> I taught many hundreds of students quite a variety of courses ranging from 'Finite Math', entry level algebra, all the way up to honors calc and linear algebra. Unique among all the courses, 'Finite Math' was so entirely abstract I, the math grad student, could not understand the point of the exercises.
Yup. The stuff we teach in trig and precalc provides some preparation for calculus, but it seems pretty broken.
Even early calc instruction tends to be broken-- we make things complicated over various historical analytic crises, but then we take enough shortcuts in explanation that we still need to relearn half of everything when we take an actual analysis class.
I have a 5th grader who's taking Algebra I now and will rapidly run out of math classes he can take while in middle school. I'm hoping I can find him a better path for preparation for calc.
I have a 5th grader who's taking Algebra I now and will rapidly run out of math classes
Have you tried giving him more challenging problems based on his existing math level? Work on his problem solving skills could benefit him a lot. Most high school math is just repeating the example solution with different numbers.
One nice source of challenging problems is the past contest page of the CEMC [1] (disclosure: my university runs these contests and I have participated in grading for them). They have contests for a variety of grade levels and both the contest PDFs and the full solutions are available for download, entirely for free!
He's doing well in the MOEMS Math Olympiad stuff and is going to be taking a class on inductive and deductive reasoning this summer.
To some extent, we have a different problem-- a lot of what "more challenging problems" tend to be is word problems with lots of distractors and often even vocabulary he doesn't know yet. He has great math intuition and ability to symbolically manipulate and spatially visualize problems, but he reads like a smart 10 year old, not a 13 year old.
I'm sort of assuming this will sort itself out as he gets older-- and he still got an A- in his pre-algebra class (which had enough shuffling of radicals, etc, that it was pretty decent fraction of Alg I).
(Also a little gripe about the online math class having a nominal 3 hour final... you can see his accuracy in the last half of the test was a fair bit lower than the first half).
Does he like to read fiction? When I was that age I excelled in math but I also loved to read fiction, mostly fantasy novels. I read Lord of the Rings at a very young age and it really expanded my vocabulary.
I'm not sure how English is taught at your son's school, but I have the sense that a lot of schools give kids material at grade level. I think this is an unambitious approach. Reading to challenge yourself is the best way to get better, just as solving challenging math problems helps you get better at math.
Admittedly, it's hard if the kid is not interested in what you're getting him to read. I am a bit dismayed at what appears to be a broad decline in fiction reading among boys. I think there needs to be more emphasis on finding books that boys will enjoy reading while also challenging them.
He's an OK reader and a bit over grade level, but not to the degree of math. Same for attention span and social skills.
I have two other boys that are much more well-rounded (further ahead in reading, better socially, but a bit less ahead in math).
I agree the decline of reading is a bummer. I read Tolkien pretty young, too, and even moved on to Homer. It's hard to even picture my kids doing this.
On the other hand, there's more high quality literature written explicitly for kids these days (a lot of what was read / is revered as good children's lit of the past is drivel).
That’s good to hear! When I was in elementary school our parents were asked to buy dictionaries for us kids. We learned how to use a dictionary and I found myself using mine all the time when I read challenging books.
Other things you might consider for your precocious son are puzzle and brain teaser books. I used to love doing challenging puzzles. The books by Martin Gardner can be very mathematically rich. I haven’t read them myself but they were recommended to me by a retired high school math teacher who taught the intro to math education class I took. I plan to eventually acquire a large library of these sorts of materials, once I’ve graduated and have more time, money, and space.
> Even early calc instruction tends to be broken-- we make things complicated over various historical analytic crises, but then we take enough shortcuts in explanation that we still need to relearn half of everything when we take an actual analysis class.
Take a look at Calculus Made Easy. It's non-rigorous - it uses infinitesimals without the machinery of non-standard analysis, instead of limits - but it presents the material in a clear and concise way such that just about anyone who can do basic algebra and computations can learn the material and develop an intuition for how calculus works. Once the intimidation factor is gone, they can go back and learn about epsilon-delta etc.
Thanks for the suggestion-- I'll pick it up! It looks like there's a recent revision with modern notation (though the reviser couldn't resist adding a chapter based on limits).
Yup. The stuff we teach in trig and precalc provides some preparation for calculus, but it seems pretty broken.
Even early calc instruction tends to be broken-- we make things complicated over various historical analytic crises, but then we take enough shortcuts in explanation that we still need to relearn half of everything when we take an actual analysis class.
I have a 5th grader who's taking Algebra I now and will rapidly run out of math classes he can take while in middle school. I'm hoping I can find him a better path for preparation for calc.