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Mathematics, as a field of study, has features that set it apart from almost any other scholastic discipline. On the one hand, correctly manipulating the notation to calculate solutions is a skill, and as with any skill mastery is achieved through practice. On the other hand, such skills are really only the surface of mathematics, for they are only marginally useful without an understanding of the concepts which underlie them.
Consequently, the contemplation and comprehension of mathematical ideas must be our ultimate goal. Ideally, these two aspects of studying mathematics should be woven together at every point, complementing and enhancing one another, and in this respect studying mathematics is much more like studying, say, music or painting than it is like studying history or biology.
In view of mathematics unique character, the successful student must devise a special set of strategies for accomplishing his or her goals, including strategies for lecture taking, homework, and exams. We will examine each of these in turn.
There you are, just you and the textbook and maybe some lecture notes, alone in the glare of your desk lamp. Its a tense moment. Like most students, you turn to the exercises and see what happens. Pretty soon you are slogging away (struggling), turning frequently to the solutions in the back of the book to check whether you have a clue. If youre lucky, it goes mostly smoothly, and you mark the problems that wont come right so that you can ask about them in class. If youre not so lucky, you get bogged down, stuck on this problem or that, while hours slide by like agonized (distressed) glaciers, and you miss your favorite TV show, and you think of all the homework for your other classes that you havent got to yet, and you begin to visualize burning your textbookexcept that the stupid thing cost you 80 bucks
Lets start over.
There you are, just you and the textbook and maybe some lecture notes, alone in the glare of your desk lamp. Relax. What are you here for? For whom are you doing this homework? Your teacher? Your parents? No, homework is for you, and you alone. It is your opportunity to learn, and to begin to gain mastery - and that is what you are here for. Not a grade - knowledge. Presumably, your instructor has just lectured the material, but have you read the material in the textbook yourself yet? You havent? Then do so. Reading the textbook is something practically no student does, yet it can make a world of difference in how difficult the material seems to you. When reading a textbook, remember that it is not a novel, nor indeed like any other kind of book. Written math is dense. Each paragraph - sometimes even each line - contains deep ideas, which may require a novel way of thinking to understand. It may take you 20 minutes or longer just to absorb and understand a single page. That is normal. Read it with blank paper available and a pencil in your hand. Work through the examples yourself, until you thoroughly understand each step. Writing things down is far more effective than highlighting or underlining. Read the footnotes. After you have done these things, then you are ready to look at the exercises.
Many instructors (but not all) encourage their students to work together on homework problems. Modern learning theories emphasize the value of doing this, and I find that students who collaborate can develop a synergy among themselves which supports their learning, helping them to learn more, more quickly, and more lastingly. Find out how your instructor feels about this, and if it is permitted find others in class who are interested in studying together. You will still want to put in plenty of time for self-study, but a couple of hours a week spent studying with others may be very valuable to you.
Most problem sets are designed so that the first few problems are rote, and look just like the examples in the book. Gradually, they begin to stretch you a bit, testing your comprehension and your ability to synthesize ideas. Take them one at a time. If you get completely stuck on one, skip it for now. But come back to it. Give yourself time, for your subconscious mind will gradually formulate ideas about how to work the exercise, and it will present these notions to your conscious mind when it is ready.
About a third of the students in any given class, on any given assignment, will look the exercises over, and conclude that they dont know how to do it. They then tell themselves, I cant do something I dont understand, and close the book. Consequence: no homework gets done.
About another third will look the exercises over, decide that they pretty much get it, and tell themselves, I dont need to do the homework, because I already understand it, and close the book. Consequence: no homework gets done.
Dont let this be you. If youve pretty much already got it, great. Now turn to the hard exercises (whether they were assigned or not), and test how thorough your understanding really is. If you are unable to do them with ease, then you need to go back to the more routine exercises and work on your skills. On the other hand, if you feel you cannot do the homework because you dont understand it, then go back in the textbook to where you do understand, and work forward from there. Pick the easiest exercises, and work at them. Compare them to the examples. Work through the examples. Try doing the exercises the same way the examples were done. In short, work at it. You will learn mathematics this way - and in no other way.
