Because especially for harder problems there were probably dozens of mathematicians who studied the problem for centuries in order to find those derivations that your teacher presents to you.
In my school math education we had the standard that everything we learn get’s proved. If you are not in the habit of proving math, students are not well prepared for doing real math in university which is about mathematical proofs.
In general the math that’s not understood but memorized gets soon forgotten and is not worth teaching in the first place.
That’s a great rule, but it still has to have limits. Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
In Germany every student has to select two subjects called “Leistungskurse” in which he gets more classes. In my case I selected math and physics which meant we had 5 hours worth of lessons in those subjects per week.
When I went to high school in Israel we had a similar system, but extra math wasn’t an option (at least not at my school).
A big part of an undergrad math (or CS) degree is spent on these subjects. I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
The German school system starts by separating students into 3 different kind of schools based on the academic skill of the student: Hauptschule, Realschule and Gymnasium. The Gymnasium is basically for those who go to university. That separation starts by school year 5 or 7 depending on where in Germany the school is located.
You have more than 3 years of math classes at school. I think proving stuff started at the 8 or 9 school year. At the beginning a lot of it focused on geometry.
At the time I think it was 4 hours of math per week for everyone. I think there were many cases where the students who were good at math had time to prove things while the more math adverse students took more time with the basic math problems.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
Might as well be a description of almost all the non-CS math content in my CS undergrad degree. (The only core subjects missing are probability and statistics). Of course, the depth and breadth and quality of treatment may still be different. But maybe an average high school in Israel is really that much worse than a good high school in Germany.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school. (And they had only 10 years of school in total, ages 7 to 17, while we had 12, ages 6 to 18.) I always thought his stories may have been biased, because he went on to get a graduate degree in applied math and taught undergrad math at a respected Russian university. So I thought maybe he also went to a top high school and/or associated with other students who were good at math and enjoyed it.
But I know there is a wide distribution of math talent and affinity among people. There are definitely enough students for math-oriented schools, or extra math classes or programmes in large enough schools, at that level of teaching. I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly. But there’s no reason these problems should be universal.
In university students often spend time in large lectures in math classes. There’s no real to expect that to be a lot more effective than a 15 person course with a good teacher.
I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly.
In our times the incentives go against teaching like this. in Berlin centralized math testing effectively means that all schools have to teach to the same test and that test doesn’t contain complicated proofs.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school.
Yes, the difference between a math education at bad school with only 3 hours per week at the end and the math education at a good school in Germany with 5 hours per week might be the freshman year of a non-CS math content of a CS undergrad degree.
What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
EDIT: Personally, I think going into computer science would have been easier if in high school I learned logic and set theory my last two years rather than trigonometry and calculus.
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 :-)
In my school math education we had the standard that everything we learn get’s proved. If you are not in the habit of proving math, students are not well prepared for doing real math in university which is about mathematical proofs.
In general the math that’s not understood but memorized gets soon forgotten and is not worth teaching in the first place.
That’s a great rule, but it still has to have limits. Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
In Germany every student has to select two subjects called “Leistungskurse” in which he gets more classes. In my case I selected math and physics which meant we had 5 hours worth of lessons in those subjects per week.
When I went to high school in Israel we had a similar system, but extra math wasn’t an option (at least not at my school).
A big part of an undergrad math (or CS) degree is spent on these subjects. I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
The German school system starts by separating students into 3 different kind of schools based on the academic skill of the student: Hauptschule, Realschule and Gymnasium. The Gymnasium is basically for those who go to university. That separation starts by school year 5 or 7 depending on where in Germany the school is located.
You have more than 3 years of math classes at school. I think proving stuff started at the 8 or 9 school year. At the beginning a lot of it focused on geometry.
At the time I think it was 4 hours of math per week for everyone. I think there were many cases where the students who were good at math had time to prove things while the more math adverse students took more time with the basic math problems.
What did the most advanced students (say, top 15%) study and prove by the end of highschool?
It’s been a while but before introducing calculus we did go through the axioms and theorems of limit of a function.
Peano’s axioms and how you it’s enough to prove things for n=0 and that n->n+1 were basis for proofs.
Your previous comment:
Might as well be a description of almost all the non-CS math content in my CS undergrad degree. (The only core subjects missing are probability and statistics). Of course, the depth and breadth and quality of treatment may still be different. But maybe an average high school in Israel is really that much worse than a good high school in Germany.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school. (And they had only 10 years of school in total, ages 7 to 17, while we had 12, ages 6 to 18.) I always thought his stories may have been biased, because he went on to get a graduate degree in applied math and taught undergrad math at a respected Russian university. So I thought maybe he also went to a top high school and/or associated with other students who were good at math and enjoyed it.
But I know there is a wide distribution of math talent and affinity among people. There are definitely enough students for math-oriented schools, or extra math classes or programmes in large enough schools, at that level of teaching. I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly. But there’s no reason these problems should be universal.
In university students often spend time in large lectures in math classes. There’s no real to expect that to be a lot more effective than a 15 person course with a good teacher.
In our times the incentives go against teaching like this. in Berlin centralized math testing effectively means that all schools have to teach to the same test and that test doesn’t contain complicated proofs.
Yes, the difference between a math education at bad school with only 3 hours per week at the end and the math education at a good school in Germany with 5 hours per week might be the freshman year of a non-CS math content of a CS undergrad degree.
What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
EDIT: Personally, I think going into computer science would have been easier if in high school I learned logic and set theory my last two years rather than trigonometry and 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.
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 :-)