A good starting point for this discussion might be this quotation from the late Israel M. Gelfand, a pioneer of writing correspondence course materials in secondary school mathematics in both Russian and English: "Students have no shortcomings, they have only peculiarities. The job of a teacher is to turn these peculiarities into advantages."
Thus far the Stack Exchange discussion includes a lot of complacency along the lines of "some students just don't have the capacity to get it" that I never found when I was living in Taiwan. Over there, even the below-average students in seventh grade are all expected to learn quite a lot of algebra and geometry, which they consolidate knowledge of during junior high school. By the end of junior high, students in Taiwan and students in urban areas of China and students in Korea, Japan, Singapore, and Hong Kong generally know United States secondary school mathematics quite well. (I have heard Chinese graduate students in the United States, students in nonquantitative subjects, deride the mathematics section of the GRE general test used for graduate school admission as a test of "junior high math," and that is literally true in terms of the standard curriculum in China.)
So how about it? What creative ideas do we have here to help the young learner learn how to solve proportions, an aspect of day-by-day reasoning that only secondary school graduate might reasonably be expected to know? How to reignite a spark of interest in mathematics after school lessons that may have been a turn-off, may have been flat wrong,
I've taken the GRE once in India, and I can confirm that the math section was RIDICULOUSLY easy. I could ace it without breaking a sweat, and I'm (was) actually a fairly average student.
> (I have heard Chinese graduate students in the United States, students in nonquantitative subjects, deride the mathematics section of the GRE general test used for graduate school admission as a test of "junior high math," and that is literally true in terms of the standard curriculum in China.)
this is true for students educated in the US as well! it's super frustrating, because you're staring at trig identities that you last looked at ten years ago.
Agreed. I always assumed the GRE must have been designed to accommodate/calibrated to humanities and social science grad students (or other majors without a significant math component) more than STEM students. Almost every CS friend I had either scored a perfect score or missed only one. It was a little frustrating, as there's no room to differentiate students who are average from those who are outstanding.
Same story here. I was actually a math major and took the general GRE (which I believe is what we are discussing) in my senior year. I only studied the night before, just to see if their prep guide included any funky vocab or other memorization questions. I looked at some vocab, spent 10 minutes reviewing geometry and trig identities, and called it a night. Turns out I didn't even need any of it.
(I did get one question wrong, though, and looking back I definitely know which one. It was an easy one in the middle of the exam questions and I think I just messed up a concept like "less than" instead of "less than or equal to".)
All my math/C.S. friends thought it was really easy and I think the worst score I heard from them was 720 / 800.
Interestingly, though, back when I took it the average score for the math section was notably higher than for the other sections. Just by missing 1 question I think I was bumped down to about the ~90% percentile.
> I always assumed the GRE must have been designed to accommodate/calibrated to humanities and social science grad students (or other majors without a significant math component) more than STEM students.
This is probably true. I mean, this is why we have GRE subject tests...
Admittedly I've not kept up with the GRE since I took it in 2004. It was computer adaptive at that point, but I found it fairly easy (and had a perfect score on the math section)
I can't say about China particularly, but many school systems route students into entirely different kinds of schools pretty early. Those with good early (elementary or middle school) math and science scores go on to STEM focused high schools and are then routed into STEM programs in Uni. Other kids will end up on a "vocational" track and go to a High School with a strong trade training and essentially do a supervised apprenticeship, if they're really good there they'll go on to a specialized trade school and focus in some kind of trade, instead of changing oil in a car they might rebuild alternators or transmissions or something.
I've heard China has done similar routing for several topics of national prestige, kids end up in an Olympic gymnastics or diving program and train in that program until they either make it into the Olympics, fail out or end up as a coach when they're in their single digit ages. I'm not sure about other areas.
gre general test has always been crap. I've had the unfortunate occasion to take it thrice (for each of my 3 graduate degrees) because the scores expire (what a scam!).
Now the subject math gre, that's fucking awesome. I once challenged my math professors to take it with me & they scored less :) Its broad enough to envelope most math specialities & deep enough to stump you if you haven't been paying attention to your speciality. Here's a copy of the subject math- If you don't know any CS or math but can quickly code some ruby/js/whatever, try #44.
otoh if you know some CS, try #33, #25,#28 & #30.
http://www.ets.org/s/gre/pdf/practice_book_math.pdf
In one article I've read about the school, instead of rote learning about pi and circles, they'll encourage the students to make their own bicycle. And through finding "projects" that the kids want to work on, the students bump into having to learn things like: "how do I measuring the circumference of a circle".
Ugh. I would have hated that. I would have much rather learned a few simple math concepts than figured out how to build a bicycle. Not everyone has the same idea of fun.
I was an unmotivated math learner, who did not want to admit it. I essentially stopped at integral calculus(failed three times), although I did try a linear algebra course at the same time and had to drop it because the effort needed to complete the proofs was beyond me.
On the other hand, I took well to immediate, CS-y applications. I liked graph theory, since I could write pathfinding algorithms with it. And I learned to like parts of Fourier analysis, since I could understand DSP with it. So I know I can enjoy the study, given the right context.
However, I think it speaks to the difficulty of math education that nobody discovers exactly the same approach to its study. But perhaps this is the wrong way to frame it. Perhaps, to put it another way, other subjects allow students to cheat themselves more easily into getting answers without understanding.
