You are going to pass all of your mathematics tests.
You are going to pass because you will believe in yourself, because you will know the mathematics, and because you will have practiced enough of the problems so that the examination will look familiar and not frightening.
Think about some long-range goals, beyond the mere passing of tests, which are within your reach with some additional effort:
You can learn the mathematics well enough so that you will think of yourself as mathematically able. Here are some things you can do so that the effort you put into passing your tests will reward you far beyond the grades you get.
Reflect on Your Work
After each problem you do, take a few seconds to think about the problem. What mathematical concepts were involved? Which part of the problem gave you difficulty? What question about the ideas behind the problem could you ask a teacher about?
Reflection is thinking about what you have done, what you have heard, what has happened to you so that you can learn from the experience. Some people reflect on each day in their lives because they want to avoid repeating avoidable mistakes. They want to continue doing the things they did which worked well for them.
Most students rarely or never reflect on what they learn at school. Taking the few seconds needed to ask, "What was the main thing I must remember from that class?" or "from that chapter?" or "from that discussion?" helps your brain to process an experience. You will be astounded at the way reflection helps you to remember what you have heard or have done.
Few students or teachers have reflected on the major concepts of their subject. What are the major concepts of mathematics? Which of these concepts applies to the current problem (or the current lesson, or the current topic)?
For your guidance, here are a few of the concepts central to mathematics:
1. Sameness and equivalence
2. Evidence and certainty
3. Measure and measurement
4. Symbols and meaning
5. Characteristics and representation of data
6. Symmetry
7. Relations and functions
8. Invariance
9. Operations
10. Inference
11. Mathematical systems and models
This is not a complete list, but you might want to reflect on where topics fit on this partial list. Probability? Probability is a measure (concept 3) of events; the study of probability is related closely to the study of length, area, and volume. Statistics? The mean, median, mode, range, and dozens of other measures (concept 3) are characteristics of data (concept 5).
Graphs and charts are also representations of data (concept 5).
Slope is a measure (concept 3) of a line and an invariant (concept 85 in a set of parallel lines.
When you solve an equation, how are you certain (concept 2) that you have all the solutions? When you solve a pair of simultaneous equations by addition and subtraction, how do you infer (concept 10) that the solution of the new set of equations is the same as the solution of the set you were given?
The act of thinking about questions such as these is a powerful mode of study. It makes connections, fixes ideas in your long-term memory (the memory that lingers after tomorrow) and it puts you in charge of what you are trying to learn.
Keep a Journal
A simple route to productive reflection is to keep a brief daily mathematics journal with four key elements:
1. The topic of the homework assignment or lesson and the date.
2. The major mathematical concept(s) involved in the assignment or lesson.
3. One question about the lesson or assignment which you would like to have answered to improve your-understanding.
4. One application of the lesson or assignment to another subject, topic in mathematics, life situation, or problem which you made up.
If you can discipline yourself to keep such a journal for two weeks, compare your grades on the material covered during those weeks to your usual grades. You will be pleasantly amazed. You may decide that the journal, requiring perhaps five minutes of time each day, is the best investment of time you have ever made to improve your grades and your retention of schoolwork.
Practice New Ideas by Making Up Problems and Solving Them
When you try to make up problems on the current assignment, you might want to begin simply by taking a problem in the book that you have solved successfully and changing the numbers, operations, or variables. The new problem is not very different from the one you solved; nevertheless, it is a start. Soon you'll be making up more powerful problems.
A far richer technique is easily available with any verbal problem. Copy down the data; omit the question. Explore several things you might be able to deduce from the data given. Write one or two questions based on the data which you could answer. You will be amazed once again. Often you will figure out for yourself what the question in the problem actually was! On occasion you will have come up with a better question than the one in the book. The resulting problem is yours in a very meaningful way. You are not only reflecting on the problem; you are turning it like a gem in the sunlight to see hidden brilliancy. You are creating mathematics.
Before a test, try to make up five to ten problems that would worry you if you found them on the actual test. Try to solve those problems. If you cannot solve them yourself, enlist a friend, family member, or teacher to help you. If the problem is solvable, reflect on why you had difficulty with it. Was it unfamiliarity with the topic, with a skill related to the topic, with the underlying mathematical concept, or with the thinking skills used in analyzing the problem?
If the problem cannot be solved by you, your friends, or your teacher, you might consider what it is that made this problem so different from those in this book. You might want to investigate problems that cannot be solved at all. Consider the following: find a rational number which is two more than itself. Such a number would satisfy the equation
x = x + 2.
That could happen if and only if 0 = 2. Thus, there is no rational number which is two more than itself.
Some of the most successful students in high school and college try to predict the questions that will appear on examinations and to answer those questions. They reflect on the way the writer of the test might be thinking. Whether or not they predict the questions correctly, they are studying for the examination in an effective way which helps them to remember their reading and thinking for a considerable period of time.
Why not try it yourself?
Copyright © 1999 by Kaplan Educational Centers.
