The trouble with trying to arrive at any single definition of Mathematics is that Mathematics is different things to different people. A research level Mathematician might see it differently (finding patterns, abstraction, theory - axioms and proofs) from an Engineer who has a purely practical interest in it (cookie cutter methods and formulas). For everyday use Mathematics is a set of algorithms for doing stuff with percentages, fractions, basic arithmetic etc.
Currently what's frustrating is that we subject everyone to the same level of training. Those whose interest lie elsewhere are subjected to unnecessary and meaningless torture, while those who are really interested in Mathematics as a subject are left to their own devices to find additional training (which thankfully is easier these days than when I was a kid, though it could be made substantially easier still).
You beautifully sum up my experience with math at the university, as a CompSci student in Germany. The absolutely most frustrating thing for me, in the beginning, were the basic math lectures (which I, as a disclaimer, still haven't finished). At first, of course, I did what most students do in their first semester: bitch about the apparent uselessness of the subject.
That's about one and a half years back, now. I realized by myself that math is actually pretty useful. The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit."
Personal eye openers for me were my internship (and continued assistant work) at a department of the university that deals with image processing (which is basically a boatload of applied math), articles like Wolfire's excellent "Linear Algebra for Game Developers"[1], and plunging myself into a lecture about Fuzzy Set Theory and Artificial Neural Networks (which again is applied math).
I actually like abstractions, generalizations and logic reasoning (I wouldn't study CS if I didn't). I still don't think that math is the only way to teach those, but it's arguably the largest field of science dealing with these things.
I don't really think the problem is math itself, it's the way it is taught to most people - in the math way. Mathematicians teach math to do more math, which in turn is used to to even more math. And that's fine - for a mathematician. For literally every other field of science that needs math as a tool, it's highly frustrating to be subjected to what you aptly describe as "torture".
I am a math professor at a large state university in the US. A comment and a question:
> Mathematicians teach math to do more math, which in turn is used to to even more math.
That is true of some mathematicians, and of some subjects. But I think most mathematicians teach and learn math because they think it's cool, not because it's useful -- either to learn more math, or for the "real world".
I might also add that I always appreciate it when teachers take a very long view of things. For example I have trained in the martial arts, and most teachers take the perspective that you want to master the art completely, rather than just learn one or two things. Some people don't like this, but I deeply appreciate it.
> The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit."
As a math instructor I am always looking for ways to improve, but I'm having trouble extracting useful criticism from your remarks. "They throw it at you and expect you to deal with it" seems to be true of challenging lessons in anything.
I would cheerfully welcome general advice and suggestions from you or any other HN readers. (But please keep in mind that both my interests and my expertise skew heavily theoretical!)
I've the same gripe. Maybe, I can elucidate the parent's point.
I can appreciate the mathematician's fancy for reducing numerical manipulation to an indivisible, atomic precision. Creating poetic abstractions is dear to many in applied sciences. Where it sticks in the craw, where it stops making sense, is where you have this beautiful construct and it's purpose is left a mystery. If it's a piece of art, fine. It's welcome to hang on the wall over there next to the others, but I'm not going to remember it in much detail without some context. If there isn't a concrete use, if there isn't a purpose, the idea won't make it out of short-term memory. "But it's the foundation for all of these other beautiful edifices of logic that we'll cover in the next chapter!" It's like being told, "This is red paint. This is green paint. This is burnt umber paint...", without ever being shown a Picasso, Renoir, or da Vinci. If you ground the presentation with messy, imprecise, real world numbers, the logical purity of the subject will not be lost or sullied, the esteem for it's power only more appreciated. Pull the pin, show us what it can do!
You may not be a physicist or engineer, and that their respective topics will be covered in their respective classes. But those same physicists and engineers probably aren't mathematicians. So what business did they have teaching me math... better than a math teacher ever did?
>But I think most mathematicians teach and learn math because they think it's cool, not because it's useful
That's exactly the point I'm making. That's math for math's sake - which is perfectly okay if you're actually studying math, because in that case it's what you signed up for in the first place.
The problem is if you study something else (for example, Computer Science, or Physics), and are thrown into the exact same math lectures, with no one telling you how all that fancy math is actually useful for anything but doing even fancier math. Only after the mentioned experiences at my internship, where I had the chance to work and talk with people who actually do applied math did I see the point of it. What I think is that mathematicians - and I don't mean this as an accusation - are often unable to understand that while math for math's sake works fine for themselves, it's not really the best approach for people who want to actually apply math to their own field of science.
Here are a few observations based on my undergraduate days as a math major.
Engineers, mathematicians, and anyone in the sciences together took a 3 term calculus sequence and ordinary differential equations. After ODE in our second year, disciplines started to split apart.
The calculus courses were taught by math professors and their TA's. Calculus was presented through exercises with a small tribute to proofs. Right here is I think where things went wrong.
The engineers really didn't care about proofs and, as important, the exercises were learning by rote. At no time were engineers given anything practical to do with calculus. They would have been better served by exercises that led to building something: a model bridge, a model circuit, anything that makes the use of calculus more concrete.
The mathematicians were concerned with why calculus works but weren't given a good grounding in proofs. Consequently, I was not prepared for the sudden jump in proof maturity when I took real analysis during my fifth term. Abstract algebra equally boggled my mind the following term. Rather than study calculus with engineers, I would have been better served by learning calculus through historical development with emphasis on the reasoning process the great mathematicians went through in developing calculus.
Engineers and mathematicians had diametrically opposed purposes for learning math yet were forced through the same curriculum that was a compromise between the two disciplines. Most likely the compromise was due to resources, but it did nothing to make math more interesting to either student engineers or mathematicians.