Everybody complains about word problems, sometimes even the instructor. One
is tempted to feel that math is hard enough without some sadist turning it into
wordy, dense, hard-to-understand word problems. But again, ask yourself: Why am
I studying math? Is it so that I'll always know how to factor a quadratic
equation? Hardly. The study of math is meant to give you power over the real
world. And the real world doesnt present you with textbook equations, it
presents you with word problems. Your boss doesnt tell you to solve for x,
he tells you, We need a new supplier for flapdoodles. Bobs Flapdoodle Emporium
wholesales them at $129 per gross, but charges $1.25 per ton per mile for
shipping. Sallys Flapdoodle Express wholesales them at $143 per gross, but
ships at a flat rate of $85 per ton. Figure out how each of these will impact
our marginal cost, and report to me this afternoon.
The real world. Personally, I love word problems - because if you can work a
word problem, you know you really understand the math.
Math teachers are a mixed bag, no question, and its easy to criticize, especially when the criticism is justified. If your own math teacher really connects with you, really helps you understand, terrific - and be sure to let him or her know. But if not, there are a couple of things you will want to keep in mind.
To begin with, think what the teachers job entails. First, a textbook must be chosen, a syllabus prepared, and the material being taught (which your teacher may or may not have worked with in some time) completely mastered. This is before you ever step into class on that first day. Second, for every lecture the teacher gives, there is at least an hours preparation, writing down lecture notes, thinking about how best to present the material, and so on. This is on top of the time spent grading student work - which itself can be done only after the instructor works the exercises for him or herself. Finally, think about the anxiety you feel about speaking to an audience, and about your own math anxiety, and then imagine what a math teacher must do: manage both kinds of anxiety simultaneously. It would be wonderful if every instructor were a brilliant lecturer. But even the least brilliant deserves consideration for the difficulty of the job.
The second thing to keep in mind is that getting the most out of a lecture is your job. Many students suppose that writing furiously to get down everything the instructor puts on the board is the best they can do. Unfortunately, you cannot both write the details and focus on the ideas at the same time. Consequently, you will have to find a balance. Particularly if the instructor is lecturing from a set text, it may be that almost everything he or she puts on the board is in the text, so in effect its written down for you already. In this case, make some note of the instructors ideas and commentary and methods, but make understanding the lecture your primary focus. One of the best things you can do to enhance the value of a lecture is to review the relevant parts of the textbook before the lecture. Then your notes, instead of becoming yet another copy of information you paid for when you bought the book, can be an adjunct set of insights and commentary that will help you when it comes time to study on your own.
Finally, remember that your success is your instructors success too. He or she wants you to achieve your goals. So develop a rapport (empathy) with the instructor, letting him or her know when you are feeling lost and requesting help. Dont wait until after the lecture - raise your hand or your voice the minute the instructor begins to discuss an idea or procedure that you are unable to follow. Use any help hours that are available. If you are determined to succeed and your instructor knows it, then he or she will be just as determined to help you.
For many students, this is the very crucial of math anxiety. Math exams represent a do-or-die challenge that can inflame all ones doubts and frustrations. It is frankly not possible to eliminate all the anxiety you may feel about exams, but here are some techniques and strategies that will dramatically improve your test-taking experience.
When studying for a Math Exam, you have two main goals:
Learn the material so you can do well on the exam.
Learn the material well enough so you will still know it next semester!!!
(Most of you will be taking more maths. These classes always depend on the material you learned the previous semester! If you don't really learn it, you'll crash and burn in your next class!
TIP #1: Have all memorizing done a couple of days before exam.
TIP #2: Use flash cards for memorization of formulas and rules!!!
Before the exam:
1) Look over homework and lecture notes.
