As good a time as any to go re-read "A mathematician's lament" [0] which begins:
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.
Very interesting read. It's not only a problem with mathematics though, it's nearly every subject in school, whether it's science, languages or math. It seems to me that the problem is that to teach well, you need good, autonomous teachers, and that just doesn't scale. Like the author said, you can't teach teaching.
I'm of the unpopular opinion that digital learning is the only possible solution for this. I'm talking about a digital tutor with infinite patience, with the best qualities of the best teachers. I'm not talking about a robot that passes the Turing test, just one that passes the test for the narrow field of teaching a specific subject.
People say you need real teachers, you need the human touch, but if the average teacher sucks (and they do), then I'd rather be taught by software. And I think it's just one of those things people say because it sounds true: "you need the human touch", just like people said they prefer real books over e-books, face to face conversations over texting, navigating by feel over using the GPS etc. These sentiments almost always turn out to be wrong.
Yes you can. I mean sure there is an explanatory gap with things like empathy and identifying scaffolding opportunities. But that just means you can't teach ALL of teaching with a book, experience and guidance is also required.
> I'm talking about a digital tutor with infinite patience...
One that can empathise with a students particular background/learning style and understand what might work to convey a novel concept better?
> People say you need real teachers, you need the human touch, but if the average teacher sucks (and they do), then I'd rather be taught by software.
It's not about the human touch. It's about empathy, and expert diagnosis of learning conditions, which the worst teacher does a better job of than the best computer/software. Just because most of our teachers suck doesn't mean software is the better solution. Better teachers are the solution, perhaps that is better education for teachers, or perhaps it's more communication from the realm of pedegogy down to the teaching curriculum that teachers are taught from.
Software can assist teachers, and can even replace certain aspects of teaching. But you will leave a lot of students behind if you try to replace teachers completely.
My prediction is that data can replace empathy, just as Watson can beat human competitors most of the time. Students struggle with new concepts, but do they all struggle in uniquely different ways? Aren't there patterns? I think the data from millions of interactions, and a curriculum guided by the best teaching minds the world has to offer will be better than the average teacher, up to a certain point. Note that I mean this in a very narrow sense, when constrained to a specific topic, and especially when explaining something intuitively to a student and answering students' questions. I agree though that for the foreseeable future there still need to be teachers.
I'm all for better teachers, but I think it's just not going to happen. They haven't become better in the past few decades or even century, rather possibly the opposite, with the need for many more teachers than there are competent people to fill those positions. The only solution I can see is to make teaching one of the most highly paid and highly respected professions. There doesn't seem to be any incentive for this, and it would require a huge cultural change. And even if that was the case, there would probably be the a concentration of teaching talent in the big cities at top schools, while rural schools will have to do with the scraps. Software at least is democratizing.
Of course what I'm suggesting would require some very sophisticated software that doesn't exist today, but I would say that for me, if I would have followed the videos on Khan Academy (for science and math) instead of the education I received, I would probably have been better off. I grew up in a rural area, and I know how bad it can be. After getting an engineering degree, I now see how little those "teachers" actually knew or understood. It's shocking really. But I'm not surprised when you look at the kind of people that go into teaching today.
You can't teach teaching in isolation. A math teacher has to know math and teaching, but that's not really how they're trained - an undergraduate Math Education curriculum diverges from the Math curriculum shortly after calculus, which is far too soon.
An undergraduate math education is more than enough to teach the current high school math curriculum.
From my point of view though, if my statement above is true, that is very sad. As I don't think it should be. The math curriculum is horribly deficient. But software doesn't seem to be the answer to fixing the curriculum.
> An undergraduate math education is more than enough to teach the current high school math curriculum.
Is that an undergraduate Math degree, or a Math Education degree? If the former, you don't get training in pedagogy. If the latter, you don't get enough proof-based classes to properly understand what math is or to fully understand the subjects you'll teach.
I just graduated high school, I always thought Math was amazingly interesting and I couldn't understand why I wasn't doing so good in school and why topics that I thought would be very interesting appeared boring after the teacher told I had to learn some rules and apply them over and over again to get good grades (I'm oversimplifiyng but you get the idea). We were thought the 'what' and not the 'why'. I was in a computer science high school so we had subjects that you'd find in a CS curriculum in any university - EE, C, OOP and a bunch of other random stuff. So for example when it was time our math teacher had to tell us about sin and cosine the explanation was merely 'oh here's a bunch of ways to calculate your angles, remember them for the next exam'. It was in some other class (something related to EE) that the teacher went deeper and and explained all the magic behind it. I mean, seriously, the Wikipedia page has animations that make it easier to understand than my math books.
I always thought I was stupid for not having good math grades, I question the sharpness of my brain to this day.
Essays like this make me want to pick up Math again and this time learning it the proper (whatever that means) way... learn to think analytical and in terms of pattern making/matching etc etc... not even sure where to start though.
If you're motivated / curious enough / have enough time, I recommend Spivak's Calculus as a way of learning calculus the, as you say, 'proper (whatever that means) way.' It's basically a treatment of calculus from the perspective of real analysis, having the 'no ad-hoc rule teaching' policy at its foundation. You prove things, build up a non-fragmented edifice slowly, and end up being introduced to analysis such that you can then pick up other things (it's a rather extensive treatment of calculus indeed.) Or so goes my narrative in the midst of frustrations regarding self-motivating to continue individually progressing through the book. :)
The 3rd and maybe the 4th editions can be found online by doing an internet search for pdf/djvu files.
I majored in mathematics at the University of Chicago and Spivak's Calculus was my first-year calculus textbook. It confirmed what I had long believed: what I was being taught in high school wasn't really mathematics.
Frankly, his parody of a "music education" sounds a lot like my music education.
My orchestra teacher used to drill us on the circle of fifths and how many sharps were in this or that key. But I didn't give a damn about any of it until a friend showed me how to improvise over the 12 Bar Blues.
"See, if you play this note in this key, and if you wiggle your finger like that, it sounds really cool." "Ohhhhh..."
The problem with this metaphor is that math is not only a form of art. Only for a very small fraction of population, math is art. For the rest, some basic maths are necessary skills for surviving in this society, like the skill to following rules/ play under rules. While music is much less on this necessity side.
I did. And part of my comments are even from the later part of his article on the criticizing of high school geometry. he talked from the real subjects of geometry side, and looked down the formal language presentation in the textbooks. But the latter is an important skill training, which is arguably more important than the mere geometry pattern recognizing itself for most of people.
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.
[0]: http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf