Of course, his main point is to advocate for using computers to teach and learn math; he's selling Mathematica/Wolfram Alpha.
But this is dangerous and a little shallow. And here is why.
Math is the language of science; just like a C -- or any other programming language -- is a language of computing; just like English is a language of humans.
There are many ways to learn a language, but I think they all have something in common. You have to spend a lot of time learning, reading and writing. The writing is quite important. You have to write the language to be good at it.
The thing about math is that currently to write as it is supposed to look like can only be done most conveniently on paper. Writing a math formula in Mathematica or Wolfram alpha requires a latex-like mini language, another thing to learn, and can be disrupting.
This fact alone means that computers can only supplement; and they can be an excellent supplement. But it can not replace doing math by hands, on papers, particularly for those who are learning this new language.
I do not disagree with you, but this is not answering his arguments.
Math is the language of science[...].
There are many ways to learn a language, but I think they all have something in common. You have to spend a lot of time learning, reading and writing. The writing is quite important. You have to write the language to be good at it.
He's not arguing that scientists should drop writing math altogether (although you have to admit that they are spending huge amount of efforts making the computers do the calculation for them). He's addressing the more general issue of "teaching math in elementary schools". Why do we do it today? not everyone is going to be a scientist!
The thing about math is that currently to write as it is supposed to look like can only be done most conveniently on paper. Writing a math formula in Mathematica or Wolfram alpha requires a latex-like mini language[...].
Yeah writing math formulas on a computer is a pain. But:
- you and I come from a generation of pen and paper math solving. The idea of markup annotation of your text is still very young, but is it so far fetched to imagine that in few generations, all text will be filled with metadata and tags? LaTeX is a pain because of our current primitive keyboards and screens.
- more people should work towards improving this. Wolfram Alpha is doing good things, they're not the only ones working on this either, but we still need to bring more attention to the issue.
Also, who cares if he sells WA. Let's talk about his idea, not his persona.
I like your analogy with learning a language, and I would use that analogy to say the current education system is only teaching grammar rules. Imagine sitting for hours a day for an entire school year conjugating verbs without ever reading a story, listening to a conversation, or trying to pronounce words.
There is so much more to mathematics than calculation. Proofs, number theory, and simulations can all be understood by children and are much more important to their education than long division.
Err. This what he saying, you focus more on manipulating equations in the language of maths such as algebra, calculus, and less time on remembering what's the answer to 7 * 8.
Is this really a win? For most small problems or subproblems, a person who has solved similar problems many times before can solve it mentally much more quickly than by consulting a computer.
Which is which? A person who works in either role will do so more quickly by virtue of not needing to constantly consult a machine over trivial problems.
Focusing on algebra, calculus, and so forth would produce more engineers/scientists/mathematicians etc . These are things rote teaching the times table can not do. His not saying let computers do all maths, just the repetitive bits.
Symbolic Manipulation is still necessary.
I learned vectors(Mostly for games), eigenvectors far more intuitively by programming them on computer as a kid, then calculating them by hand.
I knew what adding two vectors actually represented, or what the crossproducts was, rather than just calculating it(Which the computer always did).
I get top marks in calculus, linear mathematics and most college level maths classes, but my little secret is that i'm not actually very good knowing my times tables by 'art. If someone asked me to calculate a basic times table question, it would take a few seconds to work it out manually. Yet the general public thinks that's weird, but I actually found it's quite common at this level.
I think he argues from the wrong point. Yes, we need to teach math, not
calculating. No, the argument is not "because we can do calculating with
computers today". Lockhart got it right in his famous lament - the problem is
that we face a system of self-sustaining bullshit that actually thinks it's
teaching math.
My guess is that Conrad Wolfram probably plans to conveniently also sell the
accompanying software for this new education, which would make it natural for
him to focus on this point. I actually agree that computers can help us educate
people better, and that we need to create appropriate software for this. But my
concern would then be that such software must be free/libre. No compromise. We
cannot risk the education of future generations to be locked into $corporation's
proprietary products.
I recently had the experience of helping a young algebra student with her homework. She is very bright and is in an accelerated math program that teaches algebra in the 7th grade.
However, it was so disappointing to see the curriculum. I was in the same accelerated program 15 years ago and the curriculum hadn't changed a bit. It was still vague and unhelpful. They sent her home for the summer with a large packet of problems to solve without any explanation how to solve them.
I majored in math and I still had to guess at what the worksheets were asking the student to do. They used vague variables and were essentially just teaching rote memorization. It reminded me of my own confusion in mathematics when I was growing up, how one year a variable would mean one thing and the next year it would mean something else. "Solve and explain," means just as little to me now, using math on a daily basis, as it did to the 11 year old version of me.
After so many years, it would have been nice to see the curriculum focus more on application of mathematics and less on how to memorize and calculate.
