1) You need to be able to do basic calculations before you can do advanced proofs. I taught a lot of high school seniors, and I had a ton of students who were smart enough to handle abstract concepts, but couldn't follow along when I showed them cool proofs because they got caught up on the basic calculations (because they hadn't learned them well in middle/high school).
2) Good high school teachers DO do a lot of pattern recognition/abstract reasoning. That's the entire idea behind a discovery lesson and constructivist teaching - having students learn formulas by discovering patterns and reasoning about them.
3) Again, as he points out, American high schools do do proofs in Geometry. He thinks they're really pedantic, but there are good reasons why 2-column proofs are so tedious. For one, students seeing proofs for the first time freak out, so giving them structure helps. For another, if the students write out every single step, it's easier to identify who really knows his/her stuff and who's BSing.
I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples:
[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/
The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).
There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle."
Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.
I read/write slow and I have an incredibly hard time memorizing anything so I rebelled. Even after I got my act together, got my GED, and went to college I suffered through the various levels of Calculus because it was the same tune all over again. Classes like Linear Algebra were a lot harder for me to "see" but it was still faster & easier for me to take the time to visualize it.
When I went over it with him, I showed him several spots where he made careless mistakes - he added where he should have subtracted, he multiplied where he should have divided, he screwed up a decimal point, whatever. I told him, "It's easy to spot your errors now because these are easy problems. But when you get to harder math, it's going to be much harder to find out what you did wrong, and a teacher isn't going know whether you made a careless mistake or just don't know it at all. By showing your work, you show the teacher that you actually know it."
Nowadays he understands the reason why he needs to show his work but still hates it. I'm hoping that he'll be like this only while the math is easy.
The point of showing the work is to learn the mechanics of solving the problem. If you do not learn the mechanics of solving for simple problems, you will not be able to apply the mechanics to more complex problems where one can no longer intuitively see the answer.
It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them". That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.
Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.
Mathematics has always seemed the same to me. I don't really use much of it day-to-day, but occasionally I'll come across a geometry problem or something when I'm building software; maybe I end up doodling triangles, and using basic trig and algebraic manipulation to understand more or solve my problem.
Much of our teaching processes focus on skills, rather than a more abstract notion of "education." There's been much said about why this is a bad thing; I'm rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don't teach math, but we do teach the skills that are required to "experience" math at a later date.
So maybe I've convinced myself of the validity of the title, if not the individual arguments.
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I'd agree with you, if what we taught was an understanding of the concepts and established mechanisms. However, it seems to me that, most of what I saw in schools was just symbol manipulation.
For example, people didn't actually seem to understand that to get the area of a circle you took the radius, multiplied it by the ratio of the diameter to the circumference and squared it. They understood that you took the radius, multiplied it by a magic number, and for unknown reasons squared that.
The mapping of the symbols onto reality was often missing. It wasn't problem solving beyond the level of having a lookup table in your head that said 'When calculating an area do this, then this, then this.'
All that said, there are things it makes sense to memorise after you understand them - low level components where the speed gained in doing so allows you to use them in higher level abstractions. My point isn't that it doesn't make sense to teach people tools. But that to just give them the tools without the understanding of how they function seems harmful to their ability to create and adapt their own tools down the line.
The mapping of the symbols onto reality was often missing.
For the author's analogy, music is not being taught like what you describe. In his analogy it's being taught as several years of learning to read and transcribe music, without listening to or performing it.
Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching "real math" once the students reach 13 or 14 years old. And throughout this, it's rarely hinted at how this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don't realize its real potential because the solution to the "problem" is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) / 2' and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won't and will get to college thinking they're "good at math" and then fail horribly because they don't have the skill set for college mathematics, they don't realize what college mathematics entailed (so many jokes about my "modern algebra" textbooks, "We took that in 9th grade!").
EDIT: Grammar.
Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You've counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you've done that, you abstract all of arithmetic with algebra.
And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have "learned" everything up to calculus, but they don't actually understand what's going on. They just know the formulae and how to plug-and-chug.
