May

16

Refresh your High School Math skills

Filed Under Essential Math, Math Education, Suggested reading | 33 Comments

In my first article “The most enlightening Calculus books”, I argued the importance of maintaining high standards for mathematics education and suggested deep and inspiring calculus books for those of you who are interested in pursuing the joy of learning mathematics. The feedback has been overwhelming and I wish to follow up with an article that addresses a couple of remarks that I’ve received by email.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general - and mathematics, technical and scientific education in particular - are key for the development of every country and ultimately for good of humankind.

However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them.

Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as pre-calculus, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

Simple Precalculus Questions:

1) Factor the following polynomials:

  1. \displaystyle x^{2}-6x+9
  2. \displaystyle x^{2}+x-6
  3. \displaystyle x^{3}-27

2) Solve for x:

  1. \displaystyle 3x^{2}+5x-2=0
  2. \displaystyle |x^{2}-x|=3
  3. \displaystyle x^{4}-8ax^{2}+16a^{2}=0
  4. \displaystyle \frac{x^2+x-6}{x+3}=0
  5. \displaystyle 2\sqrt{x} = x - 15

3) Find the values of x for which:

  1. \displaystyle x^{2}>9
  2. \displaystyle |2x-3| \leq 5
  3. \displaystyle |2x-1| > 9
  4. \displaystyle |x-1| + |x-3| \geq 8

4) Evaluate:

  1. \displaystyle \log_{2}{1}
  2. \displaystyle \ln{e}
  3. \displaystyle \log_{2}{1024}
  4. \displaystyle \frac{4^{8}2^{4}}{2^{12}}

5) Solve for x:

  1. \displaystyle 5^{x}=10
  2. \displaystyle \log_{3}{7x} = 2
  3. \displaystyle \log_{x}{9}=2
  4. \displaystyle \ln(3x-2)=0
  5. \displaystyle 3^x+x=4

6) Solve for x, where \displaystyle 0\leq x \leq 2\pi:

  1. \displaystyle 2\sin{x} = 1
  2. \displaystyle \tan{2x} = \frac{\sqrt{3}}{3}
  3. \displaystyle \sin{3x} = 1
  4. \displaystyle \cos^{2}{x} - x = 2 -\sin^{2}{x}

7) Write the equations of the following curves in the Cartesian plane:

  1. Parabola
  2. Hyperbola
  3. Circle
  4. Ellipse

8 ) Find the vertex, focus, and directrix of the parabolas given by the equations:

  1. \displaystyle x^{2}=16y
  2. \displaystyle y^{2}+4y+12x=-16

9) Find the center, vertices, foci, and eccentricity of the hyperbola given by the equation:

\displaystyle \frac{x^{2}}{4}-\frac{y^{2}}{36}=1

10) Find the equation of a circle whose center is at (2, -3) and radius 3.

11) Determine the center and radius of the circle with equation:

\displaystyle x^{2} -4x+ y^2 - 18y = -4.

How did it go? Did you experience many struggles and the feeling that “I used to know this stuff”? If so, then it is a good idea to go for a refresher before attempting calculus right away. The following are two books that you may find useful to respectively learn and refresh basic math in a well organized manner:

  • Precalculus by Michael Sullivan: a big book, which is quite extensive and thorough. If you want an all-in-one book that covers all you need to know about precalculus and more, in a clear but college oriented manner, than this is without doubt an excellent choice. It will likely make the step up to Calculus quite easy.
  • Schaum’s Outline of Precalculus: it has a less prosaic approach but it’s still very clear and easy to read. If you were pretty good at math in high school and you just forgot a few things because you haven’t touched these topics in a while, then pick this book up. It is adequate for already mathematically inclined people who are in a rush to brush up the skills they once had.

If you feel entirely clueless and would like a “for dummies” type of book, the following two titles seem to have a good table of contents and excellent reviews:

If you would like to use some free resources available online instead, here are some lessons:

If you know of any other resources that are available for free, or if you successfully used other books for these purposes, please feel free to use the comment section to add to the discussion.

May

13

The most enlightening Calculus books

Filed Under Math Education, Suggested reading | 33 Comments

In Walter Rudin’s autobiography The Way I Remember It, he comments on a calculus book defining it as “too good to be widely used” and further states that:

Widely used calculus books must be mediocre. — W. Rudin

The temptation to discard that statement as elitist may be strong, but it is worth noticing how there is so much truth to it. Education is embracing a dangerous downward spiral towards the oversimplification of mathematics in favor of letting every student pass, despite their actual understanding of the subject.

Dumbing down mathematics is a dangerous trend which affects students at all levels, from primary school where children are no longer taught how to perform division by following the standard algorithm to a complete emphasis on anti-racist mathematics and calculus courses where delta and epsilon are not mentioned while teaching limits, because they are considered to be “too complicated” or “too confusing” for most students. Of course, this is not the case everywhere, and bright students who study on their own will always exist (despite the fact that the social satire of Idiocracy may be more accurate that we like to think).