I think, with respect to current curricula, we focus far too much on computation, and on proof. Both are answer-oriented by nature, whereas in my self-study I was motivated by developing a process, and in understanding the process I was considering things with language and hasty spatializations of ideas, long before I was ready for the formal applications. And we give almost no credence to process development as part of mathematics education. A typical lecture begins:
"Assume the following. We define..."
And this is perfectly reasonable if you've already immersed yourself in the jargon and want to get to the point as quickly as possible. But if you aren't ready for it, it's intimidating. A tiny definition, a sentence or two long, can become something you struggle over for a week, which only compounds the intimidation factor. In theory the problem sets should help you explore further. But if they're unmotivated, the students can barely try at the problem sets.
The main issue I had in the US K-12 system was both a lack of obvious applicability (there are no math labs), and a complete lack of teaching to my particular learning style (visual-tactile, i.e. watch then do hands on).
There's a boring lecture, some numbers on the board, then drills. This method works, but only when there's a huge pressure on the student to drill...something that the U.S. system just isn't structurally built to supply in the ways that East Asian systems are.
My wife grew up in South Korea and came out of High School knowing the level of calculus I wouldn't even touch until the last semester of my Sophomore year in College in a STEM program. Her 10th grade math textbook looked like an advanced version of my University Calculus course.
The problem I had, and the original question has is that there's just no connection made for the student in the U.S. system from math class to anything remotely useful. Abstract statements like "engineers need math" just simply doesn't mean anything to a bored out of their mind 7th grader. They have to actually use math to solve something they could solve with basic arithmetic and a calculator. Something with a reward at the end other than yet more notebooks full of drilled exercises: some flashing lights on a digital circuit, winning the pot at blackjack, launching a water balloon at a target, whatever.
But modern math education doesn't just fail in making it boring, it makes math scary. Calculus becomes this "special" course only the tip-top AP students take...if your High School even offers it (those special students might even have to go take it at a local college). Basic calculus isn't really all that hard to learn the mechanics of. It shouldn't be treated as special. Just "yet more rules for algebra". Even Math teachers would talk about trigonometry in hushed tones.
Modern Math education doesn't even show student how to apply math to critical reasoning skills. Something even students who go on to the most liberal of liberal arts can benefit from.
Importantly, I don't think anything I could have learned in K-12 mathematics remotely prepared me for the kinds of upper-level mathematics I eventually became exposed to at University. K-12 math prepared me to be a carpenter or a cashier, but it should have left me at a jump off point to be a scientist, or a philosopher, an engineer or a physicist, or at the very least given me a small glimpse into those worlds, because I sure as heck got a glimpse into 17th Century Japanese poetry styles, Shakespearean Theater, Classical Orchestra performance, being a member of Congress, and other career paths I wasn't interested in pursuing.
This is a very important distinction to make, however issues with this style of teaching (training rather than deep education) are difficult to see in subjects other than mathematics.
At a high school level, I doubt any subjects really build on early learnings as much as mathematics. History, biology, and chemistry for example are all easy enough at the US high school level, as little actually builds on top of early concepts. Sure, it helps to have insight into theory behind chemistry, but I can still memorize some elements, molecules, and energy levels of electrons. High school doesn't really go beyond that. Mathematics however, is incredibly abstract, and without a framework of theory to encompass everything, is just a vague sequence of techniques.
I would hope the poster knows what things the sister actually likes to do. Math is the one subject people will use to excess when it isn't seen as Math. Watch any fantasy draft or craft and see how much math is going on. Find the Math in the math she is already doing and work up from their.
We have the same problem with new students at the community college I currently work at. I am at the "searching for sources" phase of trying to put together a "just numbers" course to get students comfortable with numbers and their uses.
There is no more important thing than knowing this. What will this woman-to-be do with this math? Will she be an business manager or an accountant? Will she be a fashion designer or a fabrics engineer? Will she be a naturopath or a chemist? Each of these either/or's has a significant difference in the amount of basic math required.
Once you discover what she loves and is naturally good at, help her follow her path, and then add in math as necessary. Honestly, there are some people for whom cross multiplication will never factor in significantly in life. You may be wasting her valuable time by pushing her in mathematics.
The first and hardest step is getting them to think of math as something other than torture devices for teaching children obedience. (Un)fortunately, being a mathematically-adept child from a mathematically-adept family, I've never encountered that problem and can't offer solutions.
Many people in society value practicality. Why should I do something if it doesn't directly benefit me in the physical sense? How will this make me more likable/valuable/wealthy?
Theoretical studies like mathematics serve no purpose to people who view the world this way, except to qualify their earnings gained through other pursuits. Theory is something some smart guy with a big head does. "Normal people" are practical, and work with what is real.
This sort of viewpoint is one reason (in my opinion, the reason) why many students give no effort in learning math, because it is "too hard", when "hard" means "different", and "different" equates to "not concrete/tangible".
Actually, from a practical viewpoint, learning math will make you more valuable and wealthy; not more likable or popular though. It is actually bound to make you more unpopular, and that is a big deal in the schoolyard microcosm.