2) Make an exam for yourself (or better yet, for a study partner): Take it after a delay period - So you won't remember where you got the problems - If you take the exam too soon, you may think you know the material better then you do! (This should be done at least two (2) days before the exam - not the night before or you'll freak yourself out!)
NOTE: It is extremely important that you are able to do the problems without knowing what section they came out of!! Be sure to mix the problems up when you are practicing!
3) Go back over what you had trouble with on your practice exam. This is the stuff that you didn't absorb well enough from just doing your homework.
4) The night before exam day do something fun - But not too much fun!
5) On exam day, have breakfast. The brain consumes a surprisingly large number of calories, and if you havent made available the nutrients it needs it will not work at full capacity. Get up early enough so that you can eat a proper meal (but not a huge one) at least two hours before the exam. This will ensure that your stomach has finished with the meal before your brain makes a demand on the blood supply.
6) One hour before exam glance over flash cards and don't talk to classmates about the exam - They may say something to confuse you or make you nervous.
7) Dont cram (fill up). The brain is in many ways just like a muscle. It must be exercised regularly to be strong, and if you place too much stress on it then it wont function at its peak until it has had time to rest and recover. You wouldnt prepare for a big race by staying up and running all night. Instead, you would probably do a light workout, permit yourself some recreation such as seeing a movie or reading a book, and turn-in early. The same principle applies here. If you have been studying regularly, you already know what you need to know, and if you have put off studying until now it is too late to do much about it. There is nothing you will gain in the few hours before the exam, desperately trying to absorb the material, that will make up for not being fresh and alert at exam time.
Exam time:
8) When you get the exam, look it over thoroughly. Read each question, noting whether it has several parts and its overall weight in the exam. Begin working only after you have read every question. This way you will always have a sense of the exam as a whole. (Remember to look on the backs of pages.) If there are some questions that you feel you know immediately how to do, then do these first. (Some students have told that they save the easiest ones for last because they are sure they can do them. This is a mistake. Save the hardest ones for last.)
9) It is extremely common to get the exam, look at the questions, and feel that you cant work a single problem. Panic sets in. You see everyone else working, and become certain you are doomed. Some students will sit for an hour in this condition, ashamed to turn in a blank exam and leave early, but unable to calm down and begin thinking about the questions. This initial panic is so common (believe it or not, most of the other students taking the exam are having the same experience), that its just as well to assume ahead of time that this is what is going to happen. This gives you the same advantage as when the dentist alerts you that this may hurt a little. Since you've been warned, there's far less tendency to have an uncontrollable panic reaction when it happens.
So say to yourself, Okay, I may as well relax because I expected this. Take a deep breath, let it out slowly. Do this a couple of times. Look for the question on the exam that most resembles what you know how to do, and begin poking it and prodding it and thinking about it to see what it is made of. Dont bother about the other students in the room - theyve got their own problems. Before long your brain (remember, its a muscle) will begin to unclench a bit, and some things will occur to you. Youre on your way.
10) Math exams are usually timed - but remember, its not a race! You dont want to dally (waste time), but dont rush yourself either. Work efficiently, being methodical and complete in your solutions. Box, circle, or underline your answers where appropriate. If you dont take time to make your work neat and ordered, then not only will the grader have trouble understanding what youve done, but you can actually confuse yourself - with disastrous results. If you get stuck on a problem, dont entangle (trap) yourself with it to the detriment (loss) of your overall score. After a few minutes, move on to the rest of the exam and come back to this one if you have time. And regardless of whether you have answered every question, give yourself at least two or three minutes at the end of the exam period to review your answers. The oops mistakes you find this way will surprise you, and fixing them is worth more to your score than trying to bang out something for that last, troublesome question.
11) In math, having the right answer is nice - but it doesnt pay the bills. Show all your work on the paper.
