Calculus and pattern recognition

[fsm]

When I was studying for my final yesterday, the person I was with told me I get too wound up in the "math" and the "bigger picture" of calculus is pattern recognition. It is the ability to recognize patterns and once that is determined to carry out the appropriate steps associated with that pattern. A little light bulb went off in my head and actually calculus got a little bit easier. I was just curious if this statement is correct?

 

[DeadWolfe]

It might help if you gave us more details.

 

[fsm]

What do you mean? This wasn't for a particular problem. This was mentioned for calculus as a whole.

 

[HallsofIvy]

Most of mathematics could be said to be about "pattern recognition"!

 

[fsm]

Most of mathematics could be said to be about "pattern recognition"!

He said that too but was saying it was more "special" for calculus. Anywho I guess for me it was more like the scene in Dead Poets Society when Robin Williams tells them to stand on the desks to change their perspective. Just provided math in a new light that no one ever explained to me before.

 

[FrogPad]

Just curious, but how does thinking of calculus as pattern recognition actually help you specifically?

Just wondering about your perspective. If you could point out a specific example, it would probably be easier for me to understand. Hopefully I can pattern match the example to illuminate the entirety (or at least a part thereof) of your thoughts.

 

[matt grime]

Well, all calculus proofs that you meet at this (presmumed) level are the same (i.e. spotting a pattern) and all results are saying the same thing. There is nothing mysterious about this. I tend to choose the explanation in terms of errors and playing a game: you get to pick an epsilon and if I can find a delta I win (it is continuous) if not you win (discontinuous).

 

[FrogPad]

So a computer could presumably solve all questions at the _x_ level? No insight into the question is required? A problem can be pattern matched and then steps are given to set the state to winning or losing?

I think an argument could be made comparing the "canned" responses given when one learns a foreign language to the "uncanned?" responses that one gives using their natural language.

In one sense an individual is regurgitating a response and an action ensues.
ala, Fourier expand x, or solve $$\int \sin x \,\, dx$$

In another sense an individual is "thinking" (maybe this is merely regurgitating at a higher level) and creating ideas, discovering insight, etc...

hmm, thinking of an example here seems to be rather difficult. Maybe using math to capture a large amount of ideas into a compact form, such as Maxwell's equations. Again this can come back to "pattern" matching, since a formal system must be setup, and the problem is confined to that space and the operations within it. I guess at one point the complexity of the pattern matching becomes too great to consider it pattern matching.

Afterall, we have to feed the computer its instructions and more importantly interpret the results.

 

[matt grime]

To solve anything at this level is purely mechanical: you are given all the methods required to solve things and it is just 'plug and chug' as one of my students once said.

Now, to decide what to do to solve/prove something for which there is no method, or to even decide what to solve/prove in the first place, now there is the interest.

 

[matt grime]

No insight into the question is required?

The only insights required at the level I think we're talking about are

1. if necessary, translate words into equations etc.
2. recall which method is required to solve these things.

 

[FrogPad]

The only insights required at the level I think we're talking about are

1. if necessary, translate words into equations etc.
2. recall which method is required to solve these things.

And hence, "pattern recognition".

To solve anything at this level is purely mechanical: you are given all the methods required to solve things and it is just 'plug and chug' as one of my students once said.

Absolutely. None of these questions are new. No insight is required to "solve" these problems. One could create a book of all solved problems for various questions, and an advanced lookup table. Of course this wouldn't be feasible, so it makes more sense to teach the mechanical methods to solve these questions. In a way what is being taught is how to use the lookup table.

However, I don't think that learning these methods can be trivialized. In learning these methods, one gains insight into the questions themselves. One learns how to look at the problem as a whole, how to setup boundaries, how to break up a problem.

Calculus is trivial; differentiating is mechanical, integrating is mechanical. ODE's are extremely mechanical. PDE's? (this is where my knowledge begins to stop), but from what I can tell, there are methods for solving, but it is not as mechanical as the other subjects I mentioned.

My point is this... screw the mechanical aspects. Those mechanical methods were created from non-mechanical ideas (not always of course), and THIS is what is important... it's the creation of the plug and chug methods themselves. These methods weren't simply pattern matched, because nowhere in that giant book of solutions exists a solution. A new page has to be written. Inventing vector calculus, group theory, or proving the unknown cannot be linked to "pattern matching". Or can it?

Maybe it already exists, and it just requires someone quite intelligent (like yourself matt) to figure out what the pattern is, match it, and yield a solution.

