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How To Study Math
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- How to Study Math, Science and Engineering
- by Dr. Phillip R. Rosenkrantz
California State Polytechnic University, Pomona
- Acknowledgment: Some of the information below was adapted from a video presentation by Prof. Lillian Metlitzky
As a department chair from 1990-1997, I counselled most of the students to do not succeed in our department's engineering majors. Most students who fail academically, do so because of a few critical errors in judgement and in how they approach their studies. As a student (again) myself from 1984-1988, I became an avid student of "how to study" mathematically based subjects. I had the good fortune to learn some of the secrets of studying mathematically based subjects (such as statistics and engineering) and developed a few new techniques of my own. I have shared these methods with students on an individual basis and in numerous workshops and presentations. Students who have used these techniques report excellent results.
- The most effective way of communicating these techiques is in a workshop format. Unfortunately just reading this document leaves too much to the imagination. However, if you take the time use these methods you will see that they actually work! I look forward to sharing this information with others in a workshop format in the future.
This guide focuses on the basic strategies for being an effective math, science or engineering student. These strategies cover three phases of student life:
Phase I - Planning - Planning ahead and choosing classes
Phase II - "Inside class" activities - Note taking, listening, test taking, attendance
Phase III - "Outside class" activities - Using notes, studying, homework, term papers and projects, visiting instructor office hours, etc.
In this paper you will learn:
- How to get better grades and have more personal time (the "Big Picture") - Do's and Don'ts.
- How to organize yourself for school.
- The three basic strategies for taking notes.
- How to study.
- How to take tests.
- What policies you need to understand to succeed at the university level.
- Strategies for choosing classes and instructors.
- Reasons why capable students fail.
How to study "non-mathematical" subjects.
Survey of Available Resources - Much has been written on how to be a good student--but very little on how to be a good math, science or engineering student. On my shelf are twelve books and articles I have collected on how to be a good student or how to study. Only two of them give any special attention to "mathematics". All of the other books are directed to students majoring in non- mathematically based majors.
What Is Different About Studying Mathematically Based Subjects - The study of math (or related subjects such as engineering, science, or statistics) is a process that requires progressive, step-by- step learning of fundamentals or the "order" of things. Engineering goes beyond just learning "order" and further emphasizes the "design" process. The goal of engineering education is to learn, understand, and use laws of science within the discipline of a profession and the needs of society.
Math and science are filled with "ordered" structure. This truth makes them different from the study of humanities and social sciences where philosophy, creativity and inference may be of more concern. Unfortunately most students try to use the same methods in all classes. This approach usually leads to a deficient approach to learning math related subjects. In most cases this deficiency is compensated for by a innate intelligence, added time to the task, or persistently repeated failed courses. In many cases, however, this deficiency is not compensated for and the student fails somewhere along the way.
My belief, based on experience, is that 99% of the students who enter engineering have the brain power to graduate. Those who fail to graduate because of non-outside forces (eg. health, finances, personal problems, etc.) usually do so because of poor strategy, planning, study habits and methods.
Math Skill Self Survey
An on-line survey is linked below for the purpose of self-assessment and introducing some of the concepts that will be presented later. It is suggested you take the survey and look at your score. The lower the score, the more you may need to re-assess your methods. If your your score is very low, do not be discouraged. Finding out new strategies that allow you to succeed that actually take less of your time can be very invigorating!
Before I get into the tips for how to study math let me first say that everyone studies differently and there is no one right way to study for a math class. There are a lot of tips in this document and there is a pretty good chance that you will not agree with all of them or find that you can’t do all of them due to time constraints. There is nothing wrong with that. We all study differently and all that anyone can ask of us is that we do the best that we can. It is my intent with these tips to help you do the best that you can given the time that you’ve got to work with.
Also, let me apologize right off the bat if I offend you with some of what I’m going to say in this document. Offending you is not my intent, but in some cases there is simply no other way of saying some of what needs to be said than how I’ve said it here. I have found over the course of time that sometimes in order to get students to understand that they need to do more to be successful in a math I’ve had to be very blunt with my advice. There are portions of this document in which I will continue to be blunt. Again, the point is not to offend but to get some of the people reading this to realize that they need to do more.