Sadly I am in this camp, I say sadly because part of me thinks that I should care more about math yet I don't. I program every day and I could give a flip about the math behind a circle. Business math for me is pretty much canned routines. I know where to get the formula I need should anything advanced come up, I don't usually care about why it works, only that it does work. I trust it works because the sources I use are trusted.
There is a great benefit to teaching practical only, it engages people easier when they can see the immediate benefit. There will never be a shortage of those who want to know more on a give subject, maybe for this guy's sister he needs to find what moves her. Not everyone likes math, and even of those who do I wonder how many get to work with it
One day you will start to notice that not every source can be trusted. And you will need critical thinking skills and logic -- the underpinnings of mathematics -- to separate truth from lies. The specific formulas and structures of geometry and calculus math are irrelevant -- they are just examples that are universal across all cultures.
I feel like his sister. I've been trained to solve math problems using formulas, but I don't know the why.
Some day, in a math class, the teacher was talking about how to solve derivatives - and it required to know what infinite means. I asked to all my classmates to put their hand up if they knew what infinite means. Nobody did. But the teacher went on, omitting that basic concept.
I have just finished high school and I feel like I don't know why 2 + 2 = 4
I feel the same way, despite being 'good at math' and having completed the math curriculum for an engineering degree. I know how to use everything, I understand what is being represented and the general structure of the tools I use, but I feel like something is missing. Math used to 'flow' for me when I was in grade school, but somewhere along the way, I think I lost the plot and no longer think mathematically. I am quick to look up formulas rather than intuiting them. I have moments where I completely understand gradient descent for multi-dimensional logistic regression, but they are fleeting and pass with some retarded thought like why does 2 + 2 = 4? I then give up and just implement the parts of the algorithm without keeping track of the broad picture and it feels... dirty. Guh.
I feel exactly this way as I am trying to teach myself some more involved statistics. There are moments of clarity but the broad picture is too fuzzy. And there was a time I could get really excited about math - still love the experience of (re)learning it but it seems like several years of formula grinding with the sole intent of passing entrance exams has taken its toll in the worst possible way by diminishing some of the intuition and feel I seemed to have.
> I have just finished high school and I feel like I don't know why 2 + 2 = 4
To be fair, axiomatizing integer arithmetic has been attempted (unsuccessfully) by intellectual titans like Godel, Russel, and Whitehead, and is nowadays considered "too hard" by research mathematicians, so you probably shouldn't be so hard on your schoolteachers ;)
> To be fair, axiomatizing integer arithmetic has been attempted (unsuccessfully) by intellectual titans like Godel, Russel, and Whitehead, and is nowadays considered "too hard" by research mathematicians
That is just not true in any interesting sense. The Peano axioms[1] have been around since the 1880s. Gentzen showed in the 1930s how to prove their consistency given transfinite induction up to a sufficient height. There are certainly hard open problems in the areas of mathematics surrounding theories of arithmetic, but to say it's a problem that mathematicians have thrown up their hands and given up on is false.
I've been gradually doing a few of these as I bootstrap Gridmaths.com. You can see some sample screenshots of worked problems in blogposts at : quantblog.wordpress.com
I think we miss engaging with quite a few students, because we focus on procedures and process rather than visualization in school [ I could be wrong, this is subjective, but at least we should try and explore different ways of presenting math ]
Khanacademy videos and GeoGebra might be useful also.
I tutored in uni for extra beer money and I've tutored children, teenagers, fellow university students, grad students, and adults. What I learned from this was that there was no one single best way to teach someone things. The phrase sucks, but it really depends on the person. Some people you can teach better by not giving them the answer and trying to get them to think and that's great. On the other hand, some people actually do excel if they are simply given the answer (of course one doesn't just stop after giving the answer, you would then explain the answer, how we got there, etc, and then knowing this, solve another similar problem to see how much of it they understood). Figuring out your student's learning style is unfortunately the hard part, and it looks like this guy in the link has not figured out his sister's learning style. Even just reading that conversation he posted---he sounds extremely condescending.
Just picking something out, he wrote:
"The circumference of the glass divided by the diameter gave me pi, what does that mean?"
OK, first of all, he goes to say that she JUST now learned what a circumference and diameter is and still doesn't know what a "pi" is. Why in the world would you ask a question like this with unfamiliar words to someone who just learned these words? He was making it unnecessarily difficult to understand. Of course she said she didn't know and then he goes on and says "WELL what if I divide 10 by 2?" Which is like comparing apples to oranges.
Sometimes the brain just isn't ready to understand at the time.
I struggled with math early on because why the fuck do I care what the cicumference of a circle is? When things don't have some practical relevance, they are harder to learn.
To motivate someone, you must find a way to tie in the subject to something they are genuinely interested in.
Having grown up in public schools in the southern US, I don't understand why we focus so much on memorizing formulas but not the underlying concepts or how they apply to other things. The power of math is the ability to abstract other problems into math and then solve them with a standardized toolset. We spent the first decade of our schooling learning very basic mathematics (up to algebra), and I hadn't seen calculus until my second year of university.
