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.
Such cynicism. What do you have against Feynman's presence in the California committee to select, among other things, math books?
Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math:
> It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and intersection of sets and the special use of the brackets { } and so forth, all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or to be taught in school.
The reason new math failed is because it was organized and administered by committees (comprised of people who don't do math). So your example is humorously evidence to my claim.
What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be spent on each topic.
This is the problem. If you're looking for things to test your students on then you will invariably add pointless content.
Physicists are no exception (they invented the elaborate and pointless notation for vectors, the bra and ket). I don't propose we test students on mathematical notation, not at all. I propose we actually have them do mathematics (craft proofs, make conjectures), and I assert that any committee trying to craft a curriculum would ruin that. They have already ruined proofs in high school geometry.
"evidence to my claim"? But it's an example of a committee rejecting new math, not proposing it. And do you seriously want to argue that Feynman doesn't "do math"? Actually, it looks like you do. Why should only a certain type of math practitioner determine the curriculum? Do we only let people like Ramanujan set the curriculum? 'Cause that won't work.
What you're complaining about is called "democracy." Yes, democracy is messy. The current movement towards high-stakes testing in the US is anti-democratic in part because it coalesces more power into fewer people. If you don't want a committee then what do you want? A so-called "education czar" with absolute power to make things work the way you think it should?
That was a fun read, but no one can teach that way. You can't become a teacher by studying math, physics, chemistry, biology, literature, french, programming, history, geography or english. You become a teacher in the US by getting a teaching degree. I believe that is the biggest problem we have in the US education system.
Teachers only need minimal knowledge of their subject, and if someone discovers that they don't actually like teaching then they just keep doing it anyway because they got a teaching degree, and can't do much else with it.
Low standards for math teachers is a huge concern, and Lockhart addresses it too, I believe.
Some possible ways to help: give teachers less class time and more time to prepare interesting lessons, pay teachers more, raise the standards of training and hiring.
I know a number of math teachers who have undergraduate degrees in math, and then attended a 2 year masters program to get their degree in teaching.
I also know a number who came from a humanities background, and "learned enough math" in the single, poorly taught math-for-teaching class they had in getting a masters.
One of these groups makes better teachers (and likely has a wider career path, because of the undergraduate degree).
I also know some smart teachers that started with something else and ended up teaching high school. I have more examples of people who got a degree in something else and wanted to teach, but then found out how hard and expensive that transition is.
I really enjoyed your link too. So thanks for posting. But I do have some qualms about the article's position.
> Two weeks of content are stretched to semester length by masturbatory definitional runarounds.
That Lockhart describes trigonometry as "masturbatory" I find ironic. Lockhart argues that math is fun for its own sake. Personally, I think that math is fun per se. I.e. patterns and ideas and puzzles are fun. But historically, much of math was invented as engineering-shortcuts. E.g. trig identities for navigation, newton's calculus for friction, etc. The philosophical Greeks and their geometry I think are the exception. So I think most people would rebut that math's actual raison d'etre lies in it's tangible application. The manner in which Lockhart describes math is why physicists joke that theoretical math is nothing more than masturbation.
> In certain of the arts and crafts, we sometimes do precisely this--requiring a child to "express himself" in paint before we teach him how to handle the colors and the brush. There is a school of thought which believes this to be the right way to set about the job. But observe: it is not the way in which a trained craftsman will go about to teach himself a new medium. He, having learned by experience the best way to economize labor and take the thing by the right end, will start off by doodling about on an odd piece of material, in order to "give himself the feel of the tool." [1]
I do agree that math curriculums are excruciatingly insipid. And Lockhart makes some good points. But I think this other link makes a strong argument too, which is contradictory to what Lockhart argues. So I was thinking: for each unit, maybe teachers could explain what problems the unit was actually solved historically. This differs from the article, because 6th graders don't really need to know about Riemann Surfaces if that's not what the unit covers. This differs from your Lockhart, because I don't think it's as "masturbatory". But it still affords students a glimpse of where all this pedagogical grinding is headed, allows students to reify to something relevant, and retains that "puzzle" feeling.
Again, I like puzzles. But teachers have a responsibility to teach everyone, not just the select few who solve rubiks cubes in their spare time. How much of that would be solved by removing the educational mandate on math, I don't know.
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!).
