It is just not possible to read a math textbook like you would a read a novel and be able to understand and learn the material.
Have you ever read through a section, paragraph or even just a sentence of a math textbook and at the end you have no idea what you just read? If so, you are not alone. And most teachers don't know these techniques or they learned them by trial and error on their own and expect you to do the same.
[There could be other reasons you have never been taught to do this. See the basic learning tools page for more info.]
Using supplementary books can boost your learning significantly. For suggestions on how to select and use supplementary books, read the discussion on the college books page .
On this page, we give you specific guidelines on how you can read and learn from math books as well as recommendations of books that we have found helpful for precalculus, calculus, differential equations and math proofs. Some of the books we recommend are free and the rest of them are reasonably priced on Amazon.
Remember - - If you don't need a specific edition of a textbook, buying a previous version will probably save you some money while giving you good, up-to-date content.
Precalculus and college algebra books are quite plentiful but not all of them are helpful. Here are the ones that we think will help you the most.
Free Textbooks - - - Recently, some free calculus textbooks have shown up online. Now, these are not the usual watered down versions that are everywhere. These are full textbooks that instructors are using in classrooms at reputable colleges and universities.
The best free book we've seen so far is Active Calculus by Matt Boelkins. It is over 500 pages of good material and there is a free workbook available as well. A second book we recommend is simply entitled Calculus I, II, III by Jerrold E. Marsden and Alan Weinstein. This book is actually three books and there are student guides as well. For a list of other free textbooks, check out the American Institute of Math - Approved Textbooks.
Purchased Textbooks - - - As far as purchased textbooks go, the best we've found is Larson Calculus. If you have a choice, go with Larson. If you are looking for a textbook for reference, go with an early edition of Larson. The third and fourth edition are both good.
There are a couple of things you need to know when navigating through the list of Larson Calculus textbooks.
1. There are two main types of books, Early Transcendental Functions (ETF) and not ETF. The difference is in the structure of the material. The ETF version has the calculus of exponentials, logarithms and trig mixed in with calculus of polynomials. The non-EFT version has all the calculus of those functions separated out in later chapters. We recommend the ETF version since the flow of the material is better in our opinion and easier to learn from. However, you need to go with whatever your instructor suggests.
2. There is also the option of purchasing a copy that says just Single Variable Calculus. This is basically the first half of the full book (which contains both single and multi-variable calculus). We recommend the full version, since you never know when you might need an extra chapter or two. But, again, go with what your instructor recommends.
Here are some links to Larson textbooks, several editions. We include only the full ETF version. However, you can use these links to look for other versions, if these don't fit you needs.
Reference Books - - - For a reference book to help you learn calculus or give you extra practice, we recommend these books. The absolute best books to supplement your calculus knowledge are How To Ace Calculus and How To Ace The Rest Of Calculus. For suggestions on how to select and use supplementary books, read the discussion on the How To Save On and Use College Books page.
Books for differential equations need to be more indepth and comprehensive than for calculus or precalculus, since differential equations might be considered advanced math and is usually required for students who are actually going to use it and therefore really need to know it.
There are many books out there but these suggestions should get you started for ordinary and partial differential equations. For suggestions on how to select and use supplementary books, read the discussion on the How To Save On and Use College Books page.
Elementary Differential Equations by Boyce and DiPrima has been the standard textbook at many universities for years. New versions are still being produced but it can often be difficult to read because it can be quite terse. So you need to take a lot of notes and fill in a lot of blanks. That said, it is still a good book and will give you a good grounding in first semester differential equations, if you are willing to put in the work.
These links are to more current editions of the textbook. If you don't require a specific edition, an earlier edition will work nicely.
If you are required to have it for a class, we recommend you get a supplementary text as well.
Ordinary Differential Equations (Dover Books on Mathematics) is a great supplementary text for beginning differential equations. It has great reviews on Amazon. We recommend most Dover books because they are well written and have great content, while at the same time discussing topics with depth and insight. This book will not disappoint the serious student.
These next two books discuss partial differential equations, usually taken the semester after ordinary differential equations. Dover books are some of the best supplementary math books out there, including these.
On the How To Study Math Proofs page, we give concrete techniques on how to read and understand math proofs, as well as some links for additional help. Here are some book suggestions if you are interested in learning more.
Learn From Math Books
What are going to tell you on this page will help you skip the trial and error part of learning how to utilize math books, which most people have to go through, and jump right into being able to learn and understand math directly from textbooks. Many schools are implementing a teaching method called flipped teaching and these math classes require you to know how to read math textbooks. Part of the beauty of these techniques is that you begin to learn on your own and become a more independent student. So, if you have a teacher sometime that is not very good, you will still be able to learn.
So, what do you need to do? We have listed below the main techniques that we think will help you.
Before starting to read, scan the section to see what the important points are. It would be nice if your instructor provided an overview of exactly what you are supposed to do and learn for each section but that usually won't happen. So you need to do it and you will need to learn everything, unless your instructor tells you otherwise. Assume everything is going to be on the exam and assume you will need to know everything for your next class or sometime in the future.
1. Get out a pencil (not a pen), an eraser and a notebook. You will use these to write notes as you go along. These notes don't have to be organized or clean.
2. Slow down and read each word and sentence carefully. This a hard one since we are so used to reading quickly so that we can get through with whatever we are doing, so that we can go through the next task as quickly as possible. You need time for your mind to become aware of what the book is saying and process new terms and ideas.
3. When going through examples, carefully process each step until you understand what they are doing. If there are things that are going on that you don't understand or it seems like the book is skipping steps (which all books do), then write out the step in your notebook, filling in the missing steps. Process it by writing it out until you understand the step. Then go to the next step. If you get stuck, write a note to yourself in your notebook and see if subsequent steps help you understand. If not, then get some help from your study group, fellow student, tutor or instructor.
4. Do not skip the proofs. Although most instructors skip the proofs and do not even require you to read them, read them anyway. The proofs give you an idea of when and how to use theorems and push your learning so that you can understand even more of the material. As you go through a proof, write out each step and fill in the gaps, similar to examples. However, with proofs, it helps to rewrite the proof in your own words.
5. Do not skip graphs and pictures. They are excellent ways to help remember concepts. If you understand a graph and how it relates to an equation or concept, you now have two ways to remember the material.
6. Write down important terms in your notebook along with the definition, in your own words. So what are the important terms? When you are first learning, you don't know. So write down the obvious like bold, italicized or highlighted terms and concepts. Include theorems or anything with proofs. Then write down what you THINK are important terms. You will find out later if they are or not. Don't be afraid to write down too much.
7. Read other textbooks and supplemental math books. What?! Read more than just what I am required to read?! Yes, because the point of reading is to understand and there is not any one textbook that will help every student. We have some suggestions in the How To Save On and Use College Books page but they are just to get you started. This is important and will help you a lot.
8. Work a few practice problems even if you don't have to. As a student, your time is limited. You have a lot to do and not much time. But if you take the time to do a little bit of work over and above what you are asked to do, you save even more time later since you will not have to relearn the material. You will already know it and have some experience with it.