It's no surprise to me that kids are having issues understanding the parts of a circle, why the formulas do what they do, or how to do the related math. I've been there. I think teaching basic physics and some of the more esoteric math earlier (number type sets, limits, boolean logic, simplified of course) would help with some of this. I always wondered how to do things infinitely or how to calculate things certain ways in algebra, and it always seemed like parts were missing. Calculus filled in so much of this for me, and I wish I had known it earlier. Physics and video games (wiremod) gave me reasons to learn some of the math.
Another thing I wish there had been more of is the history and practical implications of this stuff. I never cared about the quadratic formula before I learned about electric circuits or things falling in gravity. I didn't know at the time it went back to Babylon, India, and Greece.
Kids don't get taught how to think critically and use the tools at their disposal, and that's the largest problem with our education system. We learn how to ask someone for the proper tool to use and then do the work ourselves without thinking about it. It's probably an accurate representation of what work in much of America has come to nowadays, busy work, but it's a shame that we don't do something better for the next group coming along.
When schools try to teach a holistic sense of numeracy to connect with average students, with programs like Discovery, Tiger parents* revolt with stuff like http://wheresthemath.com/
* Using Tiger parent as a trope here, not specific to any ethnicity.
You can't really force someone to learn something they don't wish to learn. And it's not about whether or not a person is capable or incapable. The value of math needs to be communicated - try using examples that speak to the girl's worldview. Show her the wonders of math and what it can do for her to solve her puzzles in life.
I never got the hang of what I learnt in school until I began working and it gave me so much reason to what I'm studying.
Generally my advice would be to go back to the most basic set of skills you can identify that she doesn't understand and work up from there - including having her construct formula to solve problems (preferably ones that she finds interesting.)
Though, honestly, if she doesn't want to learn maths she's not going to learn maths. If you're really concerned for her and she doesn't want to do something, then you're probably best off just teaching her the tricks and cheats so that she can get a good result and then forgetting about the whole thing - it's not like maths is likely to be a particularly important aspect of her life.
#
With respect to pi in specific -
> Instead of giving her the c=πd formula she wanted so badly, I wanted her to understand that π represented the amount of times the diameter "fits" into the circumference and that this is the relation between the parts of the circle.
If she's never constructed formula to solve problems for herself, she many not know that you can do anything with that sort of information. It's not a trivial step if you don't already know what = and * and / and so on really do to go from any particular instance to c = pidiam.
> I measured as accurately as possible the perimeter and diameter of the mouth of a cup I had and showed her that dividing the numbers produced approximately pi. This unfortunately didn't provide the "ohhh" response I was looking for, which signified that she didn't intuitively understand division.
Try cutting up an apple or counting little blocks or something like that. If she doesn't understand division then trying to use division in any capacity with relation to more complex problems is a waste of time.
> "The circumference of the glass divided by the diameter gave me pi, what does that mean?"
I thought you werne't just giving her the formula? If she were able to substitute letters in she'd have it. =p
Really though. It doesn't mean* anything beyond that pi is the circumference over the diameter. You need other knowledge to network it into before it becomes meaningful.
First, not everybody will have interest in Math. And that's fine. Secondly, you need to be creative for teaching math, especially at a junior level. I think Lockhart gives a very good insight on this matter - http://www.maa.org/devlin/lockhartslament.pdf
It's important to realize that jr. high algebra is largely syntactic transformations.
a - 2 = 3 is a syntactic transformation of a - 1 = 4.
Most of algebra is taught as a series of syntactic transformation rules. The skill students acquire is learning to apply the right rules to solve a larger problem.
If algebra were taught in a more abstract way, there would be an even more pronounced stratification of aptitude, such that the educational system would not be able to teach to the variety of different levels. At least when it's taught as syntactic transformation, the best students are still largely bounded by working memory rather than intellect.
There are professionals and researchers out there whose job is dedicated to this issue of improving math education (and the same with physics education, computer science education, etc.). You can find great ideas and sample lessons and great educational software.
Math is taught out of context, or completely decontextualized - that's why people hate it, and that's why it's hard to motivate people to learn it. Math knowledge is both situated and embodied. Look at mathematics education research for ideas on how to teach math more effectively.
Mathematics is arguably the worst taught subject. I think this is entirely the fault of the teachers and writers of the curriculum, who systematically fail to explain their subject in an approachable way. I wrote a rant about it here:
TL;DR: This rant identifies a major cause: math is taught as if one should magically be able to comprehend its language and syntax without explanation.
Math uses symbols heavily as its own jargon, and someone approaching formulae that have unspecified symbols, without knowing this jargon, is typically fruitless.
That said, I agree that math is terribly taught. My mother and my wife both hate math, and talking it out with both of them, I've been able to pinpoint it to a single bad/discouraging (and sometimes straight out mean) math teacher for both of them. Once the teacher is mean or otherwise bad about teaching, they have not only failed as teachers, but have sometimes caused the students to put up mental walls that can be hard to tear down.
I suspect the sister mentioned in the OP may have had such an experience.
I never got math or liked it until I had to do real-world things with it, at which point I dug and managed to teach myself things that I needed. Nearly all my math teachers were horrible. One or two of them seemed almost intentionally obtuse... like they were trying to trick you and taking pleasure in your failure to understand what they were teaching.
This is exactly how I feel about math. It's also why I love Python. Single character variable names are bad variable names. Terse method names are worse than clear method names.
