Looking back, it wasn't because the material was hard, or boring, but because it was completely unmotivated. They may as well have been asking me to organize piles of toothpicks or count ceiling tiles to fill up the class periods. There was simply no sense to it.
I think this is partially understood, and the attempted solution is to try to find applications for the math you're learning. Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. So they go home, ask their parents "what do you use Algebra for?" and get "I don't." and that's that.
I think, along with application concepts, the history of these maths...why were they created, about the creators who came up with the concepts, what were they trying to solve, let's solve the same problems with these new techniques, etc. This would have grabbed me and pulled me in. Instead you sometimes get a little box in the reading quickly going over all this (if you are lucky) before a hundred problems are dumped on you to drill through. It's rarely talked about in class, and if it is, it's by a teacher who doesn't know or appreciate the value of this history at all.
By doing this you show applicability to entire fields, even if the child doesn't understand what's involved in the field. Those kids that say "when I grow up I want to be a biologist" will encounter stories about people who learned to apply logic or math or fractions or whatever to solve difficult problems in biology or whatever, and then relive those moments as they try to solve the same or similar problem using the tools they were just given.
The problem is that if you don't fill the period with busywork? The students don't build up the intuitive, gut-level skills they need to handle the higher-level stuff.
Math requires building muscles in your head, and muscles take reps. Lots of reps. Let's look at the elementary level: it's not enough to simply introduce BEDMAS or least-common-denominator and write a few problems that get the student to check off "okay, I know that" from the list of stuff they know.
The student has to be so practiced at these concepts that they could do it while trying to juggle 9 other extremely complicated brand-new concepts in their head because that's what you need to do when you build calculus upon factoring upon algebra upon fractions upon arithmetic.
If you fail to do this, you get kids walking into grade 9 math class who've forgotten everything they were supposed to know about fractions and teachers have too much new material to cover to re-teach fractions (my wife is a math teacher. This happens way, way too much - too many students enter her class missing necessary prerequisite skills). Students struggling with applying the basics cannot properly grasp the new material - so the basics have to be rock solid.
I think you're absolutely right that math has to be motivated. A lot of the effort to motivate it is to provide problems that students understand relate to the real world. Most of them come across as contrived.
You hit on a good concept with the idea of incorporating the history of math. I'm working on a Geometry curriculum that does just that. Understanding math in the context of a grand story of mankind trying to understand the world around us is a lot more interesting than as a set of received wisdom that must be memorized.
I would love to see more math material along these lines, but there isn't much out there, and it takes a long time to produce. I've been working on mine for around 6 months and am just up through looking at how the discovery of incommensurable lengths influenced Plato's philosophy. The idea is to work through the history and philosophy in parallel with Euclid's Elements, relating points back and forth where possible.
I've tried teaching a history of math class to local homeschool kids along those lines, focusing on Egyptian - Greek history, with some success. It takes a lot of research and work though.
> Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to.
Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications.
Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?
The irony is that all mathematics up to the high school level was historically motivated by practical problems. Algebra began as a system of rules to fairly partition land, for irrigation or inheritance, for example. But most of the high school math teachers are not actually mathematicians, so they themselves have no clue what algebra and calculus are used for. There ought to be incentives for professionals of all fields to go into teaching.
Focusing on the age is probably the wrong approach. Math true then is still true now. The problem is that the curriculum is now held sacred (and I mean that fully as nastily as possible), and it is being held there by people who are now the blind-leading-the-blind unto the sixth or seventh generation. I've gotten around in the math world a lot since primary school, and what I was taught vs. any actual modern mathematical practice, across the entire spectrum from utterly practical to the utterly theoretical, bears surprisingly little resemblance to what I wasted my time with in school. The current curriculum is bizarrely overfocused on real analysis to the near exclusion of all else (a few other disciplines will be introduced, generally poorly motivated either practically or theoretically, fiddled with for a couple of weeks, then dismissed; matrix math, for instance... in primary school nobody ever gave me a clue why we cared about this, and once you're done solving polynomial sets of equations (itself an unmotivated activity), you're pretty much done), and even if we grant that real analysis really is worth crowding out everything else, it still comes with a very 19th century flavor to the whole thing, too, wasting time on some dead-ends and missing some things that would be both mathematically and practically useful, too. (For instance, numerical approximations methods are usually covered as "Newton' Method For Finding Roots", and that's it. There's several things I'd trade away in favor of more of this, and it's easy to motivate why this is useful stuff.)
I'm pretty pessimistic in the short term about any efforts to reformulate math curricula, though, due to the aforementioned blind-leading-the-blind unto the seventh generation problem; math educators aren't even aware how bad the problems are, let alone in any position to fix them themselves, or aware enough to ask for help.
I think one problem is that mathematicians would have high school students focus on solving puzzles and slowly introducing technical content and proof-technique, while the rest of the world cries out that they need real-world applications and preparation for calculus.
The truth is that the puzzles and proof techniques are what develop mathematical thinking skills. Every mathematician knows this, but no administrators are publicly willing to believe it.
As a student in my first /real/ real analysis course (called "Advanced Calculus I" by the registrar, predictably), I have a slight quibble with your characterization of high school "calculus" as real analysis. What we did in high school, which I now think of as nothing but "calculus" in scare quotes, is to real analysis as arithmetic is to mathematics proper---because that's all it was, glorified arithmetic. Even though I had a great teacher and an understanding principal who allowed me to just go and study math on my own for two years, even then I was misled into the glorified-arithmetic world of studying "calculus" for advanced standardized tests. That's all high school math ever was, and when I scored in the 99th percentile on all the standardized tests, it was because they quite literally just asked arithmetic questions (sometimes disguised as "plug this into the quadratic formula", sometimes as "take the derivative then plug in this value").
TL;DR: high school math, and even much of what's taught in an undergraduate degree now (in math!), is just glorified arithmetic.
The curriculum changed. For example, logarithm tables were a standard topic. I don't know haw many hours in a year were use to explain it. Now that subject luckily has disappeared because it's not necessary.
Note: There are some tables that are very similar, for example the voltage of a thermocouple.
As a person who does some math, I disagree with this article. I agree with the author that mathematics is the language of abstraction, but do not think that school mathematics should put abstract before the concrete. One would think that the failure of "New Math" of the 1960s would have been enough:
Just because something is abstract and more general does not necessarily imply that it is more insightful than concrete calculations. I feel that abstract mathematics should be confined to higher education. A good antidote to such articles are these by legendary mathematicians:
This is like learning art history instead of how to draw. How does the "majestic harmony of Platonic solids" help a middle schooler actually do some math? Why should the average sixth grade class learn about Riemann surfaces when most of them haven't even learned what an exponent is yet?
An occasional lecture on a 'fun' topic (or even better, an assigned reading and short essay -- expository writing should be practiced in every subject starting in middle school or earlier) could be motivational. But the author wants to spend "20% of class time opening students' eyes to the power and exquisite harmony of modern math," which he expects would "feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements." That's fantasy land. The author sounds like what he is, a UC Berkeley mathematics professor. If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems. That is, the "stale and boring" stuff.
I've taught a lot of enrichment seminars on Saturdays. They involve "clock arithmetic," or Fermat's little theorem and the basics of RSA encryption for older students, or fractals and ways of computing fractal dimension (and exploring polynomials and iteration and Newton's method), or the differences between geometry on a beach ball, a paper towel center tube, and a piece of paper, or even a bit of Mill and Frege's ideas on how to define a number... Kids love this stuff and they get really excited and they start asking questions about infinity and making connections with their other learning. They have fun, but the topics aren't fun -- they are important and useful! Learning about solids has applications (if that's what you value) to proteins and polymers and all sorts of funky stuff. Check out http://phys.org/news/2014-02-years-mathematicians-class-soli... for some new solids mathematicians have just discovered, inspired by biological research.
Seems better than just learning to hate math. Our current system is certainly not effective in teaching young people to add fractions, which I see as the single greatest indicator for competence in college mathematics (sigh).
Ok, after another moment's thought your argument seems similar to "Why should kids read stories and books? To succeed in writing dissertations they need grammar. All that extra reading of literature should be confined to the side..."
I don't see how teaching the motivations, history, context, and future uses of the current topic of study on Mondays (while using the other four days for more traditional teaching) would be significantly worse than what we do now.
It actually sounds a lot better.
> If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems
Further, the only time I ever needed it "down cold" was when college professors were insisting I not use a calculator on exams. The rest of the time, I was better off knowing about the context and motivation behind what we were doing, not calculating faster.
