Situations That Occur in Math and Techniques to Use

Contents:

• Situations that Occur in Math and Techniques to Use
• How Math is Different
• The Goals of Studying Math
• Basic Methods That Are Used in All Studying

This section discusses common study situations that students meet in math, and it suggests some techniques to make the studying more effective and efficient.  Some techniques come from the standard techniques described below, and others are especially adapted to subjects like math.

List of situations:

• You are about to start studying a math textbook
• You are reading new material in a math textbook for the first time
• You are studying the text for the second time.
• You are studying worked examples
• You have just read the text and worked problems and learned how to do a procedure
• You are planning to do math homework problems
• You are reading a problem
• You are actually solving a math problem
• You are getting stuck on a problem
• You do a few problems and find you understand them and are tempted to skip doing the others.
• You have just finished solving a math problem
• You have just finished solving a math problem and you know you got it right
• You have studied a principle and practiced it and want a handy memory trick for remembering it.
• You study diagrams where the diagram is separated from the labels
• You find symbols meaningless or hard.
• You have done all this and still cannot understand something in a text
• You do not know all the prerequisites on a certain topic.
• You are studying material that you already studied more than a year ago
• You read several lines of material, understand the first part, but by the time you are at the end you forget it and could not put the whole together. 
• During a study session your brain gets tired and it’s hard to make the material meaningful
• You have returned to a topic the next day and notice that you have forgotten some material
• You have gone 3 days without studying math and are coming back to it
• You have just finished a problem that was both easy to do and similar to others you have done
• You have just finished a type of problem that was new to you
• You have just finished a problem that gave you a lot of difficulty
• You make mistakes on problems because you do not solidly know addition facts or multiplication facts.  (Subtraction is the reverse of addition, and division is the reverse of multiplication.)  You want a reliable way to learn them.
• You have bad habits in math; once you learned a certain skill wrongly and your “bad skill” interferes with learning the right thing
• You are reviewing for atest: The day before
• You are reviewing for a test: The day of the test
• You are taking a math test: General advice
• You are taking a test: Stuck on a problem
• You do not have enough time to study adequately for a test
• You need to deal with test anxiety
• You must deal with low self-confidence that you can do a certain kind of math task
• You have to deal with wandering attention
• You will be studying a lot of similar material after math studying
• You sense that there are some weaknesses in the way your text or instructor has organized the material.

Situation:  You are about to start studying a math textbook

• Generally, review the prerequisite concepts for three to five minutes.  Often you will find that prerequisite material in the prior chapter. 
   
• The purpose is to bring your prior knowledge into your conscious mind so that (1) you will understand the new material in terms of it, and (2) you will save time in memorizing it because you can associate the new material to the correct prerequisite skills. 
   
• If you don’t review, you will find your memory is “cold” and it will take you longer to understand the new material.

Situation: You are reading new material in a math textbook for the first time

• What will you see in a math textbook?  Usually there are explanations of what a math concept means, ways to use it to solve problems, and some worked-out problems. 
   
• Goal:  When you read new material the first time, it is helpful to set the goal of getting an overview and accept that you will not understand the details yet. 
   
• Pick a page or two on one sub-topic and read entirely through the section trying to get the general idea.  You will probably find it useful to read naturally and accept it when you do not understand a passage. 
   
• If you don’t do an overview of new math material, but try to understand it all the first time, you may, paradoxically, find it even harder because you’ll lack a context to fit the new details into.

Situation: You are studying the text for the second time.

• When you read a text passage the second time, read slowly.  Pay full attention. 
   
• The purpose now is to get the exact meanings.  Read slowly enough that the meanings of the words, symbols, and diagrams come into your mind.  This slow reading is especially important for people who normally read fast. 
   
• Get the meanings in terms of separate math principles.
   
• Try to summarize a principle with a name.
   
• When you cannot understand, put a “?” in the margin so that you will find puzzling parts later and not forget to clarify them. 
   
• Let your mind also notice what the new principles remind you of, both similar math ideas you already know and prerequisite knowledge.  Let your mind go beyond the buildup given by the text.  This thinking associates the new to the familiar and speeds up building memory. 
   
• If you do not take time to notice associated ideas and think about them briefly, you will hurt your ability to remember the math later.
   
• If you find that a passage is even slightly difficult, then explain it to yourself.  Your goal is to make your mental model match the text’s ideas.  Go line by line, sentence by sentence.  Talk aloud.  Or talk in your head.  Explain each sentence to yourself in other words.  Note where your understanding is clear and where there are gaps. 
   
• If you don’t explain it yourself, you will end up with less clear ideas and will not be as aware of gaps and will make more mistakes down the line. 
   
• When it is appropriate to draw diagrams or graphs, do so.  Your act of drawing will add a second level of understanding and memory because drawing uses visual-spatial memory. 
   
• Notice situations where the text’s ideas are different from your own ideas.  Seeing discrepancies between the text and your own mental model are important. 
   
• When you notice a discrepancy, stop.  Think over which is better: the book’s or yours.
• Adjust your mental model consciously. 
• Assuming that the text is correct and your mental model wasn’t accurate or complete, then what would happen to your knowledge and memory if you did not notice a discrepancy and consciously fix your mental model?  After you shut the book and let time pass, your ideas will tend to return to your original defective mental model.  Then later when doing problems or taking tests, you will make mistakes.
   
• When you go slowly and do these time-consuming things, remember that you are saving time later when you do problems because you won’t make as many mistakes and won’t have to restudy as often.  If you have never studied before by doing self-explanations and diagrams, you will be pleasantly surprised by the mental power it will give you.

Situation: You are studying worked examples

Research now demonstrates that students succeed better when they study worked examples provided by the text and use them later as patterns for solving homework problems.