Also in mathematical notation there is no concrete flow to how you evaluate an equation. You can start just about anywhere. With programming, the order in which things get done might be arbitrary (and frustrating at first) it is at least consistent.
Mathematical notation is really good, for some people, to describe, communicate, and explore relations. They can directly see that by increasing the value of the denominator the resulting value decreases. But without a tool designed by Bret Victor, my brain does not do that. I want to follow along a set of instructions, maybe iterate through a series of inputs, and then I understand how the operations applied to the input produce an output.
Math has always seemed to me to be badly formatted code.
Seeing this is such a universal issue really saddens me.
In Israel, High School level math is divided into 5 point, 4 point and 3 point courses. 3 points is a very basic set of mathematical tools, "math for everyone" if you will, which includes basic algebra, basic geometry and trigonometry, basic statistics and what could possibly pass for the first lesson of a calculus course.
4 points is pretty much what this guys sister is studying, conceptually, in that it's a lot of memorization and very little to no understanding. This includes higher level algebra, trigonometry in 3D space, calculus and some other stuff.
5 points is where the problems start - students are required to do 4 point level exercises with "twists" which make them harder and require actual thinking. This sounds great until you understand that most 5 point students have no idea about the underlying mathematical concepts and are just memorizing the answers to thinking part as well. Instead of being elite students who understand the internal workings of mathematics they are distinguished by better memory and more cramming.
So yes - horrible math education is everywhere and no one seems to know what to do about it. Three cheers for the human race...
It's filled with a lot of history on why things are as they are and it builds up a substantial base of math knowledge from there. I can't comment on whether the additional background information would help someone who is math shy to "get it" but, from the parts I read, it certainly rounded out (and expanded) my knowledge.
I've tutored quite a few kids of varying ages over the years and in my experience, there's no harder group to teach than teenage girls - and it's doubly hard if the kid in question is fairly popular and active in other areas. There are just so many things competing for a kid's attention that it makes the most sense to (most of) them to quickly memorize a formula, get the best grade possible with minimal effort, and move on to more enjoyable things.
The best method I've found for tutoring teenage girls (and I'm saying this as a girl who hated math as a teenager in spite of a love of computers and economics) is to start off by finding out what they enjoy. Put the lessons aside, don't try to talk theory, and don't get all crazed trying to show her how awesome math is. Eyes will glaze over and you will have lost before you ever get started. You can't teach someone unless you first inspire an interest.
What does she want to do as a career? What are her hobbies? What kinds of things interest her? At that age, she's probably thought about a lot of different careers, and in pretty much any career, you can tie things back to math (especially junior high and early high school math).
If she enjoys cooking, you can talk about proportions and cross multiplication and show how you can use that to adjust the number of portions produced by a recipe. I know of one family where the parents had their daughter go on Pinterest and find recipes for a family meal, then scale them up to fit the size of their family.
If decor is her thing, you can talk about floor plans and usable area. Make a diagram and figure out how much space an 8' circular rug will leave for a bed and desk...the swing radius of a door is generally 1/4 of a circle, and you can't put most things in that space.
Mathalicious.com is a good source of more real world lessons.
Take the time to walk through the logic and show how things relate back to her lessons. It's not efficient, but in most cases, you don't have to do this for every single concept. The goal is to (a) create interest, and (b) show how math is relevant to her life.
Above all, be patient and understand that even with the best, most passionate instruction...she might still hate math. The world needs many different types of people, and not all of them have to have a deep understanding of math. She can still be happy and successful even if she never learns to like math.
100 comments
[ 3.2 ms ] story [ 186 ms ] threadThus far the Stack Exchange discussion includes a lot of complacency along the lines of "some students just don't have the capacity to get it" that I never found when I was living in Taiwan. Over there, even the below-average students in seventh grade are all expected to learn quite a lot of algebra and geometry, which they consolidate knowledge of during junior high school. By the end of junior high, students in Taiwan and students in urban areas of China and students in Korea, Japan, Singapore, and Hong Kong generally know United States secondary school mathematics quite well. (I have heard Chinese graduate students in the United States, students in nonquantitative subjects, deride the mathematics section of the GRE general test used for graduate school admission as a test of "junior high math," and that is literally true in terms of the standard curriculum in China.)
So how about it? What creative ideas do we have here to help the young learner learn how to solve proportions, an aspect of day-by-day reasoning that only secondary school graduate might reasonably be expected to know? How to reignite a spark of interest in mathematics after school lessons that may have been a turn-off, may have been flat wrong,
http://www.ams.org/notices/200502/fea-kenschaft.pdf
and at a minimum have left the learner with an excuse to not keep trying to learn?
this is true for students educated in the US as well! it's super frustrating, because you're staring at trig identities that you last looked at ten years ago.
(I did get one question wrong, though, and looking back I definitely know which one. It was an easy one in the middle of the exam questions and I think I just messed up a concept like "less than" instead of "less than or equal to".)
All my math/C.S. friends thought it was really easy and I think the worst score I heard from them was 720 / 800.
Interestingly, though, back when I took it the average score for the math section was notably higher than for the other sections. Just by missing 1 question I think I was bumped down to about the ~90% percentile.
This is probably true. I mean, this is why we have GRE subject tests...