I've noticed the same thing in all the school curricula I've seen. The word problems are often vague, and all to often completely ambiguous. They require guessing what the question writer had in mind when she wrote the question, which are usually unstated assumptions that apply only to the way math is done in schools. This is not the way to teach the kind of precise thinking and careful definition of terms that one needs to understand why math is the way it is: the “study of precisely defined objects.”
Teachers, if you want your pupils to understand how to create a mathematical problem from a description of a practical one, you must train them first to state the situation with absolute precision and without ambiguity. The best way to do that is, as in the teaching of all virtues, to fastidiously follow that habit yourself. That the questions in math textbooks fail to do that so often is another sign of the enormous institutionality of our education problems.
> how one year a variable would mean one thing and the next
> year it would mean something else
I'm probably misunderstanding what you mean, but isn't that the purpose of a variable? In programming, a variable foo in one method which represents Widget objects is completely independent from a variable foo in another method which represents Gadget objects. In math, solving for X in one problem might be to find the missing angle X, but in another problem it might be to determine the elapsed seconds X.
I'm not disagreeing that variables are meant to be variable.
However, it's important to teach context behind variables before asking an 11 year old to solve them or else it becomes a point of confusion. They are asked to use rote memorization to complete the calculations but they don't seed their memory before asking.
Well stated. For further deep thinking and a wonderful read, I recommend the original, "A Mathematician's Lament" by Paul Lockhart (2009): http://www.maa.org/devlin/lockhartslament.pdf [PDF]
I don't see why the software used is particularly important. The point should be the mathematics, rather than specific tools. I don't know how hard it is to transition from Wolfram Alpha, to, say, Matlab, but I don't think it's difficult.
In high school physics about 22 years ago, we did an experiment to calculate the acceleration due to gravity by repeatedly dropping a weight with a ticker tape attached to it that fed through a hole punch tool and measuring the spacing of the holes in the tape over time. As part of the analysis and write-up, we had to calculate the standard deviation across our sample runs. That seemed really tedious to me, so I wrote a program in BASIC that took a data file, ran the calculations and saved the results. I recall a certain to-do as the school deliberated over whether this was an acceptable way to complete the assignment, and I was asked to include the source code for my program so the teacher could determine that I understood the concept well enough to program it.
I think it would be cool if calculators weren't allowed in school unless you programmed them yourself. Now trying to enforce that might be difficult but it is an interesting idea.
I noticed a few of articles on Hacker News has been about math recently. According to Google Trends, math is also rising in popularity, especially in US.
I think math is even more appealing to programmers. You are allowed to assume that certain things happen. You may define a set on the fly and all of a sudden, the set is populated without pressing a single key. Let me try, {prime numbers}. There, I got an infinitely large data structure. I hope HN doesn't crash after posting this comment.
As programmers, we put abstract ideas into implementations. Math seems to forgo all of that. Everything just happens magically as long as your mind can wrap around it and you have a precise definition for it. No debugging. Isn't this a dream?
So I am wondering if there's a community of programmers who have turned to devote their time to math. Like the video said, computer frees you from computation. Hence, it makes sense for math's popularity to rise, especially since we got more people interested in programming. Math seems like the natural next step. I don't have any hard data about this, just my intuition.
"And, of course, as we all know, with a computer you can take a simple problem like solve 5x2+2x+1=7 but you can make it harder, for example solve 5x4+2x+1=7. The principle of the problem is still the same."
This is not, in fact, true for polynomial equations of order greater than four. From Wikipedia, "the Abel–Ruffini theorem states that there is no general algebraic solution... to polynomial equations of degree five or higher." This is often featured as one of the main results in an undergraduate abstract algebra course.
I'm all for automating calculations, but the algorithms behind calculations and their proofs are important and often fascinating. Handing students an implementation of such algorithms, e.g. Mathematica, takes away their opportunity to code up the algorithms and understand them deeply.
For that matter, asking students at the high school level to derive the quadratic formula will likely be quite challenging for even the best students in the class.
Asking students to come up with the formulas for 3rd or 4th degree polynomials on their own is probably going to be beyond anyone below the very top tier of human capability. A teacher would be extremely lucky to see one or two students that can do this in their career.
Then again, most people could easily solve these problems if they know what terms to Google :)
Anyway, as I recall the third- and fourth-degree cases require clever substitutions (whereas the derivation of QF is just completing the square with general coefficients).
On one end you will hear people equate math to calculating, this is almost exclusively how math is taught in school today, and on the other, people like to talk about math ideas, and how we shouldn't really teach people how to calculate anymore since we got computers for that. Like in many aspects of life, learning math is about balance.
In truth, people should learn from both areas simultaneously, since they are mutually connected. Calculating is a tool that makes you a better problem solver, and being a problem solver will allow you to find new tools to calculate faster. You can't separate either discipline from one another.
