That book is used in Princeton MAT 216 (basically honors introduction to real analysis), but Gunning is not necessarily the instructor (Princeton’s web site lists other instructors for this academic year).
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Prof Gunning retired 2 years ago at the age of 90 :-). He taught the course for several decades. He has been in the department long enough that he calls Andrew Wiles as young Andy. That course saw a few fields medalists over the years as they passed through Princeton. It is a rite of passage, and students subsequently bond over the fact that they’ve gone through that course. You can’t just register for the course. You go and talk to Gunning (not anymore) for a half hour, and he assesses whether you have the mathematical maturity to take the course. This is the sense in which I meant that you need to go in prepared to take some of these courses (in this case significant facility with proofs) before you enter these places. You can’t go in unprepared. It is currently taught by Pardon and someone else. Pardon is considered a fields medalist in waiting. He was a full Prof at Princeton at the age of 26. He was also a Princeton alum/valedictorian from undergrad. The department takes this course very seriously. They have very good professors teach it, because it lays the foundations for entry into the department. When Gunning was leaving, for a semester, I think they may have had Fefferman teach it.
Now here is an interesting fact. Every other year or so, there will be a brilliant high school kid from the surrounding town of Princeton, often a Prof’s kid, who registers for that course. The prof usually doesn’t put the kid on the curve. Because the understanding is that high school kids are brilliant, and they distort the curve too much for the undergrad kids, who themselves are brilliant.
To @mtmind 's point though, my son bemoans the fact that high school calculus is not taught like this.
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It should not be surprising that, in many cases, high school students taking college courses are among the strongest students in those courses, compared to the actual college students.
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Those would be the exact kids I mentioned in my earlier reply, LOL, since that’s my district. I haven’t seen a correlation with the super accel kids being disproportionately prof kids though. Seems like there’s a big contingent who went to Charter before PHS, which has an aggressive acceleration option.
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This has been enlightening. In our district, the Geometry in 9th, Alg II in 10th is the norm, but there are many kids taking AP Calc BC in 10th. A combination of acceleration in elementary school and taking summer classes in middle school. It’s a little frustrating for my kids, who were not accelerated beyond the +1 (which is the norm for their school) early on, because we said no to the summer classes for a variety of reasons. They gravitate toward STEM but feel they can’t compete because they are 2-3 years behind the kids who started supplementing with private tutors in early elementary, were recognized as advanced in grade 3/4, and continued with summer courses.
I am taking from this that outside of their school, they are not as “behind” as they think, and still have a great shot at STEM programs as long as they continue through to Calc in senior year. Anyone want to weigh in on whether that is a fair assessment?
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As several people have mentioned above, the best time to accelerate is pre-K, at which time the kids don’t even know they are accelerating. Our kids went to a pre-school which had 15 kids altogether, run by an 85 year old Hungarian woman (no more expensive than daycare) who fled to the US during WWII. She used to be strict with both the parents and the kids. The kids used to love the “granny”. She made them learn multiplication tables to 12x12, reading and writing in pre-K – 90minutes of homework a day. My son read all volumes of Harry Potter in 3 months in first grade after “graduating” from pre-k and going to a different school. At this other school, in 2nd grade, he told the math teacher, that dividing 3 by 0 did not make sense because you cannot divide anything into zero parts. This is the benefit of early strong foundations.
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There is so much variability!
AP calc senior year was the most advanced track offered at my D’s private HS. Lots and lots of depth. She was tracked in 6th grade but still needed a very high math score on her placement test to place into that track.
In our new town, hitting calculus senior year is the norm, +0. No tracking to get to calculus.
She skipped calc I freshman year in college after taking old exams and talking to her advisor.
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Earlier you posted that Calculus was just “slopes, areas and limits.”
If taking these courses earlier is the “lazy choice,” then those with any math prowess ought to be able to figure it out in a class like this, right?
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Calculus done in high school is only slopes, areas, limits. Genuinely. I taught my son 2-3 ways of taking a derivative in 6th grade. You just need to explain in a way they can understand. I talked about slopes and areas.
They do figure it out. It is intuitive.
When the kids went to play soccer in middle school, the coach used to tell us that they need to touch the ball as much as they can.
All these things are the same way. The more they touch stuff, earlier on, the more intuition they build.
I disagree. My Kids just did normal playing and coloring and being read to, for a few mornings a week in the preschool years and still learned to read on their own by 3ish and the super math-y one just always loved numbers and puzzles and definitely was not doing any real math education until kindergarten started. Yet, she still was noticed and selected by teachers to move to +3 in elementary. Gifted kids are gifted no matter what and do not need super-early acceleration or early formal education to have their brains stimulated. It is easier to have (school-driven ) math acceleration in earlier grades (ie before or early in middle school), but there is just no need for what you describe, and loads of gifted education research supports that it is not needed.
