When I was in high school, I chose to take standard-level IB math instead of higher-level IB math. The standard-level math focused on probability and combinatorics, whereas higher-level was focused on calculus (you had to take calculus first).
I appreciate the opportunity to have learned about these other subjects, which are arguably more useful in everyday life (and I went on to be a lawyer and a startup founder, where advanced calculus would not have been especially useful). It also meant my schedule was somewhat easier, since the standard-level class was not as challenging, and I was more mature by the time I took calc AB as a senior. I'm glad my parents realized that getting as far down the calculus path as possible was not the only goal, and suggested this path for me.
I've always been surprised that probability and statistics are not getting more stage time in school. It's one of the most practically useful areas of mathematics, and also one of the least intuitive so people with little training get it wrong all the time.
I think it all depends on where you went to school and possibly who noticed you while you were there.
I took 2 years of Probability and Statistics in Highschool in 1998-99 and 1999-00 at a school with 200 students. At the 4000 students school I taught at a few years ago we offered AP Probability and Statistics (had have been for at least 10 years, but probably much longer than that). In both situations, you could (and many did) take Stats without Calc.
Most times when people say "schools should teach X", many schools are (and have been) doing it (taxes, car maintenance, carpentry, gardening). Just maybe not your school ... or maybe nobody told you that it was a possibility at your school... Or maybe it's not at your school, but it is offered at another school in your district...
Or maybe it's just not offered at your school. Because there is an AP exam associated with Stats, it is fairly easy to get the class made as long as there are students that want to take the class and enough teacher slots to accommodate that. If a school is understaffed in the math department and class sizes are nearing 40, then you probably won't find a Stats class there.
I followed the same path - it served me well in life, but man did my university assume when you were taking Calc 1 that you had already taken calculus in high school mess me up.
Ah… IB must’ve been why I found the discrete mathematics subject at university so much easier than my peers did - I’d already been exposed to a lot of it.
International Baccalaureate. It's like AP, but internationally accepted. To get the full diploma, you have to take three standard-level tests and three higher-level tests.
Schools should honestly have two parallel tracks of math (which students should take at the same time). There should be the standard "analysis" mathematics of algebra, geometry, trigonometry, calculus. But there should also be a track of (what I will shorthand to) Discrete mathematics, which focuses on proofs, combinatorics/ probability, number theory, graphs, etc. Discrete Mathematics used to very much be a niche mathematical field, who needs to know about number theory!? But computers has really made these other fields incredibly important to modern society. Plus writing proofs is a good exercise for anyone.
I think you need Algebra 1... maybe I'm too stuck in the old ways.. but at some point you need to understand what a variable is and how to "solve for x". How to plot points, read and interpret a graph. Identify patterns in series of numbers, etc.. Call it what you want, but without the content of Algebra 1 you're going to have a hard time communicating ideas in the language of Mathematics. And these kids also have a Physics graduation requirement where they will need to at least solve f=ma.
Geometry is usually the "proofs" class. You're only really learning geometry so you can write proofs. You could plug&play that with a Discrete Math/Sets/Boolean/Logic class. I think Geometry is conceptually easier to understand as a 14/15 year old because you can "see" that the proofs work. Truth tables are kind of visual, but still a little more abstract than triangles and rectangles.
Combinatorics/Probability is already a half year course that's usually combined with the half year course of Statistics. I can see non-AP versions of this class split into two full year classes.
I imagine this would be something like what you're thinking of:
Algebra 1 -> Discrete Math -> Probability -> Statistics
The only thing standing in the way of something like this is politicians (state boards of education) and startup costs. For example, the graduation requirements in Texas are "4 credits of Math including Algebra, Geometry, and Algebra 2" (and the content of those classes are explicitly laid out in the TEKS). And you would also need to buy new textbooks/curriculum... which is money that schools don't really have to spend.
I agree with most of what you wrote, but it’s puzzling to me that you think that proofs are more of a feature of discrete math than continuous (analysis). My first introduction to proof-writing was in geometry, and a majority of proofs that I have read or written have been in analysis. But I was a math major in college.
I also did proofs for the first time in Geometry. But the reason I think of proofs as more in Discrete, is that you usually get proofs immediately in Discrete via number theory, graphs, or things like induction. But you usually don't really get proofs in analysis until you take, what real analysis or complex analysis? So you typically have trig, precalc, calc 1/2/3, differential equation and partial differential equations that is just calculations. That's my thinking anyways.
List of items should have Linear Algebra. It is by far the most useful topic that could be done outside of the Calculus sequence because it can help you with Machine Learning / computer science in general and even quantum computing. (Quantum computing is not really as important now ).