• Study worked examples with microscopic carefulness.
• Search for three elements of worked examples: the problem formulation (statement, description), the solution steps, and the final answer itself.
• The way to get meaning in worked examples is to follow three things at each step as you go: (1) What the current goal is; (2) what the current situation is; and (3) what action to take for the current goal-and-situation.  
• Explain to yourself what principle the text’s authors used in each step to decide how to respond to each goal and situation with the action they took.  Normally, the principles will be found in material you studied in that chapter. 
• Test yourself by covering up the last line of the worked problem and reach the solution yourself.  Then cover the last two lines and finish the problem.  And so on until you are doing the whole problem yourself.  You are likely to find it easy many times.  And when you find you make mistakes, be glad because they signal you don’t quite understand part of how do that type of problem. 

Situation: You have just read the text and worked problems and learned how to do a procedure

• Practice on a new problem right away.  Don’t delay.  As soon as you have figured out a worked problem, find a similar problem and do it yourself.
   
• The first purpose is to save time by using the new knowledge while it is still fresh in your working memory, because in a minute or two your memory will fade.  You are most likely to get problems right when you practice right away.  If you delay practicing and go on and study additional math, your memory for that procedure will fade quickly, and when you do go back to it, you may have to restudy it, thus wasting time.
   
• A second purpose for practicing new material on a new problem right away is to add a second memory—a memory for the experience of doing a problem—to the first factual memory.  By putting the information into two memories, you increase your chances of remembering the new material.
   
• It is generally not helpful to study several different new math procedures at a time before trying homework problems, because you will forget so much.  Exception: When the new material is very easy.

Situation: You are planning to do math homework problems

• Do at least some math homework problems in the same session that you study the text and the worked examples.  The purpose is to use your new knowledge while it is fresh in your working memory and has not been forgotten.  If you delay after you study, you are likely to forget some material and have unnecessary difficulty in working the homework problems.  
   
• Reviewing: Sometimes a good text and teacher will keep giving you a few problems on the same topic for several days through a term.  That helps you review. Remember that the more separate times you make contact with the same material and the more you practice, the better you learn.  When the text and teacher don’t repeat problems of the same type, you can get good results yourself by doing the most important problems over again.  Of course, you will find it easier the second and third time.  You can’t skip steps.  You must actually do the problem with pencil and paper over again.  The repetition will increase memory both for that problem and for your understanding of how to do problems in that category.

Situation: You are reading a problem

• Translate it: When you read a problem, translate each word and each sentence into what is meaningful to you.  Draw diagrams.  Make tables.
   
• Do self-explanations of the problem.  Go line by line, talk it out.
   
• Be aware that research shows that students have the most trouble with sentences that state a relationship between two variables.  An example is: “Mary is twice as old as Betty was two years ago.  Mary is 40 years old.  How old is Betty?”  Read such relationship statements slowly and analyze them carefully.
   
• Integrate it: After you have translated the problem and made it meaningful, then tie the parts together, integrate them, into a meaningful model that makes sense mathematically.  You can often tell you’ve got it when you recognize that the problem is a familiar type of problem, such as a triangle, distance-rate-time, interest rates, river current and boats, and work problems.

Situation: You are actually solving a math problem

• Planning the solution: Do such things as (1) finding a related problem as a starting point, (2) restating the problem, or (3) breaking it into smaller parts and smaller subgoals.
   
• Executing the solution: Carry out the solution.  Be careful.  Check your work.

Situation: You are getting stuck on a problem

• A major cause of getting stuck is that your thinking moves repetitively over the same situations, goals, and techniques.  You stop searching widely and fail to use the full resources available in your mind, notes, and textbook.  Therefore, you need to break up the thinking patterns.
   
• Move.  Move your body, your head, your hands.  The purpose is to give your mind some other stimulus to pay attention to so that it will break up the unhelpful circular patterns of thinking. 
   
• Breathe.  Breathe deeply several times.  Think of your abdomen and do abdominal breathing.  The purpose is the same as above – to change your unproductive mental state.
   
• The classic advice for solving problems that puzzle you is to find a similar problem that you can do.  Use it as a model or analogy to guide you in solving the new problem.  In this case, use worked examples as such models.  Prepare for next time by studying worked examples and preparing prototype problems in advance.
   
• Reread the problem again carefully.
   
• Try to figure out what type of problem it is because it’s possible you did not recognize the type.   
   
• When you get stuck or find that the text’s answer makes no sense and, yet, you seriously believe you understood the text, try to decide where the difficulty lies: in a misprint, the wrong answer in the book, an untaught application, or in your own understanding.
   
• The final advice is familiar.  Go back to the text and reread the parts that bear on the problem.  Ask someone for help.

Situation: You do a few problems and find you understand them and are tempted to skip doing the others.

• The issue you must face is to figure out if all the remaining problems are this easy.  What you do depends on that answer.
• If they are all this easy, you need to evaluate your teacher’s goals: (1) Does the teacher expect students will find the task hard and give more problems than are necessary for you to learn it?  (2) Does the teacher know from past experience that students actually need a wide range of practice to build the cognitive skill of solving these problems?  You may benefit from doing them all more than you realize.

Situation:  You have just finished solving a math problem

• Find out whether you got it right, whenever possible.  Check the answers in the back of the book, ask a friend, or try to double-check the solution.  If you skip getting feedback, your brain will tend to recall the bad methods you used as if they were right, and it will cause you difficulties when you use those skills again on similar problems.
   
• Use this right-wrong information as feedback to how effective your actions were on this kind of problem.  Use this feedback to reinforce or correct your knowledge of the procedure to follow.

Situation:  You have just finished solving a math problem and you know you got it right

• Stop for a few seconds before going on to the next problem.  Do two things: Reinforce yourself for using the techniques successfully, and mentally review the steps you took to solve the problem.
   
• Reinforce yourself for using the techniques successfully.  Let yourself feel good about success.  Talk to yourself in a natural way to praise or congratulate yourself honestly for getting it.  “I did it!”  “I got that method”!  “Yea!  I solved it with that method!” 
   
• A good variation is to praise your ability to use the techniques.  That builds self-confidence for doing math.
   
• Your purpose is to reward yourself for your actions and give your mind positive reinforcement for what you did in order to improve your memory and motivation for the future.  If you skip feeling good after success, your brain is less likely to recall the procedure and you are less likely to be positively motivated. 
   