In my view, I think students should take one calculation, and one deeper math class a year starting in high school. That wat they could delve deeper in learning how to calculate (learning different methods for multiplication that might come in handy when doing mental math, for example), and start dealing with real, thought-heavy math from an earlier grade (learning about combinatorics, graph, game, and probability theory early in high school, etc).
I would say, stop concentrating on calculation, and teach math. However, calculation is often a useful way to ensure you understand something. So, calculation will likely still be a part of learning math, but it shouldn't be the focus.
It's sad how many people think that math is arithmetic, and that if we let kids use computers or calculators then they're not "doing math". Not just the general public but PTA leaders, school board members, and school administrators.
I live in a relatively wealthy school district in Ohio where the schools generally are well-regarded. Last year, I went to the schools "parents math night". I had expected to be exasperated by, you know, those idiots who will be teaching my children (my older son is entering first grade this fall). However, I ended up impressed with the teachers and disappointed by the parents in attendance. The school had chosen the "Connected Math" curricula (connectedmath.msu.edu) for its middle school math program, and some parents were outraged that Little Johnny wasn't learning long division (or whatever). Their arguments could be summed up as, "Well, that's not the way we learned it." I angered the other parents by playing a bit dumb and asking for a short description of the curriculum. Then, I stated loudly that it "sounded delightful." The teachers barely contain their smiles. I ordered a couple of the books in used form from Amazon to see for myself, and I think I would have found the exercises fun as a middle schooler. I really wanted to ask the parents doing the complaining, "Don't you love your children? You pay more for a house just because it's in this school district, and then you lobby to dumb-down the math curriculum?"
The problem is that the textbooks are teaching calculation. So in this instance "doing math" is calculating some equation. Solve some equation is easier to write in a textbook than an unambiguous word problem that a grad student needs to solve and put the answers in the back of the book and in the teachers solution manual. Not to mention that the math teachers are not always the best.
I remember reading an article about almost the same topic by Felleisen, maybe Flatt, maybe Culpepper and some others from the PLT group, it was about using DrRacket to teach maths, there was an example involving the take-off of a rocket iirc. Is anyone able to find it? I don't remember the title nor where it was published.
I really like the idea of teaching math in tandem with elementary programming. It demystifies writing code and can be very helpful in teaching important math concepts like functions, variables and graphing. At the very least, exposure to programming at an early age will get more kids interested in the subject.
Does anyone know of good examples of schools that are already doing this with something like Dr Racket?
I remember to this day the lesson in primary school, when we first started solving word problems. It was sth like 3rd class. Teacher would ask simple question, like "how many apples we had to add to each chest, if there is 4 chests with 7 apples in each, and we need to have total of 32 apples". And kids had to "brainstorm" and translate that into mathemathic process and solve (not algebra, just "first we need to know how many apples we lack, so we substract, then we need to know how many apples per chest we lack, so we divide", we didn't knew algebra yet). If the question wasn't specific, the kid that pointed it out got praise for being observant, and teacher specified what he meant. Most kids liked that lesson, and there was fierce competition to be the first to guess. That was the moment I felt in love with math, before that I thought I'm more of a humanist (which meant it's OK I can't understand decimal fractions ;) ).
I think teaching kids to translate problem into math is very important, not only for math, but also if they want to be economists, computer programmers, etc. I cringe when I read that kids "know math, but don't know which equation to use to solve the problem". My wife taught math in secondary school for a while (in Poland, not USA, but we've recently had reform modeling our schools on those in "the West"). Many kids she taught had no idea how to translate word problem into math problem. They just pattern-matched keywords to equations. It's very sad, and I understand why they hate math - it must be completely frustrating to play this "guess the equation" game.
So - I think there should be more focus on teaching kids to understand the basic word problems, before they even know algebra. To be sure, that they understand when we need to substract x, and when we need to substract from x.
So basically what he is saying is that drilling kids into memorizing math procedures like LONG DIVISION is not a good way to get people interested in math and analytical thinking in general. Ok. Sure. He is right that modelling is probably what "people who don't get math" struggle the most with.
But saying that math is just "computational stuff, turn the crank, let the computer do it" kind of thing is ludicrous. You can use mathematics as a blackbox model to do cool demos and stuff, but at some point you //have// to open the box and see how it is done.
It just happens that the standard math curriculum (numbers, functions, sin, cos, etc.) is a very good mix of conceptual complexity and computational complexity that can all be understood with pen and paper. You won't need to trust anyone, the proof is right there on the sheet. Anyone trying to take that understand in order to replace with with a SolveEquation[.] is not on the right side of intellectualism. Demos yes, but view source (is there view source for Mathematica functions?) and explain source are //sine qua non// for deep knowledge.