How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood why the formula was used just as much as how it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they're fucked.
In any case, though, I think that the current curriculum is as good as it's going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren't going to learn calculus unless you understand algebra, and you aren't going to learn algebra unless you understand arithmetic.
You and I live in different worlds. It seems to me that the bulk of what I do writing software is related to math in some manner. I guess I have a lot of "number cruncher" sort of clients.
Programming is inherent mathematical, but if you don't progam functional, the math "behind" your software gets more abstract ... or should I say obscure?
In the end your computer is a big function with input, calculation and output. But if you write your code with side effects you can't do a fine function based split-up of your code, you have to consider bigger chunks of code, consisting of many dependent functions as one "mathematical" function.
If the math is more complex and logical reasoning about it gets harder, people start to think about it as "not-math" but something different.
Is this so, or is it true that we merely do not allow any other path?
At least in the United States, very few children attend anything other than public schools, and these public schools have a strict curriculum that introduces few concepts but in a rigid order, interspersed with months of monotonous busy-work that comprises little more than arithmetic and solving equations.
As a whole, society has never ran the experiment to see if it is a necessary step, so claiming that it is necessary is silly. It is perhaps true that if we did run the experiment, it would fail repeatedly, but only then could it be claimed to be necessary.
Skill based education is better than not learning anything. No one argues this, it's a strawman. The point is, learning a method to do something without any kind of explanation or example of why you ever would do it is woefully suboptimal. In an attempt to come up with the most absurd example of this - it can feel like learning to scuba dive in a world with no water deeper than 8 feet, especially given that most teachers I had actually could not give me examples of how to apply the "math" I was learning (and I was a very immature handful back then, very unshy about demanding an explanation of why I was wasting my time learning something that I couldn't see a use for). The author isn't arguing for eliminating all the "skills-based" education he references, he's arguing that we move it closer to how music is taught - teach all the skills, but then immediately apply them to something the students can relate to, at that age (not in 7 years when they are working and run the risk of being like me, and not recognizing the importance of these concepts until then). Math is everywhere. Most people learn basic computation fairly easily, since it can be taught in the context of going to the grocery store or splitting a check. Get beyond this, and all of a sudden, schools quit even attempting to anchor math education in reality (by this I mean a situation that could conceivably occur in reality, not simply taking a number problem and putting words to it).
The argument in favor of mandatory musical education would actually probably benefit from stealing a bit of this piece: music is one of the best forms of education in terms of immediate application of the concepts and methods you are taught, perhaps only behind physical education.
I understand the larger point about the difference between high-school math and high-level mathematics, but come on, don't be so pedantic! Everyone calls whatever it is you study in High-School & Elementary school - Math.
On a related matter, physicists and other scientists have done a pretty good job of communicating to the general public what it is they do. On the other hand, very few people actually know what professional Mathematicians actually do - something I realized when I struggled to explain it to my dad the other day.
The 1% of kids who did well in high school and then fail in college because they are so attached to their rote memorization of techniques have a profoundly broken approach to problem solving that is bigger than the education they received. I've tutored many kids exactly like that and it is very hard to pry them free of that mentality. It is part of their personality. Also, those kids were never really good at math in high school either and were battling (using tutors for help frequently) uphill to get through their entire primary curriculum.
The much bigger and real tragedy of math education in the US is the very large percentage of kids who have been labeled as "not good at math". Those kids 99% of the time are actually plenty good at math but have fallen out of the system because of frustration and a poor fit for their learning style. Those kids don't end up in universities trying to take calc for science majors at all because they believe they aren't capable and that is a crime.
> 99% of kids who were "always good at math"
> will continue to be good at math in college.
So your claim that this is "a rant against a straw man" does not match my experience. As a result, I'd be interested to know what leads you to make the claim I quote. Do you have figures, or personal experience? And are you talking about math as in Analysis, Number Theory, Logic, Topology, and similar, or are you talking about "Math Methods" such as are required in subjects such as Physics and Engineering?