It's interesting how problems get solved. There have been times where it has taken me a few days to solve problems in my PDE class. Obviously not two days straight of thinking, but subconciously something was going on, that eventually led me to a solution. Maybe some type of pattern recognition is happening in the background.

Now I'm sure you work on problems that take weeks, months, or years to solve. What's going on behind the scenes? If you can answer that question, and then create a "mechanical" process for it, we would be all set.

 

[matt grime]

Sadly there is not enough desire in the student body it seems to learn about the whys and wherefors of mathematics (I speak of things like freshman and sophmore courses in the US) except perhaps in the honours (sorry, honors) classes.

I did take the 'big picture' way to teach the chain rule (it's quite simple: if I alter these things a little, and that's a function of these then they change like so...) instead of teaching them as the book intended: four different rules for four different cases (and sod the general case). I think I was justified in my 'teach a man to fish' thinking, though the students disagreed.

I must take issue with your assertion that ODEs are very mechanical. I presume you're talking analytical solutions here not numerical. Very very few ODEs can be solved at all, and even fewer PDEs. And the same is true of integration. That is why there is a lot of emphasis on learning lots of examples in these courses: they are the only ones you can actually do.

Incidentally, where I am now, we (Bristol) do teach the ideas behind the method and not just the method; most places in the UK do.

The creation of 'group theory' arguably was 'pattern matching'. Like most maths its invented because people have worked through some examples and need to formalize their ideas. Galois had worked out the ideas, and a lot of notions like solvability, through looking at roots of polynomials. Amazingly, a lot of the ideas had come up before with Gauss working through what we would now call class field theory.

One of my old teachers, Tim Gowers, has a belief that anyon can do maths if they just put in the time to 'spot these patterns', and I recall reading someone formulate the argument that you only needed very few ideas in maths to have a career in it, all you needed was the insight to see how to apply it to any situation. Their example was, remarkably, Hilbert, though I forget the details. Opinion seems to be divided on the role that computers will play in the future in this arena.

 

[FrogPad]

I must take issue with your assertion that ODEs are very mechanical. I presume you're talking analytical solutions here not numerical. Very very few ODEs can be solved at all, and even fewer PDEs. And the same is true of integration. That is why there is a lot of emphasis on learning lots of examples in these courses: they are the only ones you can actually do.

In the ODE and integration case, I was referring to what is taught in the class. They were very "simple" (relative term) problems that allowed the "plug and chug" method to work its way to a solution. Each example just needed one to spot the pattern, determine which method should be applied, and move along.

It was interesting to me to see the ODE class being applied in a different situation. In an introductory engineering course we had to model a bungee drop experiment. Basically we had to use Newton's laws to model an object that would fall through the air and rebound. The "interesting" part was that this very simple could not use nonlinear equations. We had to use numerical methods to solve them. Yet, the entire ODE course was methods for analytical solutions.

The same with integration. How do I solve:
$$\int e^{x^2} \,\,dx$$ with my calculus knowledge?

The questions that are posed in these courses are merely plug and chug, and hence very mechanical. It would have been interesting to have a few questions on exams that asked one to solve "unsolveable" ODE's and integrals, basically where the plug and chug method falls apart. A question that the correct answer is something along the lines of: "this is a class of problem where the methods taught in this course do not apply"
This might be considered a trick question, but in some ways it's a trick to teach students all these methods and just skim over the fact that it only applies to a very small portion of problems.

I did take the 'big picture' way to teach the chain rule (it's quite simple: if I alter these things a little, and that's a function of these then they change like so...) instead of teaching them as the book intended: four different rules for four different cases (and sod the general case). I think I was justified in my 'teach a man to fish' thinking, though the students disagreed.

I found it quite interesting to watch my professor think out loud in the PDE course I took. For example, he was discussing the dirac delta function and showed us the following property:

$$\int_{-\infty}^{\infty} f(x)\delta(x-a)\,\,dx=f(a)$$

Then, he said something along the lines of: think about it, as we move from $-\infty$ to $\infty$ f(x) is being multiplied by $\delta(x-a)$.

I think that right there is what makes a good professor. He could have simply put that property on the board and never said a thing about it, but instead he offered some insight.

Incidentally, where I am now, we (Bristol) do teach the ideas behind the method and not just the method; most places in the UK do.

I wish I was getting my education there :) I would have gone into math if I grew up there.

 

[FrogPad]

Oh, I forgot to ask you. Do you have any suggestions on interesting reads that I should check out? I read godel, escher, and bach and loved it.