This is sad. Really sad. My personal opinion is that Americans just dislike academic matters so much that they are incapable of forming a coherent solution for their children's schooling.
Let me tell you some personal anecdotes. I moved to Minnesota during my High school senior year from Vietnam. It sucked: I couldn't hold on to conversations, and people couldn't understand my pronunciation; so I was bored and very lonely. You know what I do during my lunch time? Trying to re-prove Fermat's Last Theorem. Is there any "real world" application of that? Well, there may be, but I don't care. It was fun.
You know how people listen to (and rock to) puke music on buses? I rode bus a lot during my college years, and spent that time wreaking my brain to work on various NP complete problems (my favorite is the man-woman-dog matching :D). Do I know it is beyond my capacity? Yes. Do I care about any real world problem of it? No. Why did I do it then? It's fun. It killed time. I still do this from time to time during meetings at work.
When you think about it, most of math, or most of the academia for that matter, is not readily applicable. True, people use calculus to build Hoover Dam, but the effect of that information on whether or not a student likes math is equivalent of a political speech to a person political orientation: either you already like it (math/Obama) and cheer for it, or you already hate it (math/Obama) and boo/ignore it. These things, by itself, can't change a person's opinion regarding the big picture.
You know why I like doing math in my free time? Because I have done a whole load of it. Vietnamese High School graduates are required to learn around sophomore math level in American college (and I fully expect most other East Asian systems to be the same). This explains neatly why Asian students are "smart." They freaking know the things already. Furthermore, the learning of math there involved an absurd amount of drill (compared to American high schools). Oh, calculators are banned until high school, and graphing calculators are banned. Period.
Now, should that not make me abhor math? I mean, slaving myself so much time over such stupid and abstract matter must bore me to death, right? Actually, the opposite happened. It's like taste bud. You rebuild your taste bud every so often, so if you have been eating a whole load of broccoli lately, you will start to like broccoli. Yeah, after the first few days, you can't even digest to crap. However, after a few months, you can't digest without broccoli. Humans are adaptable. Do a lot of math, and you will like it.
This also speaks of the problem with calculators and computers. The problem is no so much that they dump down math. The problem is that they alienate the students from math. When you do math, all of it, the result is made of your sweat. It may be wrong, but it's yours. It may be stupid and winded and long and whatnots, but it's yours. It's personal. I remember once, during 5th grade (yes, elementary school), there was this problem that I just could not solve, and we were kicked out of class for lunch break. Walking out, I cling to the teacher and cried out of frustration. The same frustration that keeps me up at nights when some stupid bugs could not be fixed. Must be the same frustration that all artisans experience at challenges. It's the indication of the bond that I have with my work, MY math, and it necessitates the toil, the slaving, the mind-numbing work of calculation.
I hope you see my point: there is no elevator to interest in math. Math lovers must have toiled and slaved over stupid calculation over and over, until a point that they could actually produce the trace of math working, just like the zone in programming or any other art. That trace, that focus, that absorption builds the love for math.
Of course, this means that the students must fist cross the initial barrier of math, just like the first few days of broccoli eating. In Asian cultures, academia is a point of pride. Parents take enormous pride in their children's academia success. Schools take enormous pride in their students' academia achievements. Think about football in American high schools. It's something similar. Study well, perform well in competition, and a student will enjoy numerous privileges and honors that is just unattainable through any other mean. The whole culture rates academia above any other activities, and use that as the measure for potential success. Playing sport well will earn praise, but it is understood to be temporary, fast passing. Studying well, and people will talk about how you will got to such and such colleges, and will become such and such person in the future. It's just different.
Back to problem of American math schooling. Here is my prescription: just make the damn students do it. Require them to work on math. Forget about the whole, um, personality and uniqueness and whatnots. You know where these things are from? The toil of work (for students, that's studying) builds personality, values, and one's importance and uniqueness. You know where self esteem is from? It's from holding a piece of paper on which a problem has been solved and solution has been neatly presented; it's from the overcoming of frustration and challenges that the problem presents. Artisans take pride and esteem in their works. Students take pride and esteem in their tests and homework (except when you tell them that the football players will get the praise, the money, and the sex, and only these 3 things matter). Stop worrying about overworking them. Actually, overwork them to the ground. Leave them no energy and time to experiment with drugs and sex and alcohol.
It takes challenges and toil to make a person. Stop taking these things away from the youngsters!
Back to problem of American math schooling. Here is my prescription: just make the damn students do it. Require them to work on math. Forget about the whole, um, personality and uniqueness and whatnots. You know where these things are from? The toil of work (for students, that's studying) builds personality, values, and one's importance and uniqueness. You know where self esteem is from? It's from holding a piece of paper on which a problem has been solved and solution has been neatly presented; it's from the overcoming of frustration and challenges that the problem presents. Artisans take pride and esteem in their works. Students take pride and esteem in their tests and homework (except when you tell them that the football players will get the praise, the money, and the sex, and only these 3 things matter). Stop worrying about overworking them. Actually, overwork them to the ground. Leave them no energy and time to experiment with drugs and sex and alcohol.