As an aside, my math adviser asked me about continuing to graduate school. I replied that I had no confidence in my ability to understand higher math. His reply, "Oh, as an undergraduate we largely leave you to figure things out for yourself. Once you're a graduate student we start showing you how math really works." This is not a good policy for attracting potential mathematicians.
I too have been practicing martial arts for close to 20 years. You see the same difference in purpose there as well. There are people who practice martial arts and then there are martial artists. Martial arts schools have the luxury of letting the 2 groups grow apart over time. Universities have 4 years to get the 2 groups developed for different purposes. They need to become more aware of the time limitation.
Slowpoke's comment really resonated with me, so perhaps I can provide some concrete examples.
8 years ago, I graduated from my Software Engineering degree at Swinburne University in Australia. As part of my degree I had to sit through 3 years of engineering maths.
In our first programming tutorial we were taught how to write a program to display “Hello world”. It was immediately obvious to me why we might want to write a program that displays messages on the screen and what sort of problems this would be useful for solving.
In our first maths tutorial we were taught how to add, subtract and multiply complex numbers. To this day, I still do not understand why I would need to do that. Where do my complex numbers come from? What do they represent? When I add them together, what does the answer mean? The only problems that I can apply my maths education to, look exactly like those on the tests: what is (-3.5 + 2i) + (12 + 5i) ?
It is possible that in my job I am besieged daily with problems that I could use complex numbers to solve. But if so, I am totally incapable of recognizing them!
Fourier series were a particularly egregious example – the subject started, when the lecturer came in and wrote up 3 boards of dense maths, and said something along the lines of “… and this is the formal derivation of a fourier series!” It was as if somebody had tried to teach programming by explaining the algorithm a complier uses for translating source code into machine code. Then expecting the students to just figure out how to write actual useful programs, all by themselves! I believe this is what Slowpoke was talking about when he said: “The problem, as I still see it, is that nobody fucking tells you why. They throw it at you and expect you to deal with it. No further explanation - "It's math, we ain't gotta explain shit."
The way we were taught fourier series particularly hurt. Years later, I found out by myself, that the things are actually incredibly useful. As it turns out, that there are these things called fast fourier transforms, that programs use all the damn time, to do fantastic stuff!
So, to give a very concrete example of how maths education in universities could be improved: If only the first lecture on fourier series had instead explained what they are used for, and why they were so important that we were going to spend 5 weeks on them. Then perhaps I would have had a much better understanding of what they are and how to use one. It helps so much, to be able to think as the lecture is writing boards filled with formulas: “Shit! Now I can use this to do X!”
Thanks for taking the time to listen. It’s such a shame that maths is being taught this way, because mathamatics is both so very useful and very important…
Edit: Actually I reflection I sat through 3 years of maths, not 4
If you had enrolled in electrical engineering then by half way through second semester of first year you probably would have been having lots and lots of light bulb moments. Complex numbers and fourier series and transforms rapidly become like oxygen for EEs. Essential but consumed almost without noticing.
You suffered from the university's lack of bandwidth -- only so many lecturers, only so many lecture theatres, many other competing classes. They try to find efficiencies where they can -- e.g. by giving all of the engineering departments a common mathematics curriculum.
That curriculum is almost certainly driven by the needs of the EEs and ME/CivE's much more than chem and software, and I'd guess software engineering is looked at is if there's really no maths required. After all, software engineers just program and write documents, don't they? The truth is of course that there is just as much scope for maths, if not more, in a software engineering or computer science course, it's just a whole other type of maths that's needed.
Hmmm ... It is entirely possible that the EE's went straight out of that maths tutorial and into an electronics class that used complex numbers for a practical application.
To be honest all us software engineers understood that we didn't make up the majority of maths students. I would have been perfectly happy to hear about how I could have used complex numbers to design circuits or build bridges. But I guess the maths classes were designed to provide the pure theoretical foundations upon which other, more subject specific classes would build. It just so happens that for software engineering there were no subject specific classes which made use of the maths.
However, after reading your comment I still don't know what to do about it. As far as applications I'm familiar with... well, they are typically to other areas of math, as Slowpoke was complaining about.
Since you mentioned Fourier series, please imagine that I am about to teach an undergraduate course in the subject, and I want to follow your advice. However, I don't know jack shit about EE or any other practical application of Fourier series. What would you do in my shoes?
Hmmm, well its a very interesting question. I'm not entirely sure what I would do in your shoes, but some places start looking may be:
Why are Fourier series on the curriculum? There must be a reason why they are so important that a whole undergraduate course would be dedicated to them.
How and when were Fourier series first described? I understand Joseph Fourier made some pretty major contributions to maths. What was he trying to achieve? Did Fourier series help him do it? Why did other people pay attention, and how was the idea popularised?
Finally, if you really don't know anything about EE or any other applications for it may be a good idea for you to lean a bit more about the various fields that are built on top of yours. Having a better understanding about how your area of interest relates to others is always a good thing.
I know from a software point of view, even though they are hidden from day to day coding, Fourier series are hugely important. Fourier series allow us to compress audio and images down to sizes where they can easily be transferred across the internet [1]. Almost every single digital image you see, song you listen to and movie you watch will have had a fourier transform applied to it. Did you know that movies on netflix account now account for 32% of north american internet traffic? Without the fourier transform its safe to say the world wide web as we know it would not exist.
