What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
The thing that’s wrong is exactly that it would indeed have to be instead of calculus. And then students would not pass the nationally mandated matriculation exams or university entry exams, which test knowledge of calculus. One part of the system can’t change independently from the others. I agree that if you’re going to teach just one field of math, then calculus is not the optimal choice.
I do believe that for every field that’s taught in highschool, the most important theories and results should be taught: evolution, genetics, cell structure and anatomy in biology; Newtonian mechanics, electromagnetism and relativity in physics (QM probably requires too much math for any high-school program); etc.
There won’t be time to prove and fully explain everything that’s being shown, because time is limited, and it’s better that all the people in our society know about classical mechanics and EM and relativity, than that they know about just one of them but have studied and reproduced enough experiments to demonstrate that that one theory is true compared to all alternatives of similar complexity.
And similarly, I think it would be better if everyone knew about the fundamental results of all the important fields of math, than being able to prove a lot of theorems in a couple of fields on highschool exams but not getting to hear a lot of other fields.
As far as possible, we should allow students to learn more and help guide them to the sciences. But scientists are in the end a small minority of the population and some things are important to teach to everyone. I don’t think calculus passes that test, and neither does classic geometry and analytic geometry, which received a lot of time in my school.
Instead I would teach statistics, basic probability theory, programming (if you can sell it as applied math), basic set and number theory (e.g. countable and uncountable infinities, rational and real numbers), basic computer science with some important cryptography results given without proof (e.g. public-key encryption). At least one of these should demonstrate the concept of mathematical proofs and logic (set theory is a good candidate).
Interesting question. I’m a programmer who works in EDA software, including using transistor-level simulations, and I use surprisingly little math. Knowing the idea of a derivative (and how noisy numerical approximations to them can be!) is important—but it is really rare for me to actually compute one. It is reasonably common to run into a piece of code that reverses the transformation done by another pieces of code, but that is about it. The core algorithms of the simulators involves sophisticated math—but that is stable and encapsulated, so it is mostly a black box. As a citizen, statistics are potentially useful, but mostly just at the level of: This article quotes an X% change in something with N patients, does it look like N was large enough that this could possibly be statistically significant? But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
I see computer science as a branch of applied math which is important enough to be treated as a top-level ‘science’ of its own. Another way of putting it is that algorithms and programming are the ‘engineering’ counterpart to the ‘science’ of (the rest of) CS and math.
Programming very often involves math that is unrelated to the problem domain. For instance, using static typing relies on results from type theory. Cryptography (which includes hash functions, which are ubiquitous in software) is math. Functional languages in particular often embody complex mathematical structures that serve as design paradigms. Many data structures and algorithms rely on mathematical proofs. Etc.
But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
That is also a fact that ought to be taught in school :-)
The thing that’s wrong is exactly that it would indeed have to be instead of calculus. And then students would not pass the nationally mandated matriculation exams or university entry exams, which test knowledge of calculus. One part of the system can’t change independently from the others. I agree that if you’re going to teach just one field of math, then calculus is not the optimal choice.
I do believe that for every field that’s taught in highschool, the most important theories and results should be taught: evolution, genetics, cell structure and anatomy in biology; Newtonian mechanics, electromagnetism and relativity in physics (QM probably requires too much math for any high-school program); etc.
There won’t be time to prove and fully explain everything that’s being shown, because time is limited, and it’s better that all the people in our society know about classical mechanics and EM and relativity, than that they know about just one of them but have studied and reproduced enough experiments to demonstrate that that one theory is true compared to all alternatives of similar complexity.
And similarly, I think it would be better if everyone knew about the fundamental results of all the important fields of math, than being able to prove a lot of theorems in a couple of fields on highschool exams but not getting to hear a lot of other fields.
Really? I think it’s very beautiful and it’s what hooked me. And it’s the bit the scientists use. What would you teach everyone instead?
As far as possible, we should allow students to learn more and help guide them to the sciences. But scientists are in the end a small minority of the population and some things are important to teach to everyone. I don’t think calculus passes that test, and neither does classic geometry and analytic geometry, which received a lot of time in my school.
Instead I would teach statistics, basic probability theory, programming (if you can sell it as applied math), basic set and number theory (e.g. countable and uncountable infinities, rational and real numbers), basic computer science with some important cryptography results given without proof (e.g. public-key encryption). At least one of these should demonstrate the concept of mathematical proofs and logic (set theory is a good candidate).
Interesting question. I’m a programmer who works in EDA software, including using transistor-level simulations, and I use surprisingly little math. Knowing the idea of a derivative (and how noisy numerical approximations to them can be!) is important—but it is really rare for me to actually compute one. It is reasonably common to run into a piece of code that reverses the transformation done by another pieces of code, but that is about it. The core algorithms of the simulators involves sophisticated math—but that is stable and encapsulated, so it is mostly a black box. As a citizen, statistics are potentially useful, but mostly just at the level of: This article quotes an X% change in something with N patients, does it look like N was large enough that this could possibly be statistically significant? But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
I see computer science as a branch of applied math which is important enough to be treated as a top-level ‘science’ of its own. Another way of putting it is that algorithms and programming are the ‘engineering’ counterpart to the ‘science’ of (the rest of) CS and math.
Programming very often involves math that is unrelated to the problem domain. For instance, using static typing relies on results from type theory. Cryptography (which includes hash functions, which are ubiquitous in software) is math. Functional languages in particular often embody complex mathematical structures that serve as design paradigms. Many data structures and algorithms rely on mathematical proofs. Etc.
That is also a fact that ought to be taught in school :-)