STOP DOING MATH LONG ENOUGH TO LEARN IT How to Study Math –Short Version Delano P. Wegener, Ph.D. Spring 2005

HOW TO STUDY MATHEMATICS The advice contained in this document will encourage you to study mathematics differently than you probably have in the past. Take my word for it, it works. This method has worked for tens of thousands of students. It will work for you, but some effort will be required.

HOW TO STUDY MATHEMATICS It is worth observing that this is the approach used by professional mathematicians. This observation is worth while not because you want to become a professional mathematician. It is worth while because professional mathematicians have been successful in their study of mathematics. It worked for them, and with some effort it will work for you. It is worth observing that this is the approach used by professional mathematicians. This observation is worth while not because you want to become a professional mathematician. It is worth while because professional mathematicians have been successful in their study of mathematics. It worked for them, and with some effort it will work for you.

HOW TO STUDY MATHEMATICS You should be honest with yourself. You probably do not have a long history of with mathematics. In fact you might have a long history of a lack of success with mathematics. Neither of these situations is a stigma. Simply be honest and acknowledge the fact. You should be honest with yourself. You probably do not have a long history of success with mathematics. In fact you might have a long history of a lack of success with mathematics. Neither of these situations is a stigma. Simply be honest and acknowledge the fact.

HOW TO STUDY MATHEMATICS Change your attitude about how mathematics is learned. First: Change your attitude about how mathematics is learned.   Mathematics is not learned by working problems.   Mathematics is learned in order to work problems.

HOW TO STUDY MATHEMATICS If you work 100 problems of a particular type, there is a good chance that you have learned to mimic a process which leads to a solution for that one problem type. It is unlikely that you will have learned any mathematical concepts. In fact I have witnessed many instances where the student inferred a totally incorrect set of reasons for the process. If you learn the concepts then you are able to solve this particular type of problem as well as many others.

HOW TO STUDY MATHEMATICS Change your frame of mind Second: Change your frame of mind think of studying mathematics think of studying mathematics rather than doing mathematics. rather than doing mathematics.

HOW TO STUDY MATHEMATICS Listen to the words used when students, teachers, parents, and the general public speak about mathematics. Compare those words and that language with the words and language used when other subjects are discussed.

HOW TO STUDY MATHEMATICS Students “study” history, social studies, language arts, etc. They read literature. They neither read nor study mathematics. Almost universally when mathematics is the topic of discussion you hear that students “do” mathematics.

HOW TO STUDY MATHEMATICS Teachers encourage their students to “do” their math every day to avoid falling behind. Parents check to insure that their children have “studied” spelling, “read” their literature, and “done” their math.

HOW TO STUDY MATHEMATICS Students take their queue from this language and routinely skip the reading material in a math book. They seldom read and certainly do not “study” the concepts presented. Students do precisely what is expected of them, they turn to the exercises and attempt to “do” their math.

HOW TO STUDY MATHEMATICS Doing rather than studying leads to:   lack of understanding   collection of unrelated rules to memorize   collection of tricks for solving nonsensical problems found only in mathematics textbooks.

HOW TO STUDY MATHEMATICS Third: Make up your mind to concentrate on concepts rather than activities. Third: Make up your mind to concentrate on concepts rather than manipulative activities.

HOW TO STUDY MATHEMATICS If you concentrate on manipulation, your focus is so narrow that you will never understand nor be able to use mathematics. If you concentrate on manipulation, your focus is so narrow that you will never understand nor be able to use mathematics.

HOW TO STUDY MATHEMATICS Fourth: Watch for and make use of concepts that remain the same when working with mathematical objects. Fourth: Watch for and make use of concepts that remain the same when working with different mathematical objects.

HOW TO STUDY MATHEMATICS This is difficult for the student to do because you cannot see similarities until you have studied quite a bit of. This is difficult for the student to do because you cannot see similarities until you have studied quite a bit of mathematics. Your instructor will point out those that remain the same and will be useful in future work. Your instructor will point out those concepts that remain the same and will be useful in future work.

HOW TO STUDY MATHEMATICS YOU ARE NOW READY FOR THE SIX STEPS FOR STUDYING MATHEMATICAL CONCEPTS

HOW TO STUDY MATHEMATICS Step 1: Memorize all definitions and special properties as soon as you encounter them. In the following days and weeks continue to go over these items until the statements that you have memorized are “your own words” for the concepts.

HOW TO STUDY MATHEMATICS As with any other language, you must memorize the meanings of in order to understand what you read, and what you hear. Imagine trying to read, write, speak, or listen if you do not understand any nouns or verbs. As with any other language, you must memorize the meanings of words in order to understand what you read, and what you hear. Imagine trying to read, write, speak, or listen if you do not understand any nouns or verbs. You must know the meanings of mathematical terms in order to communicate about mathematics. You must know the meanings of mathematical terms in order to communicate about mathematics.

HOW TO STUDY MATHEMATICS You must be able to: communicate in the discipline communicate in the discipline

HOW TO STUDY MATHEMATICS If you do not memorize definitions, you are trying to make sense of things that look like: (1) If the same ________ is ________ to (or ________ from) both sides of an ________ the resulting ________ will be ________ to the original ________.

HOW TO STUDY MATHEMATICS or: or: (2) If both sides of an ________are (or ________) by the same ________ ________, the resulting ________ is ________ to the original ________. (2) If both sides of an ________are ________ (or ________) by the same ________ ________, the resulting ________ is ________ to the original ________.

HOW TO STUDY MATHEMATICS or: The process to ________ a ________: Start with the original ________ (the one to be ________ ) and use the above two properties to ________ ________ ________, all ________ to the original ________, until we arrive at the ________ ________.

HOW TO STUDY MATHEMATICS Step 2: Read and study the explanations provided in text, class, or website.

HOW TO STUDY MATHEMATICS Be sure you understand the concepts being explained. Be sure you understand the concepts being explained. examples provided can help clarify some items. The examples provided can help clarify some items.

HOW TO STUDY MATHEMATICS When looking at an example, always ask yourself; When looking at an example, always ask yourself; What concepts are being illustrated? What concepts are being illustrated? If you cannot answer that question, the example is useless to you. If you cannot answer that question, the example is useless to you.

HOW TO STUDY MATHEMATICS Step 3: Work a few related exercises to determine if you have learned the concepts. If these exercises give you trouble, go back to the explanations and study some more. Then try some more exercises.

HOW TO STUDY MATHEMATICS If you are unable to work a textbook problem, it means either you do not understand a necessary concept or you were unable to recall the concept. Very rarely is your inability to solve a textbook problem due to a lack of manipulative skills.

HOW TO STUDY MATHEMATICS The primary purpose of exercises in textbooks is to provide the student with a means of demonstrating to himself that he has learned the concept.

HOW TO STUDY MATHEMATICS Step 4: If a concept simply eludes you, get help.   Ask for an explanation in class or   Get help during office hours.   See a tutor

HOW TO STUDY MATHEMATICS Individual Learning Preferences have long been recognized by educators and cognitive scientists. Obtaining assistance from another source may present the concept in a manner which makes it “learnable” by you.

HOW TO STUDY MATHEMATICS Step 5: ….. Step 5: When your work on a few exercises indicates that indeed you do understand the concept, you may want to work a few more (possibly more difficult) exercises to insure that your assessment of your understanding is valid, that the concept is available for recall, and that you are able to transfer the concept to new situations. …..

HOW TO STUDY MATHEMATICS Step 5: …. Sometimes you may want to work even more problems simply to improve your speed for dealing with that particular kind of problem.

HOW TO STUDY MATHEMATICS Step 6: Constantly review all memorized material as well as major concepts. At regular intervals of a week or two work a few exercises to enhance retention and ability to transfer.

HOW TO STUDY MATHEMATICS The need for Steps 5 and 6 is justified by the last three External Conditions of Learning as specified by Gagne [1]. These three conditions of learning address feedback, assessment, retention, and transfer.

HOW TO STUDY MATHEMATICS This concludes the short version of a seminar on How to Study Mathematics. Thank You for Your Attention