This problem exists because educational changes such as these reflect negatively on the competence level of the average student, meaning that a decent chunk of the next generation of mathematicians, engineers and scientists will be facing increasingly complex, unsolved problems and challenges with an inadequate amount of preparation. Most students have a tendency to adapt their study levels around the difficulty of the examinations they will be exposed to. In other words, no matter how easy a Calculus course is, there will always be a majority of C students. Making the courses dumber has only resulted in the creation of artificial A, B and C students who wouldn’t stand a chance against a C student of the 50’s, in terms of understanding and mastering the subject.

It is therefore very important to consider what are the great classical books available that can provide a solid basis for studying mathematics and that can guide, inspire and enlighten the student who wishes to learn mathematics the way it should be. To be factual there are still institutions which adopt valid textbooks and didn’t jeopardize their curricula too much, but that’s not the general rule unfortunately. What are then the best Calculus books? “Best” is very relative to the student, but in this context I will consider what is “best” for relatively bright students who have a genuine interest in calculus/mathematics. Books that I myself have found very appealing to me, and therefore my list is declaratively biased. All of the following books do a very good job of covering the material, explain the subject matter well, have mathematical rigor, proofs for the most important theorems, challenging exercises, and are able to really teach Calculus I at a sufficiently high level. Each of the books listed below is able to enlighten, guide and inspire the reader who is willing to put the time and effort into going through them:

  1. Calculus by Michael Spivak: I can’t praise this book high enough, to me this is THE calculus book. It has a great selection of topics, careful and extremely rigorous proofs, and it goes well beyond the scope of calculus, so much so that a better title would be “Introduction to the beauty of Real Analysis”, because it really bridges between calculus and more advanced real analysis, showing how beautiful mathematics is. Spivak literally guides you in a enlightening experience, discovering Calculus, starting from the properties of the numbers and building on top of it. The examples are very meaningful, the explanations are clear, and the subject is so well presented and motived. It is, in my opinion, one of the most inspirational mathematics books I’ve ever read. The exercises proposed at the end of each chapter test different levels of understanding and don’t fail to challenge the reader on the subject. These are exercises which really make you feel like you are taking a second course on the topic. In fact, I’d say that the exercise sections of this book are especially valuable, and you’re highly encouraged to work through all of them. This book is great for anyone who is serious about mathematics and who wishes to have a very solid foundations upon which to face the challenges of more advanced courses down the road. Spivak’s book comes with a few selected solutions to odd numbered problems, but if you are self-studying or are disciplined enough, you may want to consider the associated answers book (which by the way is in print, despite what Amazon lists on their pages).
  2. Introduction to Calculus and Analysis, Volume 1 by Richard Courant and Fritz John: a classical, well thought introduction to Calculus and Analysis in one variable, where explanations are very clear and the material is covered in a way which provides a good deal of motivation. The approach is more practical and less abstract than other Calculus books, while conserving a perfect balance between mathematical rigor and intuition. This book comes with plenty of exercises that will reinforce the knowledge of the student. The abundance of physics applications, make it ideal for physics majors and engineers as well. This is the first volume of a trilogy republished by Springer, if you are serious about Calculus, you may want to also consider the other two more advanced volumes: Volume II/1 and Volume II/2. An absolutely beautiful set of books.
  3. Calculus, Vol. 1: by Tom M. Apostol: a very comprehensive book, methodical in the theorem/proof approach, adopted by many high-end universities as a first year textbook for courses in calculus or calculus and theory. Its coverage of the subject is impressive and provides a good selection of standard exercises. It is an excellent reference and textbook, albeit you may find it a bit dry and less inspirational than others at first, but you will eventually develop an appreciation of its teaching method. You may want to note that this is the first volume, and that the second volume is also worth getting: Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications.
  4. A Course of Pure Mathematics by G.H. Hardy: this is the 10th edition of a book first published in 1908 by the great mathematician G.H. Hardy. It is a classic textbook that brought much needed rigor, and reformed the way math was taught in the UK in the twentieth century. This book is intentionally written to address the brightest students of the time, therefore it is a very inspirational and enthusiastic piece of work with plenty of elegant proofs and suggestions, and exercises that are definitely very challenging. For example, there are exercises coming from the Math Tripos examinations (at Cambridge) from early part of the last century, and they will definitely keep you busy for some time. Some notations are a bit outdated and it is mostly an introduction to real analysis that may be a bit too much as a first book. But this book is a masterpiece nevertheless, and it’s a classic that as a mathematician you will want to have in your library. I think it’s ideal as a reference and as a supplement to other textbooks (e.g., Spivak).

More than just calculus, these also serve as introductions to Analysis, and in general to mathematics at an undergraduate level. They are challenging, not for the faint heart, but ultimately a joy for math lovers.

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