I've heard China has done similar routing for several topics of national prestige, kids end up in an Olympic gymnastics or diving program and train in that program until they either make it into the Olympics, fail out or end up as a coach when they're in their single digit ages. I'm not sure about other areas.
Now the subject math gre, that's fucking awesome. I once challenged my math professors to take it with me & they scored less :) Its broad enough to envelope most math specialities & deep enough to stump you if you haven't been paying attention to your speciality. Here's a copy of the subject math- If you don't know any CS or math but can quickly code some ruby/js/whatever, try #44. otoh if you know some CS, try #33, #25,#28 & #30. http://www.ets.org/s/gre/pdf/practice_book_math.pdf
In one article I've read about the school, instead of rote learning about pi and circles, they'll encourage the students to make their own bicycle. And through finding "projects" that the kids want to work on, the students bump into having to learn things like: "how do I measuring the circumference of a circle".
On the other hand, I took well to immediate, CS-y applications. I liked graph theory, since I could write pathfinding algorithms with it. And I learned to like parts of Fourier analysis, since I could understand DSP with it. So I know I can enjoy the study, given the right context.
However, I think it speaks to the difficulty of math education that nobody discovers exactly the same approach to its study. But perhaps this is the wrong way to frame it. Perhaps, to put it another way, other subjects allow students to cheat themselves more easily into getting answers without understanding.
I think, with respect to current curricula, we focus far too much on computation, and on proof. Both are answer-oriented by nature, whereas in my self-study I was motivated by developing a process, and in understanding the process I was considering things with language and hasty spatializations of ideas, long before I was ready for the formal applications. And we give almost no credence to process development as part of mathematics education. A typical lecture begins:
"Assume the following. We define..."
And this is perfectly reasonable if you've already immersed yourself in the jargon and want to get to the point as quickly as possible. But if you aren't ready for it, it's intimidating. A tiny definition, a sentence or two long, can become something you struggle over for a week, which only compounds the intimidation factor. In theory the problem sets should help you explore further. But if they're unmotivated, the students can barely try at the problem sets.
There's a boring lecture, some numbers on the board, then drills. This method works, but only when there's a huge pressure on the student to drill...something that the U.S. system just isn't structurally built to supply in the ways that East Asian systems are.
My wife grew up in South Korea and came out of High School knowing the level of calculus I wouldn't even touch until the last semester of my Sophomore year in College in a STEM program. Her 10th grade math textbook looked like an advanced version of my University Calculus course.
The problem I had, and the original question has is that there's just no connection made for the student in the U.S. system from math class to anything remotely useful. Abstract statements like "engineers need math" just simply doesn't mean anything to a bored out of their mind 7th grader. They have to actually use math to solve something they could solve with basic arithmetic and a calculator. Something with a reward at the end other than yet more notebooks full of drilled exercises: some flashing lights on a digital circuit, winning the pot at blackjack, launching a water balloon at a target, whatever.
But modern math education doesn't just fail in making it boring, it makes math scary. Calculus becomes this "special" course only the tip-top AP students take...if your High School even offers it (those special students might even have to go take it at a local college). Basic calculus isn't really all that hard to learn the mechanics of. It shouldn't be treated as special. Just "yet more rules for algebra". Even Math teachers would talk about trigonometry in hushed tones.
Modern Math education doesn't even show student how to apply math to critical reasoning skills. Something even students who go on to the most liberal of liberal arts can benefit from.
Importantly, I don't think anything I could have learned in K-12 mathematics remotely prepared me for the kinds of upper-level mathematics I eventually became exposed to at University. K-12 math prepared me to be a carpenter or a cashier, but it should have left me at a jump off point to be a scientist, or a philosopher, an engineer or a physicist, or at the very least given me a small glimpse into those worlds, because I sure as heck got a glimpse into 17th Century Japanese poetry styles, Shakespearean Theater, Classical Orchestra performance, being a member of Congress, and other career paths I wasn't interested in pursuing.
I made some suggestions here https://news.ycombinator.com/item?id=5858652
Training is what the poster's sister has.
Education is the bit that is missing.
Unfortunately education and training are confused these days.
I have the same battle with my children and mathematics. The teacher teaches them the mechanics of mathematics but no meaning, reason or application.
At a high school level, I doubt any subjects really build on early learnings as much as mathematics. History, biology, and chemistry for example are all easy enough at the US high school level, as little actually builds on top of early concepts. Sure, it helps to have insight into theory behind chemistry, but I can still memorize some elements, molecules, and energy levels of electrons. High school doesn't really go beyond that. Mathematics however, is incredibly abstract, and without a framework of theory to encompass everything, is just a vague sequence of techniques.
Having a concrete example of how PI was calculated was very easy to wrap my head around:
http://betterexplained.com/articles/prehistoric-calculus-dis...
We have the same problem with new students at the community college I currently work at. I am at the "searching for sources" phase of trying to put together a "just numbers" course to get students comfortable with numbers and their uses.
Perhaps buying some of WeWantToKnow AS's software https://itunes.apple.com/us/app/dragonbox+-algebra/id5220691... https://itunes.apple.com/us/app/dragonbox-algebra-12+/id6344... for iOS if you have that.
Indeed. Check out my irb history from playing Hero Academy (there's way more above):
There is no more important thing than knowing this. What will this woman-to-be do with this math? Will she be an business manager or an accountant? Will she be a fashion designer or a fabrics engineer? Will she be a naturopath or a chemist? Each of these either/or's has a significant difference in the amount of basic math required.
Once you discover what she loves and is naturally good at, help her follow her path, and then add in math as necessary. Honestly, there are some people for whom cross multiplication will never factor in significantly in life. You may be wasting her valuable time by pushing her in mathematics.
Theoretical studies like mathematics serve no purpose to people who view the world this way, except to qualify their earnings gained through other pursuits. Theory is something some smart guy with a big head does. "Normal people" are practical, and work with what is real.
This sort of viewpoint is one reason (in my opinion, the reason) why many students give no effort in learning math, because it is "too hard", when "hard" means "different", and "different" equates to "not concrete/tangible".
There is a great benefit to teaching practical only, it engages people easier when they can see the immediate benefit. There will never be a shortage of those who want to know more on a give subject, maybe for this guy's sister he needs to find what moves her. Not everyone likes math, and even of those who do I wonder how many get to work with it
You don't. Full stop. This is a lesson that I've had to learn the hard way.
To be fair, axiomatizing integer arithmetic has been attempted (unsuccessfully) by intellectual titans like Godel, Russel, and Whitehead, and is nowadays considered "too hard" by research mathematicians, so you probably shouldn't be so hard on your schoolteachers ;)
That is just not true in any interesting sense. The Peano axioms[1] have been around since the 1880s. Gentzen showed in the 1930s how to prove their consistency given transfinite induction up to a sufficient height. There are certainly hard open problems in the areas of mathematics surrounding theories of arithmetic, but to say it's a problem that mathematicians have thrown up their hands and given up on is false.
[1] http://en.wikipedia.org/wiki/Peano_arithmetic
I've been gradually doing a few of these as I bootstrap Gridmaths.com. You can see some sample screenshots of worked problems in blogposts at : quantblog.wordpress.com
I think we miss engaging with quite a few students, because we focus on procedures and process rather than visualization in school [ I could be wrong, this is subjective, but at least we should try and explore different ways of presenting math ]
Khanacademy videos and GeoGebra might be useful also.
http://www.gocomics.com/foxtrot/2009/01/25/
Just picking something out, he wrote:
"The circumference of the glass divided by the diameter gave me pi, what does that mean?"
OK, first of all, he goes to say that she JUST now learned what a circumference and diameter is and still doesn't know what a "pi" is. Why in the world would you ask a question like this with unfamiliar words to someone who just learned these words? He was making it unnecessarily difficult to understand. Of course she said she didn't know and then he goes on and says "WELL what if I divide 10 by 2?" Which is like comparing apples to oranges.
I struggled with math early on because why the fuck do I care what the cicumference of a circle is? When things don't have some practical relevance, they are harder to learn.
To motivate someone, you must find a way to tie in the subject to something they are genuinely interested in.
It's no surprise to me that kids are having issues understanding the parts of a circle, why the formulas do what they do, or how to do the related math. I've been there. I think teaching basic physics and some of the more esoteric math earlier (number type sets, limits, boolean logic, simplified of course) would help with some of this. I always wondered how to do things infinitely or how to calculate things certain ways in algebra, and it always seemed like parts were missing. Calculus filled in so much of this for me, and I wish I had known it earlier. Physics and video games (wiremod) gave me reasons to learn some of the math.
Another thing I wish there had been more of is the history and practical implications of this stuff. I never cared about the quadratic formula before I learned about electric circuits or things falling in gravity. I didn't know at the time it went back to Babylon, India, and Greece.
Kids don't get taught how to think critically and use the tools at their disposal, and that's the largest problem with our education system. We learn how to ask someone for the proper tool to use and then do the work ourselves without thinking about it. It's probably an accurate representation of what work in much of America has come to nowadays, busy work, but it's a shame that we don't do something better for the next group coming along.
* Using Tiger parent as a trope here, not specific to any ethnicity.
With the students that I've seen in college physics/math courses, ability to solve real-world arithmetic/word problems quickly is limited at best.
I never got the hang of what I learnt in school until I began working and it gave me so much reason to what I'm studying.
[1] http://en.wikipedia.org/wiki/Integrated_mathematics
Though, honestly, if she doesn't want to learn maths she's not going to learn maths. If you're really concerned for her and she doesn't want to do something, then you're probably best off just teaching her the tricks and cheats so that she can get a good result and then forgetting about the whole thing - it's not like maths is likely to be a particularly important aspect of her life.
#
With respect to pi in specific -
> Instead of giving her the c=πd formula she wanted so badly, I wanted her to understand that π represented the amount of times the diameter "fits" into the circumference and that this is the relation between the parts of the circle.
If she's never constructed formula to solve problems for herself, she many not know that you can do anything with that sort of information. It's not a trivial step if you don't already know what = and * and / and so on really do to go from any particular instance to c = pidiam.
> I measured as accurately as possible the perimeter and diameter of the mouth of a cup I had and showed her that dividing the numbers produced approximately pi. This unfortunately didn't provide the "ohhh" response I was looking for, which signified that she didn't intuitively understand division.