Furthermore, because it is actually practical to transfer movies and songs across the internet, mass piracy of media is possible. Millions of people are sharing (fourier transform compressed!) movies over sites like the pirate bay. This has lead to a backlash from established media corporations demanding stronger copyright protections. Currently a huge legislative battle is being fought, which will have a impact on such disparate areas as the future of censorship, what rights people have over the things they create and the role of money in politics.
All because of what the fourier series lets us do!
He taught high school math between 2004 and 2010, but became frustrated and is now studying at Stanford on a doctoral fellowship, with interests in curriculum design and teacher education.
I'm just glad that there's now a public debate about better ways to do math education.
I really wish I'd had a teacher like him back when I was in high school. I had an algebra 2 teacher call me an idiot in front of the class ... and when I went for after school help, he simply made fun of me the whole time.
I agree that matching the variety of possible assignements by some generic method is very difficult, though in my humble opinion, we already have a lot of the tools required. There is a ton of improvements to do(lots and lots of usability), but the base is there - and a project of such a scale would not be unheard of.
> Its hard to communicate a fraction to a computer
> "Explain whether 4/3 or 3/4 is closer to 1, and how you know."
You just did that. Twice.
Square root ? "(3/4)^(1/2)" or maybe "sqrt(3/4)". There's no complexity in parsing that. I do agree it is not as natural as on paper but maybe tablets will find a way to improve that. Thats what innovation is here for after all.
>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."
I am not familiar with the domain, but dont we have some automatic theorem-proving tools? Validating the answer to such a question would look like a perfect use case to me.[edit : clarified]
> [the description example]
Im not sure about this one. On the one hand i used to have a project in college about reconstructing a picture from an incomplete description - and its hard. On the other hand we are expecting a perfect description. Hence it would be pretty much isomorphic to the code of a program used to draw the picture. Matching the two images is also doable. Pseudocode would actually be the best way to transmit this image .
I agree it is not as natural as on paper it's only unnatural because you did not spend 12+ years doing it that way. I spent enough time with my old ti86 that it became vary natural to express math on it despite the poor interface.
I have no problem using this machine to say:
abs(4/3 - 1) = abs(3/3 + 1/3 - 1) = abs(1/3)
abs(3/4 - 1) = abs(4/4 - 1/4 - 1) = abs(-1/4)
abs(1/3) > abs (-1/4), so 3/4 is closer to 1 than 4/3.
Keep in mind that this is a toy example. The original essay used it because it was a toy, and therefore didn't break up the flow of the writing or scare away the mathematically unsophisticated (this is a common problem faced by essay-writers who are trying to explain complicated things).
Try something harder - at least, say, the quadratic formula. Or, if you really want to appreciate what we're getting at, try a chewy example, like Maxwell's equations:
These are expressible on the calculator command line, and first-year grad students with Mathematica licenses type them out fairly often, but you're going to discover that Knuth spent a decade of his life inventing TeX for a very good reason. Without proper notation it is hard to reason about math.
You can, of course, use the better class of notation editors to get your computer to display math in proper notation, just as the Wikipedia authors did, but it's a bunch of fiddly work, rather more work than writing out the math by hand. And then you find that you can't make very faint tickmarks or cross-outs or circles on your math. You can't easily draw arrows connecting one line to another.
A useful side-effect of pencil-written mathematics is that the intermediate steps are there in front of you. Watch a professor talk you through a derivation on a chalkboard. Observe that, despite the fact that chalkboards can be erased, using technology that has been available since prehistory, the professor rarely transforms equations by erasing the old ones and replacing them with new ones in-place. That's because recopying the equation after every one or two transformational steps leaves behind a changelog that represents your train of thought. When the problem is almost done but you're trying to track down the sign error you'll be grateful for that.
And, yes, computers have undos, so you can rewind and fast-forward your math. But that's inferior technology for thinking about problems. Ask Tufte: The secret to reasoning about data or logic is to spread it out in front of you in as flat a manner as possible, so that you can move from step to step using nothing but your eye muscles, or defocus your eyes a bit and view the whole problem space in the abstract.
Giant desk-sized iPads may one day render pencils and paper obsolete for serious math, but I'm not convinced that will happen in my lifetime. The hardware is expensive, the software is far more expensive, and paper is cheap, and scanners for digitizing paper are cheap.
People complained just as hard when transitioning from slide rules to calculators and they have some valid points, but the net gains where also clear. It's easy to make a bad interface but there are plenty of great Math interfaces that keep a listing of all previous steps above what your working on both how you enter it and show you how things would look like on a blackboard. There are a also plenty upsides like the ability to cut and paste lines so you can avoid a lot of stupid mistakes like dropping signs. But, far more important from and educational perspective is a students improved comfort using a computer to do advanced math vs. the near phobia that you and many others apparently have.
I don't doubt that one can, and that we will, build a computer system for manipulating higher mathematics that's so much better than a stack of paper, a pencil, and a decent eraser that you won't even own the paper. What I doubt is that it's done yet. But I haven't exactly been looking for it, so maybe I'm wrong. Certainly, once it's done there won't be any problem selling it to me (except for the sad fact that I no longer manipulate equations on an everyday basis).
He is writing about math education. The characters (3/4)^(1/2) make sense to all of us who have already learned math and know some programming languages, but that syntax is pretty confusing to students who are just developing a real understanding of exponents.
There is plenty of room for automation in math education. But in a really good math education, the automated tools need to be balanced with more socially-oriented approaches to education. Students need to talk to each other and to good teachers about their work. Students need to see each other's approaches and hear each other's ideas, and have face-to-face conversations about math.
One could make the argument that any mathematical syntax is equally confusing for the novice--so why not start them on something they'll be using later anyways?
I think we presume a great deal in suggesting that a simple flat array of characters and operators is somehow less understandable than a nicely typeset equation (especially when you've never written one before!).
I was thinking the same thing. It might be better to learn this stuff at a console from the get-go.
One advantage would be that you could try invalid syntax and operations (i.e. x/0) and see the errors that result in real time as opposed to an hour or a day later after the teacher marks up your test. Then you're more likely to stick with it until you get it right, which in turn means the answer is more likely to stick with you.
I'm not saying that Matlab, Maple, and LaTeX have the whole "How do we represent numbers to computers well?" thing down, but we sort of do. A simple subset of LaTeX for use in classrooms could be useful (something like MathML or the like).
"Pseudocode would actually be the best way to transmit this image ."
I wholeheartedly agree. A graphics routine or a LOGO program would do a great job of describing that.
The author seems to have an irrational dislike of people trying to use computers in this fashion, which I find strange. I certainly agree that something like a geometric proof (in absence of a good modeling language) is difficult to automatically check, but at the same time I question whether or not the human element would be any more useful here. Math teachers, especially at lower levels, are not infallible.
I would almost venture that a better test, one that examines both critical thinking and ability to logic about a problem, would be a battery of small programming problems to solve some kind of geometric or graphical challenge. It's a bit all-or-nothing, but it would show that the student can both interpret a problem and also describe the steps to solving it.
This is the exact attitude that the blog post is saying is wrong. For mathematics, computers are tools. Computers don't create the answers, they assist the user in finding the answer. They're time-saving and error checking devices, which are useful after the student learns the concepts inside and out. They are not supposed to solve the problems directly.
>>"Explain whether 4/3 or 3/4 is closer to 1, and how you know."
>I am not familiar with the domain, but dont we have some automatic theorem-proving tools? This would look like a perfect use case to me.
Theorem proving tools would work if the students wrote their answers in a format that the tool would work in. In this case, it would be a natural language proof instead of a formal proof, which simply isn't possible to parse right now. Perhaps it will be in the future.
> Theorem proving tools would work if the students wrote their answers in a format that the tool would work in.
Theorem proving tools would work if you were writing formal proofs. At the level he's talking about, students are not writing formal proofs -- they're writing explanations.
Right. It seems to me that the despair is that it is very hard, from knowledge like "student checked box B in this test, which was correct" to deduce "student actually understands the concept".
Perhaps a partial solution to this problem can be found in... wait for it... programming! That is, it is really hard to write a program to find general answers to problems unless you understand the basic idea. This definitely doesn't work for everything, but I wonder how big the domain is where this is a really good way to mechanically assess understanding, and whether the language that'd be required would itself become a bigger component of the measure than what you're trying to assess -- mathematical reasoning.
You're asking people to do math in a non-mathy way. How would you feel if schools insisted, for the purposes of learning, that all programs be written out by hand in plain English?
Or, heck, just think of how annoyed some developers get when asked to write code on a whiteboard during an interview. "That's not how we code! If you want me to write code, give me a computer."
I see that question as being useful because it assesses several different understandings at the same time; the key ones I see are the understanding of fractions (including the idea of improper fractions) and the ordering of the values represented.
A human instructor asks for the explanation to explore a bit if the answer was just a guess or if it was reasoned out, and whether the reasoning was correct. An automated evaluation system would have an easier time factoring in results from previous evaluations (it can perfectly remember an arbitrary number of tests across an arbitrary number of students...) and could check understanding by presenting several questions (it won't get sick of looking at the answers).
I guess if the proposition is that human instruction can be replaced by automated systems that is crazy. Really, I expect most people are looking to supplement it and to make it more effective.
I wrote this post. To summarize my argument in three lines:
There ARE different ways of defining mathematics, and some of them contradict each other.
Silicon Valley companies wrongly assume their platforms are agnostic on those definitions.
For better or worse, if you're trying to make money in math education, the Common Core State Standards are the definition that trumps your or my preference for recursion, computational algebra, etc, and those standards include a lot of practices for which, at this point in history, computers aren't just unhelpful, but also counterproductive.
Wrong. There IS software that presents math using Guided Discovery. In a visual manner that doesn’t hide the math. It also stresses introducing the conceptual BEFORE the computational as you do. The software is the online “ST math” program that has shown a respectable effect size and statistically significant gains under randomized controlled trials. It is definitely worth discussing this program and not blanketing all math ed tech as bad. Can you comment on this?
Elsewhere on this page, someone comments: "I would love to see one of these 'complaining about Khan Academy' articles with an attached example of the author's Right Way to teach whatever subject."
I don't think that the article was too strident at all. (I was a high school math teacher for years, and computers seem utterly useless to me when it comes to teaching kids to think mathematically.)
Math is presented as several pointless tasks that need to be memorized, in order for the test in a week. For example, every exam tests if we "know" the theorums from the tested sections. An example of a theorem we were tested on is, "Each diagonal of a rhombus bisects two angles of a rhombus". Students can see that that statement is true, but students don't know why (including me).
As a result, we don't see any applications of it besides the test, the final, and future math classes.
I haven't seen any software that doesn't just make what teachers already do faster. Everything I've seen for math is just incremental -- things like putting a scantron online, or letting me find out what my homework is online. At this point, technology isn't helping me understand math better.
An example of a theorem we were tested on is, "Each diagonal of a rhombus bisects two angles of a rhombus". Students can see that that statement is true, but students don't know why (including me).
The way to construct a rhombus is left-right and up-down symmetric. Because of that, you can pick one up, flip it along a horizontal or vertical axis, and put it back exactly where it came from. That operation keeps the diagonals where they are, and puts those to-be-proven-equal angles on top of each other.
He's basically disappointed SV is attacking this problem with the methods they understand best?
I don't see how one woud expect SV-type startups to adress maths education without leveraging their strength --which is in computing. I would expect experts in education to use other more pedagogical approaches.
It's like going on about a carpenter who wants to approach a problem with wood in mind.
I would love to see one of these "complaining about Khan Academy" articles with an attached example of the author's Right Way to teach whatever subject.
I don't think there's really a route to improving education merely by complaining about the approach of others.
I have routinely offered my own versions and contrast them with Khan's video lessons. Watch this MSNBC clip about Khan Academy and my classroom at about 1:30 :
http://www.youtube.com/watch?v=GwE6iWEhtRk
That looks pretty great. I guess your approach only has a few downsides:
* you can only teach (I'm guessing) 300 or fewer students per year, so you can't really have the same reach as Khan
* your method costs orders of magnitude more (someone is paying your salary, and for facilities)
* you can only reach students who are physically near you
* students can only access this education on a very rigid schedule
I don't think your approach is bad, but I don't think you're really solving the same problem. How could you scale your method to teach the entire world for a few dollars per year per student?
If Khan's method scaled (at lower cost) without sacrificing other features of high quality face-to-face instruction, then I would expect people who pay private school tuition (K-12) to push their schools to implement such an approach in order to lower tuition.
But there's a reason why the elite pay $30K/year for K-12 private schools: Small classes, strong student-faculty relationships, high-quality facilities, and the exceptional educational experience that comes along with those features that cannot be replicated on a large scale.
Dude - seriously, you are like the Khan Academy Stalker. I have taken an interest in KA and have started to use it in my class. But every article, web posting and message board that discusses KA, you show up to criticize KA and promote your own website. Do you spend all your time stalking this guy online? It's kinda creepy.
I teach hs math and physics and have gone to having my students learn at their own pace, using KA videos. KA provides the lecture, which provides a framework and road map for students to follow. I provide the project learning and guidance - which is where the students spend 80-85% of their time.
I have also gone to a "tutorial" method of teaching. I assign work/projects to students to work on independently and in groups. While they are doing this, I will pull 3-4 students out to have a tutorial session, where they are given a problem and must solve the problem. Sometimes it will take 3-4 classes for the small group to solve the problem. Once they do, they go to the next concept. I have students that are learning linear equations, quadratic equations, and solving polynomials at the same time. Because I have changed how long a student has to learn, I can teach the students "where they are". A student will struggle with learning quadratic equations if they don't even understand linear equations. So I let the student take as much time as they need to learn linear equations. Once they master the concept, they move on to quadratic equations.
This is not just a bunch of students watching videos and then answering questions online. In fact, most of their work is done offline, with projects and other applications. I have also found that most of my students will attempt to try and solve the problems before watching any videos or asking for my assistance.
I have found KA to be an invaluable tool that allows me to teach more effectively. Being able to do 'tutorials" with my students has made such a huge difference in the understanding and learning of my students. (we just took our practice state test and 92% of my students passed. The norm in my school has always been around 68%). I also have found the majority of them are actually engaged in the learning process. This is a big change from what I used to do..lecture to a bunch of glazed over and half asleep students. Once I would finish explaining a concept, I would ask, "who does not understand"? 3/4 of the class would raise their hand.
Sal Khan is much better at the lecturing, I am better at the one-on-one give and take between student and teacher. I am better at this because it allows me to better understand how my students process information and solve problems, which helps me diagnose when they get stuck. I would not be able to set up this system if not for KA, so I personally think they guy is genius and greatly appreciate everything he is doing.
KA does not replace or become the teacher. KA allows teachers to use as a supplement so they can work with students in the trial and error of learning.
Now that organizations like Khan Academy have traction and resources, I think they'll be able to bridge the gap. They'll be able to iterate as they receive feedback from the educational community.
It would be nice if when teachers gave them feedback, they'd act on it. Unfortunately, it seems that any criticism of the Khan Academy is seen as an attack on the whole of the Silicon Valley, and every geek comes out to lead the charge against the educators suggesting that they have entirely the wrong approach.
Instead of viewing mathematics as a series of problems to be solved, each of which has a solution, maybe it would be neat of the Khan Academy actually spoke to some educators (and then publicly announced the results of this consultation)?
What would you suggest calling running pilot studies at schools (both affluent and lower-income), like Khan Academy has done with schools in Los Altos and Oakland (http://educationnext.org/can-khan-move-the-bell-curve-to-the...), and getting feedback and suggestions from the teachers who are implementing the pilot programs? Of course they could they do even more (although who of us really knows who they've spoken to and how much/what about), but it's a bit unfair to suggest that they haven't been working with educators.
The lessons they're learning from the massive amounts data they've been collecting from real students (ones outside the pilot programs, who are using their material in an uncontrolled, natural manner) also shouldn't be discounted, and I would argue that it might be as useful as talking to educators can be. It's definitely a Silicon Valley thing to put data up on a pedestal, but should it be valued any less than education research that can be hard to generalize from due to problems with experimental design (like giving extensive training to teachers in the experimental conditions when it's unlikely that most teachers who'll have to implement the same experimental curriculum will have the same sort of training/enthusiasm) or anecdotal evidence from teachers working with one or two classes?
Are you talking about the feedback in this article? Its sort of hard to act on "Using computers to teach math is just stupid".
As to why many people might want to defend Khan Academy, well, its because I think I would have been much happier with Khan Academy than the math education I actually had, and I would very much like it to be available to children like myself. I was bored stiff in math class in middle and high school, and being able to work at my own base, not bound by the slowest person in the class, would have been amazing.
The problem here is that everybody is wrong and everybody is right. There's no magic method that will help everyone learn everything. The answer is that we need to figure out the right mix of things, and empower students and teachers.
My knowledge of Khan Academy-type methods mostly comes from some NPR features about experiments in New York City. In the featured experiment, there was a classroom of kids that were in a workshop setting. Some using computers, others working at desks with access to a remote tutor, others talking to the teacher.
The teacher was in a role more like a coach vs. a talking head at the front of the room. Different students need help at different points, and using the computerized system enabled the teacher to provide more individualized help.
Educational establishment types are afraid of change -- because their power is derived from the cash flow of union duespayers whose jobs may be at risk. Silicon Valley types see a way to tap into a ridiculously large stream of money.
Learning math from a human in a classroom works because it forms a local culture that cares about math. Students have a chance to be motivated by what the teacher cares about, and this is a very social phenomenon.
It is harder to express sqrt(3/4) on a computer in a meaningful way because there is no ubiquitous and intuitive interface for doing this -- the article has a good point here. But that doesn't mean it's a problem we can't address.
I like the idea of thinking of students in a classroom as young mathematicians, because it respects the students and inclines us to teach in alignment with real-world use cases -- it addresses the 'when will we ever use this' question. From that perspective, it would be a mistake to exclude technology based on its shortcomings. If it is genuinely useful for doing math, then it is worth learning.
I see a lot of concern about math education -- I think most of it misses the main problem, which is that there's zero motivation for people who are great at math to become teachers. If I have a PhD in math, and I become a math teacher, then (i) my math friends will think I will never publish a paper again, (ii) they'll think I am no longer good enough for serious math research, and (iii) I will take a huge pay cut vs typical math-PhD jobs. I might as well add (iv) I would suddenly be working with many people who don't respect or care about what I'm good at (students). I'm an optimist - I think all of these are addressable in the long run. Questions about the virtues of technology are secondary to the ecosystem and cultural perceptions around teachers.
Fine. Express the homological proof of the Brouwer fixed-point theorem on a computer.
Ideally in a way that is easily understood by both a computer and a mathematician. Bonus points if it can be used as a teaching tool, too, so that a (sufficiently mature) math student could read it and follow the proof.
Assuming you agree the above is Really Hard™, then we've established there's an upper limit on how effective computerized math education can be (for now).
This leads to a bunch of interesting questions. Where is that limit? Can we design a computerized version that satisfies the design requirements for that system? Do we have the technology to make it happen? Is it even a technology problem to begin with?
We have arrived at a point where innovation in education is just beginning. Khan Academy doesn't have all of the answers, but they have made a significant contribution to the marketplace of ideas and tools, which will continue to evolve and grow.
That type of innovation is the key to significant progress. And this sort of discussion will drive it forward.
So many people seem eager to pick the winners and losers in this space right now. It is a bit early for that.
As lovely as symbolic mathematics is, it does have one major problem: it can not be parsed.
I do not mean the symbols cannot be drawn etc, but even a simple expression like "ab + c" can not be disambiguated. Is it "a*b + c"? or is there a variable "ab" added to c.
This requires the idea of "closures" .. i.e. history/state in which the expression appears. (Wolfram discussed it at a conference once, but I don't have the reference alas.)
So the solution is either to simply introduce new symbols to facilitate parsing (i.e. require multiplication symbols .. but there are more) or introduce heuristics that use history as the human does.
One promising technology is touch screens and sophisticated trackpads (Magic Pad for example) which lets us "draw" mathematics.
For me, traditional maths education in my country fails me (although my country is ranked very highly in maths and the implications of failing maths is huge as schools reject you)
I would say, what don't work for you doesn't mean it won't work for others. Khan Academy, at the very least gave me hope that I can actually do some maths, which is more important than giving up on maths totally.
Awesome! I was just talking about this with a friend the other day. My formal education is in mathematics, but I've been writing software since high school.
I wish Dan made his argument in a less pointed way, though.
The underlying question is this: what does education look like on the web? Can every subject be taught there?
In design, a skeuomorph is a derivative object that retains some feature of the original object which is no longer necessary. For example, iCal in OS X Lion looks like a physical calendar, even though there's no reason for a digital calendar to look (or behave) like a physical calendar. The same goes for the address book.
This is what I see happening in online education. I don't think it's a case of "lol, Silicon Valley only trusts computers," but rather starting off by doing the most literal thing.
Textbooks? Let's publish some PDFs online. Lectures? Let's publish videos online. Homework and tests? Let's make a website that works like a multiple-choice or fill-in-the-blank test.
These are skeumorphs. There's no reason for the online equivalent of a textbook to be a PDF, it's just the most obvious thing.
For me it's 1000x more interesting to ask "On the web, what's the best way to do what a lecture does offline?" than to say "Khan Academy videos are the wrong way of doing it."
I think sites like Codecademy point the way when it comes to programming. The textbook is the IDE.
What does that look like for math? It's much harder because, like Dan says, computers aren't the natural medium for mathematicians, so there will always be a translation step from math-ese to computer-ese.
Once you're past basic math and are working out of a higher-level textbook, the exercises becomes very awkward to express on a computer in a way a computer can understand.
And then grading -- oh boy!
Take this, for example. Here is the first exercise from my first-year college calculus textbook (Michael Spivak's Calculus).
Prove: If ax = a for some number a != 0, then x = 1
If you see that and think "That's easy enough to express in a way a computer understands, and there are proof verification systems" you're missing the point. No mathematician does his work that way. It would be like asking someone to learn programming by reading through a book and instead of writing
for(i = 0;i < n;i++) {
printf("%d\n", i);
}
you force students to write it down by hand in plain English instead of C. If you're an engineer, think of all the ink spent on whether whiteboard interviews are good or not. Asking mathematicians to do their work on a computer will get the same kind of response.
(Empathy, brother!)
The above was just my stream of consciousness, but I've been thinking about this for a bit now. Love this topic!
*: Yes, I've seen his videos. They're fantastic and do a great job illustrating his educational philosophy.
There are areas of mathematics for which proof assistants do a much better job; here is an interesting presentation using the theorem prover Coq as a virtual TA for discrete math: http://www.cis.upenn.edu/~bcpierce/papers/LambdaTA-ITP.pdf
I want to take objection to the presentation of your example. It should be contextualized with the first chapter of Spivak's textbook, where he talks about algebraic properties of addition and multiplication for "numbers" (which aren't particularly well defined, but ostensibly since this is a Calculus textbook they are actually real numbers.) You can do these proofs fairly straightforwardly with rewrite rules, and proof assistants actually have rather good support for this. (The real difficulty shows up when you get to more complicated theorems, when we'd like to sweep rigor a little bit under the rug.) Look at the end of this book chapter for an exercise which asks the user to teach a proof assistant how to automatically show many simple properties of groups http://adam.chlipala.net/cpdt/html/LogicProg.html
I'll read that presentation when I get back. First impression: the author needs to learn about luminance as it applies to readability! Blue text on black background == headache.
When you say "do a better job" what do you mean? I'd love to see papers about it.
And yes, in that chapter you learn the field axioms and it's fairly straightforward for a computer to verify the statement given you present it to the computer in the correct way.
That's my point, and I think Dan's point, too. Mathematicians don't do math that way.
Like I said: "If you see that and think "That's easy enough to express in a way a computer understands, and there are proof verification systems" you're missing the point."
As for http://adam.chlipala.net/cpdt/html/LogicProg.html, it looks relatively foreign to me. I'm going to read through it and figure it out, but would it helpful for a first-year college student who went from AP Calculus the previous year to Spivak's Calculus the next to be spending their time on logic programming?
They've probably never even heard the word "lemma."++ It assumes you're at least a little proficient with the language of mathematics.
Pointing to things like Coq as a solution is like telling an average computer user, "Why use Dropbox when you can just use rsync and some shell scripts?"
It might be the basis for something better designed, but it's only part of a larger, hitherto undiscovered, solution.
++: I'm describing my experience. Spivak was the first time I was introduced to the proof-as-exercise paradigm that's the cornerstone of every higher-math textbook.
Proof assistants have close ties to logic and type theory, so it's not much surprise that they're quite good for doing this small, subarea of mathematics. (Indeed, I think in the not too distant future we will be teaching logic using something like an interactive theorem prover. It makes a bit of the formalism quite clearer.) There has been some push, especially in Europe, for using these programs to prove other mathematics; it is certainly possible, although few would say it is pleasant or resembles how ordinary mathematicians work. But I think that, for specific areas of mathematics, these tools are useful today. Especially when trying to teach students how to think formally; you'll find that formal corresponds closely to how the computer thinks about things (and the whole problem is that mathematicians are not very formal at all.)
I just went through the presentation. It's an interesting experiment, but seems completely unsuitable for teaching the example I gave from Spivak to a first-year college student with no prior exposure to CS or formal mathematics.
It might be useful to someone who has a extensive background in functional programming, but it's in a completely different universe than the one most mathematicians live in.
I agree that at this point, it's completely unsuitable for the staples of college mathematics (e.g. calculus, differential equations), although it was not clear non-mathematicians were being subject to any sort of rigor anyway.
That said, these tools have been used to teach Freshmen CS majors, and to reasonably good effect.
It's interesting that many of the points raised in the article demonstrate the opposite point - that computers should have a greater role in mathematics, and that the current system needs to be changed.
Currently, despite what they claim, schools basically just teach students to follow set methods for solving problems with little thought, basically like an algorithm. These processes can all be solved by a computer, so instead the student can learn the actual thought processes involved and how to use computers to solve real-life problems. Math education has not fully adjusted to the existence of computers. Some of these skills themselves are difficult, so teaching a large group of students in one class would probably not be so effective. Instead, the best way to learn would probably be to use an interactive tutorial combined with the option to ask someone for help.
I think the author is fussing around the wrong problem. I guess the situation is something like this.
Incompetent/bored math teacher < khan academy < better online learning platform < Good math teacher.
The number of bad math teachers across the whole planet is staggering considering that we are nearly 7 billion people on the planet especially more so in the developing and underdeveloped countries. Khan academy sets a reasonable good lowest common denominator education that our next generation of students can receive both in quality and coverage. Competition will just push this bar even higher.
To replace good teachers we may have to wait for deep AI even if wanted to do that. Till then KA, Udacity etc. can produce more autonomous students who can make better use of a great teacher's time.
That second problem is interesting. I'm gonna give it a try, though I know essentially nothing about geometric proofs.
Inscribe 8 squares inside a circle such that point A intersects the circle, point B intersects point C on the adjoining square, and point C intersects point B on the proceeding square. For every square, draw a line from point D to the center of the circle.
I might be missing the point, but math is just working with symbols - we don't need to bend to fit the symbols, but we can make the symbols bend to fit us.
The way math was written 200 years ago won't be the way we write it in a few decades, will it?
Mathematical notation has been chosen, or developed over time, because it is highly efficient at communicating a huge amount of information in a short amount of space. It is almost as if the natural tendency of mathematical notation is to act a as a space-saving algorithm; the information content to notation ratio in mathematics is extremely high.
Most attempts to communicate this notation to computers has been difficult at best, with MathML being much too complex for humans to actually write, and LaTeX often requiring much looking up of the various short cuts that been developed, particularly for beginners.
Here's an example of some LaTeX to produce a mathematical diagram.
What juiceandjuice said. The transition costs (in terms of re-educating people and re-writing text books) is far too high for the moderate benefit of bending our notation to the limitations of 2012 computers. Instead, this problem will be solved by things like mathematica, tablets, and the math handwriting recognition software which appeared on HN the other day:
would "u r wrong" have been understood as easily 30 years ago as today? Language change, even 30 years is enough for some changes to happen. Now I'm not saying that 30 years from now we will be talking in im speak but you can't deny that english of today has changed in the last 30 years.
Mathematics has also changed over time. Trying to read mathematics documents from Fermat's period would be rather hard today. In the 20th mathematics saw some pretty drastic changes in the way it's expressed (someone can correct me if I'm wrong on this). Check out this group who had some pretty big influence on how math is expressed today. http://en.wikipedia.org/wiki/Nicolas_Bourbaki
Being symbol agnostic is acceptable, even expected, once mastery is accomplished. Dealing with both forms (computer and paper) while learning can be challenging - possibly enough to be a turn off altogether.
Funny that I was just teaching that lesson about fraction comparison last Saturday morning, but I was using fraction pairs that are a bit more challenging (from the excellent textbook Algebra by the late I.M. Gelfand and Alexander Shen).
I just put problem 40 from the book, which I taught last week to children of third-grade to fifth-grade age, into Wolfram Alpha's natural language interface.
The Wolfram Alpha input and output is convenient for making the teaching point, and could spark a discussion about problem 41, which is
41. Which is greater, 12345/54321 or 12346/54322?
Of course a sensitive mathematics teacher is supposed to recognize at once that what is really being asked for by the second problem is a way to generalize when a/b is greater than (a+1)/(b+1) and when it is not. I will wrap up that part of the lesson next week.
I have posted recently here on HN that "There will continue to be an important role for in-person teachers,"
even after online teaching tools become much more fancy. I recommend some good tools (I don't think Khan Academy is the best available online mathematics teaching tool, but its price point is appealing) in that comment. I also include links in that comment to thoughtful recent articles about improving mathematics education. A skillful teacher will teach learners how to use tools, when tools are suitable for getting the answer, and how to use the unaided human brain and speech when that should be enough to get (and EXPLAIN) the answer. A big part of mathematics learning is learning how to use appropriate tools and methods in different circumstances. I don't decry online mathematics learning tools; I model in the classroom using the good-old human brain, sometimes with some help from pencil-and-paper or whiteboard-and-marker calculations, to puzzle through challenging mathematical problems
and get reality checks on whether the procedure used to reach a solution is correct or not.
AFTER EDIT: After posting this comment, I asked my Facebook friends (who include a number of professional and amateur mathematics educators, including homeschooling parents who have brought up International Mathematical Olympiad gold medalists) about the blog post submitted here, and one of those friends suggested that the blog post author look closely at the Art of Problem Solving
model of online mathematics education. "The medium is not the message, because the medium is only stepping in to do (interactively) what you would do in person if you could, and instead distributing the teaching resource more widely, but basically in the same mode." I agree with that suggestion, and with that comment on whether or not the medium is the message if online mathematics education is well done.
This problem was enough hard for me that I want to post my solution.
(a+1)/(b+1) = (a * (1 + 1/a))/(b * (1 + 1/a))
Which in turn, is the same as (a/b) * ((1+1/a)/(1+1/b))
This can only be greater than a/b if (1+1/a) / (1+1/b) > 1.
A fraction is greater than 1 if the numerator is greater than the denominator, which means 1/a > 1/b. Dividing by a greater number leads to a smaller result, so this happens if a < b.
12345 < 54321, so 12346/54322 is greater.
Did I overlook a much easier way to solve this, or is third grade much better at math than I am now?
I suck at pure math, so I tend to think of this in some sort of analogy.
Suppose the fraction a/b is some statistic you are trying to measure, say, a batting average or percentage of correct notes played in Guitar Hero. (a+1)/(b+1) would be the new fraction after you got the next one right. By getting the next one right, did you improve your score?
Of course, if you have a perfect record already, getting an additional 1-for-1 won't change anything. And if a>b, then you'd have to somehow score more than 1 point per attempt in order to maintain the same ratio, so (a+1)/(b+1) would be lower.
I used to be a mathematician, and I think your comment is everything that math education in school should aspire to be :-) Another example in the same vein is Terry Tao's airport puzzle (http://terrytao.wordpress.com/2008/12/09/an-airport-inspired...), scroll down to Harald Hanche-Olsen's comment for the best explanation.
That is a very nice explanation. It shows why intuitively we expect (a+1)/(b+1) to be greater. But note that there are some unstated assumptions. Namely that both a, b > 0. And so we see that intuition at times carries unstated assumptions that can be a trap in a generalized situation.
Tokenadult and codehotter didn't mention the case when a or b < 0.
Currently what's frustrating is that we subject everyone to the same level of training. Those whose interest lie elsewhere are subjected to unnecessary and meaningless torture, while those who are really interested in Mathematics as a subject are left to their own devices to find additional training (which thankfully is easier these days than when I was a kid, though it could be made substantially easier still).