I would guess only a small fraction of students will do something in real life where they need linear algebra beyond the simple stuff taught in an Algebra 2 course.
True though but the article does say alternative math classes and I don’t see why to not have an understanding of Linear Algebra when things like number theory and graph theory are on the list.
> you as an adult get asked to “stop helping them get ahead of the class”
Whether by parental training or self study, some teachers treat it almost as a crime to understand the material before the class starts.
In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff.
I had friends who did that, so taking senior/graduate seminars in history instead of sleeping through U. S. History 101, where you could get a B if you remembered Washington was president before Lincoln.
>> you as an adult get asked to “stop helping them get ahead of the class”
> Whether by parental training or self study, some teachers treat it almost as a crime to understand the material before the class starts.
I remember a 7th grade math teacher confiscating my workbook when she found out I had done the entire workbook in the first week. I never really understood why. I spent most of that year telling the teacher she already had my homework and refused to do it again on principle. I ended up taking a 0 on every homework assignment and had a C in the class instead of an A+. Fuck that teacher.
> In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff.
I did this for first year history, computer science, and calc classes. Highly recommend giving it a try.
> The College Level Examination Program is a group of standardized tests created and administered by the College Board.[3] These tests assess college-level knowledge in thirty-six subject areas and provide a mechanism for earning college credits without taking college courses. They are administered at more than 1,700 sites (colleges, universities, and military installations) across the United States. There are about 2,900 colleges which grant CLEP credit.[4] Each institution awards credit to students who meet the college's minimum qualifying score for that exam, which is typically 50 to 60 out of a possible 80, but varies by site and exam.[5] These tests are useful for individuals who have obtained knowledge outside the classroom, such as through independent study, homeschooling, job experience, or cultural interaction; and for students schooled outside the United States.[6] They provide an opportunity to demonstrate proficiency in specific subject areas and bypass undergraduate coursework. Many take CLEP exams because of their convenience and lower cost (price varies by institution, though typically $89) compared to a semester of coursework for comparable credit.
CLEP is great! But it allows you to skip requirements.
My friend in the 400 level seminars was frighteningly ambitious and very very fast learner. I think he would have had a nervous breakdown sitting through History 101, Biology 101, etc etc.
My high school math ended with algebra. I placed out of geometry and trig and went into calculus my first semester in college. I was not prepared. Do not recommend.
I ended up going the opposite direction. My 11th grade trigonometry teacher forced me to repeat trig despite my having gotten a B. I was angry at the time! She did me a favor because by the time I got to Calculus in University it was very intuitive and I didn’t have much of a problem with it.
I had something similar at uni. Guidance plan had me goto Calculus without doing Trig, despite not having studied Trig in years. I had to withdraw from Calc twice.
Another great topic is infinite set theory (not just naive set theory as the author mentioned) touching on the transfinite cardinals. Maybe a little advanced, but if you can just get to the part where you see how there are different "sizes" of infinity, it can help a lot to understand how infinity is not a number but instead more a way of describing trends in a mathematical process over time. Sort of adjacent to Calculus in the sense of wrangling with infinity, but decidedly different since it's dealing with discrete mathematics.
I suppose it is. But, to me, that's actually kind of the point. I think a meta lesson of infinite set theory is that logic can lead to strange places. So a practitioner should think of logic as a way to get from point A to point B in conceptual space and not so much as a way of exploring what conclusions are reachable from point A. Actually maybe it's fine to do the latter, but just take what you discover with a grain of salt.
I think the context of mathematics is more important than what is taught. I took AP Calc with AP Physics, and I found I understood calc much better when it was in the context of basic physics. Instead of just teaching math, we need to put it in contexts that are more interesting to students that helps encourage them to learn.
One more anecdote: I took Discrete Math in high school because I was already taking Calculus, had space in my schedule, and didn’t want to take another elective like French or video production. Discrete Math was the “bottom-tier” class for people who weren’t interested in higher-level math and just needed to fulfill math requirements. Which is ironic because the stuff I learned there I’m using today much more than calculus or statistics.
Math competitions are fun for recreation but the topics they cover tend not to be very productive for educational purposes. Enrichment topics actually broaden a student’s mathematical foundation, preparing them for a career in pure mathematics much better than the “calculus path”, which is focused on applied math for science and engineering.
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I appreciate the opportunity to have learned about these other subjects, which are arguably more useful in everyday life (and I went on to be a lawyer and a startup founder, where advanced calculus would not have been especially useful). It also meant my schedule was somewhat easier, since the standard-level class was not as challenging, and I was more mature by the time I took calc AB as a senior. I'm glad my parents realized that getting as far down the calculus path as possible was not the only goal, and suggested this path for me.