Try cutting up an apple or counting little blocks or something like that. If she doesn't understand division then trying to use division in any capacity with relation to more complex problems is a waste of time.
> "The circumference of the glass divided by the diameter gave me pi, what does that mean?"
I thought you werne't just giving her the formula? If she were able to substitute letters in she'd have it. =p
Really though. It doesn't mean* anything beyond that pi is the circumference over the diameter. You need other knowledge to network it into before it becomes meaningful.
a - 2 = 3 is a syntactic transformation of a - 1 = 4.
Most of algebra is taught as a series of syntactic transformation rules. The skill students acquire is learning to apply the right rules to solve a larger problem.
If algebra were taught in a more abstract way, there would be an even more pronounced stratification of aptitude, such that the educational system would not be able to teach to the variety of different levels. At least when it's taught as syntactic transformation, the best students are still largely bounded by working memory rather than intellect.
Math is taught out of context, or completely decontextualized - that's why people hate it, and that's why it's hard to motivate people to learn it. Math knowledge is both situated and embodied. Look at mathematics education research for ideas on how to teach math more effectively.
For example Jo Boaler's work. She is teaching a MOOC on this topic of how people learn math next month: https://class.stanford.edu/courses/Education/EDUC115N/How_to... http://joboaler.com/
NCTM - National Council of Teachers of Mathematics http://www.nctm.org/
RUME - Research in Undergraduate Mathematics Education http://sigmaa.maa.org/rume/Site/News.html
As someone else mentioned, the app DragonBox helps with understanding algebra: http://dragonboxapp.com/
The Adventures of Jasper Woodbury taught math in context, in the form of complex challenges: http://viking.coe.uh.edu/~ichen/ebook/et-it/ai.htm
https://news.ycombinator.com/item?id=5819935
TL;DR: This rant identifies a major cause: math is taught as if one should magically be able to comprehend its language and syntax without explanation.
That said, I agree that math is terribly taught. My mother and my wife both hate math, and talking it out with both of them, I've been able to pinpoint it to a single bad/discouraging (and sometimes straight out mean) math teacher for both of them. Once the teacher is mean or otherwise bad about teaching, they have not only failed as teachers, but have sometimes caused the students to put up mental walls that can be hard to tear down.
I suspect the sister mentioned in the OP may have had such an experience.
Also in mathematical notation there is no concrete flow to how you evaluate an equation. You can start just about anywhere. With programming, the order in which things get done might be arbitrary (and frustrating at first) it is at least consistent.
Mathematical notation is really good, for some people, to describe, communicate, and explore relations. They can directly see that by increasing the value of the denominator the resulting value decreases. But without a tool designed by Bret Victor, my brain does not do that. I want to follow along a set of instructions, maybe iterate through a series of inputs, and then I understand how the operations applied to the input produce an output.
Math has always seemed to me to be badly formatted code.
4 points is pretty much what this guys sister is studying, conceptually, in that it's a lot of memorization and very little to no understanding. This includes higher level algebra, trigonometry in 3D space, calculus and some other stuff.
5 points is where the problems start - students are required to do 4 point level exercises with "twists" which make them harder and require actual thinking. This sounds great until you understand that most 5 point students have no idea about the underlying mathematical concepts and are just memorizing the answers to thinking part as well. Instead of being elite students who understand the internal workings of mathematics they are distinguished by better memory and more cramming.
So yes - horrible math education is everywhere and no one seems to know what to do about it. Three cheers for the human race...
http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...
It's filled with a lot of history on why things are as they are and it builds up a substantial base of math knowledge from there. I can't comment on whether the additional background information would help someone who is math shy to "get it" but, from the parts I read, it certainly rounded out (and expanded) my knowledge.
This is how π was made clear to me when I was younger.From there I understood it was some kind of ΅natural and constant ratio" of circles.
The best method I've found for tutoring teenage girls (and I'm saying this as a girl who hated math as a teenager in spite of a love of computers and economics) is to start off by finding out what they enjoy. Put the lessons aside, don't try to talk theory, and don't get all crazed trying to show her how awesome math is. Eyes will glaze over and you will have lost before you ever get started. You can't teach someone unless you first inspire an interest.
What does she want to do as a career? What are her hobbies? What kinds of things interest her? At that age, she's probably thought about a lot of different careers, and in pretty much any career, you can tie things back to math (especially junior high and early high school math).
If she enjoys cooking, you can talk about proportions and cross multiplication and show how you can use that to adjust the number of portions produced by a recipe. I know of one family where the parents had their daughter go on Pinterest and find recipes for a family meal, then scale them up to fit the size of their family.
If decor is her thing, you can talk about floor plans and usable area. Make a diagram and figure out how much space an 8' circular rug will leave for a bed and desk...the swing radius of a door is generally 1/4 of a circle, and you can't put most things in that space.
Mathalicious.com is a good source of more real world lessons.
Take the time to walk through the logic and show how things relate back to her lessons. It's not efficient, but in most cases, you don't have to do this for every single concept. The goal is to (a) create interest, and (b) show how math is relevant to her life.
Above all, be patient and understand that even with the best, most passionate instruction...she might still hate math. The world needs many different types of people, and not all of them have to have a deep understanding of math. She can still be happy and successful even if she never learns to like math.