This experience is shared by friends of mine who are now pursuing PhDs in the hard sciences or working in finance: the computer does the calculating, but you need to know how to write mathematics, not how to manipulate numbers (by hand).
I'm very much a math person, but I think this author is being rather naive. In the current political climate surrounding education, it seems like madness to say, "Teach math more like we teach art!" After all, art classes in the US are pretty much first on the chopping block for the many, many people who think education should be all about test scores and checklists.
I would love to see more kids exposed to "real math" somehow. But there's a valid question of how to make sure they wind up with practical, applicable skills in the process, and I found the arguments here about "the value of abstract thinking" too vague to feel convincing (at least to folks who aren't already immersed in real math). I don't know the answers to those questions, but I certainly hope we can find them.
Not to mention, "The curriculum is 1000 years old" is a terrible argument from a logical perspective. I have no idea if that means math is timeless or dated.
There are other funny gems.
"A mathematical statement is either true or false"
The problem is that they already don't end up with practical applicable skills, and this method (teaching with an emphasis on discovery over lecture) is known to have results.
See, for example, this lengthy technical document describing (in more scientific terms) teaching math like art [1]
I agree, and I don't see the relevance of 1,000 old findings to an bad math curriculum. The beauty of math is that findings 1,000 years ago are still relevant today.
I think first steps in improving math education are teaching concept instead of calculation, and focus on understanding math principles instead of memorizing.
As a math/CS major, when I wanted to fully understand the basic concepts of pi, pythagorean theorem, I looked back at how they were originally discovered. It gave math a story behind it, and these concepts stuck with me because I understood them. I think we could improve people's understanding of math through telling the story of many of the concepts, rather than throwing concepts at students and hoping that they stick. Stephen Wolfram discusses the concept will on Concept vs. calculation: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...
Memorization is the road to failure in math, and many teachers push memorization rather than understanding.
I read that article over the weekend. I agree completely, but it's downstream of a bigger problem. Think of it this way - when we file the bug fix, the report will reference a different bug fix that solved this one downstream.
We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically.
One thing I've noticed is that the US system loves a "plan". This might be a result of the career ladder - a teacher is on the bottom rung, whereas someone who sets the curriculum for all teachers has moved beyond that lowly spot. But what we really need is an armada of accomplished math majors in actual teaching positions.
Without that, the plan won't really matter. With it, the plan? It matters, sure, but I'd personally be inclined to give quite a bit of latitude and autonomy to people who were in the top tier of their class and majored in math and who are inclined to teach.
> We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically.
Then don't pay me 25%-33% of what I'd make working in industry.
I don't think that you can appreciate this stuff until you actually have the mechanical background in math.
Unlike most subjects in K-12, you can't BS math. You're right or wrong. There's no subjectivity in evaluation, and no opportunity for lazy students like me to BS a tired teacher with flowery prose that doesn't say much.
Why not make math accessible in easier or more practical ways? My trigonometry teacher walked us out into the schoolyard and gave teams of students assignments to measure the heights of various objects. It was a simple thing, but it made the not-so-interesting mechanics of looking up sine/cosine/tangent in some table real.
This is the viewpoint of an engineer. The salient difference in thought is that the goal is rarely to be right (initially), but to get insight about mathematical stuff.
Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can be unsatisfying and ugly. Your goal then is to revise it (or completely rewrite it) to make it more beautiful and insightful.
Here is an example: the 7 Bridges of Konigsberg problem is one of the most famous (solved) problems in all of mathematics. One can easily give a proof by exhaustion, but that is wholly unsatisfying and perhaps the ugliest possible proof. A much better proof involves some insight into graph theory, and a satisfying and elegant proof would give you a characterization of these kinds of trails in graphs.
You don't need a mechanical background to start trying to solve such puzzles (I know because I've taught it that way), and you don't need any mathematical background to appreciate the very elementary proof. But as you let students flounder with puzzles like these, you can slowly introduce technical matter. The point is that it has a context they care about (people like working on puzzles!).
Which may also be someone's work, but it was one I read recently.
Second, I learned some Calculus in school, how about you? That is not a subject a 1000 years old. So we haven't been teaching it that way for 1000 years.
Third, did this guy not have story problems? Applied Math is basically what this author is claiming we don't have.
Fourth, Graphing is pretty new. We didn't use to visualize plots because we didn't have the resources, I can't tell you when that became part of algebra, but it was in the past 100 years.