12) Finally, place things in perspective. Fear of the exam will make it seem like a much bigger deal than it really is, so remind yourself what it does not represent. It is not a test of your overall intelligence, of your worth as a person, or of your prospects for success in life. Your future happiness will not be determined by it. It is only a math test - it tests nothing about you except whether you understand certain concepts and possess the skills to implement them. You cant demonstrate your understanding and skills to their best advantage if you panic through making more of it than it is.
When you get the exam back, dont bury it or burn it or treat it like it doesnt exist - use it. Discover your mistakes and understand them thoroughly. After all, if you dont learn from your mistakes, you are likely to make them again.
1. Slow down!! The flow of a math book is not like the flow of a novel. A novel should be read fluently, but math books cannot. If you are reading a novel and are somewhat distracted, you can still get the idea of the story. When you are not concentrating on math, you will get very little out of it, and itll seem more difficult that it really is.
2. Every word counts. Math books are usually not repetitive, so
there is little chance of picking up missed information from reading on. Writers
of math texts believe that extra words and repeats get in the way of clarity.
Never start in the middle of the book, the chapter, or the page. Each page
assumes you have mastered the previous pages.
3. Understand each sentence before you go on. Reread as many times
as necessary for you to master an idea. Mastery can take minutes, hours or days.
4. Do not skim diagrams and other kinds of illustrative material.
5. Words and symbols of math have very specific meanings. If you are at all uncertain about the meaning of a term, look it up or ask someone to explain it.
6. Write as you read.
A. Work out proofs, derivations, and sample problems. After reading an example, cover it up and try to work it out yourself.
B. Write key recall words in the margins.
C. Mark formulas, definitions, cautionary notes (with an asterisk, check mark, etc.)
D. Note questions on concepts you need to have clarified.
7. Record key points on a separate sheet.
8. Make 3cm by 5cm review cards with formulas, properties and facts.
9. Keep testing yourself on a separate sheet of paper.
10. Without looking back, write out and say aloud the important points.
11. Create tasks for yourself as you read the text.
After reading an example and working it out for yourself, try to think of other examples that would fit the idea being discussed. Think of other relevant problems and try to solve them.
12. Use more than one math book. Use these other math books as reference texts to clarify or better explain a topic you are studying. Pick books that appeal to you. If you are very verbal, a book with long explanations is likely to be most helpful. If you are very visual, you might choose a book that has more illustrations.
13. Read the chapter before, and again, after class.
You will get the most out of class if you have read the material before the instructor presents it. Even if you felt you understood the material in class, read the chapter over, soon after class. This second reading will help you store the information you've learned in your long term memory.
1. Attend all classes and take full class notes. Research has shown that successful students never cut class and usually take down at least 64% of what is discussed in class. Failing students write half as much and often miss class.
2. Make it a practice to read over the topic or chapter before going to your math class. This will give you a much better understanding of what is being discussed in class and as a result you'll learn more from lecture.
3. Organize your notes into a notebook devoted only to math. Use the first half for class notes and the second half for homework. Take a complete set of class notes and add any helpful clarifications to your notes that you hear in class. Mentally follow all explanations and try to understand the concepts and principles. Then write down the main points, steps in explanations, definitions, examples, solutions or proofs.
4. Date each day's class notes. Write the topic or chapter heading on top of the page. Leave a 2 cm margin on the left side for comments.
5. It's important to stay current. Do not allow yourself to fall behind or the entire course will become an effort and a struggle for you.
6. Review immediately after class and again eight hours later. Research shows that most of the information is lost within the first 20 to 60 minutes after learning. However, if you review immediately after class and again within the same day, and then do weekly and monthly reviews, the information you have learned will remain in long term memory.
7. Ask questions. Always remember you have the right to ask questions before, during and after class. Never avoid asking a question out of fear of looking stupid. Don't allow a question to go unanswered. Get help fast.
8. When you feel "lost" ask your teacher to explain the first step that you did not understand; then question any later steps that you still do not follow. When you can't see the overall picture of what the teacher is doing, ask questions.