For reference, this:
> The much bigger and real tragedy of math education
> in the US is the very large percentage of kids who
> have been labeled as "not good at math". Those kids
> 99% of the time are actually plenty good at math but
> have fallen out of the system because of frustration
> and a poor fit for their learning style.
Admittedly, this is anecdotal, but it does seem to support the argument/rant in the article.
I feel that I, personally, was very lucky in my high school mathematical education in that my teachers exposed us to the concepts/meanings of all of the operations we were doing long before they exposed us to the procedural trivia of finding integrals/derivatives and the like. It is unfortunate that not all schools are like this.
TLDR: Take the maths for the sake of knowledge, even if you don't 'have to'. It can't hurt.
I remember being taught about derivatives with the formal limit formula, then all the tricks for finding them (power rule etc), and finally finding function extrema using derivatives; all before developing much intuition about the concept.
I literally had a professor who read from the book for lecture and then did his tests out of the book. Granted, it's probably one of the best physics text books ever written (David Griffiths) but it's very, very difficult to learn this way. Students resort to memorization when they have no other options. Nearly every student could solve problems within a range of 'like' problems (change a variable or two, a power, etc.) but had no actual ability to solve truly unique equations in the subject or even formulate their own questions.
This was a 3rd year physics course at a top-50 school.
I hit a wall in calculus 2 in University, because previously I had been able to rely on simply showing up and not putting in much effort to still do "pretty good". That was the first time I did poorly in any math/science course, and was a pretty big wake-up call. As a result, I ended up having to study more to fill in the gaps.
I was naive, came from a poor background with crappy public schools, and didn't think of math as particularly important to focus on until I saw how useful it was in physics. I don't think it's an all too uncommon story.
"Math" up through high school in the US, and through that freshman calculus course, is mostly about calculating the right answer to a problem. Math is a much bigger subject; calculations are a trivially small part of the whole.
On the other hand, "The much bigger and real tragedy of math education in the US is the very large percentage of kids who have been labeled as 'not good at math'" is also very true. And many of those people might not be good at calculations but would be fine with the rest of the subject.
http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf
EDIT : Oops, I hadn't seen that he already linked to this text. That'll learn me to post too quickly. Oh well.
IMO this is a pedagogical insanity, flooding young kids with formalisms that took centuries to emerge without any explanation about their background and enforcing form over content, which is what cuts many super talented people and forces them to focus at different fields.
There are many problems with contemporary math that are conveniently avoided (binary logic for example - most of the population doesn't believe it has any connection to thinking due to weirdness of material implication and teacher's insistence that this is the right way to think, never mentioning that its distant father Aristotle was so discontent with it that he immediately developed a first proto-modal logic), etc. If some constructionists and intuitionists weren't going against the scientific current, we wouldn't have had computers for a long time.
The last 2 years of high school, however, I had picked the 8 hour math options (25% of total course time) and the fun was quickly beaten out of it by having to learn formal ways to write a proof. Saying the same thing in plain language was 'invalid'.
From that point on math felt more like learning a foreign language than about doing logic.
Learning rigorous mathematics is frustrating at first because the veracity of a statement can seem intuitively obvious but difficult to prove. However it is a key stepping stone to modern mathematics and will considerably sharpen your intuition after you've gone through the process.
But the trend toward (excessive?) formalism at the expense of intuitive understanding is much larger in scope than Bourbaki.
The tests were actually applying those equations, being able to pull apart a problem into what bits of information you had, what bits you needed, and being able to combine and substitute out formulas to get the missing bit you were asked for.
In hindsight, I can vaguely see how the instruction might have helped on the test (I'd have gotten used to substituting out and deriving new equations from the old), and I can see how I might have done better on the tests (write out what equations I remembered. Write out what data I was given. Write out what data I was missing. Start substituting things out until I found a way to calculate the thing they asked for), but at the time, the only thing that helped me pass was actual instruction and practice outside of class on solving problems.
Halfway through Calc III it was interesting to watch the visual students, who normally would do very well, have a rough time, while other students, who just treated it as an abstract system had more overhead/trouble/were slower when learning initially, but it paid off when getting to un-visualizable systems.
