Math and Science Learning Techniques
You CAN Ace Calculus
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On this page we discuss techniques, tools and ideas to help you learn math and science. Most of the discussion refers to math but the techniques can be applied to physics, engineering and most hard sciences.
Math and science are not difficult, they just take some different study techniques than you might not use in other subjects and some persistence.
The topics are listed below. Click on the button to open or close the discussion.
'I have a terrible math/science teacher. What can I do?'
If you have a bad or difficult teacher this semester, I'm sorry to say that your situation happens to every single person that has ever been in school and it seems to be more common with math teachers than with any other subject. I don't know why. Also, I am sorry that you have to deal with this. This situation is not fair and should never happen. But the reality is that it does. But you have control here and you can decide what to do. Here are your options.
1. You can run. You can drop the class, change sections or just quit.
2. You can let it get to you, complain all semester and then blame the teacher when you get a poor grade.
3. You can deal with it and get the best grade possible under the circumstances.
If you choose options 1 or 2 above, most people wouldn't blame you. Lots of people have done one or both and have gone on to succeed in life (although not as successful as they could have been). You are not a bad person if you choose one of them. However, choosing 1 or 2 will not help you in the long run and you will lose an opportunity to learn about dealing with difficult people. There will be a day when you have to deal with a difficult co-worker or, even worse, a difficult boss. And you have the same choices now that you will have to make then. Also, right now only your grade is the issue, not your career. So doesn't it make sense to choose option 3? If so, read on. I will give you some ideas on how to work through this.
Okay, so if you are still reading, you have chosen (or are at least thinking about choosing) option 3. Let me start out by telling you that it will be challenging and you won't always like what you have to do. But I can tell you that it will be worth it. So keep reading.
First, what you need to do is change the way you think. Once you do that, you will be able to do some specific things that will get you through the class. Every action you have ever done has started in your mind. Your thoughts are extremely powerful and they affect everything you do. Are you ready? Okay, let's get started.
1. Realize that your teacher is human and makes mistakes. They were young once. They have a life, a family and, for them, teaching is job. Some teachers teach because they are good at it. Some teachers teach only because they don't know how to do anything else. Maybe their parents are teachers and didn't give them any other choice when they were young. Maybe they are natural musicians but they were not allowed to pursue that. Maybe they are new and have a lot to learn or maybe they are old and are just coasting to retirement. Whatever the reason, they ended up as your teacher after making many decisions along the way, maybe not always the best decisions. But your first job is to give them a break. Try to see them as a fallable human who is doing their best, even if it doesn't look like they are trying.
Now, understand that this is one of the hardest thing you will do and you will probably have to do it repeatedly every day you are in this class. But this is the key to everything that comes next. Do not skip this one and expect the other things to work because they won't. So, it's time to suck it up and just do it.
2. Okay, now that you have a little bit of perspective on the teacher, you can let go of the expectations you have of them and shift your attention to learning math. This is an important shift that is absolutely necessary if you are to succeed. You need to think about what you say to yourself about math. Do you say it is hard, evil and/or difficult or impossible? Do you tell yourself that you will never succeed, that you are not good at math, that you will never get it? If you are saying those negative things to yourself, they will become your reality and you will not succeed. That's the bottom line. You need to stop saying those things and start saying things like you can learn math, you are able to succeed, you have all the tools you need to master this class . . . things like that. I am not talking about positive thinking. I am talking about changing your reality by starting with what you think. What you think, will become reality. Do you want to fail? Then think about failing. Do you want to get good grades? Then start thinking about it.
3. Take charge of your own learning. It is way too easy to expect someone else to teach you and, honestly, it just won't happen, especially if you have a bad teacher. But even with a good teacher, you can learn much more than they will teach you by taking charge of your learning. So what does this look like? Taking charge means that you take responsibility for learning including reading the material before class, asking questions of the teacher and, if that doesn't work, finding someone else to answer your questions or, even better, figuring it out on your own. Reading additional material and researching outside of class so that you will truly understand the material. Turning in assignments complete and on time. Finding out what the teacher expects and following their rules and guidelines. Working within the system but not in a way to be limited by it. Doing more than expected inside and outside of class. Learn to be a good student.