I can't believe LA Times ran such a poorly informed piece.
Math is like knitting. It's not useful to read about it. You must do it.
When reading an advanced math book for the fist time, I can spend 1 hour per page, because I must copy the proof, try a few "improvements" that are dead ends, do some exercise, and understand what is the main idea of that part.
It's very difficult to mixing some modern subjects without calculation. Most of the time, the idea is to just hide the technical details so now it's totally unintelligible and the only possible way to learn it is by memorization. It's possible to teach magic and religion in the same way.
I like to add some comments about applications in my classes, but they are marked as off topic, and they are short because usually there is no time to spare.
Some of the proposed subjects are plausible to teach, for example module arithmetic. I'd like to see a discussion in school about the module 2, 9 and 10 arithmetic. (2 is almost intuitive, 10 is easy to see, and 9 is the base for the "rule of nines" test, that is usually teached without proof)
But, for example, Riemannian geometry is difficult. It's possible to pick a sphere surface and show that the geodesic intersects in two points. Explain the sum of the angles in a triangle? And then what? I don't have any intuition about hyperbolic geometry. It's difficult to students to see what is happening with the geometry in a plane in spite it's easy to draw, and we have paper everywhere to tray.
> I recently visited students in fourth, fifth and sixth grades at a school in New York to talk about the ideas of modern math, ideas they had never heard of before. [...] I used a Rubik's Cube to explain symmetry groups: Every rotation of the cube is a "symmetry," and these combine into what mathematicians call a group. I saw students' eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group.
It's an interesting topic, but the problem is not how much you can explain but how much the students understand. Did he make any test to measure it? Can the students solve the Rubik's Cube?
I'd like to see a test for this idea. Pick 10 classes and divide them in two groups. Give 5 of them the traditional education and 5 of them the "modern topics" proposal. Pick the classes at random. Pick the teachers at random!!! Don't compare one class with a standard underpaid "traditional" teacher with 30 students and a class with a specially selected for the experiment teacher with 10 students. And compare them with multiple choice exercise, not teacher opinion or self report interest.
The biggest problem in education isn't the curriculum or method of teaching or the teachers, it's how motivated the student is (which is usually a product of their home environment). Trying to solve this social issue with "technical" fixes is not going to succeed.
> By hiding math's great masterpieces from students' view, we deny them the beauty of the subject.
The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.
As a mathematician, how much time do you spend studying proofs written by other people?
I'm genuinely curious, because I think that if I wanted to be a novelist, I would spend a lot of time reading novels before trying to write one. I feel like we never have children read the great "literature" of mathematics, but then expect them to write their own proofs and are surprised when they hate it.
Teachers should have the place of manager and coach with specialists brought in who are subject matter experts. We had this in art (painting,music), PE and computer programming when I was in public school.
To force or demand that teachers be skilled and have deep interests in everything they teach is not realistic.
Math curriculum may be rote and boring, but if you don't master arithmetic, fractions, exponents, etc., you're going to have a lifetime of poor decisions regarding investments, financing, etc., which will be very costly.
It sounds like a return to the "new math" of the 1960s. Compare this article:
> If we are to give students the right tools to navigate an increasingly math-driven world, we must teach them early on that mathematics is not just about numbers and how to solve equations but about concepts and ideas.
> It's about things like ... clock arithmetic — in which adding four hours to 10 a.m. does not get you to 14 but to 2 p.m. — which forms the basis of modern cryptography, protects our privacy in the digital world and, as we've learned, can be easily abused by the powers that be.
> Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra.
[+] [-] bane|12 years ago|reply
Looking back, it wasn't because the material was hard, or boring, but because it was completely unmotivated. They may as well have been asking me to organize piles of toothpicks or count ceiling tiles to fill up the class periods. There was simply no sense to it.
I think this is partially understood, and the attempted solution is to try to find applications for the math you're learning. Except that once you hit even basic algebra, you quickly run out of applications a young student can relate to. So they go home, ask their parents "what do you use Algebra for?" and get "I don't." and that's that.
I think, along with application concepts, the history of these maths...why were they created, about the creators who came up with the concepts, what were they trying to solve, let's solve the same problems with these new techniques, etc. This would have grabbed me and pulled me in. Instead you sometimes get a little box in the reading quickly going over all this (if you are lucky) before a hundred problems are dumped on you to drill through. It's rarely talked about in class, and if it is, it's by a teacher who doesn't know or appreciate the value of this history at all.
