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.
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.