4. Work with other students in the class. Get a study group together with one person in the group that knows the material less than you and another person that knows the material better than you, if possible. You will get a tremendous boost in learning if you have to teach another person. And you will also have someone to ask questions of. Don't make the group too big, max 4-5, less if you can't work together, more if you need it.
1. Go to the bookstore page and get several books. First, get a book on how to be a better overall student. There are different techniques to study math, english and history. Learn the differences and implement the techniques. Second, get another book on calculus. The best I've found is How To Ace Calculus and it's sequel. Next, if you have a bad textbook, get a used copy of a good one, like the ones in the bookstore. The best I've found is Larson Calculus. Since you don't need it for a course, you can get a previous edition, sometimes for as little as $10. Use them and study them.
2. Read the rest of the learning techniques page giving study techniques and implement them, especially the section on how to teach yourself ( in the section entitled The Number One Study Concept Every Student Needs ).
3. Do not skip anything that you don't understand while you are learning the material. Be diligent and persistent. The number one personality trait that will help you the most is persistence. Develop it and use. Don't give up on anything. Also, if you skip something, there is a very good chance that that concept will be on the exam. Good teachers ask for the hard stuff on exams. Poor teachers just ask you to regurgitate the easy stuff.
There are three keys to doing well in your class, (1) to change your thinking, (2) taking responsibility for your own learning and (3) persistence. Develop those three areas and you will learn how to do well in any class with a bad (or even good) teacher.
'Why should I learn math/calculus when I will never use it?'
This is a great question! I mean, why study 'useless' material that you will never use, probably don't even like and is not that interesting?
Well, here is why. When you read, hear, see or learn something, the chemistry of your brain changes. These changes help you process other things in other areas that you may not expect or even ever know. For example, learning to play a musical instrument helps you learn more easily. That's right. We discuss it in more detail on the music and learning page. Your brain is changing constantly, controlled by what you put in it, including music you listen to, movies you watch, friends you hang with and books you read. Most of the time, you will not be aware of the changes but they happen.
Here is a good article by one of our favorite bloggers, Scott H. Young entitled Why Learn 'Useless' Things? that you might enjoy. This article applies to language learning but toward the end he discusses other areas of learning.
Learning math, calculus and science, is training your brain to think in a more structured way. The information may or may not be important. But the activity of learning this type of material changes how you think, not just what you think. You will be able to learn other topics more easily and more deeply if you take the time to learn math now.
None of us likes to waste our time. We want to be doing meaningful, fun and fulfilling things, including activities that will get us somewhere. So, do not fret. Learning math, calculus and science is not a waste of time. You are training your brain and how you think. It's hard but it's worth it.
General Math and Science Study Techniques
In this section, I'm going to give you some ideas on how you can learn math and get a better grade. You may read it here or download the pdf document.
Most people don't know that they need to study math differently that other subjects, like history or English. As you read through this, I think you will see that learning math (especially calculus, higher math and science) requires different study techniques.
In general, do not fill your brain with useless special cases. It is MUCH easier to remember the general rule and then apply it to special cases. Unfortunately, mathematics textbooks are filled with special cases. So you will need to weed out what is a special case from what you REALLY need to learn. Most teachers teach special cases as well, since it is easier for them. You need to watch for this and learn the concepts, not the special cases.
Consider learning math like you would learn a foreign language. I know, I know, you are already thinking it IS like a foreign language, right? Well, you are correct! Math has a language of its own and learning it is more like learning a foreign language than you might think. If there are words you don't understand, look them up in a math dictionary. Look up terms or concepts online. See how other books and teachers teach the same concept. If you are going to spend a substantial amount time learning math this semester, you may as well do it right.
Use reference materials, help labs and other students liberally (but ethically). Ask your teacher lots of questions. If you skip material that you don't understand, it will come back and bite you, I guarantee it. Math builds on itself. (This is one big difference between Math and History or English.) So if you miss one small concept, you will miss the next one that builds on it and on and on it goes. It's kind of like driving a car, going somewhere with directions. If you miss the first turn, the rest of the instructions will not make sense and you will never find your destination. So, get help often.
If you are nervous about taking calculus, you are perfectly normal! Most people are nervous. I mean, look at the name of the class CALCULUS. Even the name is scary. However, you can do it. Most of the problems you will have when trying to understand this material are not that the material is difficult. It's that you have decided in your mind that it is difficult. Convince yourself that you can do this, that you will do well in this class. Don't let the past, other people or anything tell you you can't!
