I’ll try not to make this post too rambling. I learned to read pretty quick. I cant remember back to a time when all words were just gibberish symbols. I do remember probably kindergarten or first grade sitting in reading circle and phonetically making the sounds of a word I hadn’t seen before while reading a paragraph or sentence. But at some point it became effortless. Reading wasn’t something that was an effort, you just moved your eyes over the page and pictures were created in your mind.
In math, i’m barely literate. I was terrible at it in school. It never clicked in the way reading does. I was able to maintain middle of the road grades but it was through following a specific set of rules. I never had an ‘AHA’ moment in math.
Does such a moment exist? I’ve been told by math savy friends that it appears I never got past the equivalent of having to sound out words and am therefore like a first grader scratching his head at how someone could possibly enjoy reading Proust. Does math ever become something beyond memorizing a set of rules to follow? I would feel sorry for a person that didn’t know how to read. Aside from the day to day difficulties, missing out great novels, histories etc. A huge part of what brings me joy in life.
I worry that my math illiteracy makes me as the person who cant read. There’s all these amazing experiences available if I can just learn the rules.
What I found as I worked my way up to a Bachelor’s was that I’ve pushed back the boundaries of ignorance. There is still a vast sea of mathematics that I don’t understand and probably will never understand, but it’s now a lot harder than the maths I didn’t understand some years ago. Things like applying the quadratic formula, deriving ditto, differential calculus from first principles, proving Pythagoras’s Theorem and so on are now in the “so well understood I don’t have to think about them” camp, or “unconscious competence” as it’s called.
It’s not so much “learning the rules” as building understanding and realizing how many different parts of it fit together, and why. Reading English is stuffed full of arbitrary rules about spelling and pronunciation (even school headteachers can get that one wrong and say “pronounciation”; there’s no such word), and that’s before you get on to simile, metaphor, hyperbole… there’s none of that clutter in mathematics, which isn’t cursed with a dozen different ways to pronounce “-ough”. You can learn some of maths by learning rules by rote, but if you can learn why the quadratic formula is “minus bee plus or minus root bee-squared minus four aye cee, all over two aye” you will never forget or misapply it again.
(aye is not a homonym for eye. Arbitrary pronunciation rules again.)
For me it never did click either. I stayed with my peers in the normal class for that grade but it never, ever, ever clicked. It does for some though. I didn’t notice till junior year in highschool when I was taking…hmmm I think back then it was advanced algebra, gemotry then what they called algebra 2, but a lot of friends were even past the regular classes and getting college credit for the classes they were taking. I think regular in my school at the time just went up to calculus. I was always a bit jealous but everyone has thier weaknesses.
Dosen’t matter in the long run. When I have time I may try again but higher math dosen’t come into play ever unless you are in a field that requires it. I would like to understand it and would be interested now as opposed to when I was in school.
I would say it’s more accurate to state that high level reading becomes more like math. As you move away from the realm of recreational reading into dense texts, even well written ones, that are trying to communicate complicated topics, reading becomes equally as bogged down and a hard slog.
The commonality between both is that they both are trying to communicate something interesting and novel and thus, are necessarily going to be hard. Math is much more terse than English though so a given page of Math tends to pack in more ideas.
But yes, once you’re familiar with high level math, it’s pretty easy to breeze through what a layman would think of as intimidating math because it’s actually relatively simple concepts.
I wonder if OP is also following the nearby current thread:
where we are also talking about the teaching and learning of math.
Although the thread is somewhat rambling, it revolves around a discussion of what topics should be taught, and how they should be taught. There differing opinions about the role of “rote” learning vs. the importance of “deep understanding”.
It does seem that some people’s minds are much more “wired” for picking up on advanced math than others’ – But I am convinced that the skill of the teachers and the textbooks makes a big difference too. I am sure that most of us would agree with that. The way you are introduced to math, especially in the earlier grades, I am sure, can make a big difference in how math-phobic or math-philic you are.
Myself, I tend to be rather “visual” about it. My 2nd (or 3rd?) grade teacher taught multiplication by drawing pictures of rectangles, with n cells across by m cells down. But when the time came to learn “the five fundamental laws” (3rd or 4th grade?), that same diagram was shown to demonstrate why multiplication is commutative.