[1] E.g. http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.ht...
To be fair to the prof, you cannot actually picture 11 dimensions. That looks like a joke to me (hey, I laughed).
I was never able to adjust to the recitation-style math lectures in University and ended up having to, as you say, teach myself by reading the book and posing questions to myself as I went along.
I was and still am angry that quality of education I was receiving in exchange for tens of thousands in tuition was considerably poorer than what I was used to receiving for free in public High School.
I agree with the majority of the author's points, but I despise his quick judgement on freshman students complaining about calculus.
I also said the same 'ironically stupid thing' in my freshman year, but that's because I _dreaded_ doing calculus as it's traditionally taught. It's much harder to find elegance in calculus than it is in say, algebra or geometry. (Mostly because the 'grunt' work behind it is so much more tedious.) Those are similar to programming in the sense that coding has elegant, extensible solutions and quick, dirty hacks. With calculus, I always felt like I was a inadequate human version of Mathematica.
More simply put, I could always solve problems using shortcuts in high school both to save time and to give myself more of a mental challenge. In intro calculus classes, there is no such thing.
By analogy, it's like choosing to use options / maybe monad instead of null in a service that takes input. The end service's functionality is the same regardless! (maintainability might be a different matter though, hehe)
Programming is about problem solving first, and teaching programming is about teaching someone to look at a problem analytically, and how to use an abstract flow of logic to solve a problem, and how to diagnose issues with your own logic when things go wrong. It's about critical thinking. It's about attention to detail.
It's about all of these things first, and about slapping keys second. But too many people see learning programming as learning the act of typing code, and focus too much on rote memorization of syntax, or teaching tools instead of thinking. Someone should come out of a course saying that they learned how to think in this new way as programmer, not that they learned a new programming language.
I didn't pursue a career in mathematics, like a lot of my classmates, but these lessons gave me more anything else I did in all years spend on 'education'.
It describes a 4-year program, from 8 to 11 grade of russian school, for kids from 13-14 to 16-17 years old respectively.
It's a part of the prevailing attitude "Everyone should go to college." High schools focus on rote learning instead of critical thinking to improve their chances of admission to a good college. Then those same colleges frown on those mechanical methods. The worst part is we're training people to be spoonfed knowledge rather then seek it.
If you want an education in math, you don't memorize multiplication tables, just use a calculator. If you want training in math, you don't talk about applications and word problems and proofs, you memorize certain symbolic manipulations.
If people learned programming like they were taught math, they would be forbidden from knowing the concept of a quicksort exists until they successfully memorized and recited a 3SAT proof, "just because". It would be interesting to ace an automata theory class before writing your first line of code. Probably not required, and probably not a good idea, but it would be interesting.
You can educate a kid about what a derivative is in grade school as soon as they can slap a ruler up against a graph, what maybe 2nd grade or so? But you can't train them what a derivative is until mid to late high school, at least post-geometry and post-algebra (and some aspects of 1st year calc require a post-trig background, but not all)
This is aside from the meta issue that math is usually weaponized into a tool of filtration, because anyone can master it given enough innate skill or sweaty effort, so it seems "fair" to use for filtration. If you eliminate its usefulness as a filtration system, that doesn't mean as a culture we're not going to filter, it just means we're going to torture undergrad aged kids by filtering them on a new criteria, perhaps how well they've memorized historical names and dates, or how well they've memorized geographic maps, or how well they've memorized the bones of the human body or the electron configurations of the periodic table.
The main factor that I think led to some science happening anyway was that a "good" middle-class suburban school felt it had to own some fancy technology to come across as modern and well equipped. So we had computer labs decked out with Macs, and pretty decent chemistry equipment. But once you spent some money on some fancy stuff, you need to put it into the school day somehow.
The computers were mainly used as quasi-free-time, where you had some self-directed time to play on the Macs, as long as you were doing it in one of the "edutainment" applications. Some of those were designed by educators with a constructivist approach to free-form, experimentation-based education, which produces a kind of virtual-science environment (other applications, of course, were badly designed and not useful for learning anything). And then the chemistry lab had to be used for something too; that one was a bit more directed, generally going through some standard experiments.