• If you feel grim and negative and determined after you finish a problem, you may be sending yourself a signal to avoid future math homework.  If you go further into negativity and call yourself stupid and slow after solving a math problem, you will create even more negative feelings towards math. 
   
• Do not focus only on the end result of success because that will merely associate praise with success.  You want your brain to associate the techniques and procedures you used with praise and rightness. 

• Mentally review the problem type and the steps you took.  Your review creates a summary.  Just a few seconds is usually enough. 
   
• Your purpose is to organize the separated steps of problem-solving into an organized bundle, a chunk, so that your mind will remember it better. 
   
• If you skip doing a review and summary, you lower the chances of recalling later how to do this kind of problem.  If you delay doing a review of a problem until later, your memory will fade.  Take advantage of the time right after problem-solving when all the information is fresh and hot in your working memory.

Situation: You have studied a principle and practiced it and want a handy memory trick for remembering it.

• Exemplar problems: You can deal with the problem of forgetting certain procedures by creating an exemplar problem and memorizing it.  An exemplar is a real problem that is typical of a category of problems.  You can also use one of the text’s worked examples as an exemplar. 
   
• Here is an example.  Suppose you have trouble doing percentage problems.  Create one specific example of a percentage relationship, lay it out and solve it and memorize it.  It could be:  25% of 60 = 15.   That is the exemplar.  Then you know that it translates into a decimal: 0.25(60) = 15.   Next you practice swapping the terms:  60 = 15/.15 or .25 = 15/60.   Once you’ve got that solid, you’re ready for new problems, no matter what part of the percentage relationship is given and what is missing.  A future problem might ask: A department store is selling sweaters at 18% off and you paid $36.90.  What was the original price?  So you go back to your exemplar that was 25% of 60 = 15.  You plug the new problem in.  You know you have to figure that “18% off” means that the price you paid is 82% of the original price, i.e. 100% minus 18%.  So, 82% of X = $336.90.  Next: 0.82(X) = $36.90.  Next: X = $36.90/0.82 which = $45.

Situation: You study diagrams where the diagram is separated from the labels

• When the worked example has a diagram (a graph, a geometric shape), sometimes you will experience the difficulty that the diagram is printed up above and the definitions of the parts are printed below.  Words and diagram are not integrated.  That forces you to go back and forth from the picture’s parts to the words below.  Research shows that separated diagrams and words cause trouble to students.  It is better to integrate the diagrams and words. 
   
• You can put one finger on a part of the diagrammed object and put another finger on the right label and then look back and forth, telling your mind what the part is.
   
• You can write in your own book.  Write the label by the part.  Use arrows.

Situation: You find symbols meaningless or hard.

• Since math uses a lot of symbols, it can be hard to read and understand.  This is normal.
   
• You can expect that the two hardest times will be (1) when you read and use the symbols on your first contact with them; and (2) when you have been away from the math for a day or more and return to the topic and experience the normal forgetting that occurs after delays.
   
• Do the normal things to increase understanding: Slow down; pay close attention; talk to yourself about what the symbols mean; treat the symbol as if it is a word and treat the equations and functions that use symbols as if they were sentences, so that you look for meanings both at levels small and large (you treat each symbol as a chunk and treat the whole equation as a chunk); use the symbols in problem-solving; review frequently.  Expect the learning to occur within a few days.
   
• When slow speed of reading equations bother you, check if you find your mind is slow to retrieve the meaning of each symbol.  If that is the case, then give yourself speed training.  Here’s how: Prepare flash cards with the symbol on one side and the definition of it on the other side.  Look at the symbol side first and within one-half of a second turn the card over and read the meaning.  Next look at the symbol and try to think of the meaning as fast as you can.  Practice over several days. 

Situation: You have done all this and still cannot understand something in a text

• I don’t need to tell  you that you can ask someone: a friend, tutor, or teacher.
   
• Define as clearly as you can how much you understand and when you don’t understand and finally where your understanding resumes.  Mark it.  That will let you make specific what the issue is. 
   
• The classic next step is to put it aside, move on to another topic, then sleep on it and return to the topic the next day.  By then your unconscious mind may have put it together and help you out.
   
• If next day’s studying does not clarify it, back up and go through the topics and procedures that are prerequisite to the puzzling topic.  Such information may give clues.
   
• Once you finally understand it, then do something so you remember it.  Protect yourself from the danger of forgetting it.  Mark it; review it; practice especially hard on problems that use that topic.

Situation:  You do not know all the prerequisites on a certain topic.

• Lack of knowledge of prerequisites will hinder you from successfully understanding math concepts and problem-solving.  You need to decide whether to take a prerequisite course or whether you can do well enough to get by.
   
• Here are some criteria to think about: Do you want high grades?  If so, take the prerequisites.  Do the missing prerequisites affect a small area, a few topics, or lots of topics?  The more your gaps are, the wiser it becomes to take a prerequisite course.  The fewer the gaps, the more it is possible to get by.  How much time for study do you have?  If you don’t have a lot of study time, the lack of prerequisites will hurt you more.
   
• Ask your teacher what to do.  A teacher has a broad overview of the role prerequisites play.
   
• Ask if there are review materials on computer or in a math resource center that can fill in some basics.  Perhaps it will be enough to partly fill the gap.

Situation: You are studying material that you already studied more than a year ago

• Many college students take courses that review partly or completely material they had in the past.  Your question will be deciding how carefully you need to study and how much of the homework you need to do in order to refresh your skills.
   
• First, expect that relearning will happen faster than the original learning did.
   
• Second, notice that there are two kinds of learning involved: (1) factual knowledge, meaning the math concepts and facts; and (2) procedural knowledge, meaning knowing how to follow math procedures and the degree of speed and accuracy you have in solving problems. 
   
• It is possible you will remember more factual knowledge than procedural knowledge; although you feel you know the principles, you may not be able to solve many problems.  That means you should test your knowledge both of the text’s explanations and the assigned problems.  You might know one and not the other.  You may need to do a lot of the problems, even though they are review, in order to build up your speed and accuracy.
   
