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Comment by jules | Hacker News Reader

This is even true in college, where for example a large part of a physics education is spent computing integrals and solving differential equations in special cases, using approximation methods by hand. It would be more appropriate for an intuition of the physics to use symbolic and numerical methods on a computer a lot more. You realize that doing arithmetic by hand is a largely pointless exercise when you get a calculator, and we should similarly recognize that doing integrals and differential equations by hand is largely futile when you have things like maple and scipy.

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I can tell you right now I'd never hire someone who wasn't able to compute the result of a linear differential equation without using a computer, because it's something that every person I hire needs to do very frequently while designing circuits. The difference in quality of design between one who can analyze a circuit on paper or in the head and one who relies on simulation to discover basic properties of the circuit is massive. Iterated simulation is not a tenable approach to the design of any sufficiently complex circuit.

Moreover, the nature of innovation in my corner of the mixed-signal circuit design world is such that said innovation rarely (if ever) comes as a result of a computer simulation. Much more likely, a person with a deep understanding of the fundamental underpinnings of his/her particular problem gains insight into its solution as a result of the same experience and intuition that leads to the aforementioned understanding.

I can have a computer calculate Fourier transforms for me all day, but it's vanishingly unlikely that any amount of such calculation will lead me to the kind of insight that sparked the invention of CDMA.

It is definitely useful to be able to do linear differential equations by hand. It's not useful to keep doing these things by hand. Just like it's useful to know the algorithm for multiplying two numbers, and it's not useful to keep doing multiplication by hand.

What lets people invent new circuits is their good intuition about circuits, not their ability to solve linear differential equations quickly by hand or to compute integrals by hand. When an expert is analyzing a circuit on paper he is thinking about "what happens if the input to this circuit is a sine wave with high frequency", he's not going to solve the differential equations by hand.

Rather than circuits look at how electromagnetism or quantum mechanics is taught in college. In my case it was integrals, integrals, integrals. Doing these by hand provided approximately zero intuition into the physics. We could have covered more ground if the instructor would just type these into maple, instead of doing them on the board or in the book by hand. Or how many times have I not had to compute eigenvalues of 2x2 or 3x3 matrices. How many times have we not applied crude approximations in class because doing it by hand was too difficult, when typing it into a compute would give you 100 digits of precision in a couple of milliseconds. One time one of my maths teachers how to compute tan(2) or something like that by hand. After half an hour of calculation he had 2 digits. Computing the integral of something to a crude approximation in an edge case strikes me as futile as computing tan(2) by hand.

Maybe, but maybe not. Just like driving through a city doesn't give you the same understanding as walking through a city, immediately solving (say) the Schroedinger Wave Equation for a finite square well doesn't give you the same appreciation for the underlying physics as struggling with the terms one by one.

Lets be clear, no one isn't saying that you shouldn't struggle with the terms one-by-one. Or to not solve something by hand ever -- but teach it and then find useful engaging (and appropriate) ways to use it. But its the current nostalgia over doing things "mind numbingly" repetitively. And "mind numbing" really is the operative phrase here.

I went through this as a child, where you do the same problem over and over again. Some of the symbols change, but the operations are identical.

It reminds me of when I was in 1st or 2nd grade doing long subtraction. And I had informed the teacher that I wanted to create the problems for the class, so she let me (on the chalkboard). But I snuck in some problems where the result would be negative. The kids struggled with these problems, but not a single student in the class said anything about the structure of the problem. And why would they? They learned nothing about what they were actually doing. It was a sequence of subtractions and carrys and other magic.

Pretty much nothing works if it's done badly. many things work if they're done well. I claim that when done well, exercising skills by hand that can be done by the computer yields insight in some cases.

the fact that when done badly it rarely yields insight does not necessarily mean it shouldn't be done.

I'm about to make a sweeping generalisation.

The problem is virtually no one in the world can teach math properly. Those who love teaching kids usually hate math and transfer their lack of skills and negative attitude. Those who really, really understand math are usually the type of people who would never teach 3, 4, 5 year olds.

People end up doing math despite the system, rather than because of it. They can usually point to one specific inspirational teacher. My specific inspirational teacher gave me lots of mind-numbing exercises and helped me find the insights.

Rote learning of mechanical processes will rarely, of itself, lead to anything other than a hatred of math and a lack of understanding about what math is. Repetition of mechanical processes, guided by someone who really understands stuff, and asks prompting questions, can lead to discoveries and genuine excitement and engagement - I've seen it happen.

How will we ever get really good math teachers?

There I have no answer.

Not to minimize the importance of teaching, but the content is important too. Instead of the content consisting of memorizing and performing algorithms, hoping for students to discover meaningful patterns/concepts/insights or rely on teachers to provide them, maybe the content should be doing this explicitly. One topic in the high school syllabus which actually does this is Euclidean geometry, where everything is coherently derived from basic principles as opposed to a grabbag of techniques. Maybe, one could present arithmetic in the same way, not focussing on the how to do calculations but the patterns in these calculations and a few basic principles too discover/prove them. Alternatively, one could focus on applications of mathematical techniques in toy versions of real world problems.

Arithmetic algorithms still have value - for the insight they give on arithmetic and because following a complex algorithm is itself a skill with value.

But there is no need, as we do now, to insist on performing them so many times, or to do them very quickly in exams.

The main problem with this new approach, I feel, is that it makes learning mathematics harder. Building richer conceptual models which is necessary for both applications or theory is more interesting and meaningful, but also more difficult than following prescribed algorithms. It is harder to test in an exam, and where testable the problems are much harder.

This issue of algorithms vs conceptual understanding, is important at the undergrad level too. Eric Mazur has a nice video about this where he also talks about his way of testing conceptual understanding by asking very simple but illuminating questions - http://www.youtube.com/watch?v=WwslBPj8GgI

Thank you for that Eric Mazur talk -- it really captures what I see. In particular the part where he talks about how his students, at Harvard, were solving triple integrals of complicated bodies to calculate the moment of inertia -- yet they didn't have basic high school level intuition of Newtonian physics.

We're so "drill, drill, drill" focused that we lose sight of why we're doing the drills (Eric Mazur calls it "plug and chug"). It's possible that these drills will in rare instances create the likes of a Colin or an Andrew Wiles, but for virtually everyone else you have a group of students that can solve triple integrals, multiple 5 digit numbers in their head, factor matrices into any form desired -- yet not have a clue why.

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