I found both parts to be pretty disconnected from what we were tested on, which was maybe precisely why they were interesting and educational...
At least Computer Science was all right. But that's only because I was lucky enough that my school's CS department was large enough to afford to be taught by some seriously smart people truly dedicated to both the study of computer science and the art of pedagogy, but small enough such that the principal let the CS dept operate as it wants without interfering with the College Board's awful way of treating every subject. Part of a course (AP Comp Sci A) absolutely required you to at least dip your toes into the College Board's bullshit, and skimming over the Barron's book/taking the actual test, it seemed like the College Board had planned a lot of tedious stuff like Java/Java's standard library details, manual loop evaluations, and that infuriating GridWorld bullshit (a complicated, but still incredibly awful simulation program; the test assesses your knowledge of GridWorld's actor types and which Bug goes which way rather than assessing ... computer science, which is honestly what I fucking signed up for). The stuff in my school's course that really intrigued me and got my mind jogging (working through sorting algorithms, data structures, and big O analysis on your own after you've been taught the absolute basics) was the stuff they cut out of the AP Computer Science AB program, which was an earlier program that was deemed too difficult, I guess. As if the College Board was intentionally avoiding stuff that required analysis or actual thought.
Mathematics are the same way. Yes, you need to solve problems, but you also need to solve problems in ways that can build on your past knowledge and be shared with others.
(FYI, I'm a lifelong amateur musician, programmer, and data analyst. My formal education consisted of a double electrical engineering / mathematics major.)
x + 5 = 10 the equals sign is a magical mirror so when you take operations across it it changes them to the opposite of what they were
so adding five becomes subtracting five, multiplying by two becomes dividing by two, etc.
x = 10 - 5 by way of magical mirror
If you have an intuitive understanding like "both sides represent a number, as long as I manipulate it the same way the equation stays true" then you wouldn't have problems with logic when dividing by zero
Isn't naive pattern recognition the basis of deeper dimensional understanding (ya know, the 'theory')? Isn't this how intelligence is built?
It seems pretty easy to make rag on the lack of 'true understanding', when you've spent 25+ years recognizing the patterns.
Arithmetic, two column geometry proofs, and even transposing a sheet of music to another key, are all math.
I think the author is talking about education, not really math. Jeremy wants kids learning math to think about, be interested in, and search for meaning in math. He's right that teaching rote mechanics doesn't lead to curiosity for most people. But there are no high school classes that reliably lead to curiosity.
Yes I did. Your claim is BS.
Your claim is based on no knowledge of me and what math I learned in high school and, thus, is incompetent.
Your claim is an insult to me and the math I did learn in high school.
In the article, you imply that a student's claim "I was always good at math" is poorly founded. But for some students, that student claim is correct. Your implication is based on no knowledge of me and is incompetent. Moreover, for me you claim is flatly wrong -- In high school I always was good at math. E.g., my plane geometry teacher was severe in the extreme, likely the most competent in the city, and I commonly toasted her. Your implication that the student's claim is poorly founded is an insult to my abilities at math.
The claim in your title is guaranteed to be wrong for some thousands or tens of thousands or more good US high school math students present and past.
From research in applied math, I hold a Ph.D. degree from one of the world's best research universities and have published peer-reviewed original research in applied math and, thus, know what the heck I'm talking about.
You owe many thousands of good math students a profound apology.
I've participated in a few of these discussions so far (apologies if you're tired of hearing me repeat myself), and I truly believe that almost all these issues are downstream of a single and very fundamental problem: math teachers are rarely drawn from top math students.
Here in the US, we love a plan, and we have a potentially harmful concept of a career ladder. A classroom teacher is on the bottom run, and it's considered career progress to set the curriculum for all teachers.
My take on it is this - if the US drew it's math teachers from the top 10% of math graduates, the "plan" wouldn't be nearly as important. Yes, it's a good idea to have a general standard for where students should be, it's a good idea to have some kind of training for math teachers, and it's a good idea to check in every now and then. But think about the relative importance of "a great curriculum" vs "teachers drawn form the top 10% of math graduates".