I took 2 years of Probability and Statistics in Highschool in 1998-99 and 1999-00 at a school with 200 students. At the 4000 students school I taught at a few years ago we offered AP Probability and Statistics (had have been for at least 10 years, but probably much longer than that). In both situations, you could (and many did) take Stats without Calc.
Most times when people say "schools should teach X", many schools are (and have been) doing it (taxes, car maintenance, carpentry, gardening). Just maybe not your school ... or maybe nobody told you that it was a possibility at your school... Or maybe it's not at your school, but it is offered at another school in your district...
Or maybe it's just not offered at your school. Because there is an AP exam associated with Stats, it is fairly easy to get the class made as long as there are students that want to take the class and enough teacher slots to accommodate that. If a school is understaffed in the math department and class sizes are nearing 40, then you probably won't find a Stats class there.
I'm curious to think through what a second track like this would look like.
Assuming the "normal" 8-12 track is:
Algebra 1 -> Geometry -> Algebra 2 -> Trig/PreCalc -> Calculus
I think you need Algebra 1... maybe I'm too stuck in the old ways.. but at some point you need to understand what a variable is and how to "solve for x". How to plot points, read and interpret a graph. Identify patterns in series of numbers, etc.. Call it what you want, but without the content of Algebra 1 you're going to have a hard time communicating ideas in the language of Mathematics. And these kids also have a Physics graduation requirement where they will need to at least solve f=ma.
Geometry is usually the "proofs" class. You're only really learning geometry so you can write proofs. You could plug&play that with a Discrete Math/Sets/Boolean/Logic class. I think Geometry is conceptually easier to understand as a 14/15 year old because you can "see" that the proofs work. Truth tables are kind of visual, but still a little more abstract than triangles and rectangles.
Combinatorics/Probability is already a half year course that's usually combined with the half year course of Statistics. I can see non-AP versions of this class split into two full year classes.
I imagine this would be something like what you're thinking of:
Algebra 1 -> Discrete Math -> Probability -> Statistics
The only thing standing in the way of something like this is politicians (state boards of education) and startup costs. For example, the graduation requirements in Texas are "4 credits of Math including Algebra, Geometry, and Algebra 2" (and the content of those classes are explicitly laid out in the TEKS). And you would also need to buy new textbooks/curriculum... which is money that schools don't really have to spend.
Whether by parental training or self study, some teachers treat it almost as a crime to understand the material before the class starts.
In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff.
I had friends who did that, so taking senior/graduate seminars in history instead of sleeping through U. S. History 101, where you could get a B if you remembered Washington was president before Lincoln.
I remember a 7th grade math teacher confiscating my workbook when she found out I had done the entire workbook in the first week. I never really understood why. I spent most of that year telling the teacher she already had my homework and refused to do it again on principle. I ended up taking a 0 on every homework assignment and had a C in the class instead of an A+. Fuck that teacher.
> In college you can sometimes talk your way out of the Intro To XYZ classes and go directly to the upper level stuff.
I did this for first year history, computer science, and calc classes. Highly recommend giving it a try.
I'm curious why/if your parents didn't step in to correct the injustice.
Brought it up to his teacher.
Lucky for him the teacher recognized this was no ordinary pupil and contacted the Duke of Brunswick, who paid for his university education later.
Today, he'd be lucky not to end up handcuffed in a squad car.
That's another reason to pay 50K for a private school for your kids if you can afford it.
Also, for getting into upper division classes, I wish I'd had the sense to do it, looking back.
There's a formalized way to do this:
https://en.wikipedia.org/wiki/College_Level_Examination_Prog...
> The College Level Examination Program is a group of standardized tests created and administered by the College Board.[3] These tests assess college-level knowledge in thirty-six subject areas and provide a mechanism for earning college credits without taking college courses. They are administered at more than 1,700 sites (colleges, universities, and military installations) across the United States. There are about 2,900 colleges which grant CLEP credit.[4] Each institution awards credit to students who meet the college's minimum qualifying score for that exam, which is typically 50 to 60 out of a possible 80, but varies by site and exam.[5] These tests are useful for individuals who have obtained knowledge outside the classroom, such as through independent study, homeschooling, job experience, or cultural interaction; and for students schooled outside the United States.[6] They provide an opportunity to demonstrate proficiency in specific subject areas and bypass undergraduate coursework. Many take CLEP exams because of their convenience and lower cost (price varies by institution, though typically $89) compared to a semester of coursework for comparable credit.
My friend in the 400 level seminars was frighteningly ambitious and very very fast learner. I think he would have had a nervous breakdown sitting through History 101, Biology 101, etc etc.
Dead Comment
This is exactly what math competition problems are for, no?
There are books available with past competition problems, e.g. from Math Kangaroo.