I guess I have an engineering viewpoint as well--which may not be surprising because I, well, was one. I never especially connected to geometry with its proofs in high school. What I did learn was lots of basic trig and algebra and the beginnings of calculus (which I took more of in college). The result is that while I've never had much of an interest or flair for advanced math, I did develop most of the math skills I needed both for school and, more importantly, through career and life.
I've had stats as well. As I was just discussing with someone in a different context, I think stats, accounting, and similar applied math would be far more useful to most students than either most calculus (past simple differentiation which is used in a lot of contexts), proofs, or set/group theory. And, arguably, could be linked a lot better to real-world applications.
The question is what are you trying to develop in your students: factual knowledge or mathematical thinking skills? I think it's the latter, and if that's the case then proofs are by far the best way to do it.
And do you really think students would be any more motivated to learn about accounting than they are about current math? I mean, come on, nobody wants to do that stuff. It's a chore even for adults.
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.
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.
And yet, we somehow think that being able to draw an accurate picture of an ellipse given its equation is useful, along with knowing the derivative of every inverse trig function, and countless other things.
I agree that the exact form of an ellipse is not useful, but some general idea is.
Other example is the method for long multiplication. I think that it's not 1000 years old, but I can't find a source. One of the formed methods was to use square tables and the identity xy = ( (x+y)^2 - (x-y)^2) )/4. It guesses it was explained in schools, but now it's almost a curiosity. To use it, you must have a book with the tables of the squares, and the last edition of those books is from 1888: http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarte...
Don't worry, Heisenberg discovered the matrix formulation of quantum mechanics without knowing matrices http://en.wikipedia.org/wiki/Werner_Heisenberg#Matrix_mechan... :). More seriously, if you ever want to study quantum mechanics, you should learn matrix algebra before.
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.
"I have a slight quibble with your characterization of high school "calculus" as real analysis"
Yeah, I know, but if that's not what high school math is, what is it? As hostile as I am I'm not quite ready to write it off as "entirely not math"... there's some math content in there, after all.
I do want to point out that I am in contact with reality and do not expect students to start with number theory, nor do I realistically expect high school to ever cover what you covered in "real" real analysis. (Real numbers are terribly complicated things and I don't think high school should cover all the endless, endless corner cases of those either; time is short and the fields are vast.) What I'm really looking for is dropping a lot of stuff that is both practically and theoretically pretty useless, and replacing it with a lot of stuff that is, well, both more practical and more theoretically useful. Some of the core will survive (integration and differentiation aren't going anywhere, those really are fundamental to any reasonable math education, though we might spend several fewer weeks walking through all the special cases of each), but there's a lot of room for improvement.
> TL;DR: high school math, and even much of what's taught in an undergraduate degree now (in math!), is just glorified arithmetic.
Is this a bad thing? You can design and build airplanes with what doesn't amount to anything more than glorified arithmetic. I don't see the value in teaching kids more than that.
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.
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.
Try motivating art students to paint fences. And never show them any "art" except different colored fences. That's what this article is about. If you teach real math, you can actually motivate people to want to learn it.
>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)
While I get what you're saying, the motivation of the student is highly influenced by the curriculum and the teacher.
Having a meaningful curriculum content with a teacher who presents it in a way which the student can relate to is surely possible (but woefully rare). Good teaching can change things around.
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.
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]
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"
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.
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:
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.
Since you mentioned knitting (and I agree with your analogy), check out the great intuition you can get about hyperbolic geometry via crochet and knitting.
Hyperbolic baby pants also convertible into a 2-holed torus (not while on baby I guess), with added note that on 2-holed tori you can make maps that require 8 colors instead of just 4 like maps on the plane or sphere: http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2
Geometry, topology, map-coloring theorems, crafts: all things many kids never get to do anymore.
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).
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.
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.
> 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.
This comes from my interactions with colleagues at my own undergraduate institution. So it's biased evidence, but my institution was also one of California's largest math teaching credential programs, so I think it's a reasonable sample.
I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math."
All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused).
This is what they mean by the "blind leading the blind."
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.
I literally study the proofs of others every working day, for at least a few hours a day. But I am a young mathematician and still inexperienced with research.
I imagine that once you are a professor you can spend less time studying the details of proofs because you can generate them on your own once you understand the big picture. But even so this depends on your field (combinatorics folks, for example, REALLY spend a lot of time studying proofs).
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.
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.
> 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?
> 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.
It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic.
I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days.
>Trig
Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.
What motivated trig functions for me was writing 5 line BASIC programs as a kid to draw circles, spirals, etc. Realizing that COS was the X coordinate and SIN was the Y coordinate made it all clear.
Unfortunately there's no easy, low-friction way for kids today to do that with their PC.
With that kind of attitude we are just going to make more average Joe and Carlas. If you take what you are saying to a logical conclusion it leads to the exclusion of the many in fear of over education which I would argue can never occur.
There's a really, really simple way to get kids to love trig. Get them to write a video game.
Seriously, as a video game programmer, you use trig _every day_. If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem. If you want to find the angle between two points, use arc tangents.
Once kids have mastered all of that, you can teach them law of cosines and get into linear algebra; simple rotational matrices which can let you do collision detection and a whole bunch of other really, really cool stuff.
So yeah, Bob the builder might be boring as shit as a word problem, but Bob the builder might be a pretty cool video game. Without trig, as a game, Bob the builder is probably going to be boring as shit too.
"If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem."
Using cosine and sine and then Pythagorean theorem to get the length of the hypotenuse doesn't make sense - in order to get the length of the X and Y components with cosine and sine you need to already know the length of the hypotenuse.
You're right, I did state it a little awkwardly, but the statements are correct. I'll give a more concrete example:
I'm firing a projectile at 52 degrees and it needs to move 10 units. I'm at (5,7), where does the projectile end up?
delta x = cos(theta) * 10
delta y = sin(theta) * 10
So my final position is (5 + 6.16, 7 + 7.88), or (11, 15).
If I already knew how far the delta X and delta Y were, I could get the length of the hypotenuse with sqrt(6.16^2 + 7.88^2). That equals roughly 10, of course. If you want to try this out yourself, just remember that most trig functions are done in radians, so convert theta to radians with pi * theta (degrees) / 180.
Yes, I certainly wasn't disagreeing that trig is useful for games. The "if you want to try this out yourself" comes across a little patronizing in this context...
It was more trying to be helpful. A lot of people's heads explode when they see trig, and they stumble on things like radians vs degrees. You can do some really cool stuff with just a little bit of knowledge.
Yeah, I don't get why more people don't get trig - it's always come very naturally to me, but I have to work to recognize that's not the case for everyone, particularly when confronted with experiences from my past like working on homework with math majors and finding them confused when I applied a trig identity to the problem...
> 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?
You can also throw in some economics: Bob would be fired and/or get better equipment.
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.
To a certain extent that's a product of a bad curriculum though. If you teach something, tick it off, teach something else, tick that off. And build a large number of things ticked off in that manner - that don't reuse the things taught earlier - then of course you're not going to get the reps in.
As compared to, say, the syntax in a programming language. Where you don't write out hello world sixty times to get the bracket rules, you just reuse the rules in other things you're doing often enough that they become ingrained.
personally, busy work just never sticks for me, i don't remember almost any of the math i did in HS and uni that was just "do these problems over and over till you remember it" because it just wasn't all that interesting, and this goes for any subject.
However some concepts, limits, derivatives and integrals i remember perfectly well (5+ years later) because my math teacher explained how the all fit together, what they represented wrt the real world and how they can be useful and that actually got me interested.
Yea you need to know the basics, but i daresay busy work isn't the best way to teach them.
It's the difference between skills and knowledge. Limits/derivatives/integrals are interesting knowledge so you can discuss them verbally, even if you don't have extensive practice in using them.
The basic, core stuff is the opposite. Fractions, bedmas, etc. you don't care enough to talk about, but if I handed a problem that used those concepts you could do it in your head, because they're skills that have been developed down to an instinctive level.
Common factoring is dull. There's nothing to say about common factoring. But if I show you an equation where you can apply it, you'll spot it instinctively because you've had common factoring ground into your reptilian brain.
Dull work can be made fun with a little effort. One of my high school teachers had us form lines, we refactor on the fly, said the factors of a quadratic out loud, and then were able to shoot the opponent (like a duel).. After a few shootouts everyone got very quick at refactoring. That little fun boost made the grind feel meaningful.
There is quite a lot to say about 'common factoring.'