Now, I figure that there are two groups of people here reading this document, those that are happy with their grade, but are interested in what I’ve got to say and those that are not happy with their grade and want some ideas on how to improve. Here are a couple of quick comments for each of these groups.
If you have a study routine that you are happy with and you are getting the grade you want from your math class you may find this an interesting read. There is, of course, no reason to change your study habits if you’ve been successful with them in the past. However, you might benefit from a comparison of your study habits to the tips presented here.
If you are not happy with your grade in your math class and you are looking for ways to improve your grade there are a couple of general comments that I need to get out of the way before proceeding with the tips. Most people who are doing poorly in a math class fall into three main categories.
The first category consists of the largest group of students and these are students that just do not have good study habits and/or don’t really understand how to study for a math class. Students in this category should find these tips helpful and while you may not be able to follow all of them hopefully you will be able to follow enough of them to improve your study skills.
The next category is the people who spend hours each day studying and still don’t do well. Most of the people in this category suffer from inefficient study habits and hopefully this set of notes will help you to study more efficiently and not waste time. Also, you will probably find that as your studying gets more efficient you will not need to spend as much time as you once had to.
The final category is those people who simply aren’t spending enough time studying. Students are in this category for a variety of reasons. Some students have job and/or family commitments that prevent them from spending the time needed to be successful in a math class. To be honest there isn’t a whole lot that I can do for you if that is your case other than hopefully you will become a more efficient in your studies after you are through reading this. The vast majority of the students in this category unfortunately, don’t realize that they are in this category. Many don’t realize how much time you need to spend on studying and hopefully reading this document will help you to realize that you do need to study more. Many simply aren’t willing to make the time to study as there are other things in their lives that are more important to them. While that is a decision that you will have to make, realize that eventually you will have to take the time if you want to pass your math course.
Now, with all of that out of the way let’s get into the tips. I’ve tried to break down the hints and advice here into specific areas such as general study tips, doing homework,, studying for exams, etc. However, there are three broad, general areas that all of these tips will fall into.
Math is Not a Spectator Sport
You cannot learn mathematics by just going to class and watching the instructor lecture and work problems. In order to learn mathematics you must be actively involved in the learning process. You’ve got to attend class and pay attention while in class. You’ve got to take a good set of notes. You’ve got to work homework problems, even if the instructor doesn’t assign any. You’ve got to study on a regular schedule, not just the night before exams. In other words you need to be involved in the learning process.
The reality is that most people really need to work to pass a math class, and in general they need to work harder at the math class than they do with their other classes. If all that you’re willing to do is spend a couple of hours studying before each exam then you will find that passing most math classes will be very difficult.
If you aren’t willing to be actively involved in the process of learning mathematics, both inside and outside of the class room, then you will have trouble passing any math class.
Work to Understand the Principles
You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. While there is certainly a fair amount of memorization of formulas in a math class you need to do more. You need to understand how to USE the formulas and that is often far different from just memorizing them.
Some formulas have restrictions on them that you need to know in order to correctly use them. For instance, in order to use the quadratic formula you must have the quadratic in standard form first. You need to remember this or you will often get the wrong answer!
Other formulas are very general and require you to identify the parts in the problem that correspond to parts in the formula. If you don’t understand how the formula works and the principle behind it, it can often be very difficult to use the formula. For example, in a calculus course it’s not terribly difficult to memorize a formula for integration by parts for integrals. However, if you don’t understand how to actually use the formula and identify the appropriate parts of the integral you will find the memorized formula worthless.
Mathematics is Cumulative
You’ve always got to remember that mathematics courses are cumulative. Almost everything you do in a math class will depend on subjects that you’ve previously learned. This goes beyond just knowing the previous sections in your current class to needing to remember material from previous classes.
You will find a college algebra class to be very difficult without the knowledge that you learned in your high school algebra class. You can’t do a calculus class without first taking (and understanding) an Algebra and a Trigonometry class.
So, with these three main ideas in mind let’s proceed with some more specific tips to studying for a math class. Note as well that several of the tips show up in multiple sections since they are either super important tips or simply can fall under several general topics.
General Tips for Studying Mathematics
These are some general tips that where either important enough to single out or just didn’t seem to fit into any of the other sections.
- Go To Class. Remember that math is cumulative. If you don’t go to class you will miss important material that will be used in later sections.
- Get to Class On Time. Sometime important notices are only given during the first few minutes of a class.
- LISTEN During Class. In order to get something out of the class you need to listen while in class. Often this can be difficult to do but it is very important. Sometimes important ideas will not be written down on the board. Watch for things the instructor emphasizes, even if just in words. This often means the instructor thinks it’s important. The more important that an instructor thinks a topic is, the more likely that it will show up on the exam!
- Take Good Notes. Try to write down everything that instructor puts on board. It may seem easy when watching the instructor, but it often is not so easy when it comes time for you to do it. A good set of notes will help remind you how to do these problems. For some instructors writing down everything may be difficult. In these cases you should try to write down as much as possible. Note as well that this seems to contradict the previous tip. It is often hard to both listen and take a good set of notes. This is something that one often only gains with practice. You need to be able to listen while you are writing down the important parts of the lecture.
- Ask Questions. If you don’t understand something then ask your instructor. Chances are you are not the only one who doesn’t understand.
- Listen When Others Ask Questions. When other students ask questions make sure you listen to both the question and the answer. It may be that the student asking the question thought of something that you didn’t think of.
- Review Notes After Class. After each class you should review your notes. Note the topics that you found confusing and formulate questions that you can ask your instructor or tutor to help you understand the topic.
- Make a Set of Index Cards. Make a set of index cards with important formulas and concepts on them. You can carry these around with you to look over when you’ve got a few spare minutes. Use them to help you memorize the important formulas and concepts.
- Learn The (Proper) Notation. Instructors will assume you know it so you’ll need to and many instructors will take points off for bad notation.
- Get Into A Study Group. It is often helpful to study in groups. People often look at things differently so someone else may see how to solve a problem that can’t do or understand a topic that you find confusing.
- Note Due Dates. Write down the due dates for homework and dates for exams someplace you’ll see them so you don’t forget about them.
- Budget Adequate Time For Studying/Homework. It often takes more time studying mathematics to learn the subject than you may require in other classes. Homework will often take more time than you had originally thought it would. Keep this in mind as you budget time.
- Do Homework After Each Class. At the end of each class budget some time to look over the homework from that days lecture and attempt to do it Doing this will allow you time to really work at understanding the concepts covered that day. Do not wait until the last minute to do the homework as this often results in an incomplete homework set and/or an incomplete understanding of the concept.
- Do Homework Without Notes and Book. After the first few homework problems, put your notes and book up and try to do the remaining problems without referring to your notes and/or book. In most cases you will not have these during your exams so get used to doing problems without them.
- Do More Homework. Do not limit yourself to just the homework that your instructor assigns. The more problems that you work the better off you’ll be.
- Practice, Practice, Practice. Practice as much as possible. The only way to really learn how to do problems is work lots of them. The more you work, the better prepared you will be come exam time.
- Persevere. You will not just instantly get every topic that is covered in a math class. There will some topics that you will have to work at before you completely understand. The only way to really grasp some topics is to go home and think about it and work some problems. You will often find that after a little work a topic that initially baffled you will all of a sudden make sense.
- Keep Old Homework and Exams. Do not throw away homework and exams once you get them back. The homework is a good source of study material for exams and both the homework and exams is a good source of study material for comprehensive final exams (if you’ve got one).
- Don’t Forget Your Textbook. If you get stuck on a topic that was discussed in class do not forget that you do have a textbook. Often the text book will contain examples not worked in class and/or a different approach to a problem.
- Seek Help If You Need It. If you are having trouble with your math class you have many options open to you and you should take advantage of them. You can go to your instructor’s office hours, go to the tutoring room or hire a tutor to get help.
- Have the Proper Attitude. Always do the best that you can. Do not do try to do just enough to get by. Doing this can lead to major problems if you aren’t careful. If you are trying to do just enough to get by then all it takes is one bad exam and you are now failing the course. You should always do the best that you can and strive for the best grade that you can possible get.
Here are a couple of tips for taking notes in the class.
- Listen in Class. Do not just write down what you see on the board. No instructor is going to write down every word they say and sometimes the important ideas won’t get written down.
- Write Down Explanatory Remarks. Make sure you write down any explanatory remarks the instructor makes. These often won’t get written down by the instructor, but can tell you how to work a particular kind of problem or why the instructor used one formula/method over another for a given problem.
- Note Important Formulas/Concepts. If an instructor emphasizes a particular formula or concept then make note of it. This probably means the instructor feels that it’s important and important formulas and concepts are much more likely to show up on an exam.
- Question Your Instructor. If you are unclear on something ask questions.
- Note Topics You Don’t Understand. If you are having trouble understanding something being presented note that in the margin and at least write down the key words. Leave yourself a couple of lines so you can fill in the missing details later once you’ve gotten help to understand the concept.
- Review/Edit Your Notes. As soon you can after class go back over your notes. Look for any errors and/or omissions. Fill in any information you didn’t have time to write down in class.
- Review Regularly. At regular intervals sit down and review your notes so that you can learn and retain the information. Remember, that this information will probably be required down the road so it’s best to learn it as soon as possible.
Getting help when you are in trouble is one of the most important things that you can do in a math class. Here are a couple of things that you can do the get help.
- Get Help When You Need It. Do not wait until the last minute to get help. When you start running into problems it is time to get help. Remember that math is cumulative. If you don’t get help right away you will only be making it all that more difficult to understand future material.
- Ask Questions in Class. This way you get the help you need AND stay actively involved in the class.
- Visit the Instructor’s Office Hours. The instructor has office hours for a reason, so use them!
- Form a Study Group. Many people find it convenient to study in a group. Different people will see things differently and may see a way to work a problem that you don’t know how to do.
- Go to the Tutor Lab. If your school has a free math tutoring lab (many do and many don’t) then make use of it! That’s what it’s there for.
- Get a Private Tutor. You can always hire a private tutor for some help. If almost every school you will find people how are willing to tutor you for a fee. Most math departments keep a list of tutors or can tell you where you might be able to find one.
- Ask Good Questions. Saying “I don’t understand this section” is not the best way to seek help. It just doesn’t imply what you’re having trouble with and so will probably not get your questions answered. Be specific with your questions. What exactly is it about this section don’t you understand?
- Have Attempted Work With You. When you get help make sure and bring the attempts that you’ve made on the problem. This will help the person helping you to understand just where you’re having problem.
Note that this section contains some general tips on making the most out of your homework. The next section contains tips on actually working homework problems.
- Understand the Purpose of Homework. Instructors do not give you homework assignments to make your life miserable. Homework assignments are given to help you to learn the material in the class and to develop good reasoning and problems solving skills. Remember that it almost always seems easier to watch and instructor doing problems on the board than it actually is. You won’t know if you truly understand the material and can do the problem if you don’t attempt the homework.
- Have The Actual Assignment. This may seem like a silly tip, but make sure that you accurately write down the assignment and due date. This is one of the more common mistakes that students make with homework.
- Do the Assignment Promptly. You should always do the assignment as soon after the lecture as possible while the lecture is still fresh in your mind. Do not wait until the last minute to do the whole assignment.
- Be Organized. When you start working on homework make sure that you’ve got all the materials that you’ll need to do the homework such as notes and textbook.
- Review. Go back over the lecture for each section and review any examples that the instructor worked to make sure that you understand the ideas from that section. Make note of any common errors that your instructor may have mentioned. Do the same with the text book. Read the section and note examples worked and common errors mentioned in the text book.
- Read/Follow the Directions. Make sure that you read and follow all the directions for both the homework set and the individual problems.
- Be Neat. Make sure that you write neatly. This will help the instructor as he/she is grading the assignment and you when you are going over the assignment in preparation for an exam.
- Show All Work. Make sure that you show all of your work. Do not just give the answer. Many instructors will not accept homework that consists only of answers and no work. Showing your work will also help you when you are reviewing for the exam.
- Check Your Work. Always go back over your work and make sure that you’ve not made any simple arithmetic/sign errors.
In the previous section there were some general tips in regards to homework sets as a whole. Here are some tips to help you actually work the problems. Note that some of the ideas were important enough that they are actually in both sections.
- Read the Problem. Read the problem to get an idea of what you’re being asked to do.
- Read the Problem Again. Now that you know what you’re being asked to do, read the problem again. This time around make note of what you are given and what you need to find. Also make sure that you understand just what you’re being asked to do.
- Clearly Note What You’re Asked to Find. Clearly write down somewhere what you’re being asked to find.
- Clearly Note What You Know. Clearly write down somewhere all the information you’ve been given.
- Draw a Diagram. If appropriate draw a diagram and label what you know and what you need to find. Often diagrams will suggest the solution technique, so it’s a good idea to get into the habit of drawing them.
- Devise a Plan. Try to figure out what you’re going to need to work the problem. Identify formulas that may help you. See if there are any intermediate steps/answers that will be needed in order to arrive at the final answer.
- Work a Similar Problem. If you can’t figure out how to work the problem find a similar problem that is simpler. Work this then go back and compare what you did in the simpler problem to the problem you’re asked to do.
- Work the Plan. Once you’ve got the plan, work it out to get the answer.
- Check Your Solution. Is the answer in proper form? Does your answer make sense? If possible, plug your answer back into the original problem as a check.
- Go Back Over the Problem. Once you’re satisfied that you’ve got the correct answer go back over the problem. Identify concepts/methods/formulas that were used for the problem. Try to understand why these concepts/methods/formulas were used on this problem. Look for identifying characteristics that will help you identify this kind of problem in the future.
Studying for Exams
Here are some tips on studying for exams.
- Start on Day One. You should always be studying for the next exam. Do a little each day, or at the very least start studying 2 - 3 days before the exam. Do NOT start studying the night before the exam. Cramming, while a time honored college tradition, just doesn’t work as well as spending time each day studying.
- Get a Good Nights Sleep. Get a good nights sleep the night before the exam. It is important to be well rested and mentally sharp when you take the exam.
- Make a List of Important Concepts/Formulas. Review your notes and make a concise list of important concepts and formulas. Make sure you know these formulas and more importantly how to use them!
- Rework Homework Problems. Do not just read over the homework problems. Actually rework them. Writing down the steps will help you to remember them. Make sure that you try to do the problems without looking at the solutions.
- Rework Book/Notes Examples. Cover up the solutions to book or note examples and try to rework them. When looking for problems from the book don’t forget that most books have a review section at the end of each chapter that usually contains more problems.
- Look for Identifying Characteristics in Problems. While doing your homework you knew which section it came out of. This provided some clues as to the solution process. During an exam you won’t have this to help you. So, while reviewing your homework look for identifying characteristics that will give you clues on how to identify that kind of problem.
- Take a Practice Exam. Find some problems and treat them a practice test. Give yourself a time limit and don’t use your notes or book.
Taking an Exam
Taking exams is probably one of the most important things that you’ll do in a math class and so it’s important to do the best that you can. Here are some ideas that will help you while you’re taking an exam.
- RELAX!!! This is the first step to successfully taking an exam. Unfortunately, it’s also one of the hardest things to do. The more worked up and nervous you are during the exam the more likely you are to forget something or blank out. The worst thing that you can do in an exam is panic.
- Be Smart. By this I mean be smart as you take the exam. You should go over the exam three times. First, go over the exam and work all the problems that you KNOW you can do. Second, work all the problems that you think you can do, but aren’t sure. Last, go back and work the remaining problems. In this way you will get all the points that you know you can get.
- Be Time Efficient. Watch the clock. Don’t spend a huge amount of time trying to get the points for one problem! If you spend too much time trying to get the points for one problem, you may not be able to finish the exam and lose more points than you gained by taking all that time for that one problem.
- If You’re Stuck Move On. If you find that you’re stuck on a problem, move on to a different problem and come back later to finish the problem. Don’t waste time trying to get 10 points and then not be able to finish and miss 20 points because you ran out of time. Note that is really the same as the previous tip, but it was important enough to make again and in a different way.
- Show All Your Work. Make it as easy as possible for the instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the instructor will assign partial credit based on the work you show. Don’t leave it to the instructor to read your mind trying to figure out if you knew what you were doing or not.
- Never Leave a Problem Blank. You should never leave a problem blank. Even if you don’t know how to finish the problem, write down as much as you do know. The instructor can only give partial credit if there is something wrote down for the problem. Writing something down is not a guarantee of partial credit, but not writing nothing down is a guarantee of no partial credit!
- Read the Problem. Make sure that you read the questions carefully and completely before you answer the question. If the instructor asks for the answer to be written in a specific form then make sure that you do that.