By doing this you show applicability to entire fields, even if the child doesn't understand what's involved in the field. Those kids that say "when I grow up I want to be a biologist" will encounter stories about people who learned to apply logic or math or fractions or whatever to solve difficult problems in biology or whatever, and then relive those moments as they try to solve the same or similar problem using the tools they were just given.
[+] [-] Pxtl|12 years ago|reply
Math requires building muscles in your head, and muscles take reps. Lots of reps. Let's look at the elementary level: it's not enough to simply introduce BEDMAS or least-common-denominator and write a few problems that get the student to check off "okay, I know that" from the list of stuff they know.
The student has to be so practiced at these concepts that they could do it while trying to juggle 9 other extremely complicated brand-new concepts in their head because that's what you need to do when you build calculus upon factoring upon algebra upon fractions upon arithmetic.
If you fail to do this, you get kids walking into grade 9 math class who've forgotten everything they were supposed to know about fractions and teachers have too much new material to cover to re-teach fractions (my wife is a math teacher. This happens way, way too much - too many students enter her class missing necessary prerequisite skills). Students struggling with applying the basics cannot properly grasp the new material - so the basics have to be rock solid.
[+] [-] SamuelMulder|12 years ago|reply
You hit on a good concept with the idea of incorporating the history of math. I'm working on a Geometry curriculum that does just that. Understanding math in the context of a grand story of mankind trying to understand the world around us is a lot more interesting than as a set of received wisdom that must be memorized.
I would love to see more math material along these lines, but there isn't much out there, and it takes a long time to produce. I've been working on mine for around 6 months and am just up through looking at how the discovery of incommensurable lengths influenced Plato's philosophy. The idea is to work through the history and philosophy in parallel with Euclid's Elements, relating points back and forth where possible.
I've tried teaching a history of math class to local homeschool kids along those lines, focusing on Egyptian - Greek history, with some success. It takes a lot of research and work though.
[+] [-] wisty|12 years ago|reply
Algebra is problem solving. You have a formula describing the problem, and want to re-arrange it to find one of the variables. It has a lot of useful and logical applications.
Trigonometry though ... screw that. "Bob the builder has made a truss, and wants to figure out the angle A. He doesn't own a protractor, and can only measure a few of the lengths." Seriously?
[+] [-] eliteraspberrie|12 years ago|reply
[+] [-] jerf|12 years ago|reply
I'm pretty pessimistic in the short term about any efforts to reformulate math curricula, though, due to the aforementioned blind-leading-the-blind unto the seventh generation problem; math educators aren't even aware how bad the problems are, let alone in any position to fix them themselves, or aware enough to ask for help.
[+] [-] j2kun|12 years ago|reply
The truth is that the puzzles and proof techniques are what develop mathematical thinking skills. Every mathematician knows this, but no administrators are publicly willing to believe it.
[+] [-] belovedeagle|12 years ago|reply
TL;DR: high school math, and even much of what's taught in an undergraduate degree now (in math!), is just glorified arithmetic.
[+] [-] gus_massa|12 years ago|reply
Note: There are some tables that are very similar, for example the voltage of a thermocouple.
[+] [-] sn41|12 years ago|reply
http://www.youtube.com/watch?v=8wHDn8LDks8 [Tom Lehrer, "New Math"]
Just because something is abstract and more general does not necessarily imply that it is more insightful than concrete calculations. I feel that abstract mathematics should be confined to higher education. A good antidote to such articles are these by legendary mathematicians:
http://pauli.uni-muenster.de/~munsteg/arnold.html [V. I. Arnold]
http://ega-math.narod.ru/LSP/ch1.htm [L. S. Pontryagin]
[+] [-] bhixon|12 years ago|reply
An occasional lecture on a 'fun' topic (or even better, an assigned reading and short essay -- expository writing should be practiced in every subject starting in middle school or earlier) could be motivational. But the author wants to spend "20% of class time opening students' eyes to the power and exquisite harmony of modern math," which he expects would "feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements." That's fantasy land. The author sounds like what he is, a UC Berkeley mathematics professor. If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems. That is, the "stale and boring" stuff.
[+] [-] kaitai|12 years ago|reply
Seems better than just learning to hate math. Our current system is certainly not effective in teaching young people to add fractions, which I see as the single greatest indicator for competence in college mathematics (sigh).