Um, no, this has nothing to do with hard work, and everything to do with the way mathematic is taught. I am a programmer and I would love to understand mathematics the way I understand programming. There got to be far more to mathematics than simply learning how to follow steps.
As a programmer, I write, debug, fix and improve code. With math, I don't understand why am I merely following steps when computers can do it at far higher reliability than I could.
I agree with what you are saying. I've been contemplating the reasons why I don't understand math the way I understand programming for some time, and opened up my old Calculus book a few days ago and started rereading it from the beginning. I intend to work through the entire book over the next couple of months. I believe because of the way math is taught, I never developed a truly intuitive understanding of what was going on after algebra. When I taught myself how to program, I learned then immediately applied what I was learning, and continued applying it continuously in all the software I wrote after. Algebra and arithmetic are pretty analogous in that regard.
For me, it is necessary to develop an intuitive understanding of something before I can really appreciate it, and more importantly, manipulate and apply it to arbitrary situations. The way math is taught, intuition is never really delivered. In programming, it's possible to look at an algorithm and have trouble understanding what it does. However, I have never implemented and debugged an algorithm or data structure and not developed a thorough understanding of it in the process. In Calculus, the rules were given, but there was never any effort spent to foster an understanding of why they are the way they are, or the bigger picture. To me, this would be like learning merge sort by running through the steps, but never actually implementing the algorithm as a whole to truly understand what is going on. In going back through it, I intend to relearn it the way I learned programming, so I can truly apply and reason around such a powerful tool.
Here is a famous quote where he references a passage from the book's prologue:
"Right. I don't believe in the idea that there are a few peculiar people capable of understanding math, and the rest of the world is normal. Math is a human discovery, and it's no more complicated than humans can understand. I had a calculus book once that said, 'What one fool can do, another can.' What we've been able to work out about nature may look abstract and threatening to someone who hasn't studied it, but it was fools who did it, and in the next generation, all the fools will understand it. There's a tendency to pomposity in all this, to make it deep and profound." -- Feynman, Omni 1979 (http://c2.com/cgi/wiki?FeynmanAlgorithm)
Learning Haskell is a great back-door way to learn about math, specifically abstract algebra and category theory. You'll learn about why it matters and make tons of connections between subjects you never thought were related. Plus, coming from a programming environment gives you the context you need to understand why people care.
There got to be far more to mathematics than simply learning how to follow steps.
Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely disconnected from the real world. You memorize rules of syntax and look up function definitions and follow all of the rules exactly correctly, and even after you've done what feels like a ton of that you're still just writing a dumb program that you would never use anyway. What's the point?
But I bet you didn't feel that way about programming when you were first learning, right? Things like foldl and dlopen seemed kind of neat, if not mind-blowing. People who become good at programming get there because they take delight in the simple things that make up the basics. Is there anything intrinsically beautiful about loops? There is the first time you encounter one. Oh, man, the possibilities! The feeling passes quickly, but I can't imagine how anyone who never enjoyed loops could endure the boredom long enough to become skilled at programming.
It's the same thing with math. Like programming, it really is just symbol manipulation and nothing more. The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules. Your appreciation gets deeper the more you learn. But to get there you have to enjoy the basics, the simple things. How many times did you have to write
for (i = 0; i < m; ++i)
before it became second nature? You must have derived some satisfaction from your initial fumbling with loops and functions or your interest would have died before you learned to write more complex programs. (The kids who don't enjoy the basics quit programming as soon as they realize they won't be making a state-of-the-art video game in a weekend.) Learning math is the same process; you start by enjoying where you are, even if you're starting at the mathematical equivalent of
for (int i = 0; i < 10; ++i) {
System.out.println("I love cake " + i + " times!");
}
That may not sound very attractive to an educated person with many sophisticated things to think about. It's entirely possible that there are some people who are not simple-minded enough to enjoy math. However, I think those people do not enjoy programming, either ;-)
Like programming, it really is just symbol manipulation and nothing more. I strongly disagree. Arithmetic, perhaps... but not "math."
Some of the best math proofs require little-to-no direct symbol manipulation. You might be able to reduce a mathematical statement down to the symbolic level (eg. using sets), but so much of the beauty is in the simple intuition. One simple example: the pigeon hole principle -- you can teach it to a 3 year old, and yet its incredibly powerful.
EDIT: If you read about Lockhart's Lament (as mentioned elsewhere in this thread), he specifically writes:
The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.