• Third, be aware that a college class may include more applications of math principles to different situations than your high school courses did.  That means you should look at the full range of problems before deciding you know the stuff.
   
• Test yourself to see if your quick review is working by doing certain problems two or three days after going to the class that taught how to do them or after looking at the text’s explanations.  That task will make you pull the material from cold long-term memory.  If you can do problems, under those conditions, including tricky problems with unusual applications of principles, know that you’ve got it. 

Situation: You read several lines of material, understand the first part, but by the time you are at the end you forget it and could not put the whole together. 

• Sometimes, people can understand parts but cannot put it all together.  Although slow readers experience more often than fast readers, it can happen to anyone when a textbook crams a lot of material into a long passage.
   
• Method #1: Read it bit by bit.  Assemble two related bits together and look at what they mean together.  Then add a third bit and look at the meaning of the three.  Then add another and so on.  Purpose: It builds up understanding.
   
• Method #2: Do a self-explanation.  Talk to yourself, out loud if possible where you are studying, and explain in your own words what the author means, line by line.  It associates the material to what you already know and increases memory, too.
   
• Method #3: Do what you can today, put it aside, and come back later.

Situation: During a study session your brain gets tired and it’s hard to make the material meaningful

• The sense that new math symbols and statements are meaningless can come often, especially when you have worked for awhile and are tired.  What causes it is fatigue of the process of taking sensory information into your mind and then looking up the meanings and transferring them to working memory.
   
• Method #1: Give yourself a complete mental rest.  Take a half hour off and then return.
   
• Method #2: Give yourself a partial rest by turning to a different math topic or a different subject.

Situation: You have returned to a topic the next day and notice that you have forgotten some material

• It is normal for a time gap in studying to lead to fading of your memory for what you learned earlier.  You won’t be able to do today some skills that you could do yesterday.  You’ll forget information you had known.  You aren’t stupid.  It’s normal with memory.
   
• Method #1: Review the earlier material.  Review math procedures by actually solving a problem that you solved yesterday.  You  can expect it to go more quickly on the second day.  And when you get stumped and look up what to do, you can expect it easier to find what you need and go on.
   
• Method #2: Prepare for a future gap in time by over-learning current material today by extra studying.  Prepare also by inserting a brief review in a just a few hours, so that the spaced learning effect helps you.

Situation: You have gone 3 days without studying math and are coming back to it

• A 3-day gap or longer will lead to forgetting.  Expect to feel that your knowledge has gone cold.  You will need a longer review than after shorter study gaps of 1 or 2 days. This time gap varies with different people. Learn for yourself how long a gap without a review after a study session produce dangerous forgetting.
   
• Don’t skip reviewing because otherwise you will have trouble learning any new material that depends on knowing the earlier topics.
   
• When possible, try to plan your work so that you work every day or two on math.
   
• An exception allowing longer gaps in studying would be when you’ve come to the end of a major section and know that the next topics are independent of the prior ones.

Situation:  You have just finished a problem that was both easy to do and similar to others you have done

• Do the procedure of stopping, reinforcing and reviewing.
• Notice whether the problem had any new feature.
• If there was a new feature, do a mental review that incorporates the new feature.
• If the problem had nothing new, go on to the next problem.

Situation:  You have just finished a type of problem that was new to you

• After solving any new types of problems, it is especially important to stop, reinforce and review.   The reason is that your brain perceived the steps as separate; you probably worked slowly because it was new for you, and the slow speed may have prevented your mind from seeing the whole picture.

Situation:  You have just finished a problem that gave you a lot of difficulty

• After doing a hard problem, it is also very important to stop for a few seconds, to feel good, praise yourself, and do a review of the steps you took.  The reason is different.  While working on a difficult problem, your mind has experienced doing some wrong steps, feeling frustrated, doing some right steps, and getting some success.  All these things are mixed up in your mind and you may remember the bad with the good and thus create new habitual mistakes.  There is a danger that you might later forget what steps were good and what were bad and just approach a new version of that problem in a wrong way.  If, however, you review what the right approach was from start to finish and then let your emotions label those correct steps as “Really good!”, then you send a correct memory into your mind.

Situation: You make mistakes on problems because you do not solidly know addition facts or multiplication facts.  (Subtraction is the reverse of addition, and division is the reverse of multiplication.)  You want a reliable way to learn them.

• Memorizing basic math facts: You can deal easily with certain errors by memorizing basic math facts, such as multiplication tables.  You may groan because many students fear they cannot memorize easily.  But give me a chance as I explain what to do. 
• The laws of how our minds learn are clear: The more we practice, the better we get.  You can use that law simply and with few mistakes by following this procedure. 
• First, choose what you want to learn.  Make either flash cards or make two-column lists of questions and answers that you can cover up.  For example, put 8 x 7 on one side and the answer, 56, on the other.  (Flash cards are better because you can shuffle them and can discard ones that you have learned.) 
• Second, take just one and look at it and look away and ask the question and give the answer.  Practice until you’ve got it. 
• Take the second and do it by itself until you’ve got it.  Then go back to the first and repeat it.  Then the second, until you’ve got both of them perfect. 
• Then add a third and learn it alone.  Then do all three together until perfect.  Continue learning more math facts and testing yourself on all of the ones you’ve learned so far.  Keep adding them until it makes sense to quit. 
• Do it several days in a row.  (Purpose:  To benefit from the spaced learning effect.)  Continue until you have absolutely got them. 
• When you do real homework problems, if you notice you cannot recall a math fact used in a problem, shut your eyes and imagine seeing it on a flash card and see if that brings it back to memory.  If not, it’s a signal you need to review on it.

Situation:  You have bad habits in math; once you learned a certain skill wrongly and your “bad skill” interferes with learning the right thing.

