There are several reactions that one can have upon seeing the text Advanced Calculus by Sternberg and Loomis. (Those unfamiliar with this book can find it online here: http://www.math.harvard.edu/~shlomo/)(adsbygoogle = window.adsbygoogle || []).push({});

My reaction was simply to be surprised at the extreme discrepancy between the material in Sternberg and the material taught at my university. For a book intended to be a two- to three-semester honors sequence in analysis for entering college students, Advanced Calculus seems like it would be beyond most senior math majors at my school and perhaps even a few first- or second-year graduate students. Indeed, I would say that if one worked through Advanced Calculus, then one would find the Ph.D. qualifying exam in analysis here trivial.

Now, I don't go to a particularly prestigious school. It's my state's flagship campus. It's a good school. It's probably very representative of the majority of universities around the country.

But it makes one wonder how different things could be. Personally I believe that most schools could offer sequences at the level of Sternberg to their entering freshmen if math education were reformed properly. The reform would consist almost exclusively of simply raising the grading standards in high school math classes and introducing critical thinking far earlier. There's no reason that math classes can't start emphasizing logical reasoning, proofs, and creative arguments as early as eighth grade.

I believe that such reform would work because I've observed that there's essentially a quantum leap in ability between those math students who do the bare minimum and those who dedicate themselves to their studies. Obviously a large gap is to be expected, but the gap in ability isn't just large--it's extremely huge. The students who really study math seriously could simply eat the other ones alive who still, even in college calculus, will write a / (b + c) = a/b + a/c and who think that simply giving a numerical example of a theorem constitutes a rigorous proof.

And this really doesn't have much to do with intelligence. It's simply that "better" students (the dedicated ones) have pretty much always been dedicated to studying decently from junior high on. The effect of this is cumulative and profound so that upon reaching college the gap between those who studied seriously and those who didn't is miles in length.

The reform would have to do with schools making sure that students who get As in algebra are actually very good at algebra and that students who get As in calculus actually understand why, for example, monotone functions are Riemann integrable. If getting straight As in math courses in high school signified that one had learned calculus with at least a decent bit of rigor and that one had exposure to proofs (including basic techniques like induction), then I don't see why it couldn't be a nationwide standard to have students interested in math starting college in sequences at the level of Sternberg's Advanced Calculus. Less theory-intensive sequences could also be offered (and should be offered for those who don't want to study analysis intensively), but even these could be at a far higher level than most calculus sequences at colleges these days.

What do you think? Is it realistic to reform math education so that first and second year math courses could actually be at the level of Sternberg's Advanced Calculus?

Part of the reason I'm writing this is that I personally feel a bit short-changed by the whole situation. Students who are rather serious about studying math and who are rather dedicated to the pursuit are left with a fairly poor curriculum that progresses too slowly and covers material for the most part with insufficient rigor. Most math majors at my school don't take analysis or algebra until their junior year. The result is that people like me, who need and, I think, deserve a more rigorous undergraduate curriculum but who aren't ready to skip on to graduate level courses, are stuck in the middle where nothing really fits.

[Note: there are of course other books similar to Advanced Calculus. I just picked one to use an example.]

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Sternberg's Advanced Calculus and math education

Loading...

Similar Threads for Sternberg's Advanced Calculus |
---|

A Derivative of argmin/argmax w.r.t. auxiliary parameter? |

I surface area of a revolution, why is this wrong? |

I A rigorous definition of a limit and advanced calculus |

I Partial derivatives in thermodynamics |

I Euler’s approach to variational calculus |

**Physics Forums | Science Articles, Homework Help, Discussion**