Ok, after another moment's thought your argument seems similar to "Why should kids read stories and books? To succeed in writing dissertations they need grammar. All that extra reading of literature should be confined to the side..."
[+] [-] DerpDerpDerp|12 years ago|reply
It actually sounds a lot better.
> If students are going to succeed in his college-level courses then they need the basics down cold, which means memorization, repetition, and application to concrete problems
Further, the only time I ever needed it "down cold" was when college professors were insisting I not use a calculator on exams. The rest of the time, I was better off knowing about the context and motivation behind what we were doing, not calculating faster.
This experience is shared by friends of mine who are now pursuing PhDs in the hard sciences or working in finance: the computer does the calculating, but you need to know how to write mathematics, not how to manipulate numbers (by hand).
[+] [-] Steuard|12 years ago|reply
I would love to see more kids exposed to "real math" somehow. But there's a valid question of how to make sure they wind up with practical, applicable skills in the process, and I found the arguments here about "the value of abstract thinking" too vague to feel convincing (at least to folks who aren't already immersed in real math). I don't know the answers to those questions, but I certainly hope we can find them.
[+] [-] jcampbell1|12 years ago|reply
There are other funny gems.
"A mathematical statement is either true or false"
[+] [-] j2kun|12 years ago|reply
See, for example, this lengthy technical document describing (in more scientific terms) teaching math like art [1]
[1]: http://www.ams.org/notices/201308/rnoti-p1018.pdf
[+] [-] acknowledge|12 years ago|reply
I think first steps in improving math education are teaching concept instead of calculation, and focus on understanding math principles instead of memorizing.
As a math/CS major, when I wanted to fully understand the basic concepts of pi, pythagorean theorem, I looked back at how they were originally discovered. It gave math a story behind it, and these concepts stuck with me because I understood them. I think we could improve people's understanding of math through telling the story of many of the concepts, rather than throwing concepts at students and hoping that they stick. Stephen Wolfram discusses the concept will on Concept vs. calculation: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...
Memorization is the road to failure in math, and many teachers push memorization rather than understanding.
[+] [-] geebee|12 years ago|reply
We need to draw math teachers from the top tier of math majors (or physics, stats, etc). And if we do that, it's very likely that people with real insight into math will teach it very effectively but less mechanically.
One thing I've noticed is that the US system loves a "plan". This might be a result of the career ladder - a teacher is on the bottom rung, whereas someone who sets the curriculum for all teachers has moved beyond that lowly spot. But what we really need is an armada of accomplished math majors in actual teaching positions.
Without that, the plan won't really matter. With it, the plan? It matters, sure, but I'd personally be inclined to give quite a bit of latitude and autonomy to people who were in the top tier of their class and majored in math and who are inclined to teach.
[+] [-] DerpDerpDerp|12 years ago|reply
Then don't pay me 25%-33% of what I'd make working in industry.
[+] [-] j2kun|12 years ago|reply
[+] [-] Spooky23|12 years ago|reply
Unlike most subjects in K-12, you can't BS math. You're right or wrong. There's no subjectivity in evaluation, and no opportunity for lazy students like me to BS a tired teacher with flowery prose that doesn't say much.
Why not make math accessible in easier or more practical ways? My trigonometry teacher walked us out into the schoolyard and gave teams of students assignments to measure the heights of various objects. It was a simple thing, but it made the not-so-interesting mechanics of looking up sine/cosine/tangent in some table real.
[+] [-] j2kun|12 years ago|reply
Mathematicians do (and students should) focus on proofs over computations, and there is an extreme amount of subjectivity there. If you produce a false proof, it can nevertheless be beautiful and yield fantastic insights. Likewise, a correct proof can be unsatisfying and ugly. Your goal then is to revise it (or completely rewrite it) to make it more beautiful and insightful.
Here is an example: the 7 Bridges of Konigsberg problem is one of the most famous (solved) problems in all of mathematics. One can easily give a proof by exhaustion, but that is wholly unsatisfying and perhaps the ugliest possible proof. A much better proof involves some insight into graph theory, and a satisfying and elegant proof would give you a characterization of these kinds of trails in graphs.