9. To get the most benefit from a help session:
A. Use question marks to identify confusing material in your notes or textbook.
B. Write down specific question you will ask.
10. Always remember the "say and do" principle. Research shows that we remember only 10% of what we read, 20% of what we see, but a full 90% of what we say and do. So whenever possible say and do.
11. Work out lots of sample problems. Practice, practice, practice. Do assigned problems and lots more. Get sample problems from other books. Work with a classmate and explain aloud what you are learning and how to solve problems. Remember the more you "say and do" the more you will be able to recall what you're learning. You must always be actively involved in the learning process.
12. The best time to do your homework is the same day it is assigned. This will help reinforce what you have just learned.
13. Read and study all your textbook explanations of each type of problem. Whenever possible use additional textbooks and study guides as resources. Each book will discuss your topic differently and offer different examples. This is an excellent way to clarify difficult concepts and to give you more practice problems.
14. Know and understand your math terminology. This is one of the keys to success in any field. Use 3cm by 5cm review cards to study math's own unique vocabulary. Put the term on one side and the definition on the other. Carry these cards with you everywhere and review them at odd moments throughout the day. You won't even feel like you're studying.
15. Never attempt to memorize a formula (rule, proof, procedure) until you've attempted to understand it first. Make sure you can illustrate the definitions, theorems and the use of the symbols. You may want to use 3cm by 5cm cards to help you memorize some formulas for convenience and quick recall.
16. Write up summary sheets of math terminology and formulas and review them often.
17. Successful math students study math two hours per day at least 5 days a week. In addition, they work out 10 new problems and five reviews problems during each study session.
18. If math is your most difficult subject, make sure to study it before all other subjects. Do not leave it until the end. You must study math when you are most alert and fresh. It will go better for you and you'll recall more. Research also shows that you'll retain more information if you take 5 to 10 minute study breaks every 20 to 40 minutes.
19. Act as if you have control of your level of success in math. Act as if you are really enjoying it. Eventually, your habit of pretending and resulting success will make your feelings match your behavior.
Go over your class notes before starting the homework problems.
If the teacher is making some sort of computation, work along with him/her.
If you do have to miss a class, always call a classmate (before you return to class!) to find out what you missed and if your teacher assigned anything that will be due when you return.
Don't put off doing homework until the last minute; you'll just get behind. You need time to think about the stuff youre learning, not just time to work the problems. Do homework yourself.
Take very neat class notes. Write down everything the teacher writes down and try to write down most of what he/she says. Put stars by problems or points that your teacher stresses or get excited about. These are good potential exam questions!
Try to study for exams in an environment as much like the environment where you'll take the exam as possible. If you usually study in your underwear with the stereo on, having to take the test in an uncomfortable, stuffy, quiet classroom in one of those little desks may make it just that much harder for you to remember the things that you knew the night before. Since you probably can't change your test environment, you must change your study environment.
Make sure you keep up with your assignments. Math isn't like some subjects where you can "catch up" on homework over the weekend. It is necessary to complete all assignments each day. If you fall behind, the time spent in class will be less productive and it will snowball on you. You'll get to a point where it is impossible to catch up!
Arrive on time to class each day.
Whenever you try to communicate mathematical statements and ideas, you must start speaking and writing the language of mathematics. Exams and homework are also supposed to show the teacher how you think, whether you understand the concepts, and whether you've put in enough work into mastering more mechanical skills. Here are a few pointers which should help you write better papers, and - hopefully - get better grades.
Like many other points I will try to make, this one is almost as important in everyday speech as in writing mathematics. But what may be understandable when talking to a friend ("hand me that doodad over there") can be unacceptable in mathematics ("that variable is kind of small, so there").
Imagine that you are giving directions to someone you don't know, and that your life depends on it. You would speak clearly, or write in block letters, use basic terms, and you'd rely on well-defined, universal concepts rather than on notions specific to your culture or environment.