• Unlearning errors in basic procedures: It takes three times as long, on the average, to start with a “bad skill”, to unlearn the poor responses and to relearn correct responses than it does to learn a skill correctly the first time. 
• Here’s how: Get a correct explanation of what to do and a worked example as an exemplar.  Study and memorize them.  Learn both the abstract general procedure and learn to do the exemplar problem perfectly. 
• Over a period of several weeks keep doing problems of that type, even redo the same problem correctly. 
• In the meantime, train yourself when you see that type of problem to pause and think of your exemplar problem.  Then slowly follow the exemplar’s procedure on the new problem.  The purpose of doing a pause and thinking of your exemplar is to interrupt your old bad habit from taking control.  (You will not always have to pause.  Once your new skill is fully established, you can go back to normal speeds.)
• Expect to need to go slowly at first, then to have a phase of going faster, with occasional times that the old behavior pops up, and then to be able to go faster and have very few or no intrusions of the old behavior. 

Situation:  You are reviewing for a test: The day before

• Practice recognizing the different types of problems that come from different sections of the course.  The reason to do recognition practice is because tests add a new task.  On a test you must pull knowledge from a wide range of topics.  In contrast, when you do a typical day’s homework, you just have to remember and use a narrow range of math concepts, symbols, formulas and procedures.  When taking a comprehensive test, you must learn to associate the stimulus of each problem to the responses of recalling what type of problem it is and recalling the knowledge of how to solve it.
• One method of recognition practice is to look at each page of problems you had to do and read typical problems and recall the methods you used.  Stimulus – response.
• A deeper method is to write down key problems on file cards and write on the other side the page or section where the problem is from, reminders of how to solve it, or the entire solution process.  Then shuffle the cards.  Look at the front of each card and try to recall how to solve it.  That method will bring together the material that you learned in separate lessons and train your mind to discriminate the different types of problems.
• It is helpful to test yourself on everything you review.  At this phase of your learning, test yourself by looking at information and letting it fade from your working memory for a minute or so by reading something else.  Then ask the question and give the answer and check whether you got it right.  If so, good.  If not, restudy and test again.  (Do not test yourself by looking and testing within 15 seconds while the information you just saw is fresh in your working memory.  Otherwise, you cannot be sure you can pull it out of memory cold.)  It is better still to prepare little tests the day before so that you check whether you can pull it from deep long-term memory.
• Test yourself on vocabulary and definitions, on symbols and their meanings.
• Test yourself on being able to recall and understand the explanations of how to do procedures.  Review any exemplar problems you have tried to memorize.
• It is dangerous to learn math without it making sense.  Notice if you are just memorizing formulas and the steps in the procedure without actually knowing why you are doing them.  The danger of rote memorizing comes when you forget small things, because then you would not be able to figure them, out on the spot during the test.  Instead, you want to study in such a way that you could figure out forgotten things.
• Test yourself on how to solve actual problems.  It is okay to solve again important problems that you have solved before.  You should actually think through the process of solving familiar problems in order to create a procedural memory that will help you on new problems.  Otherwise, if you rush through familiar problems, it will leave too weak an impression to help.

Situation:  You are reviewing for a test: The day of the test

• If possible, review shortly before the test the most confusing and troublesome material.  The purpose is to give you a recent contact with the material, because recency of contact is a major influence on memory.  Memory is better for recently contacted ideas.
• Review by testing yourself, so that you actually pull it from memory. 

Situation:  You are taking a math test: General advice

• Use general good test-taking strategies.  Use time effectively; don’t hurry; don’t waste time on a hard problem as long as there are other problems to do; read problems carefully (research on student errors shows that their hasty reading and misunderstanding lead to a large portion of errors); check your work.
• As yourself what type of problem you are dealing with.  Often just being able to see that a problem is in a category you have studied will speed up your work.
• Bring to mind similar problems you have seen or solved and use them as clues.  Recall the exemplar problems you have memorized when it would help.
• When you see a problem and you know you studied how to solve it and you need to jog your memory, think of what the book looked like in that area, think of the teacher’s notes on the board, think of what the room you studied in that day looked and sounded like.  These thoughts are associated with the math knowledge and will help you recall the needed math.
• If you cannot go directly to an answer, see if you can use the values in the problem to calculate other values.  This is called forward-chaining.  It may give you clues.

Situation:  You are taking a test: Stuck on a problem

• When stuck on a problem in a test after trying normal ways to recall information, it is important to do something different.  Move and breathe, as advised in the earlier section on getting stuck on homework problems.
• Search for “resources” in your memory.  Resources are knowledge that you can use.  Recall principles, explanations, worked examples, formulas, definitions, and procedures for tackling problems. 
• As you prepare to temporarily leave the problem, start thinking of other things associated with it when you learned the knowledge and skills in class, text, and homework.  The purpose is to start your mind making associations to what you know so that your unconscious mind may retrieve what you need to know.  (Even random associations may jog your memory.)  After doing other problems, check back on the problem.  When you check back, do something to make it look different: Use a new section of scratch paper.  Organize your solution efforts differently.
• Assess how much time you have.  If you haven’t done other problems, do them first before investing more time into the one you are stuck on.

Situation:  You do not have enough time to study adequately for a test

• Set priorities.  You will need to figure out what are the most important things to pick out and study and which are less important.  Your purpose is to prepare yourself as well as possible by selecting the most important topics.
• One strategy is to expect that the teacher will include something basic from every chapter and every major topic on the test.  Therefore, it might be useful to learn the basics on every topic and skip the complications.  You would actually memorize basic concepts and symbols and problem-solving procedures.  Make sure to actually learn them.
• Another strategy is to minimize the time you spend in solving homework problems and to compensate for your undeveloped skills by over-studying worked examples and by creating exemplar problems.  Then you would approach problems on the test by using your memory of the concepts and exemplars to figure out on the spot how to solve them.  However, try to solve at least one problem of each type.
• Do not study everything shallowly because then you will not have much chance of getting at least some problems clearly right.
• Do not study only one chapter or topic thoroughly and abandon whole chapters, because, in emergency studying, it is usually better to get some fundamentals on everything.  (Note: Different teachers have different approaches to setting priorities.  Ask.)
• Prepare yourself emotionally for accepting a lower grade.  Remind yourself that it is better to learn part of it than to not learn any at all.

Situation:  You need to deal with test anxiety

• See the Study Tips on test anxiety on the Lane Community College Testing Office’s web pages.  Check out 2011sitearchive.lanecc.edu/testing.  Go to the bottom of the menu page for the links.

Situation:  You must deal with low self-confidence that you can do a certain kind of math task

• Taking a math course challenges most people’s self-confidence at one time or another.  After all, you learn one thing and have to move on to something else new.  It’s detailed and hard; you have to solve problems.  People have to fight the fight of believing they can succeed.
• Our unconscious minds send us negative feelings.  We catastrophize; we feel disaster is ahead.
• Staying positive requires that we use our conscous minds to think positive.  Take deliberate charge of your thinking.  Here are some tips.
• Keep thinking over the knowledge and skills that you do have.  Give yourself credit for learning.  Keep saying, “I am learning.” “I have the ability to learn.”
• Remember that it is normal in math to have to learn something new almost every day and normal to make lots of mistakes.  When we learn new things, we revert to a slow learning mode, halting steps, and needing lots of time.  It’s normal; it happens to everybody except for a few high aptitude people.  Keep saying, “Just because I’m going slowly and making mistakes is not a situation that predicts I’m going to fail.”
• Set beginner’s goals.  Do not set goals and expectations for yourself that are too high.  You’ll feel better.
• Study worked problems; they are your friends.  Use them as models for solving homework problems.
• Remember that it is normal when taking math to feel strong feelings.  People feel frustrated, angry, and worried; they want to escape.  Do not confuse strong negative feelings with a sign that you are going to fail.  Say to yourself, “Yes, I hate math.  But I can still do it.”  “Yes, I’m angry at the teacher, the text, and myself.  But I can still learn math.”  “Yes, I’m worried that I’ll fail, but even worried people can still succeed.”

Situation:  You have to deal with wandering attention

• It is very rare for people to be able to focus their attention on something for a long time.  Our minds wander constantly.  When people’s attention wanders, they often pull it back to their math in the wrong way.  Here is how to handle a wandering mind.
• First, intend to pay attention.  The reason for setting the purpose intentionally is to help you remind yourself later that you really want to focus on your math.
• Second, when you notice your attention has wandered and you are thinking about something else, turn it back to your math.  Pull it back without criticizing yourself or feeling bad.  The reason is that our minds take what we are thinking about and start thinking about additional memories.  You may say to yourself something like this: “There I go again.  Idiot.  I have no self-discipline.”  Your nice obedient mind will start calling into your working memory some more associated thoughts about this bad tendency, memories of other times your mind wandered, and negative thoughts about failing.  It fills up your working memory with junk and slows down your work.  You want instead for your working memory to have space for math information and problem-solving goals.  So just quietly turn your mind back to the subject.  Be at peace.
• Third, return your mind back to a point a little before where you were when your mind wandered.  If reading, go back a sentence or so.  If solving a problem, go back a step or two.  The purpose is to help you reinstate in your working memory the information you were thinking about before your attention got diverted.
• That’s all.  One research team gave low-achieving students these directions for dealing with wandering attention and did nothing else.  Their grades rose significantly!

Situation:  You will be studying a lot of similar material after math studying

• When you study a lot of material in long study sessions, it is possible for the various chunks of information and skills to interfere with each other and lessen your memory.  Psychologists call it interference. 
• Interference is worse when people study a lot of similar material together.  It lessens if they break up the studying by doing something different between study sessions.
• Try to study in relatively short sessions.  For example, study math for 20 or 30 minutes and then read some English or History for a bit and then return to math.  It’s important to change to quite a different topic.  Research shows that people can study in long sessions with low interference as long as the topics are very different.
• Use spaced study.  Study the same topic several times over a period of several days.
• Use techniques that help you discriminate and compare concepts that look similar.  You have probably noticed that certain math concepts, symbols, procedures, and formulas seem alike.  Study by gathering them all together; look back and forth and notice how they are similar and how they are different.  That technique will lower interference.
• When you know you have to learn something hard, study it the last thing before you go to bed.  Sleeping means that no other knowledge will enter your mind and interfere with the last learned knowledge.  This is a serious research finding.

Situation: You sense that there are some weaknesses in the way your text or instructor has organized the material.

• Spaced practice: It is common that there is not enough spaced practice provided by text or teacher.  Instead, a different topic is given each day.  When that is the case, give yourself some periodic practice sessions on old topics and problems.  It will greatly increase your memory.
   
• Discrimination practice: Some textbooks do not provide enough practice provided in discriminating topics that look similar but are different.  You can handle this problem by looking at each new topic to see how the concept is unique, distinct, and its own self.  Keep thinking: “This is different.  How is it unique?”
   
• Generalization practice:  Some textbooks do not give enough practice in generalizing knowledge from instructional examples to far-flung cases that use the same principles.  It is important to be able to transfer your knowledge from the teaching examples to unusual problems that still use the same principles.  Often tests and science classes will use these extended applications of basic principles. 
   
• You can help handle this situation by trying to grasp the underlying principle and by looking at problems and getting in touch with the underlying them in them.  Deep analysis of the patterns under the surface helps you generalize.
   
• Ask your teacher if the procedure you are learning has far-flung applications.
   
• Isolated small chunks:  Sometimes texts teach topics in little bits and you cannot see the overall whole.  Some textbooks look well-organized but a reader ends up not being able to see the woods for the trees.  They are good at separating, but weak at bringing together.  It can hinder your ability to handle homework problems. 
   
• First, you can help by surveying chapters to pick up the main theme; try to see how each of the parts is linked to the others.
   
• Second, it is helpful to ask teachers to talk about the overall pattern, because they will see it clearly.
   
• Worked examples:  Sometimes texts give too few worked examples.  Sometimes texts do not explain fully why each step is there.  Since you will use worked examples as models to use when you work on homework problems, poor ones hinder your transfer to problems.
   
• You can help by doing self-explanations of worked examples.  Go line by line and explain to yourself what is new, why it is there, what it shows you.
   