- Does Your Answer Make Sense? Make sure you’ve gotten an answer that makes sense.
- Recheck Your Work. If time permits go back over the exam and check every problem. This means completely rework the problem from scratch.
Learn From Your Errors
This is probably one of the more important sections here and also one of the most over looked. Learning from your mistakes can only help you.
- Review Homework. When you get your homework back review it looking for errors that you made.
- Review Exams. Do the same thing with exams.
- Understand the Error. When you find an error in your homework or exams try to understand what the error is and just what you did wrong. Look for something about the error that you can remember to help you to avoid making it again.
- Get Help. If you can find the error and/or don’t understand why it was an error then get help. Ask the instructor or a classmate who got the problem correct.
- Rushed Errors. If you find yourself continually making silly arithmetic or notational errors then slow down when you are working the problems. Most of these types of errors happen because students get in a hurry and don’t pay attention to what they are doing.
- Repeated Errors. If you find yourself continually making errors on one particular type of problem then you probably don’t have a really good grasp of the concept behind that type of problem. Go back and find more examples and really try to understand just what you are doing wrong or don’t understand.
- Keep a List of Errors. Put errors that you keep making in a “list of errors”. With each error write down the correct method/solution. Review the list after you complete a problem and see if you’ve made any of your “common” errors.
How to Study Math & Physics
Physics and Math are Problem-Solving Disciplines. You must learn the underlying principles and connecting themes to solve the problems.
1. Where you sit in the class may be important. Try sitting in the front half of the class.
2. Preview the book before the lecture to see what is in the book. If you know the formulas are in the book you won't have to write them down during the lecture and can listen more attentively.
3. Read the introduction and summary of the relevant chapters and look at the section headings and sub-headings.
4. Check over notes of the last lecture while waiting for the lecture to begin.
5. Take the book to class. The professor may refer to something in the text.
6. Make notes of new words, new units of measure, statements of general laws, etc. Study notes and related text material IMMEDIATELY after the class to reinforce your learnings. This should be done within 24 hours.
7. During the lecture, question what is being said continuously whether or not you verbally ask those questions.
8. Look for important themes and principles.
9. If note taking leaves no time for thinking in class, copy only the key steps and fill the remaining steps in after class.
1. Examine the information given in the course syllabus carefully before studying.
2. Build up ability to read this kind of material. Do a little bit every day (mind building is like body building).
3. Set aside one hour daily per course in this area and read. When the hour is up STOP. You may have only read 5 pages but you will know what is in them. You can then leave your bigger time blocks for your reading courses.
4. If you do one problem a day for each course you won't be left with 40 problems to do on Sunday night.
5. Go by and see the professor during office hours. Usually they welcome this. When the professor gets to know who you are from the sea of faces in a lecture class, your questions will be looked at as valid and your interest will be noted.
6. Rather than skipping the sample problems in the middle of the chapter, work them. They help in understanding the logic of the chapter.
7. Read the assigned problems before reading the chapter. That way you will know what to focus on. Typically when problems are assigned, it is the concepts in the problem which need to be learned.
8. If you work a problem and get it wrong, it is just as important to know what you are not looking for as well as what you are looking for.
9. Answers in the back of the book may be a bad crutch to use. Often there is more than one way to do a problem and your answer may be just as legitimate as the one in the back of the book.
In the week prior to the test:
1. Quickly review notes and check syllabus
2. Reread quickly your solutions to the homework problems
3. Quickly review assigned chapters.
4. Generate a list of themes, principles, and types of problems you expect to have covered in the test.
5. Review actively. Try to look at all possible ways a principle can be applied.
6. Get as much information about what is important from the professor. Look at the way the professor works through problems.
7. Don't cram the hour before the test.
8. Go into the test thinking you are the Greatest Mathematician or Physicist alive and that you have been called in to solve these problems.
9. Look through the whole test first and do problems you can answer first. Make sure and check the point value of the problems in the test.
10. Don't look at the test as a measure of your ability, "The world has been stumped for hundreds of years on this topic so why should I be able to solve it on the first try?" or "I could have gotten another set of problems which I could have answered: I didn't know these but I know how to do others," or "It's not that I am not cut out for physics, it is that I didn't know these particular problems." If you blow a test go to the professor and ask, "What is it that I'm missing?"