That seems to imply that the pleasure and beauty of mathematics is in the way one perceives symbolic manipulation. I actually think the pleasure and beauty of mathematics is in the stage before one gets to manipulating symbols; the stage where you have an idea about something and are thinking about ways to communicate it or prove it. People should be given problems ('how much of this box does this triangle take up?' to paraphrase Lockhart's example) and allowed to drum up their own solutions/syntax.
Programming is about architecting superior solutions to problems.
How then is it purely symbol manipulation? There is an intelligence required to imbue them with semantics and combine them into the right order to match the semantics of a correct solution.
Loops, variables, constants, etc. These are building blocks. One doesn't have to find them inherently interesting. The real challenge of programming comes from imbuing them with the right semantics.
The same can be said for mathematics. Sigma notation isn't interesting. It's what you use it to represent that is interesting.
Apparently I fail at irony :-) Since we are mostly programmers here, I meant people to read the second sentence, say, "Wait... what? That doesn't do it justice," and then reread the first sentence in light of their understanding of the difference between what computers do and what programmers do. Symbol manipulation is what we design our computers to do, and the power of mathematical systems can be described by mathematical logic, which reduces mathematics to symbol manipulation, but what goes on in our heads as we work with these systems is very different.
>this has nothing to do with hard work, and everything to do with the way mathematic is taught. I am a programmer and I would love to understand mathematics the way I understand programming.
How much of your understanding of programming has to do with:
a) teaching/exercises in a class
vs
b) time you've spent hacking on programs, either for a job, or in your spare time
?
Now, how much time have you spent on mathematics compared to time you've spent on programming?
The grandparent's point seems to be that fretting about not having "royal road"[1] is less productive than just spending time working with the ideas.
Good pedagogy can make a difference in how far you get in the time you spend, but there's just not a substitute for putting time in.
For me it had a lot to do with who taught my math classes. Out of all of my high school math teachers, I'd say less than half really understood what they were teaching beyond the most superficial depth. (It was even worse in my science classes.) I love teaching and would have preferred that as a career but the pay scale is really discouraging even at the university level. So now I build educational apps and hopefully can be part of the solution.
Yikes, I completely disagree. There's nothing wrong with hard work, but I definitely think that doing oodles of calculations definitely has a quick point of diminishing returns. I was also raised in a pacific-rim Asian household. Growing up, I spent lots of time doing routine calculation. Looking back, I wonder if that time was really that well spent. In elementary school, I spent an entire summer doing nothing but practicing basic arithmetic from sun up to sun down. Did it take me an entire summer to understand how to do basic arithmetic? Obviously not. Now I think the time would have been better spent tackling substantial problems with what I'd learned, rather than doing routine calculations. Does doing lots of routine calculations have an upside? Certainly. I can factor polynomial equations faster than anyone I know. While that's cool and all, I now feel like calculating speed is of questionable value beyond a certain point.
Now that I do math for fun, I spend most of my time doing recreational math style problems. I can't help but think that these would have been a better way to learn. Since I started, my problem-solving skills have improved immensely, with nary a routine calculation problem in sight.
I agree with you, but in the context of American education, the parent is right. Whenever anyone talks about improving American education, unless they specifically say otherwise, they are talking about students who are struggling to learn. When they talk about relieving the students of the burden of calculation, they are not talking about the problem of kids who have mastered calculation problems being forced to do them over and over again ad nauseam. Nobody cares about those kids, because those kids get good grades and good test scores. What are the measurements that people talk about in education reform? The percentage of students with low scores. The percentage of students who drop out. The percentage of students who are not prepared for college. This is typical for discussion of American social issues; everything is problem-oriented and not success-oriented. Certain outcomes are labeled as failures, and reform is aimed at preventing them. Education reform is aimed at kids and schools who are "at risk" of producing undesirable outcomes. Kids who can do calculations accurately aren't even part of the conversation.
So, when anyone talks about relieving kids of the burden of calculation, they aren't talking about people such as yourself who have learned beyond the point of diminishing returns. They're talking about the kids who still struggle to do the calculations correctly. They want to give up and move on with the kids' education before the kids have learned. This is linked to a perennial fantasy in elementary math education: that intuition is valuable and easy, while computation is hard and unimportant. I think this fantasy is wrong on both counts. Intuition is acquired only through hard work and practice, and intuition is only valuable if you are competent to subject your intuition to scrutiny. Intuition and calculation have to be yoked together like a pair of oxen, each pulling the other along when it falters. But as long as teachers believe that intuition is easy, there will be people willing to exploit the fantasy by promising to deliver easy, intuitive learning experiences without the useless drudgery of calculation.
This was very interesting to read. Very different perspective, thanks for writing.
I have a number of questions/thoughts:
1. With the type of schooling you went through, how many people actually like math at the end of their schooling? I realize you did, but then again there are people in American schools that also like math. How many, really?