In that other thread, I discussed “the Grid method” of multiplying polynomials, which I find to be very visually intuitive, as it shows a picture that you can actually see with your eyes, that illustrates the Distributive law. There was also a link of a diagram that shows a visual demonstration of the Pythagorean Theorem, and another link to a drawing that shows how the formula for the area of a circle can be imagined. My 7th grade math teacher (who really turned me on to math) showed us a lot of those same pictures, and they are ingrained into my math mind to this day.
A friend once posed this problem for me to think about: Prove that if the expression (2[sup]n[/sup] - 1) is a prime number, then the exponent n must be prime too. I thought about it, and came up with a proof in my head in about 30 seconds – and it entailed an almost-geometric visual image, and essentially NO algebra!
In studying more advanced college-level (lower division) math, I have sometimes had a hard time with a complicated formulation, if I couldn’t develop some kind of visual model of it in my mind. The process called “convolution” comes to mind, which is utterly arcane and hard to visualize (I think). Although, with practice, one gets better at working with math entirely by manipulating abstract algebraic expressions.
It sounds like you’ve learned what one of my early teachers used to call “monkey math”. Monkey math is just monkey see, monkey do. There’s no understanding. You just memorize formulas and when you see something you recognize you put the formula in that you memorized. It’s a very difficult way to learn math, and if you learn math this way you will never be good at it.
Math isn’t about rules. It’s about concepts. If you learn the concepts, they can help you remember the rules.
To take a really simple and stupid example, consider 4 x 3. Monkey math is you memorize your times tables, and you just remember that 4 x 3 = 12. But to actually understand it, it’s four things, three times. For example, you could have a group of four people, three times. If you forget your times table and all you learned was monkey math, you’re done. You don’t know the answer. But if you understand the concept and you forget your times table, you realize that you are just taking four things three times, so four once is four, plus a second group of four is eight, plus a third group of four is twelve. Because you actually understand what you are doing, you can derive the monkey math formula that you forgot.
That’s a really stupid example of course, but the same principle applies to more complicated math. If you learn calculus, you usually spend the first three days or so with the fundamental theorem of calculus. At the end of those three days, you don’t have a fricken clue what you are doing. It’s just this god-awful mess of equations and limits and weird math symbols. Calculus was invented for a reason. Some things change relative to the rate of change of other things, and some things accumulate over time. Integrals are just a way of summing up totals and derivatives are just a way of figuring out the rate of change. In simple geometry, an integral is the area under a curve and a derivative is the slope of a curve. It’s actually pretty simple, but I’ve never seen anyone teach it that way. A lot of times if you try to imagine geometrically what you are doing with formulas in calculus that will give you enough of an understanding that you reach your “aha” moment and aren’t just slugging through the formulas. Once you understand it, it’s much easier to figure out which formula you need to solve what you are actually doing. If you never understand it, you’re just overwhelmed with hundreds of formulas.
The key to being good at math is to always understand the concept first. Then memorizing the formula becomes easier. Understand what you are doing and why. If you can do the formula parts but can’t do the word problems, you’re doing monkey math and you haven’t actually learned what you are doing. When you can apply the concept to the word problems, then you understand what you are doing.
Have you looked at engineering textbooks as well as mathematics texts? My books always showed pictures for convolution, and I thought of a picture as soon as I read that word.
BTW, I know that “learning styles” is considered passe now, but when I studied teaching, I was with another student who always thought of how to act-out theorems. He was studying to be a dance teacher.
I have been trying to tach myself math at the khan academy on line. I think I am doing real well as I just buzz right through the lessons and advance at a very rapid rate, but then I come back after a little break for a few weeks and forget everything I learned. I have always been good with work problems and what I call mechanical and building math. I do terrible when I have to start memorizing symbols when I really don’t totally understnd or comprehend what they mean.
This sure sounds like relevant stuff for that other current math thread too. I sure do remember being taught multiplication as “repeated addition”, e.g., 4 x 3 = 4 + 4 + 4 (or maybe 3 + 3 + 3 + 3), along with a picture of a 4 x 3 rectangle alongside a 3 x 4 rectangle, to illustrate why 4 x 3 = 3 x 4. ETA: And yes, we still had to memorize the times tables too. (My father helped, by drilling me with flash cards. Does anybody ever still do THAT any more?)
