Reply to talking multivariable cal/linear algebra

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[quote=pettifogger][quote=Anonymous][quote=pettifogger][quote=Anonymous][b]1. What makes you think OP's kid is currently in 5th or 6th?  They could have done IM in 6th several years ago[/b] and now be a sophomore debating between AB/BC or BC/MV.

2. I'm a college professor, and I've never seen anyone regret knowing *too much* math.  Those who are weak in math are often stifled-- they avoid taking courses/majors/graduate programs that they would otherwise prefer because they feel constrained on the math front.

It's true that many kids who take BC or MV will repeat the course in college, but that doesn't mean it wasn't worth pushing themselves at the start.  They'll come away from the college course with greater mastery and understanding than if they were learning every topic for the very first time.  They will find the courses easier, too, which is likely to mean they won't fall down to a lower math trajectory because they feel like they can't keep up with peers.

Don't get me wrong-- no one NEEDS MV before college.  But it's baffling that people would discourage kids from doing it if it is within reach.[/quote]

No parent of a high schooler would care in the least bit to discuss an early middle school math class from years back... IM has no bearing on post calculus math electives in high school, such as Linear Algebra or MV. While I agree with you that it would be nice to see more kids get through more advanced topics before college, in reality it isn't likely because many stop after calculus due to multiple reasons (including a shaky foundation in math, and/or not interested enough in taking higher level math senior year, etc).

The real question is why are other mathematics courses offered in high school? For example courses such as discrete mathematics, number theory, probability, combinatorics, learning how to write proofs in mathematics, etc. These are so critical to developing solid logic and reasoning skills, (in addition to being fundamental for computers and computer science). One of the main barriers to hopeful computer science majors is discrete math, and it's one of the big reasons they end up switching away from CS in college.[/quote]

Could you speak more to your second point? I have a child who is in Calc BC in 10th and wants to go into CS. There will be flexibility in deciding what to take after Calc, so what do you recommend and why? (thanks, from a  liberal arts mom ;))[/quote]

A good grounding in discrete math (currently none of which happens in high school, outside of programs like Blair), is great to have to comfortably succeed in a competitive CS program. It's not always the case, but it makes it much easier to do really well in CS. It's hard to say what to recommend based on your given info, but it's promising that your child is done with BC calc with 2 years of high school to go. If you're not at Blair, you're unlikely to get any discrete math in school (multivariable calc is not helpful and probably not even well done/understood at the high school level. Linear algebra if offered is useful, but I also suspect it's not that well covered in a high school course).

Have they taken any CS classes yet, such as programming, or AP CS? CS classes try to cover some discrete math, but again there is no time to do it well because they have to focus on the programming skills. They will have to fill these gaps outside of school. There are numerous resources online, here are some of the ones I can highly recommend based on experience and a few notes about them.

[u]Intro courses:
[/u]
- AOPS Introduction to Counting/Probability (online course from AOPS, excellent introduction to discrete math, very interesting problems)
- AOPS Intro to Number Theory (same as above, excellent)

[u]Intermediate/Advanced courses:
[/u]
- AOPS Intermediate Counting/Probability (This has fantastic coverage of many discrete math topics one encounters in college/CS/math, along with many challenging problems. This would be a follow up to the first intro course)
- edX Introduction to Probability (https://www.edx.org/course/probability-the-science-of-uncertainty-and-data) This is an applied probability theory course with really fantastic lectures and instructor, however it can be a bit theory heavy thus feel a little dry. This is equivalent to the live MIT course, and definitely requires a very heavy work effort, 10-12+ hours a week.

[u]Self study:[/u]

- Try working through math contest problems such as the AMC 10 and AMC 12. All the past contests with full solutions are available for free on the AOPS website. Try all counting and probability questions you see as a diagnosis. Math contest problems are generally fantastic in developing creativity and reasoning skills, and the problems are very interesting.

- [b]Learn how to think and come up with and write proofs.[/b] This is a very important skill to have to develop mathematical maturity and quantitative reasoning, (and critical for going into a competitive math or CS program). While they do teach it from the ground up in college, many students feel overwhelmed and/or behind if they don't have any experience having done it in high school or earlier. It can take time to develop intuition for how to prove things and it definitely takes practice. Sadly, it is not taught well at all in K12 (It's supposed to be taught in school in geometry class, but it's generally not done right in most geometry classes. As a result, kids have no idea how to prove things, even if they're in the highest level of math, i.e Calc BC, etc. This is because K12 mathematics is almost completely focused on calculations vs building mathematical reasoning).

The AOPS books tied to the above courses I mentioned have enough proofs in them, but here are a few others that I liked (and there's obviously way more online resources, I'm sure others can chime in).

- Bridge to Higher Mathematics (https://www.amazon.com/gp/product/055750337X/ref=ppx_yo_dt_b_asin_image_o04_s00?ie=UTF8&psc=1#customerReviews) - Very interesting problems that draw you in, and very readable and accessible.

- Book of Proof (https://www.amazon.com/Book-Proof-Richard-H-Hammack/dp/0989472132/ref=sr_1_1?crid=8M9RABNTZY3H&keywords=book+of+proof&qid=1580849223&s=books&sprefix=book+of+proof%2Cstripbooks%2C162&sr=1-1) - Has excellent coverage of all types of proof techniques, with many practice problems.

- AOPS Intro to Geometry book (https://artofproblemsolving.com/store/item/intro-geometry) - Probably the best geometry intro book I've ever read. If done well, not only do kids learn geometry, but also really learn how to reason well and prove things (which is arguably way more important for life than the actual geometry itself, even though the geometry problems in the book are beautiful). One critical thing to note is that the proofs in this book are all in essay/free form format, which is how proofs are written in real life (not the fabricated "two column" proofs that are found in almost all K12 geometry books used in school). The two column format is used as a "crutch" to help kids latch on to proofs, but actually really hinders their thinking because it prohibits them from creatively expressing their argument (imagine having to write an essay in a constricted two column format). A perfect analogy is how a calculator over reliance in schools hinders the development of number sense.

- Lockhart's Lament (https://www.maa.org/external_archive/devlin/LockhartsLament.pdf) - This last short essay is for you and other parents. I cannot even begin to express how it poignantly describes the (rather sad) state of mathematics educations in K-12 schools and I think it really should be required reading for everyone.
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