2. How much of that math that everyone has to learn is actually useful for the average person?
3. You say "Forget about the whole, um, personality and uniqueness and whatnots." -- This is basically the approach that has been taken throughout most of history, grounded in the idea that we're all empty cups to be filled. However, the "empty cups to be filled" idea is flawed. It's true that in America people hate to be called the same, enjoy their uniqueness, etc. -- however that idea is more than just a skin-deep preference. Young people are different, have different hobbies, different values, personalities, and have different ways of learning. One can apply one method and ignore all that (ie. your prescription), or another is to realize that differences do exist and structure the learning environment to cater for it. The former is a proven way of producing decent outcomes (though very varied); the latter is in many respects unproven/unsolved, and so I believe has much greater potential. Just as HN'ers pursue innovation through technology (in most cases), the latter pursues innovation with learning outcomes.
This reminds me of http://www.math.rutgers.edu/~zeilberg/Opinion71.html, where a number of prominent mathematicians and computer scientists are traced to an Israeli mathematical magazine for young people. These are certainly not kids who had no calculation skills. They wrestled with difficult problems and developed great problem-solving skills that have served them in their careers.
I'm all for giving kids challenges, but doing hand calculation isn't hard, it's just tedious. Computers do these calculations faster and with fewer mistakes, and everyone working in the real world uses them every day to do even simple calculations. If we stop spending as much time on calculations we can move kids more quickly into advanced concepts - with the side benefit of fewer kids being turned off math by how boring the first years can be.
It's the same thing for teaching kids cursive writing. Why are schools wasting time with cursive when they should be teaching typing/keyboarding skills? Cursive writing is very quickly going the way of calligraphy.
I, for one, have never had to "sanity check" my word processor/text editor against my inputs; the letters and figures appear in pretty much the same order even if the glyphs are different because, regardless of the mechanics of the writing, in the end it's just me putting characters into the system one at a time.
Those who cannot do calculation at least well enough to do estimation have no idea if something has gone wrong, or where it might have gone wrong, when they see a result. In that regard, it is less like failing to learn to write currently (cursive simply means slanted, not connected; current comes from the French "courant, or running*) before composition and a lot more like failing to learn to read. I do decibel (common log to one decimal place) calculations in my head not because I feel the need to exercise arcanities I learned when four-function calculaors cost hundreds of dollars, but because it allows me to spot errors in mechanical/electronic calculations (including those in the analog realm) quickly. When you rely entirely on the machine, you have no way of knowing whether or not the machine is sane.
> You know what I do during my lunch time? Trying to re-prove Fermat's Last Theorem.
How many other students did you know who did the same? There are some people (you, in this instance) who like Math in the current form it is taught. But the sad truth is a large majority of the world's student population (that includes Asia) find Math a boring, dry subject that forces too many drills upon you for not much practical application later.
> The problem is no so much that they dump down math. The problem is that they alienate the students from math. When you do math, all of it, the result is made of your sweat.
Wolfram advocates using programming to drive the concepts better by making students write programs to solve the problem. You can't do that without really understanding the concepts behind it.
> I hope you see my point: there is no elevator to interest in math. Math lovers must have toiled and slaved over stupid calculation over and over, until a point that they could actually produce the trace of math working, just like the zone in programming or any other art. That trace, that focus, that absorption builds the love for math.
Again, that definitely has worked for some percentage of the population. But it in no way is a guarantee that there aren't ways to better it. The 'elevator' to interest in Math might just exist.
The point of the talk was that Math the way it is being taught only teaches students to do hand-calculation, but doesn't give enough importance to other crucial aspects (asking right question, modeling and math formulation). It also brings in Computers as a great tool to help students understand Maths better: by procedurizing learning and using programming to help students understand concepts better than just handsolving problems repeatedly.
> You know where self esteem is from? It's from holding a piece of paper on which a problem has been solved and solution has been neatly presented; it's from the overcoming of frustration and challenges that the problem presents.
It's also from running the program you wrote which solves a challenge well. That is an important idea that Wolfram presents in the talk (it is by no means new, but nothing much has come of it yet). How is understanding 'how' to solve a Mathematical problem and translating it into executable code less 'frustrating/challenging' than learning by rote a specific method to attack a specific pattern of mathematical problems and using that technique mindlessly to solve them?
I find that the comment fails to address those actual points in the talk, but instead raises rote learning and hand-calculation drills as something worth aspiring for, and something that separates the chaff from the grain. I disagree. There are many students out there who find Math (the way it is being taught now) boring and worthless, but would have loved the subject if it (the teaching style) appealed to their intelligence better.
(also, please don't overwork the students any bit more. its hard enough already for most)
>You know what I do during my lunch time? Trying to re-prove Fermat's Last Theorem
The people who are genuinely interested in math would still be doing that. We've tried teaching computation but the retention rate just isn't there, and isn't practical anyway. Why not do something useful with that time instead?
I think you're right about the "slaving" away practice part. After all, one of the world's greatest minds Von Neumann put it this way: “In mathematics you don't understand things. You just get used to them.”
I didn't really see much specific stuff here, to be honest. There was some program that you could use sliders to figure the best life insurance policy ... I mean, ok ... that's great ... What's the big deal again?
What are some specific examples. There's all this "wah schools only teach calculation." Well, what specifically are they teaching that is bad and what specifically should they teach that is better?
So, are we saying, no reason for a student to know how to take a derivative, they should know how to turn the knobs on some program to figure out the best fuel efficiency of some device in some really neat simulator?
The example about limits ... ok yeah that's cool but if you aren't actually teach them how to "calculate" (is the word "calculate" used as a boogey man?), how is that you are not doing anything other than teaching "math appreciation?"
Though it isn't worth much, I'd like to put my two cents in as a student currently going through high school.
In the district in which I'm partaking in public education, there's a dead even split between contemporary and traditional math courses. Our school offers both the Core-Plus Mathematics Project (http://wmich.edu/cpmp/) and some variation of what I'd consider traditional (Geometry 1, 2, Algebra 1, 2, etc.).
The students more often than not will prefer the Core-Plus Mathematics Project over the traditional courses early on in their education - but once the math switches over to a primarily calculating class (take, for instance, Calculus), the opinions shift and favor the other one.
What we honestly have in my district is a case of a system that's trying to split itself down the middle politically - as we're attempting to switch to a modern, context driven approach to mathematics, whilst maintaining the link to processes like Advanced Placement, which only judge on the traditional aspects (http://www.collegeboard.com/student/testing/ap/sub_calab.htm...).
While it might be a better route for the future, the present is demanding math to be calculation based for the near future. It'll only start switching into a better realm when colleges and pre-college programs like those offered by CollegeBoard make the switch to context-based systems, rather than calculation based ones.
(I'm very well aware of how rambling that is, sorry!)
Mathematicians are basically just accountants, right? And the IRS has gazillions of accountants who only really work during tax season...so, if our math programs suck, let's move our tax due date to overlap with schools' summer break, fire all the math teachers and replace them with those IRS guys!
It might be better than the existing system and it could hardly be worse. There's already been some work done on a curriculum (warning: PDF link): http://www.cs.amherst.edu/~djv/irs.pdf
These how-to-apply-math skills are valuable, but they aren't math itself. Math is about building more general abstractions, keeping strict rigor, being careful with axioms, and developing techniques that are really, really broad as a result.
Granted, calculation isn't a very good way to get into this either.
Added to the reading list. I've been saying this for quite a while. I think we were able to capture this in our 9th grade math games we have been working on.
Enjoyed skimming your article. If only more geeks actually wanted to teach kids... unfortunately most of us will only be able to do that once we've retired!
If I went back into education for 2 more years, piled more student debt on top of myself, not to mention living like a student again, PLUS having to have a part-time job while studying (while the loan is bigger for a PGCE, the costs of living are much much higher), and then got a job as a teacher, worked my ass off for 10 years, I would, if I was lucky, be earning the same salary I am today, a couple of years after leaving university.
This cannot be upvoted hard enough. Education would reform itself overnight if teachers were hired from the top third of high school classes, not the bottom two thirds, and paid salaries that started ~$70k and reached ~$100k in five years if they were good.
Students struggle because you have to be a saint or an idiot to teach under the conditions currently imposed. We don't have nearly enough of the former, and way too many of the latter.
I have a really hard time imagining that brain-dead curriculum boards and mediocre administrators would sustain their death grip on a profession suddenly flooded with really sharp, well paid, professionals.
Imagine the change in classroom atmosphere if teachers had the same social credibility as professors did (or more aptly, if high school teachers had the same respect and admiration from students as kindergarten teachers)
To be fair, I disagree with his position, but he is willing to call it something else, but the end of the day it is to teach his proposal as opposed what we now teach.
But this is dangerous and a little shallow. And here is why.
Math is the language of science; just like a C -- or any other programming language -- is a language of computing; just like English is a language of humans.
There are many ways to learn a language, but I think they all have something in common. You have to spend a lot of time learning, reading and writing. The writing is quite important. You have to write the language to be good at it.
The thing about math is that currently to write as it is supposed to look like can only be done most conveniently on paper. Writing a math formula in Mathematica or Wolfram alpha requires a latex-like mini language, another thing to learn, and can be disrupting.
This fact alone means that computers can only supplement; and they can be an excellent supplement. But it can not replace doing math by hands, on papers, particularly for those who are learning this new language.