# Problems With Calculus

## Review of the Art of Problem Solving

I was recently introduced to The Art of Problem Solving, which is a series of math textbooks aimed at gifted middle and high school students, especially those who are interested in math competitions. I only looked through one of the books, Introduction to Algebra, by Richard Rusczyk , and this review will be about that text only. Other books in the series delve into topics such as geometry and probability.

The Art of Problem Solving bills itself as a book for 6th to 10th grades. This evaluation is quite ambitious! It is, however, in character with the rest of the book, as ambitious is the best word I can come up with to describe the overall tone. (According to the author’s biography, he was a high level math competition champion as a child, and I think it would be fair to suggest that he wrote this book with his younger self in mind.) I would warn parents and teachers to take the pre-test provided on the website very seriously. If your student(s) can not get a perfect score on the pre-test without your help, they are not ready for this book, regardless of their age.

As an adult who is comfortable with math, I loved this book. Both the text and the problems are thoughtfully written and very interesting. The explanations provided are lucid. If time was not a constraint, I would joyfully devote an hour or two a day to methodically working through this book- it would probably take me a year or so to finish, and I have no doubt that I would learn a great deal. However, while my endorsement of this book is strong, it is also very limited and specific. So that you can understand, let me tell you a little bit about myself.

As a child, I was an insanely conscientious student. Not surprisingly, I did well in school and was placed in an accelerated math program in middle school. Nevertheless, I found no joy in math (and always had the nagging feeling that my success on tests and report cards was due to some sort of cosmic mistake rather than real achievement on my part.) In high school, I stopped pursuing math as soon as I decently was able. I never took pre-calculus, never mind calculus. I chose my college partly based on where I would be able to major in biology without taking higher math classes. Fortunately, I experienced an epiphany at the age of 21.

My epiphany was the result of a research project that I was perusing- I was researching certain aspects of ancient salt marshes, and my advisor told me that if I could successfully do a statistical analysis of my data, it could most likely be published. With that enormous inducement, I began studying elementary statistics, and with almost no instruction except from a textbook, I soon understood statistics well enough to analyze my data. My paper was published and, much more importantly, my fear of math was conquered.

Years past, and I became a tutor. I teach test preparation and science as well as math, but I spend the largest portion of my time teaching math to 8-14 year olds. (I’ve hired other tutors to teach more advanced math.) I’m very good at what I do, and I think it is in large part because I have a very thorough understanding of math through high school algebra, a genuine affection for the subject and, simultaneously, a clear memory of a time when math was not my friend.

All of this history is a roundabout way of explaining why I feel like I have a lot to learn from this book- although it starts out with basic algebra, it ends up covering topics normally reserved for pre-calculus. Furthermore, when I look at The Art of Problem Solving, Introduction to Algebra through the lens of my remembered childhood feelings about math, I see a terrifying tome. It does not gently lead the student forward, first with easy problems and then with gradually more challenging ones. Instead, it dashes ahead and dives straight into hard problems. This approach is great for a motivated, interested person with a solid background in the pre-requisites, but it could easily prove miserable, frustrating, and ultimately counter productive for students who do not meet that description.

I intend to begin using The Art of Problem Solving, Introduction to Algebra, but only with a select group of students who are already robustly successful in math and who are coming to me for enrichment. For example, I will incorporate Art of Problem Solving problems into my work preparing students for the Hunter College High School and Anderson School entrance exams.

I wish to offer one further warning about The Art of Problem Solving, Introduction to Algebra, specifically to homeschool families. If your child is ready for this textbook and eager for the challenge it presents, then that is a wonderful thing. However, if you plan on integrating your child into a school environment, you should be aware that the book does not touch on topics that are important in both middle and high school curriculums (primarily geometry and probability) and you might therefore want to provide supplementation in these topics.

**About the Author**

New York Academics offers

Hunter College High School Entrance Exam

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Integrated Algebra Regents Prep

. Please inquire about the availability of tutors for specific times, locations, and subjects.

**Limits Practice Problems 2 – Calculus**