For some extra help, get the book How To Ace Calculus (for Calculus 1) or How To Ace The Rest of Calculus (for Calculus 2 and 3). These are good books on condensing the concepts down to what you need to know and presenting ideas in a readable form. I have read them and found them helpful. You may be able to find them at a local library. I know you can get them on amazon for about $10. They are two of the books in our recommended booklist.
Books NOT to get: I do not recommend books that have the word dummy or idiot in the title. You are not dumb or an idiot. Don't put that message in your brain. I also do not recommend books that talk about making math easy. It's not easy, but it is understandable and you can ace it if you decide to.
The best way to learn math is ... are you ready for the secret? ... do you REALLY want to know? ... okay, I will tell you. The best way to learn math is to practice, practice, practice. There are plenty of practice problems in your textbook and on this site.
Another key to doing well in math is to decide to learn the material. Now that sounds kind of obvious but think about it. Are you in the class to just get through it so that you can get on with your life? If so, you need to rethink that. Decide to learn the material. Don't just do the minimum and try to figure out the problems you are required to do, so that you can get a good grade. If you focus on the grade, you won't do as well as if you focus on learning the material. If you learn the material, the grade will be there. If you focus on the grade, you may not learn the material and, I guarantee you, your grade will be lower. Now, come on, I'm not that naive. I know you think the goal is the grade. But just try it, okay? What do you have to lose?
Oh, another thing. Cramming for math (especially upper level math) exams doesn't work! Oh, it might have when you were in high school or maybe with algebra. But it won't work here. I have found a lot of similarities between studying math and training for a marathon.
Marathon or Math?
This graph is a visual representation of what preparing well for a math exam looks like. The horizontal axis is time and the vertical axis is intensity of preparation or how much time you spend preparing.
Basically, the raw data, which I downloaded from the Internet, shows how many miles you need to train to prepare for running a marathon. However, studying math is a lot like preparing for a marathon. Some people just work a little bit until right before the exam and then they cram and expect to do well. This works when all you have to do memorize equations or something boring like that. Not so with math. You need to train.
Imagine a marathon runner waiting until the week before the marathon and running 50 miles a day to cram in the training. It doesn't work, does it? (And it's kind of silly to think that he can do well!) It’s the same with math. If you have prepared, done the homework, worked practice problems, worked the practice exam problems, then you can relax, go into the exam with no stress and do great. Sounds good, doesn't it?
Here are some more study suggestions.
Work on math 6 days a week and take one full day off to rest your brain. If you can, take the day off from all school, work and responsibilities. Go jet skiing. Lay out in the park and get a tan. Watch movies. It's not about what you do. It's about what you don't do on that day. Again, I do this (on Saturdays). I know it works.
Allow larger blocks of time periodically during the week to work on Math. A minimum of 1.5 hours will work. However, 2-3 hours at a time is best. You need to study 6 days a week. I suggest you take one day off to relax and refresh yourself. Even though you are not working on math, your mind is still working without you knowing it. So rest is very important. If you workout or do weight training, you know what I mean. Your mind works the same way. If you can't study every day, plan ahead to the next time you can study so that you get the homework and exams done well before the time they are due.
Work the assignments and try to get them at least one day before they are due. When you work hard on something, your mind continues to work on it when you are sleeping or doing something else. Sometimes you will have an insight or realize you did something wrong on your work. Getting it done early gives you time to correct things.
Keep up on your homework, your studying and your exams. If you don't, you will fall too far behind to catch up. Don't skip any work and if you don't understand something don't skip it! This is critically important to your success in this math.
Ask for help often. Unlike other types of courses, math builds on itself. If you don't understand something, chances are that you will need it later to understand something else. If you get lost early, you will be struggling later.
And don't give up!
Calculators and Math (Especially Calculus)
Calculators are incredibly helpful when you need to do calculus. Most graphing calculators have the ability to not only graph but also do derivatives, integrals, limits and many other operations. And the calculators are very reasonably priced, most between $100 and $150. If you consider the processing power and technology involved, that is incredibly reasonable. I am a big fan of calculators. I prefer the HP RPN calculators. But the TI graphing calculators are easier to use for graphing, in my opinion.