 

[matt grime]

The same with integration. How do I solve:
$$\int e^{x^2} \,\,dx$$ with my calculus knowledge?

exp(x^2} is continuous, hence Riemann integrable. Define F(x) to be the Riemann integra from 0 to x of it. F is the answer (actually 'an' answer). That is it.

Try reading these:

www.dpmms.cam.ac.uk/~wtg10/equations.html

www.dpmms.cam.ac.uk/~wtg10/mathsindex.html

 

[FrogPad]

exp(x^2} is continuous, hence Riemann integrable. Define F(x) to be the Riemann integra from 0 to x of it. F is the answer (actually 'an' answer). That is it.

So because we have defined F(x) with properties that satisfy the integral we have therefore defined a solution? Since every aspect of the definition of solution is satisfied with these properties...

My logic could be off here, I think I'll need a day or so to let that soak in. It's definitely a different way to think about solutions (but it's not at the same time).

Try reading these:

www.dpmms.cam.ac.uk/~wtg10/equations.html

www.dpmms.cam.ac.uk/~wtg10/mathsindex.html

Thankyou. The ...equations.html was a VERY interesting read. Looking forward to going through the other links as well.

 

[matt grime]

This is one of the problems when we sometimes omit the explanations of what we really mean.

Why for instance is sqrt(2) a valid symbol as a solution to x^2-2=0 and yet the F i defined above troubling?

When you're asked to 'solve', simplify, or find something in class what you're really being asked to do is demonstrate some ability with the material to express the solution in some way that is acceptable. But there is no real way of specifying what is acceptable for everyone. In calculus for instance solve really means express as 'elementary' functions, ie polynomials or (hyperbolic) trig etc.

 

[FrogPad]

Why for instance is sqrt(2) a valid symbol as a solution to x^2-2=0 and yet the F i defined above troubling?

Excellent point. It shouldn't be.I guess it comes down to a measure of some level of comfort. The number 2 is comfortable. I can assign some physical meaning to it. sqrt(2) is somewhat more abstract then the natural number, yet I can think of it in "natural" terms. sqrt(2) is two objects that when multiplied together it equals one object. The F that you defined loses that comfort of the numbers.

I remember when I first took an introductory physics class and we had to work with variables, NO NUMBERS. Everyone (I use the term loosley) was freaking out and longing for the use of numbers. We wanted to get a solution that dropped down to a number with some units, thinking this would yield an intuitive idea that the answer was correct. After some time, our comfort level increased with the new abstraction and we became ok with purely algebraic manipulation (and then plugging in numbers at the end).

So the F you defined is troubling because it doesn't boil down to a neat little number, thus it stays away from my comfort zone. However, just like you pointed out with the sqrt(2) example, it should NOT be troubling.

It took me a day or so to grasp what you said. Not because it was necessarily hard. I could have easily just accepted it. But now I believe it, and it's pretty cool.

I wish I had the mathematical background to understand each of those
articles in the links you passed on to me. I found the " Does mathematics need a philosophy?" (http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html) very interesting.
I'm unfortunately at the level of mathematics that I need to be at for my major (unless I go to grad school). Oh well, I guess there is always PF and self study :)

 

[MalayInd]

When I was studying for my final yesterday, the person I was with told me I get too wound up in the "math" and the "bigger picture" of calculus is pattern recognition. It is the ability to recognize patterns and once that is determined to carry out the appropriate steps associated with that pattern. A little light bulb went off in my head and actually calculus got a little bit easier. I was just curious if this statement is correct?

In the words of Feynman:"Mathematics is looking for patterns"
Some sort of it is given in the article "What is science" available on tha website www.feynmanonline.com

 

[reilly]

Another way to look at this is to replace "pattern recognition" by the more general notion of intuition. A good way to develop such is to try to work the problems in your head, or guess the answer. Often, people first encounter the use of intuition in Euclidean geometry. People who actually make their living doing math are often highly intuitive, particularly if they have lots of experience. It's all a matter of personal style and comfort level. I've known how to do calculus for 50 years, I'm usually pretty good at guessing the right answer. But, an important but, you clearly must then use hard-core logic and analysis to demonstrate your intuition is correct.

All of this is quite well known to many mathematicians and physicists, some of whom -- Einstein -- tend to think in images, others use words, or symbols or combinations thereof. All of this is brilliantly discussed by Jaques Hadamard in his book, The Psychology of Mathematical Invention. Hadamard, by the way, was a world-class mathematician, greatly admired by his peers. The there's Enrico Fermi who could do highly complex physics on the back of an envelope by virtue of his extraordinary intuition. John VonNeuman could compute faster than computers in the early days-- 1940s.

Good luck.
Reilly Atkinson