• Look for three things in each step: (1) What the current situation in the problem is; (2) what the current goal is; and (3) what the right action or step to take is.
  

How Math is Different

Math is different from most other fields.  Many students find that using their ordinary methods of learning and studying does not produce the grades they can get in other subjects. 

It is different in these ways:

• Students have to both learn abstract principles and build up skills in following procedures, many more procedures than required by other fields.  Studying to build procedural skills is somewhat different than learning factual knowledge.
• Students need to learn math sequentially; the later skills build on earlier skills.  Students who haven’t learned earlier skills fully often have trouble learning later skills.
• Students practice by solving problems; they are tested by solving problems.  They will rarely be tested by essays or factual recognition items.  Learning to solve problems skillfully requires different techniques than the ones used in learning facts, theories, science and literature.
• Students must read math information that is presented in symbols.  The symbols are packed with information and require a different reading style than used in reading ordinary books, one that is very slow and careful.
• Students often find that math symbols and formulas are abstract and meaningless to them.  Different techniques are required to study abstract material and to make it meaningful.
• Students usually find that a lot of math is new to them; they don’t have a built-up body of knowledge to link math to.  In contrast, many students already know a lot of information that they can associate to material in literature, history, social science, and much of science.  The task of learning new material makes a lot of memory demands.

The Goals of Studying Math

How do you know when you are finished with work?  It depends what your goals are.  Since math is usually tested by giving students a series of problems to solve without the use of notes, students’ math goals are to learn the knowledge and skills to solve problems, without notes, and within the time limits.

Goal #1. To learn some ordinary knowledge and facts.  Some examples are multiplication facts, formulas, the meaning of math symbols, the logic of equations, how to factor equations, what logarithms are, what graphs mean.  One’s goal is to build ordinary memory for information.  This kind of memory can be built by ordinary review and memorization and associating the different chunks of information to each other.  Each chunk can be learned in a few minutes.

Goal #2.  To learn skill in procedures for using math knowledge to solve math problems.  One also needs to build up speed and accuracy in following them.  We cannot learn procedural knowledge by ordinary methods studying to learn information.  We must solve problem after problem by using the knowledge and procedures, and we get faster and faster, more and more accurate.  The time taken to achieve solid procedural learning can involve 40 or more repetitions of a procedure over a period of many days.

Goal #3. To learn how to approach problems as a category in themselves.  Students need to learn the normal problem-solving tricks.

Goal #4. To deal with the normal obstacles that occur: coping with gaps in knowledge, handling feelings of math anxiety and low self-confidence in one’s math ability; dealing with imperfect teaching by the textbook and teacher; handling the time demands; and reacting sensibly when stuck on a problem.

Basic Methods That Are Used for All Studying

When people study for the purpose of building knowledge, memory, and procedural skills in any subject, there are certain standard methods that people often use.  The brief descriptions below will remind you what they are.  For further details see this writer’s set of study tips or other books on studying and memory.

Overview

• Study each new chunk of knowledge until it is familiar and well-learned.
• Study the chunk of information or the procedure carefully and repeatedly until it is very familiar.  You want it to be clear and solid in your long-term memory so that you can recall it even when you haven’t thought about it for quite awhile.  (Psychologists call it “having a high base-level activation”.)
• Associate the new chunk firmly to other information  (make associative cues).
• Association: Whenever you learn some target chunk of information, also associate it to something else that you expect will be present at a time you need to recall the target.  Then practice until you can think of the associated thing and be reminded of the target.  By associating one item to a second one, you guarantee you can retrieve the studied item.  For example, it is common in math teaching to teach students to associate addition to problems presented in words, in numbers given horizontally, in numbers laid out vertically.  Those three associations should trigger memory for addition facts.
• Practice the knowledge or skill shortly before you will need it (within the hour before).
• Recency of contact: When you expect to need information or a skill, practice with it shortly before the task occurs.  It is known that the more recent a person’s contact with a bit of knowledge, the more activated it is and the more likely the person can recall it.  With math this fact suggests that you start a study session on new material by reviewing relevant older material.  When you are going to take a test, it suggests you review the harder material shortly before to make it fresh in your mind.
• Pay close attention. 
• Attention: When you think about information, pay close attention to it in order to give your memory image of it the strongest activation possible.  Don’t give it half your mind or else the learning will be weaker.  When you pay attention to an idea, your mind will automatically associate to other ideas it is associated with.  If you pay weak or divided attention, your mind will not associate as well to the related topics. 
• When your attention has wandered, return it to the material without self-criticism.
• Inevitably, your attention will wander.  That is normal.  Return your attention to the topic very gently without condemning yourself and without talking to yourself.  If you criticize yourself as stupid and weak for having distracted attention, then your obedient mind will start thinking of other reasons why you are stupid and weak and will put them into your working memory.  You don’t want that extra stuff in your working memory along with the topic you are studying.  Rather, you want as much as possible of your attention and working memory to contain the new information.
• Assemble very small bits of new material into chunks.  Break large amounts of new material into smaller chunks.  Work with chunks.
• Chunks: When you are studying new material, break it into small chunks of new information.  The first purpose is to fit it into your working memory, which is limited.  It can only hold a few chunks of information; and without review they will fade within a minute.  The second purpose is to organize information into sensible parts that are bite-size.  Don’t read too much material and then try to practice with it.
• While you can still remember new material in working memory (15-20 seconds), think about it and make associations to it and practice it immediately. 
• When you are building memory for brand new information, do your thinking about it and use it in practice within a few seconds of receiving the information.  If you don’t think immediately about it, it will fade from your working memory, and then you’ll need to look at it again, thus wasting time. 
• Practice recalling material that you have already partly learned after it has faded from working memory (1 to 2 minutes later).  This advice is opposite to the method you use on brand new unlearned information.
• Medium-learned information:  After you have successfully begun to learn new information and have it partly memorized, then you change your method.  You look at it and intentionally let enough time go by so that it has faded, and then try to recall it cold.
• To learn new factual material: ask yourself how it makes sense and give an answer.
   
