helps with Algebra. They are not absolutely unrelated. Next pair: Chemistry relies on and benefits from Physics. Some people might say that Physics benefits from Chemistry but I would not say that. There is some overlap of skills and at least a little overlap in some technologies.

 

[Ucha]

I find that we don't like or fear the unknown

 

[symbolipoint]

One answer to the title topic is the time restrictions of the courses and the need to earn a grade.

 

[symbolipoint]

Hate Mathematics? Or just learn too slow?

 

[Kevin the Crackpot]

I hate to bring this up, but it may be relevant to your math issues.

How are your parents and family when it comes to homework and learning?

I have Asperger's Syndrome (a kind of high-functioning autism), and schoolwork was difficult because my family made it difficult.

There was a lot of fear attached to schoolwork because of the constant threat of punishment, screaming, yelling, and so on because I'm "worthless" and will "never amount to anything" and so on . . . which is how my learning disabilities were treated. My mother even used to tell me that she wouldn't love me anymore unless I did my times tables perfectly without any mistakes.

If your parents (with the best intentions in the world) do things like this to you and/or have expectations of perfection when you're trying to learn when you have ADD, then it's not surprising that you may have problems with math.

A good way for some people with ADD to study is to lay out your math homework on a table, and next to it your history homework, and then your English homework in a row.

Then, work on math for five minutes, then move to history for five minutes, then English, and then back to math (and so on).

If your parents are authoritarian, they may have a problem with this ie: " . . . the real world doesn't work this way, and you have to do it the right way!" and so on.

I don't have an answer.

With the understanding that my ethics and morality aren't based on conveinence . . . I tend to be biased in favor of pragmatism and doing what works. I am--after all--a paramedic.

 

[Auto-Didact]

Background: I live in the U.S. and I am in 7th grade, 13 years old. I have mild ADHD
I don't really hate math, per se, but I can't help thinking that I'm unfit to do anything related to mathematics. I enjoy geometry and abstract scientific problems, but struggle in areas such as basic equations and terms. Everything just seems like it flies over my head, and despite people praising me as a good thinker and philosopher, I feel deep envy for people who understand math and people who can be in high-level classes, because I find physics fascinating (even though it takes mathematical prowess). I always panic when confronted with math homework, and my mind goes blank.
I feel like I need motivation to do better.

How you describe yourself and your dislike of math, you sound a lot like my younger self... I think you may have just had bad (or merely uninspiring) teachers. I myself definitely had mostly bad ones as a teenager. Incidentally, when I was 13, outside of geometry, I sucked at math, especially algebra, which made me not like the subject either.

I was simply never interested in doing the problems, only in learning and thinking creatively about the concepts. In fact, the only 'good' math teacher I ever had growing up, was also when I was 13; he also liked to talk about concepts and applications, but his main focus was having us finish the problems and pass the tests.

In fact, he knew how bad I was at subject, even though it interested me conceptually, yet he almost single-handedly managed to kill my interest in the subject in the following way: he turned me down in a creative moment, in which I had finally also mustered up the courage to walk up to him and ask him to give me his opinion on something quite curious I had seemed to have discovered.

You see, instead of doing the problems in class (graphing functions), I was trying to understand the properties of different kinds of graphs. Geometrically, I rediscovered something about points in such graphs (which later I would learn was called 'a derivative at a point'); the teacher looked at my work for a bit, remained silent and then told me "forget about it... just focus on finishing the homework problems".

This response practically killed my interest, and as a result I actually failed math class that year. Actually failing math bothered me so much (I had never failed anything before), that the next school year, at 14 years old, I forced myself to confront the subject head on and kick the crap out of the subject by finishing all the assignments as quickly as possible after we started a new chapter in class; usually this took a week or two to do.

As a consequence I had so much free time over in math class that I had extra time to think about math conceptually, eventually even philosophically from first principles, purely from my own thoughts (NB: this subject is called 'foundations of mathematics'). My math grades also started to improve slowly from failing grades to passable levels.

However after about 6 months of this, something clicked in my mind: I had a revelation, it was as if the floodgates had opened and suddenly I understood almost everything that was thrown at me mathematically. My math grades jumped from average to exceptional; curiously, simultaneously almost all my other grades dropped from exceptional to very good.

To make a long story short, at the end of high school my abilities in math made me fall in love with physics (read about it here if I haven't bored you to tears yet). To make a long story short, I went to university in order to study medicine and become a medical doctor, but before long ended up getting a degree in physics as well.

So, young man, my message to you is to never give up the hope and don't be afraid or cautious, neither in the face of difficult math or of having to handle failure! Endure and who knows, you might even end up getting rewarded in ways that you cannot even imagine yet.

 

[symbolipoint]

Auto-Didact,
Interesting post, #55.

Sometimes a teacher takes the wrong way to understand how to teach the subject and what to promote and what not to promote. Algebra in high school saved me, but not before strong discouragement about supposed pre-requisite knowledge from an inexperienced teacher.

Any "Mathematics" student, whether he hates it or not, NEEDS to have concepts AND skills (application of the concepts for practical or potentially life-like purposes).

 

[Auto-Didact]

Indeed, of course one also needs to put some work in, but I think that there is more wrong than just that kids aren't putting in the work. I seriously think math needs to be presented in a fashion more relatable to those not necessarily good at algebra, in a way that captures their imagination without it becoming dull. Here is what Edward Frenkel says about this.

I tutored math for quite a while; usually the kids (and adults) don't understand some practical analytic/algebraic aspect, like division, especially that fractions can be seen as equivalence classes, or what they are doing in trigonometry, or why they are doing something in algebra and they often end up getting stuck.

After they fail, they get discouraged; and bad teachers or tutors aren't able to get through to them because they essentially approach the learner in an ineffective manner. The learners then tend to end up avoiding the subject as much as possible for the rest of their careers.

This clearly seems to be the case to me, since I have sadly heard this tragic ending repeated endlessly from friends, kids, students, patients and other people (including scientists, physicians and artists) who are clearly very much interested in math conceptually, not just at a pop level either.

The same thing is for example true with physics: when people like Walter Levin and Richard Feynman speak, everyone listens on the edge of their seats. The same thing applies to Roger Penrose to some degree but he then tends to very quickly go way too far above their head, while most others (Michio Kaku, Brian Greene, Brian Cox et al.) quickly devolve into pop sci.

Interestingly, Penrose of all people was also literally very slow at algebra as a kid (but good at geometry); his teacher however was good and noticed this and therefore gave him more time to solve the problems. He speaks about this subject at length in one of his interviews posted here, the Heidelberg one I think.

I would also like to mention that Henri Poincaré, mathematician extraordinaire, wrote about this topic at length in his masterpiece, The Foundations of Science, saying that there seemed to be (at least) two type of mathematical thinkers: geometers and analysts. To close, here is Feynman weighing in on this very subject: