What Silicon Valley gets wrong about math education again and again

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?

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

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.

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.

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.

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.

I'll take a look at the videos, as you suggest.

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?

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.

Today at 3:30 I'll be at our Los Altos district teacher feedback session for hours.

As a dev, I meet with our pilot teachers regularly.

Our implementations team does it literally every day.

I have multiple email threads in my inbox of long back-and-forth conversations between our developers and our fearless teachers.

Ok, back to work!

* 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?

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.

CPDT is not a good first text; Software Foundations http://www.cis.upenn.edu/~bcpierce/sf/ does better for people with less math experience.

That said, these tools have been used to teach Freshmen CS majors, and to reasonably good effect.

Tokenadult and codehotter didn't mention the case when a or b < 0.