• When you learn factual information, use the method of thinking of how it makes sense in terms of other information that you already know.  Example: The capitol city of Oregon is Salem.  Ask yourself: “Why does it make sense that Salem is the capitol?”  One answer might be that it is in the center of the populated Willamette Valley where pioneers first settled.. 
   
• To learn new procedures and build skill: study worked examples; break each step into the current goal, the current situation, and the right action; practice many problems over many days.
   
• When you learn procedural knowledge (for example, how to divide fractions), study an example of the procedure and then do a new example, new problem.  Study worked examples and generalize them to new problems that you solve yourself.  Break each step into three phases: the current subgoal, the current situation in the problem, and the right step to take.  You’ll repeat this 3-part pattern for every step in the procedure.  Solve many problems.
   
• Space your study of factual knowledge and your practice of skills over many sessions.
   
• Don’t cram study and practice into one long session because it is not nearly as effective as spaced learning.  This applies to learning meaningful and not-so-meaningful information like names and dates, to practice of cognitive skills like writing with correct grammar and using new math procedures, and physical skills like playing instruments or playing a sport.
   
• To learn material that is relatively meaningless to you (proper names, symbols, dates, arbitrary facts) you should use special methods and test yourself to make sure you’ve got it.
   
• Although it is comparatively easy to learn things that are already meaningful,  people typically find it hard to learn and recall such meaningless information as new symbols, proper names, historical facts and dates.  A lot of math also seems that way.  Here are several possible ways to cope with material that is not too meaningful:  Use mnemonic methods; prepare flash cards;  find ways to make it meaningful;  use even more repetition than usual; test yourself often; and use  spaced learning over a longer period of time.
   
• Translate verbal material into visual and spatial representations.  Also use representations that are kinesthetic, emotional, smell, taste, lists or sequences, and story formats.
   
• Use visual and spatial representations of material in order to get two results: (1) to make meaningless material more meaningful; and (2) to build associations to the new material that will help you recall it better later.  In math draw diagrams and graphs and look at them while also thinking of the general principles.  You can also use other representations.  Examples are kinesthetic (how your muscles feel in movements), emotional feelings, smell and taste, lists or the sequences that  things occur in, and story formats.
   
• Don’t read or work so fast that you outrun your mind’s natural working speed in giving you meanings.
   
• Our minds have a natural speed of working.  Don’t outrun it..  Some people read faster than their minds can figure out the meaning of what they are reading, and the result is poor understanding and poor memory.  People who can read normal fiction and non-fiction quickly will also try to read math at the same fast pace and run into trouble.  Because math is heavily symbolic and condensed, one needs to slow down and decipher each symbol and then put the whole thing together.  How can you tell whether you are reading at the right rate?  Notice whether everything is meaningful.  If it’s not, something is wrong.
   
• Correct misconceptions and bad habits.
   
• When you have inaccurate knowledge or misconceptions about material, then you must unlearn it and replace it with  new and accurate knowledge.  When you have learned a “bad skill” so that you do a procedure wrongly and have unfortunately done it so often that it is stamped in, you will need to explicitly unlearn it.  You cannot usually substitute new knowledge very quickly for well-learned old knowledge.  One expert found it can take people three times as long to unlearn and replace their bad skill habits as it took to learn a skill new from scratch.  In math it can show up as habitual mistakes.  Examples: You may think that 9 x 7 is 56 instead of 63;  or when subtracting you may forget to borrow when you subtract a bigger number from a smaller number in that column.  Since your old knowledge and skills will “fight” the new knowledge, be sure at first to practice very slowly, to pay attention, and to practice away from real life situations.

• Set specific goals for your learning.  Define the accuracy and speed of learning you want to attain today.  Define the amount of material you want some memory of today.
   
• After trying to learn and remember something, test yourself by trying to recall the knowledge or do the skill, notice your result (feedback), and compare the result to the goal you set.  Notice how well the result and your goal match (feedback).  If they do not match, study again and try again with a correction.  And so on.
   
• When setting goals and getting feedback, be sure to call to mind what you did so that you can compare what you did to the feedback.  It is easy to do when feedback comes instantly.  It is harder to do when you do homework and there follows a delay of a few days until the teacher grades it and turns it back.  After such delays look at the missed problem and retrace your mental steps so that you can label the errors as errors and so that your mind will remember to do what is right next time.  Do not passively look at feedback because then it cannot help you learn.
   
• Intentionally learn concepts in several ways.
   
• Our minds abstract “from specific experiences to general categorizations of the properties of that class of experiences” (Anderson, John R., Cognitive Psychology and Its Implications,  4^th ed.   New York: W. H. Freeman, 1995, p. 131).  We call these concepts.  We see many examples of dogs, apples, carrots, houses, fast motion, upward movement, and so on, and we stop treating them as unique experiences and start creating concepts of dogs, apples, carrots, etc.  In addition, people and books who teach us things will directly teach us concepts. 
   
• We do several things to identify and remember concepts: we use positive examples and negative examples, memorize exemplars and prototype examples, learn definitions, and follow procedures for measuring the values of a concept.
   
• Learn schemas and associate new information to useful schemas.
   
• Our concepts are often organized into larger sets of concepts regarding events and complex things.  These sets are often called schemas, scripts or frameworks.  These large sets of concepts include standard things that always are present and the varying things and events.  Knowledge of baseball schema would include knowing about batters and 3 strikes, 4 balls, hits, fouls balls, and so on.  It would include knowing that different hitters have different experiences.  Some other examples: knowledge of what happens during a typical visit to a restaurant; knowledge of how to use e-mail and the world-wide web; basic ideas of levels in biology from molecules, parts of cells, cells, tissues, and so on up to ecosystems; knowledge of football games; and understanding of the typical structure of stories and novels.  
   
• Teachers and tutors who can find good schemas, complex metaphors, and structured frameworks can greatly speed up and deepen their students’ learning.  In fact, one of the very few fast ways to learn is to turn on the relevant schemas while studying new information.

(file: Math Studying.  5/10/2004.  Daniel L. Hodges)