It is much the same with mathematics. When you use words, think about their meaning. When you use symbols, make sure they will mean the same to both you and the teacher. This is easy in math, because almost everything you will be writing about has a precise, agreed upon definition. A "continuous function", a "definite integral", or an "open interval" mean the same to everyone with mathematical training.
Mathematics is all about implications, statements of the form "assuming that X, we have Y", or "if X then Y", or "since X is true, Y must hold", or "because we know X, Y follows". An implication has a premise - X, and a conclusion - Y. Some weell-known rule of reasoning or a theorem must "fit" the particular situation for the implication to be accepted.
For example, "1/(a - 4) is positive since a is a real number greater than 4" is a perfectly valid statement; it follows from basic properties of inequalities (we must have a - 4 > 0, and reciprocals of positive numbers are positive). There is not much of a chance that the teacher will mark this with a big red why?, unless you are taking an upper level number theory course in which you are supposed to derive such basic properties from scratch.
But in a Calculus course a statement "1/(a - 4) is positive because a is a real number" will be rejected as a `non sequitur': the conclusion does not follow from the premise. It may very well be that in the context of the problem a is indeed greater than four, and that the conclusion is true, but you have not explained this!
As with spoken language, math has certain ground rules which have to be followed when you form sentences.
For example, a relation such as "=" can only be placed between two quantities which have a chance of being equal (e.g. when solving an equation), or when you are asserting that the quantities are in fact equal. One of the most common problems we see in student papers is the lack of respect for equality and other relations.
Consider this example: "x2 - 1 = 0 = x = 1, -1" and look closely at all the equalities separately. This is unintelligible (incomprehensible) babbling (talking nonsense). It can be fixed by using symbols, or simply short English words and phrases to indicate what is happening: "if x2 - 1 = 0, then x2 = 1, which means that x = 1 or x = -1".
We often use relations "chained together" to make a point; for example: x > y - 1 > 5 - 1 = 4, so that x must be greater than 4. This is perfectly OK as long as the relation is `transitive' (as > happens to be). But mixing disparate relations in this fashion can lead to nonsense: x > y - 1 = 4 < x + 2 > 3 doesn't really say anything and resembles a banana split with an anchovy (a fish) topping.
More than with other sciences, in mathematics "what you see is what you get". There are few hidden meanings, little should be left unsaid, and there isn't too much room for interpretation or context. Equality means equality, 5 means 5, the Riemann integral is a Riemann integral whether the person reading what you wrote is a Romanian, a freemason, or a vegetarian. That's because we have precise definitions of things. This is also what makes mathematics difficult: it isn't enough to be "close" - you must be right on the money for your argument to count.
So if you write x = 1.4, you'd better mean 1.4, and not the square root of two! When you are told to find an integral of a function and you compute its derivative instead, don't be surprised if you don't get any credit. If you write about an isosceles triangle but your reasoning relies on it being equilateral, you won't get many points...
This insistence on accuracy is not some type of malice (cruelty), or the teacher's way of having fun; mathematics simply cannot exist without the rigor (rigidity) of meaning exactly what you say, and your papers cannot be fairly judged in any other manner. So don't get frustrated or angry, don't say that "I had it almost right". Just get it absolutely right on the next test.
After all this you may think that you should write 2-page justifications for everything you do. Not so; on the contrary! It takes very little to put your mathematical thinking down on paper. All you need is some commonly used symbols and a handful of words such as "if ... then ...", "therefore", "because", "this means", "that's why". In fact, most mathematicians prefer to see that rather than read long essays which use common English, because the terse (short) mathematical way of expressing things is usually much more precise.
If you can, write only as little as necessary to make the meaning very clear. It isn't easy to judge exactly "how much is enough", but you can try the following experiment: imagine that you have a little brother in the class, that he is a few weeks behind in his studies, that you are explaining the problem to him, and that he wants to know why you are doing whatever you are doing.
But if you have doubts, it is generally better to say more than to say too little.