For example, nobody knows whether there is an efficient algorithm to factor integers (where efficient depends on the number of digits in the integer). However, there is an efficient algorithm to factor polynomials, and you can write all irreducible polynomials of a certain degree as factors of one mother polynomial.
Moreover, being able to factor integers efficiently would allow you to break a lot of encryption schemes.
The problem with your comment is that busywork is not (and never has been) mathematical weight lifting. You don't gain thinking skills by memorizing multiplication tables or practicing adding fractions, you gain it by solving problems you don't know how to solve.
Of course it's not enough to state something, but making the students reiterate what you've stated a hundred times is only marginally better.
Another problem is that no student really learns ANYTHING until they need to use it to do something else. No student learns fractions until they need to use it to do algebra, and they don't learn algebra until they need it to do calculus. And they don't learn calculus until they need it to do engineering/physics or higher mathematics (real analysis, which they don't really learn until they do measure theory...).
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.
Yeah, absolutely. In fact, I'd say that it's just as important that students receive confirmation of this history and motivation in their other subjects. For example, in various history classes, we learn about kings and wars and important philosophers. But we rarely learn about mathematicians and why their contributions were important, of if they were polymaths, we discuss their non-mathematical contributions, but omit their mathematical ones entirely.
Then in writing and literature we spend endless hours on appreciating tiny non-relevant symbols, but don't read something like Ringworld and spend part of a session calculating and relating the size of it. Conversely, in Math class, writing a paper on the history and influence of Platonic solids or similar would have been an interesting break and let kids struggling with crunching numbers take an interest and shine a little.
Or imagine a math course that analyzes things like agricultural output from Roman times to modernity, involving geometry, arithmetic, algebra, percentages (taxes to the king!), nutrition, etc. Extend out to how greater agricultural output per unit land produces surpluses that enable people to take up other labors like politics or art or music.
Off the cuff examples that would need more careful thought, but hopefully the idea is sound.
Bringing math into other non-math subjects, and bringing arts into non-art subjects I think helps prop up what students see as isolated subject "silos" that have nothing to do with anything.
Literature and history courses mutually reinforce each other, and math and science course mutually reinforce each other, but there's a tremendous gap between these two groups of courses where they don't reinforce each other.
This really needs to end because it all is connected and builds relevance for students.
I'm still struggling with how to handle this. On the one hand, it makes a lot of sense for it to be public. On the other, I've given up a lot to spend time developing it (including going part time on my day job), and I would love to be able to devote more time to it, which would require making some money off of the material in some way.
I've been thinking about doing a kickstarter, but I'm interested in exploring other models. I don't have a ton of time, and I'm always torn between spending time trying to figure out how to publish/market/exploit the material versus spending the hundreds of hours reading history and math and trying to actually write the curriculum. So far I've just been punting on it till I get this first course done.
I don't have a lot of knowledge, but there are websites that allow for the monetization of lesson plans. Maybe workbooks and videos to help teachers present the materials?
I do think a kickstarter would definitely help as would getting it out in other teacher's classrooms to get earlier feedback.
Just checked out your site, it is like you are brother from different mother. Awesome stuff. I have a project I am working on that I will ping you with to get your feedback, I think you will like it.
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.
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.
You can teach exponential growth, compound interest and ratios before mastery of the mechanics. To many, mastery of the mechanics inoculates them not from darkness but from a scientific mind.
I know many people with a mastery of arithmetic and fractions yet still make poor mathematical decisions. Conversely, I see people make projections and use algebra w/o even knowing they are doing it.
Learning assembly isn't is a prerequisite for making useful higher level software. Knowing all the words doesn't make us a better essayist. Yes the low level stuff can make the result better, the spark still has to exist for the flame to grow.
I've talked with people who sell finance, and they make bank off of people with a poor grasp of mathematical fundamentals. And that's just a small bit of what it will cost a person to not master it.
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[ 5.0 ms ] story [ 189 ms ] thread> 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.
Ahh, I see. If you are the person behind "Math ∩ Programming" then I can see what you have against that committee. Feynman wrote, arguing against new math:
> It will perhaps surprise most people who have studied these textbooks to discover that the symbol ∪ or ∩ representing union and intersection of sets and the special use of the brackets { } and so forth, all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, in business arithmetic, computer design, or other places where mathematics is being used. I see no need or reason for this all to be explained or to be taught in school.