However, using a calculator extensively while you are learning calculus concepts can be detrimental. I am going to depart here from the general consensus about using calculators while learning calculus. Many teachers believe that we should integrate technology in the classroom as much as possible. I agree ... to a point. I think we reach a point where the technology gets in the way of learning concepts.
I believe the use of calculators in calculus should be limited to a graphing calculator that is used only for graphing. All other use should be restricted until you learn the concepts. This usually involves pencil and paper combined with repetition and feedback. So, put down the calculator! I know it's harder that way right now but by exam time you will know the material better than the other students in the class and you might even get an A!
However, if you are required to have a calculator, here are some suggestions.
How To Study Math Proofs
When learning math, you don't want to skip anything. This includes theorems and proofs. Theorems tell you under what conditions the results can be applied and proofs show you how to use the math you already know to produce new uses and applications. Here are a few book suggestions for those of you who want to go into more depth. But read through this section first to see if it helps you.
Let me say up front . . . most students just read theorem statements and skip proofs. If you work through each theorem statement and study the proof, you will begin to actually understand math and know when to use it. This will put you ahead in your class and in future classes. Some instructors will also give problems on homework and exams that the other students will miss because they didn't know when to apply which theorem. But you will get it right. This could potentially increase your grade. So, let's get to it.
How do you actually study proofs and theorems so that you can understand them? As you read on the page about how to read math books, you can't just read theorems and proofs like you would read a novel. Mathematicians write theorems and proofs as elegantly as they can and this will usually obscure what is going on. Even Ph.D.'s have to work through theorems and proofs, so you are in good company.
We will separate our discussion into two parts, the theorem statement and the proof. Start by getting out a pencil and several pieces of paper.
Understanding The Theorem Statement
Theorems are written in concise, very compact language which inevitably obscures what is being said. So you need to dig out the pieces in order to understand them. Theorem statements have two main parts, the conditions and the conclusions. The conditions tell you what has to be true in order to apply the theorem. The conclusions are what you can know for sure is true as long as all the conditions hold. The nice thing about theorem statements is every single condition that is required will be stated somewhere in the theorem. So you don't have to guess about a condition.
Your main task is to separate out the conditions and the conclusions. If the language of the theorem (like English) is a language you are very familiar with, then this will not be hard. Mathematicians are very concise, so every word is important. Do not skip any word, no matter how small.
At this point, it won't help you understand what to do if we just give you a bunch of generic rules to handle theorems. So we will use an example to show you how to do this, from which you should be able to extrapolate what you need to do. Here is a theorem from differential equations. You should know some math but don't worry if you don't understand all the advanced math, that's not the point. The point is to extract the information we need.
If the functions \(p\) and \(g\) are continuous on an open interval \(I: \alpha < t < \beta\) containing the point \(t=t_0\), then there exists a unique function \(y=\phi(t)\) that satisfies the differential equation \(y'+p(t)y=g(t)\) for each \(t\) in \(I\), and that also satisfies the initial condition \(y(t_0)=y_0\) where \(y_0\) is an arbitrary prescribed initial value.
This is a theorem from differential equations that determines under what conditions a solution exists and is unique, i.e. it is the one and only solution.
Almost all theorems have an if-then format, where the if and/or then may not be explicitly stated but the structure of the sentences will tell you which section is which. In the theorem above, we do have if-then sections. So, first, we write down all the conditions covered by the if section.
\(p\) and \(g\) are functions which are continuous
\(p\) and \(g\) don't have to be continuous everywhere, just on the open interval \(I\)
\(I\) is an open interval \(\alpha < t < \beta\)
\(I\) must contain the point \(t=t_0\)
Although not stated up front, \(p\) and \(g\) are functions of t and can be written \(p(t)\) and \(g(t)\). You can see this from the differential equation \(y'+p(t)y=g(t)\).
1. a solution is guaranteed to exist
2. the solution is unique, i.e. there is only one
They actually name the solution \(y=\phi(t)\). This name may be used in the proof or the subsequent discussion in the text but it doesn't come into play in the theorem statement.
Understanding The Proof
We are working on a detailed discussion on how to read proofs and we will post it here when it is finished. In the meantime, here are a couple of links to get you started and some book recommendations for more information.