> and proving or disproving theorems based on rules, exactly like in math, example: "if my assumption about initial conditions and the function implementation is correct then 'a' ought to be 42. If it is not, either my reasoning is false or the function implementation or the initial condition is wrong. Ok so it was not 42..."
Chemistry is like this. "Cadmium would be consistent with the valence electrons required for the bond to work. If the mystery element is Cadmium, then the mass would be xxx, but it's yyy, so we can rule that out..." Of course I've forgotten the specifics, but the method is just like it is in programming.
I have never been asked to do anything like this in a math class. Equations are sometimes called theorems, but that's the extent to with this sort of analytical thinking ever factored into a math class up through Calculus. (There was a little of this in geometry, and a really fun two weeks in propositional logic, but geometry was still mostly formula-driven - numbers of vertices and such.) I gather that's the sort of thing Analysis is about, but I'm not talking about math for math majors, I'm talking about math as most people experience it.
You memorize some formulas. You're given some formulas. You recognize the information you're given and apply the procedure you were taught to convert it to the information you're asked for. That's it. (Geometry is an exception, as you state. However, geometry is one year of many, and was extremely easy for me. I'm talking about Algebra, Algebra II/Trig, and Calculus.)
Memorizing and drilling steps is not how you become good at debugging - thinking is. Holding the program in your head, tracing it, asking yourself how you would design it and then figuring out where the problems in the logic are. This is nothing close to recognizing the situation as belonging to a particular category and then blindly cranking the computation for that category of exercise to reach the answer.
> I have never been asked to do anything this in a math class.
This is a tragic comment. It is also true. American children waste their time and their brains in American math classes for about 8 years each. They learn to multiply and maybe solve a quadratic equation. "Equations are sometimes called theorems," and that's it. And it's not getting better; as a culture, we don't understand or respect math so we don't support it.
Modern mathematicians don't memorize or drill steps. We don't just find a bigger number each day by adding one. We hold our problems, trace them, ask ourselves how the proof should be designed and figure out where the problems in the logic are. If you've got 2n points in a line, how many ways can you draw a curve from one point to another such that you match them all up and none of the curve intersect? What about if they're in a circle -- is the answer different? How do you characterize a straight line if you're in a non-Euclidean surface? What's the right definition of straight -- should straight mean "direction of derivative is in direction of line" or "shortest distance" or what? And if we're talking about points on a grid, how should distance be measured anyway? Do we have the right definitions to solve our problem? Distance from streetcorner to streetcorner by cab in Manhattan is different than distance as the crow flies is different than "distance" through time and space; how are the concepts consistent, then?
Formulas are just shorthand for relationships. We don't teach students the relationships or the thinking in the US. That's why my Romanian advisor would ask our graduate algebra class, "You should have learned this in high school, no? No... Hmph." American school math is like prison, a holding cell until release that merely increases the likelihood you'll do poorly in the future.
Chemistry is like this. "Cadmium would be consistent with the valence electrons required for the bond to work. If the mystery element is Cadmium, then the mass would be xxx, but it's yyy, so we can rule that out..." Of course I've forgotten the specifics, but the method is just like it is in programming.
I have never been asked to do anything like this in a math class. Equations are sometimes called theorems, but that's the extent to with this sort of analytical thinking ever factored into a math class up through Calculus. (There was a little of this in geometry, and a really fun two weeks in propositional logic, but geometry was still mostly formula-driven - numbers of vertices and such.) I gather that's the sort of thing Analysis is about, but I'm not talking about math for math majors, I'm talking about math as most people experience it.
You memorize some formulas. You're given some formulas. You recognize the information you're given and apply the procedure you were taught to convert it to the information you're asked for. That's it. (Geometry is an exception, as you state. However, geometry is one year of many, and was extremely easy for me. I'm talking about Algebra, Algebra II/Trig, and Calculus.)
Memorizing and drilling steps is not how you become good at debugging - thinking is. Holding the program in your head, tracing it, asking yourself how you would design it and then figuring out where the problems in the logic are. This is nothing close to recognizing the situation as belonging to a particular category and then blindly cranking the computation for that category of exercise to reach the answer.