And we’ve been talking about how the teaching has been changing over the years. Your description of your first 3 days of calculus bears NO RECOGNIZABLE RESEMBLANCE to any part of my calculus education. My calculus I class spent the first month or two just learning about limits and differentiation, with occasional mention of “anti-derivatives” (i.e., given the derivative of a function, find the original function). THEN integration was introduced, using the sum of rectangles pictures to find area under a curve. THEN it was shown that differentiation and integration are none other than inverses of each other, and the “fundamental theorem of calculus” is brought up, now that it’s relevant – note, this is about mid-way through the semester!
OTOH, my class, while doing derivatives, we learned about the epsilon-delta definition of limits, and even did a few simple e-d proofs. I’m told that this is no longer taught at all until more advanced (upper division) classes. If that is true, then color me disgusted.
So: I’m interested in knowing WHEN you had a class that worked the way you describe? I got mine, circa 1982.
Understanding the concept first, vs rote rules, is the primary topic of that other thread, where we started by debating the merits of learning the FOIL rule.
No, this is news to me (about convolution). Thus inspired, I just did a Google image search for convolution. No time to study them now, except to note: There seem to be a LOT of illustrations out there. Now I want to look at some of them and see if they are talking about the same kind of convolution that I kinda-sorta-half-remember (having studied that stuff 10-some years ago).
I wonder how many math rules or theorems could be illustrated well by dancing them? That’s an interesting thought. Definitely, there is plenty (especially when you get into calculus) that can be illustrated by animated videos, as opposed to still pictures. Even those “Uniform Motion Problems” we did in Algebra I might be well illustrated by moving pictures, say of trains colliding head-on
What the hell are we all doing up at 3 in the morning, talking about this stuff?
The thing that comes up too rarely is that we practice reading, even subconsciously, all day, every day. We’re surrounded by words and need to read at a low level to adequately function in society.
We’re not surrounded by high level reading, though. Hence, the need to write news articles to a 6th grader level.
We’re not surrounded by anything like even low level math on a continuous, daily basis. There’s just no opportunities to practice.
No wonder we think one is fundamentally easier for humans when it’s just a matter of orders of magnitude more exposure.
No doubt there’s some level of innate ability, but people seemed to be tripped up at low level arithmetic or basic algebra, not advanced math.
There was a blog article in the Atlantic recently that discusses this issue very well. The gist of the post is that, at least for low level high school math, a combination of hard work and more exposure pays off massively. Letting people believe they’re innately bad at math is incredibly damaging.
And it’s something we’re all clearly more familiar with at high level math. At least for higher maths, there’s no doubt people need to work hard and practice at it, so nobody bats an eye. But because of the difference in exposures, I guess it doesn’t cross people’s minds that even for lower level maths, you still need either repeated daily incidental exposure or prolonged and deliberate practice.
No, learning styles are still a recognized, accepted, and valued concept. Your friend would be what’s called a kinesthetic learner. And I’d be very curious to know what “dances” he came up with for various theorems: It’s always good for a teacher to have more teaching methods available.
It certainly should. Hardly any of the rules in math are arbitrary; they’re things you should be able to figure out for yourself, or at least to see the logic of once it’s been explained to you. Asking whether math ever becomes something beyond memorizing a set of rules to follow may be a bit like a tone-deaf person asking a concert pianist or orchestra conductor whether music ever becomes something beyond memorizing a set of notes to play.
When my daughter took calculus in high school last year they did a similar thing, focusing on limits and formulas without giving them any understanding at all of what they were doing. (ETA - southeastern PA)
Interestingly, the academic tutor on my last PGCE placement was adamant that there is no such thing as a “kinesthetic learner”, and that the study that purported to show there was such a thing had been comprehensively debunked. Doesn’t rule out the possibility of kinesthetic learning, of course, only to deny that there is a whole class of people for whom this is the preferred, even only effective, model of learning. But I’m more into math than pedagogy, so take that for what it’s worth.