You don't need a mechanical background to start trying to solve such puzzles (I know because I've taught it that way), and you don't need any mathematical background to appreciate the very elementary proof. But as you let students flounder with puzzles like these, you can slowly introduce technical matter. The point is that it has a context they care about (people like working on puzzles!).
[+] [-] hessenwolf|12 years ago|reply
[+] [-] dabrowski|12 years ago|reply
[+] [-] drakaal|12 years ago|reply
http://j2kun.svbtle.com/you-never-did-math-in-high-school
Which may also be someone's work, but it was one I read recently.
Second, I learned some Calculus in school, how about you? That is not a subject a 1000 years old. So we haven't been teaching it that way for 1000 years.
Third, did this guy not have story problems? Applied Math is basically what this author is claiming we don't have.
Fourth, Graphing is pretty new. We didn't use to visualize plots because we didn't have the resources, I can't tell you when that became part of algebra, but it was in the past 100 years.
I can't believe LA Times ran such a poorly informed piece.
[+] [-] gus_massa|12 years ago|reply
When reading an advanced math book for the fist time, I can spend 1 hour per page, because I must copy the proof, try a few "improvements" that are dead ends, do some exercise, and understand what is the main idea of that part.
It's very difficult to mixing some modern subjects without calculation. Most of the time, the idea is to just hide the technical details so now it's totally unintelligible and the only possible way to learn it is by memorization. It's possible to teach magic and religion in the same way.
I like to add some comments about applications in my classes, but they are marked as off topic, and they are short because usually there is no time to spare.
Some of the proposed subjects are plausible to teach, for example module arithmetic. I'd like to see a discussion in school about the module 2, 9 and 10 arithmetic. (2 is almost intuitive, 10 is easy to see, and 9 is the base for the "rule of nines" test, that is usually teached without proof)
But, for example, Riemannian geometry is difficult. It's possible to pick a sphere surface and show that the geodesic intersects in two points. Explain the sum of the angles in a triangle? And then what? I don't have any intuition about hyperbolic geometry. It's difficult to students to see what is happening with the geometry in a plane in spite it's easy to draw, and we have paper everywhere to tray.
> I recently visited students in fourth, fifth and sixth grades at a school in New York to talk about the ideas of modern math, ideas they had never heard of before. [...] I used a Rubik's Cube to explain symmetry groups: Every rotation of the cube is a "symmetry," and these combine into what mathematicians call a group. I saw students' eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group.
It's an interesting topic, but the problem is not how much you can explain but how much the students understand. Did he make any test to measure it? Can the students solve the Rubik's Cube?
I'd like to see a test for this idea. Pick 10 classes and divide them in two groups. Give 5 of them the traditional education and 5 of them the "modern topics" proposal. Pick the classes at random. Pick the teachers at random!!! Don't compare one class with a standard underpaid "traditional" teacher with 30 students and a class with a specially selected for the experiment teacher with 10 students. And compare them with multiple choice exercise, not teacher opinion or self report interest.
[+] [-] w1ntermute|12 years ago|reply
[+] [-] j2kun|12 years ago|reply
The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.
[+] [-] SamuelMulder|12 years ago|reply
I'm genuinely curious, because I think that if I wanted to be a novelist, I would spend a lot of time reading novels before trying to write one. I feel like we never have children read the great "literature" of mathematics, but then expect them to write their own proofs and are surprised when they hate it.
[+] [-] sitkack|12 years ago|reply
To force or demand that teachers be skilled and have deep interests in everything they teach is not realistic.
[+] [-] arbitrage|12 years ago|reply
[+] [-] WalterBright|12 years ago|reply
[+] [-] dalke|12 years ago|reply
> If we are to give students the right tools to navigate an increasingly math-driven world, we must teach them early on that mathematics is not just about numbers and how to solve equations but about concepts and ideas.
> It's about things like ... clock arithmetic — in which adding four hours to 10 a.m. does not get you to 14 but to 2 p.m. — which forms the basis of modern cryptography, protects our privacy in the digital world and, as we've learned, can be easily abused by the powers that be.
to the Wikipedia entry at http://en.wikipedia.org/wiki/New_Math :
> Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra.
[+] [-] j2kun|12 years ago|reply
[+] [-] spenrose|12 years ago|reply
[+] [-] AutoCorrect|12 years ago|reply
[+] [-] SixSigma|12 years ago|reply
just like painting and decorating isn't called interior design
[+] [-] reallysmart|12 years ago|reply
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