We could change up the curriculum to reflect the problems Mr Kun has identified, but without an armada of top math teachers, it would make absolutely no difference. If we drew our math teachers from the top ranks of math students and allowed considerably autonomy (along with general guidelines), I suspect many of the improvements Mr Kun talks about (along with so many other complaints about math instruction) would happen on their own.
So... how do we get very strong math majors who are inclined to teach into the profession, and how do we keep them there? To me, this is the upstream bug fix that will be referenced (perhaps in one line) when all these other bugs are closed.
And let me tell you those are two different worlds. My first year professor actually began the year by saying "What you have been doing so far is not math. I'll show you math". At the first test, he said: "I like you guys, you make a nice gaussian curve. Ahem, well, the curve is centered around 20%". The class was full of people who had been in the top 3 all years so far. It was not pretty. It was not surprising either, as his tests were designed so that when a student gets 90% or more, he makes it more difficult for the next year.
My second professor was a huge control freak. He had been in the same class as someone who is now a Fields medallist. He knew where you were in the understanding of his course. And he would be relentless in pushing you, in a gentle but unforgiving manner. Oh the self-guilt when you were not studying. I had nightmares/dreams about him or his tests up to three years later.
All in all, it was worth it. I met people I know I will respect and be respected by for the next 40 years.
I don't think that teaching necessarily needs to pay as much as what strong math majors can earn elsewhere, but it does need to clear the "comfortable middle class" barrier. Here's the thing - the comfortable middle class barrier includes owning a pleasant house in a safe neighborhood, travel, the ability to afford child care, and so forth... in SF, that probably takes $200k or more a year. Honestly, I'm not sure $200k even covers it. So this is a tall order in some places. And while SF is insanely expensive, this is still true of a lot of urban areas (los angeles, boston, certainly new york...)
Autonomy is also critical. Strong math students will not be willing to engage in this profession if they are excessively constrained by a bureaucratic curriculum that prevents them from applying their knowledge, skill, and passion for teaching. They'll leave.
Now, combine the two - low pay and low autonomy, and there's no way.
I'm not sure people understand just how much money and autonomy it would take to get really strong math and related students to go into teaching. Incremental changes are welcome, but not close. This is an order of magnitude difference.
The one token "numerical methods" class was all about solving ... wait for it ... continuous functions.
The problem is that to fix this weird situation we have to start teaching iterative methods of solving problems at all levels of the educational system. I have the distinct impression that people at those various levels tend to assign blame to those at other levels (this article could be an example of that).
So does it really matter that high school students don't learn the wrong math? There is a much bigger question here.
But why single out math in particular? I look back on pretty much all of my K-12 education as fairly trite and superficial, in terms of "doing real work in the subject". My experience with, say, college-level history was much more intense (and seemingly more true to the field of history) than anything in high school. On the other hand, I'm not sure I would be able to do well at "real math" like calculus or graph theory or whatever you wish to deem "real math" if I was struggling with adding numbers and solving equations, and I attribute getting past such struggles to doing bountiful rote exercises...
I say this because in my experience, the first symptom of ignorance is a feeling you know a lot.
I'm depressed since childhood because since a very young age, I had read about great minds. How could I ever feel I'm "good at maths" after reading about Gauss, or Galois?
It made me feel like the lowest form of life.
That is akin to the way Military Generals feel towards Alexander the Great: You can be a great General, but you probably will never be Alexander the Great.
Maybe this should be done freshman year: Before even a single "maths" course is dispensed, a session on the achievements of Gauss and Galois, at age 17 or 19.
Maybe a brief discussion on who Lagrange was, and what he did in his teens.
This should take out any feeling of being "good at maths", and make students shut their mouth and open their ears.
After failing that class, I still took enough from it to pass subsequent proof-based math classes with good grades. Ultimately I left a top PhD program in math after 4 years and did well for myself since that failure, but it's interesting to note that transition from easy & menial calculations to full-on hard logic that challenges you at the highest level mentally. Our education system does a poor job of preparing us for it.