What does a committee do when they are tasked with designing a curriculum or standards? They identify what should be tested, and then they pick the order that the topics are taught, likely outlining a time constraint on how long should be spent on each topic.
This is the problem. If you're looking for things to test your students on then you will invariably add pointless content.
Physicists are no exception (they invented the elaborate and pointless notation for vectors, the bra and ket). I don't propose we test students on mathematical notation, not at all. I propose we actually have them do mathematics (craft proofs, make conjectures), and I assert that any committee trying to craft a curriculum would ruin that. They have already ruined proofs in high school geometry.
What you're complaining about is called "democracy." Yes, democracy is messy. The current movement towards high-stakes testing in the US is anti-democratic in part because it coalesces more power into fewer people. If you don't want a committee then what do you want? A so-called "education czar" with absolute power to make things work the way you think it should?
Teachers only need minimal knowledge of their subject, and if someone discovers that they don't actually like teaching then they just keep doing it anyway because they got a teaching degree, and can't do much else with it.
Some possible ways to help: give teachers less class time and more time to prepare interesting lessons, pay teachers more, raise the standards of training and hiring.
See http://www.statisticbrain.com/iq-estimates-by-intended-colle... and note that, relative to other college majors, education majors tend to be dumb.
I also know a number who came from a humanities background, and "learned enough math" in the single, poorly taught math-for-teaching class they had in getting a masters.
One of these groups makes better teachers (and likely has a wider career path, because of the undergraduate degree).
> Two weeks of content are stretched to semester length by masturbatory definitional runarounds.
That Lockhart describes trigonometry as "masturbatory" I find ironic. Lockhart argues that math is fun for its own sake. Personally, I think that math is fun per se. I.e. patterns and ideas and puzzles are fun. But historically, much of math was invented as engineering-shortcuts. E.g. trig identities for navigation, newton's calculus for friction, etc. The philosophical Greeks and their geometry I think are the exception. So I think most people would rebut that math's actual raison d'etre lies in it's tangible application. The manner in which Lockhart describes math is why physicists joke that theoretical math is nothing more than masturbation.
> In certain of the arts and crafts, we sometimes do precisely this--requiring a child to "express himself" in paint before we teach him how to handle the colors and the brush. There is a school of thought which believes this to be the right way to set about the job. But observe: it is not the way in which a trained craftsman will go about to teach himself a new medium. He, having learned by experience the best way to economize labor and take the thing by the right end, will start off by doodling about on an odd piece of material, in order to "give himself the feel of the tool." [1]
I do agree that math curriculums are excruciatingly insipid. And Lockhart makes some good points. But I think this other link makes a strong argument too, which is contradictory to what Lockhart argues. So I was thinking: for each unit, maybe teachers could explain what problems the unit was actually solved historically. This differs from the article, because 6th graders don't really need to know about Riemann Surfaces if that's not what the unit covers. This differs from your Lockhart, because I don't think it's as "masturbatory". But it still affords students a glimpse of where all this pedagogical grinding is headed, allows students to reify to something relevant, and retains that "puzzle" feeling.
Again, I like puzzles. But teachers have a responsibility to teach everyone, not just the select few who solve rubiks cubes in their spare time. How much of that would be solved by removing the educational mandate on math, I don't know.
[1] http://www.scribd.com/doc/36325362/The-Lost-Tools-of-Learnin...
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.
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!).
I've had stats as well. As I was just discussing with someone in a different context, I think stats, accounting, and similar applied math would be far more useful to most students than either most calculus (past simple differentiation which is used in a lot of contexts), proofs, or set/group theory. And, arguably, could be linked a lot better to real-world applications.
And do you really think students would be any more motivated to learn about accounting than they are about current math? I mean, come on, nobody wants to do that stuff. It's a chore even for adults.
I'd guess most kids hate puzzles.
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.
Note: There are some tables that are very similar, for example the voltage of a thermocouple.
Other example is the method for long multiplication. I think that it's not 1000 years old, but I can't find a source. One of the formed methods was to use square tables and the identity xy = ( (x+y)^2 - (x-y)^2) )/4. It guesses it was explained in schools, but now it's almost a curiosity. To use it, you must have a book with the tables of the squares, and the last edition of those books is from 1888: http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarte...