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You may be right. I don’t know. I don’t look at research. I am a skeptic of a lot of educational research. Partly because there is often an agenda in many of these efforts at research (not keen on this part of the discussion 
). I am sure both nature and nurture are at play in most cases. We never forced the kids to do anything. We are very loosey goosey.
Calculus in 12th grade (+1 math track) meets or exceeds math level requirements and recommendations for frosh admission to all colleges in the US.
If you look at schedule templates for engineering and science majors (example, example, example), they almost always start with calculus 1 (i.e. assumption that the incoming frosh student has had precalculus but not calculus). Only at a very few colleges in the US is having calculus prior to frosh entry required or expected, and none states that math beyond single variable calculus is required prior to frosh entry.
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This is an interesting statement. You want to be careful what happens with that kind of statement when the word “education” is subbed out and something else is put in. Anyone’s research can be accused of having an agenda. Most recently large swaths of Americans have accused mainstream science/medicine of exactly that. I have to confess that line of logic leads somewhere scary in my opinion. It’s not that I don’t think you have a point (of sorts) and possibly even one I might agree with. But be careful with the words.
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I am skeptical of a lot of research in my own field :-). i have a PhD. This is equal opportunity skepticism …
I agree, and would add that, unfortunately, our education system is geared to view participation in an accelerated math curriculum as a sign that the student is “gifted” mathematically, when the reality is that participation in the accelerated program is often more a product of circumstance (and/or aggressive parenting). This isn’t to deny that a few kids truly are gifted in math, but most “advanced” math kids have just done it longer, at a faster pace, and with more guidance.
An example of what I am talking about is expressed in @Kls post . . .
I am familiar with schools like that, and I understand your frustration. Those kids aren’t necessarily more talented than your kids, and they don’t necessarily have more potential than your kids, they just started earlier and/or did more on the side.
While @ucbalumnus correctly points out that at most schools/majors there is no requirement to go beyond Calc I (or even preCalc) in high school,there is also the admissions recommendation that students ought to take as rigorous as courses as are offered in their high school. So, kids push farther earlier, so as to on a more rigorous track compared to their peers at the same school. I think the whole thing is absolutely crazy, but unfortunately I think it is the inevitable reality given the trend toward more and more acceleration at a younger and younger age. That said, there are a large number of tremendous colleges for STEM students, and some of them will appreciate what your kids have to offer, even if they aren’t taking MVC and Linear Algebra in high school.
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Spivak is a popular textbook used by many colleges. My favorite calculus textbook is the one by Apostol, which has been a classic since the 70s. However, these textbooks aren’t meant for HS students, even mathematically advanced students. These students can discover their mathematical talent better in more “elementary” math topics such advanced geometry, combinatorics, etc.
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Some parents here expressed and interest to go deeper. So I thought I’ll mention a resource that they may find useful. Frankly kids will study whatever they want to study. I don’t think combinatorics is elementary. My son has 3 combinatorics courses so far with a prof that is well very known in the field (the guy that proved the necklace theorem), and at least one of them is a grad course. And he tells me he is barely scratching the surface.
In 6th grade he wanted me to buy him Graham’s book on Ramsey theory. I didn’t dissuade him. It is a branch of combinatorics. He said he understood nothing. I didn’t push him. He put the book away until high school. Kids do what they want to do.
The reason I mentioned geometry and combinatorics is because they, unlike calculus, don’t require many prerequisites, so the students can “discover” their natural mathematical talent without too much preparation in the other math topics.
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That’s also fine. You should touch the ball as much as you can :-). I am not doctrinaire about this. For example, there are kids that come into Princeton that are exceptionally strong in Geometry. There are some that are off the charts in Algebra. Some kids love analysis. I am happy as long as they love something. Doesn’t even need to be math.
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I think the real issue is the incredibly slow pace of elementary school math as well as the “fun” and “engaging” way it is currently being taught. Some kids may truly gain something by three weeks of drawing duckie arrays as an introduction for multiplication and two more weeks for everybody to master partitioning a triangle as your fourth multiplication concept. But many kids will be bogged down by multiplication and
/disengage without learning the tables. When they advance the don’t have the technical skills. This is just one example of imagination-based math that results in conceptual understanding but a low skill level.
My kids are at pretty high level in math but mostly they advanced very quickly to Algebra and then proceeded to a more normal pace. They easily could have moved through elementary school math at 2-3x and I do not understand the intense gate-keeping that allows sixth grade is the magical year when they let kids accelerate, even in gifted programs. They never had tutors except one formal graded course to show mastery before middle school.
I don’t really care what level they complete in high school but slow, boring math is deathly.
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