Edit: For further context, I was a top math & science student in NY, having placed top 50 in competitions nationwide and similarly competitive state and region-wide
I am ashamed to admit that even when I got to university, I preferred the handful of maths and physics lecturers who followed a similar approach - work through the homework, memorise the answers, and pass.
no shame in admitting that you've been thru a broken system. Tertiary education isn't about setting a bar, it's about learning and discovery - setting a bar should be left for vocational training institutions, where you get certified that you are capable of doing such and such. Universities _should_ ostensibly be about personal learning and inherent motivation. I would garner that you shouldn't even be rewarded with any sort of formal certification from a university. Those who need such a formal cert ought to take an exam from a vocational training and certification institution.
Many high school math teachers actually have no idea what math is about. It is illogical to expect them to teach what they don't know.
There is no incentive for people who would make great math teachers to go into teaching in America. No social recognition. Ridiculously low salaries. Internal pressure to conform and avoid making bad teachers look bad. Who in their right mind would willingly do that, when they could be doing mathematics?
"Math" in high school is about calculation. Math is about useful abstraction. Students are expected to jump that gap on their own, without any outside help. At the same time as being expected to learn some concepts that are actually fairly difficult in their own right.
God help any students that have a full math professor teaching freshman calculus---the lectures will be about proofs while the homework and tests will be about calculated answers.
Is arithmetic not maths?
The question of what mathematics might actually be is like the question of what pornography is. I can't define it, but I know it when I see it.
That's a fairly vague and and non-rigorous statement. Making it rigorous and specific could possibly involve mathematics, if desired.
So in this respect I see arithmetic is a bit like an instance (in the OO sense) of the act of mathematics, but mathematics is a class of thinking and, especially, expressing. In particular, the fact that most arithmetic has a fixed set of rules and you don't generally invent new rules consistent with some meta-rules - the algorithm being performed could be trivially done by a computer - suggests to me that it doesn't really require much mathematical thinking.
When I was younger (mid-teens, and aspiring programmer), adults used to ask me if mathematics was important for programming. I would reply that it is not, that it is very rare for complex arithmetic or calculus etc. to be useful in most programs. That statement was true for both how I and those I was talking to saw maths at the time. But now, looking back, I think programming does actually use some mathematical thinking - real mathematical thinking - albeit not requiring anything like the same level of rigour. You invent your symbols, and compose them to solve the problem, and try to ensure invariants are preserved, and convince yourself that all cases are handled and the result won't have holes in the proof - i.e. bugs.
1. Mathematics is the collection of topics for which there is a mathematician who considers the topic part of mathematics.
2. Mathematicians are those humans who devote their lives to expanding human knowledge of mathematics.
3. Mathematics includes geometry and arithmetic.
The idea struck me yesterday that there's no real reason one couldn't teach this in elementary school or Jr. High and maybe that would be amazing for students? These kind of courses shape the way you reason.
Also some of the crazy stuff we did for physics/chemistry required a lot of math.
If not going into the nitty gritty detail, but an overview of how derivatives/integrals are so applicable in the real world.
Yes I did. The claim is false.
"I was always good at math."
From the ninth grade on, yes. The main means of measurement were standardized tests of math ability and/or knowledge.
I'll compare 'math' aptitude, knowledge, and accomplishments with you any time, any day, for money, marbles, or chalk. I'll give you a head start and big odds, and I will totally blow you away.