TL;DR: high school math, and even much of what's taught in an undergraduate degree now (in math!), is just glorified arithmetic.
Yeah, I know, but if that's not what high school math is, what is it? As hostile as I am I'm not quite ready to write it off as "entirely not math"... there's some math content in there, after all.
I do want to point out that I am in contact with reality and do not expect students to start with number theory, nor do I realistically expect high school to ever cover what you covered in "real" real analysis. (Real numbers are terribly complicated things and I don't think high school should cover all the endless, endless corner cases of those either; time is short and the fields are vast.) What I'm really looking for is dropping a lot of stuff that is both practically and theoretically pretty useless, and replacing it with a lot of stuff that is, well, both more practical and more theoretically useful. Some of the core will survive (integration and differentiation aren't going anywhere, those really are fundamental to any reasonable math education, though we might spend several fewer weeks walking through all the special cases of each), but there's a lot of room for improvement.
Is this a bad thing? You can design and build airplanes with what doesn't amount to anything more than glorified arithmetic. I don't see the value in teaching kids more than that.
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.
While I get what you're saying, the motivation of the student is highly influenced by the curriculum and the teacher.
Having a meaningful curriculum content with a teacher who presents it in a way which the student can relate to is surely possible (but woefully rare). Good teaching can change things around.
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.
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
There are other funny gems.
"A mathematical statement is either true or false"
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.
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]
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.
See how "parallel" lines act and the angle sums of triangles end up smaller than 180 degrees: http://theiff.org/oexhibits/oe1e.html
A TED lecture is linked: http://geknitics.com/2010/11/sunday-knitting-science-hyperbo...
Hyperbolic baby pants also convertible into a 2-holed torus (not while on baby I guess), with added note that on 2-holed tori you can make maps that require 8 colors instead of just 4 like maps on the plane or sphere: http://www.ravelry.com/projects/smbelcas/hyperbolic-pants-2
Geometry, topology, map-coloring theorems, crafts: all things many kids never get to do anymore.
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.
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..."
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).
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.
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.
Then don't pay me 25%-33% of what I'd make working in industry.
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.
[1] http://jeremykun.com/about/
I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math."
All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused).
This is what they mean by the "blind leading the blind."
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.
I imagine that once you are a professor you can spend less time studying the details of proofs because you can generate them on your own once you understand the big picture. But even so this depends on your field (combinatorics folks, for example, REALLY spend a lot of time studying proofs).
To force or demand that teachers be skilled and have deep interests in everything they teach is not realistic.
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.
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?
It does, but for average Joe and Carla, who's day job involves making sure the box of sprockets sitting on the loading dock matches the P.O., and who's most pressing math problem is balancing their check book every few days and using turbo tax once a year, they'll never encounter anything more complicated than very basic arithmetic.
I think the hardest math problem most people encounter is figuring out percentages so they can fill out the "tip" on the credit card receipt when they eat out. Yeah it's "algebra" but it's a very limited class of problem and people just use a calculator or a tip calculator to do it these days.
>Trig
Yeah, trig really requires good motivation. The answer to most trig questions really is "buy the appropriate measuring tool" not "do a bunch of calculations and look up the result of the trig functions in a table". I don't ever remember having the calculations for cos(), sin() explained to me, they're just magic functions, number goes in, number comes out.
Unfortunately there's no easy, low-friction way for kids today to do that with their PC.
Being able to animate and play with your math helps a lot. GeoGebra is a good example, but it's not the holy grail.
Seriously, as a video game programmer, you use trig _every day_. If you want to find the distance between two points when you have the angle (eg. I'm firing a projectile at a certain angle), use cosine for the length of the x axis and use sine for the y axis. You get the hypotenuse from using pythagoras' theorem. If you want to find the angle between two points, use arc tangents.
Once kids have mastered all of that, you can teach them law of cosines and get into linear algebra; simple rotational matrices which can let you do collision detection and a whole bunch of other really, really cool stuff.
So yeah, Bob the builder might be boring as shit as a word problem, but Bob the builder might be a pretty cool video game. Without trig, as a game, Bob the builder is probably going to be boring as shit too.
Using cosine and sine and then Pythagorean theorem to get the length of the hypotenuse doesn't make sense - in order to get the length of the X and Y components with cosine and sine you need to already know the length of the hypotenuse.