F'get about my opinion. Instead, (1) in the ninth grade I was sent to a math tournament, (2) twice I was sent to NSF summer programs in math, (3) I was a math major in college and got 'Honors in Math' with a paper on group representation theory, (4) my MATH SAT score was over 750 both times (strong evidence of being "good at math"), (5) my CEEB math score was over 650, (6) I never took freshman calculus, taught it to myself, alone, started in college with sophomore calculus and made As, (7) got 800 on my Math GRE knowledge test (means I knew some math), (8) used the differential equation
y' = k y (b - y)
to save FedEx (a viral growth equation for revenue projections that pleased the Board and saved the company), i.e., an original application of math, (9) used the statistics of power spectral estimation of stochastic processes to 'educate' some customers and win a competitive software development contract, (10) did some original work in stochastic processes to answer a question for the US Navy on global nuclear war limited to sea, (11) studied solid geometry in high school and later used it, the law of cosines for spherical triangles, to find great circle distances in a program, I designed and wrote, to schedule the fleet at FedEx, a program that pleased the Board, enabled funding, and saved the company, (12) my Ph.D. research was in stochastic optimal control, complete with measurable selection, that is, 'math'.
The claim is false, badly false.
Finally, as you hint, we will end with original work done, and I will pull out two of my peer-reviewed published papers and my Ph.D. dissertation, and with what you wrote you won't have the prerequisites to read any of them. Then, you lose the bet.
I feel sorry for your students. Go back to teaching the quadratic equation and binomial coefficients and f'get about your broad views of 'math'.
Some problem, while accurate in that the process to get to the answer worked, was a word problem. Part of this problem read "The Earth has two suns. One is blue, the other is green." He stops this early in the problem, since even here, they have made the problem about something that is completely unrelated to reality. Think back to high school - you have not lived long enough (in most cases) to be able to mentally reapply a process to a problem you have had in your own life, so education should be going far out of its way to present problems in a way that the kids encountering them can understand. The Wire had a great example - using gambling to teach probability (there's a moral argument there that I won't touch - the point is the kids could understand why they were learning math, since it solved a problem or gave them an advantage that they could immediately relate to). Programming could help with this (I haven't taken a math class since my senior year of high school, and always did well in all of them, but I feel like I understand why I would use algebra better now than ever, since I can actually relate to the concept of variables in reality, rather than "make the numbers into letters"). Baseball has taught me more about statistics and data analysis than any class I ever took.
I remember learning about finding the slope of an equation in 7th grade. I had a huge argument with my teacher (I was a bit of a handful back then), because she actually could not give me a single example of why I would ever need to care about the slope of a line beyond future math classes. That's a problem.
TL;DR - The author is likely more frustrated with American education's tendency to remove all relatability from a subject (and then not arming teacher's with good examples of how to reapply the methods they learned to a problem that encourages more investigation) than he is strictly accurate about students not actually doing math. And I agree.
Sometimes, it is necessary to create unrealistic scenarios and problems because real world problems may be too complicated for beginners. So you end up oversimplifying. In this process, sometimes the essence of the material is lost.
A solution is to pay equal attention to the problem sets and examples as the rest of the course content.
His point is that you're not doing "math" in Algebra class because "math" (as practiced by mathematicians) is actually a highly creative process, and there's nothing creative about, e.g., word problems.
I'm not trying to be impolite here; I just want to know how far my views are from the mainstream.
I know I felt that mathematics was completely devoid of context during my high school and undergraduate years.Now that I have problems to solve, I have a real purpose when I go back and relearn what I was taught years ago.
Perhaps we should not have mathematics classes at all. Instead, we should just expect to encounter mathematics every subject, and the mathematics is taught where appropriate.
(Thinking otherwise is like thinking that invading Iraq was about WMDs and the US gov't was noble-but-fumbling. Contorting ourselves into logical pretzels to preserve an illusion.)
I wish people like Edward Frenkel well. (Author of "Love and Math: The Heart of Hidden Reality", who tries to undo the damage done by math education.) But they're fighting an educational system which is inherently opposed to supporting critical thought, which fires effective teachers who refuse to work in the correct ideological framework.
The converse side of the idea - which I'm presently experiencing - is to definitionally heavy and try and show the proofs for all sorts of things. From my personal experience, this is equally as bad - I have real difficulty following proofs, where applying the results of said proofs is straightforward - easier to memorize, and apply, and slowly from that work backwards to figure out what the ramifications are elsewhere. But that's a process which takes years - I have 1 semester so in a practical sense memorization is the key.