I'm firing a projectile at 52 degrees and it needs to move 10 units. I'm at (5,7), where does the projectile end up?
delta x = cos(theta) * 10 delta y = sin(theta) * 10
So my final position is (5 + 6.16, 7 + 7.88), or (11, 15).
If I already knew how far the delta X and delta Y were, I could get the length of the hypotenuse with sqrt(6.16^2 + 7.88^2). That equals roughly 10, of course. If you want to try this out yourself, just remember that most trig functions are done in radians, so convert theta to radians with pi * theta (degrees) / 180.
You can also throw in some economics: Bob would be fired and/or get better equipment.
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.
As compared to, say, the syntax in a programming language. Where you don't write out hello world sixty times to get the bracket rules, you just reuse the rules in other things you're doing often enough that they become ingrained.
However some concepts, limits, derivatives and integrals i remember perfectly well (5+ years later) because my math teacher explained how the all fit together, what they represented wrt the real world and how they can be useful and that actually got me interested.
Yea you need to know the basics, but i daresay busy work isn't the best way to teach them.
The basic, core stuff is the opposite. Fractions, bedmas, etc. you don't care enough to talk about, but if I handed a problem that used those concepts you could do it in your head, because they're skills that have been developed down to an instinctive level.
Common factoring is dull. There's nothing to say about common factoring. But if I show you an equation where you can apply it, you'll spot it instinctively because you've had common factoring ground into your reptilian brain.
For example, nobody knows whether there is an efficient algorithm to factor integers (where efficient depends on the number of digits in the integer). However, there is an efficient algorithm to factor polynomials, and you can write all irreducible polynomials of a certain degree as factors of one mother polynomial.
Moreover, being able to factor integers efficiently would allow you to break a lot of encryption schemes.
Of course it's not enough to state something, but making the students reiterate what you've stated a hundred times is only marginally better.
Another problem is that no student really learns ANYTHING until they need to use it to do something else. No student learns fractions until they need to use it to do algebra, and they don't learn algebra until they need it to do calculus. And they don't learn calculus until they need it to do engineering/physics or higher mathematics (real analysis, which they don't really learn until they do measure theory...).
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.
Yeah, absolutely. In fact, I'd say that it's just as important that students receive confirmation of this history and motivation in their other subjects. For example, in various history classes, we learn about kings and wars and important philosophers. But we rarely learn about mathematicians and why their contributions were important, of if they were polymaths, we discuss their non-mathematical contributions, but omit their mathematical ones entirely.
Then in writing and literature we spend endless hours on appreciating tiny non-relevant symbols, but don't read something like Ringworld and spend part of a session calculating and relating the size of it. Conversely, in Math class, writing a paper on the history and influence of Platonic solids or similar would have been an interesting break and let kids struggling with crunching numbers take an interest and shine a little.
Or imagine a math course that analyzes things like agricultural output from Roman times to modernity, involving geometry, arithmetic, algebra, percentages (taxes to the king!), nutrition, etc. Extend out to how greater agricultural output per unit land produces surpluses that enable people to take up other labors like politics or art or music.
Off the cuff examples that would need more careful thought, but hopefully the idea is sound.
Bringing math into other non-math subjects, and bringing arts into non-art subjects I think helps prop up what students see as isolated subject "silos" that have nothing to do with anything.
Literature and history courses mutually reinforce each other, and math and science course mutually reinforce each other, but there's a tremendous gap between these two groups of courses where they don't reinforce each other.
This really needs to end because it all is connected and builds relevance for students.
I've been thinking about doing a kickstarter, but I'm interested in exploring other models. I don't have a ton of time, and I'm always torn between spending time trying to figure out how to publish/market/exploit the material versus spending the hundreds of hours reading history and math and trying to actually write the curriculum. So far I've just been punting on it till I get this first course done.
I do think a kickstarter would definitely help as would getting it out in other teacher's classrooms to get earlier feedback.
I totally support this.
I know many people with a mastery of arithmetic and fractions yet still make poor mathematical decisions. Conversely, I see people make projections and use algebra w/o even knowing they are doing it.
Learning assembly isn't is a prerequisite for making useful higher level software. Knowing all the words doesn't make us a better essayist. Yes the low level stuff can make the result better, the spark still has to exist for the flame to grow.
just like painting and decorating isn't called interior design