Study Math

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Successfully studying and reviewing for math classes or testing requires your dedicated work and practice, to achieve understanding. Calculators and shortcuts can help you, but only if you use them properly -- as when you are show your work in a certain way, as required, not just writing the answer.

Steps

  1. Learn the rules and concepts of math at each level (what does it mean, not just saying words and writing/drawing symbols). Don't just use the facts of math. Learn them deeply (both backward and forward, where possible and logical). That means do not be lazy, such as counting on your fingers. Memorize the rules and facts of math at your level. (So for addition, subtraction, multiplication, and division tables, forms and formulas, know them instantly, not working them out each time.) Not bothering to really absorb/learn such facts, applications and meanings properly will make more advanced math difficult or impossible, until you go back and learn the basics.
  2. Accept learning new mathematical definitions (vocabulary) so they become natural to you. Accept the math. "Agree with math. Love it." Math is very much like a new language, at times. So, you have to study it as a special kind of language and make sure it becomes part of your basic language, like common words. Study examples in your math text. Have your teacher explain any words or concepts you don't understand. Give them time to become clear. Even if your current teacher doesn't use the terminology often, you can be certain that other teachers will in other math courses.
    • You probably know that a number can be Squared or used in a Square Root, Cubed or Cube Root. Numbers can be expressed as algebraic "Terms" (oddly, that use of "term" means numbers that are added, or subtracted). Numbers can be Factored, expressed as Factors. Knowing the definitions, postulates, and theorems involving such terms will make solving and understanding many problems possible. Learn the concepts and words "deeply" when they come up in your course. Don't just wait for them to go away and hope you won't see them again. You probably will, year after year.
      • Eventually, you may need to use "factors" in the process called "Factorial" where the "!", the exclamation sign, is read as "factorial". So n!, or like 4! = 4x3x2x1 = 24 by multiplying which means "4 factorial" is 24. So 5x4! is 5x24=120; so that means 5!=120, you see? This is used in statistics, for finding the number of "combinations" of things. One kind of combining is called "permutations" into different sequences or orders, etc.
  3. Try to work ahead in the assignments of math problems by your teacher. This may seem like extra work, but it will be an advantage.
    • Work some of the problems (odd and even) from your textbook before they are assigned.
    • Some teachers always assign the even questions, if those answer are not in the back of the book, so the students can't just write the answers. Some teachers always assign some numbered odd questions, so the students can check their own work -- since the odd numbers are answered/solved in the back of many texts. Some teachers assign one set of problems for homework, and use some of the others for tests!
    • Ask the teacher for help for any problems troubling you, even problems that were not assigned. Remember, you are trying to learn. Problems that are unassigned often end up on tests. And the extra difficult problems give you a chance to earn extra credit.
    • When the teacher discusses the subject (probably before assigning the problems), ask questions that may occur to you (because of the work you have already done.) This is the most useful aspect of doing the work early: All of the other students are thinking, "Huh?" while you are thinking about a specific question you need answered.
    • Some college mathematical professors teach their classes entirely by answering questions from the students. The students are expected to come to class having completed some of the work.
    • Another benefit of completing work early (including extra work) is that if you need to turn something in late, your teacher will know that you are not trying to take advantage of his/her goodwill and will "give you a break".
  4. Have a plan of attack for each worded (story) problem. Decide what the problem really means and what kinds of operation(s) and steps will be needed. Don't just start adding, subtracting, multiplying, dividing or other operations without deciding what is needed.
  5. Draw multiple representations when possible to understand the math in diagrams, sketches, graphs of parts of the problem to be solved.
  6. If there is a sample/example problem for the section in your book, work through it yourself, and use the sample in your book to guide you on the simpler problems. Sometimes the more difficult problems will involve the example plus other things you should already know from previous sections or chapters.
  7. Check your work, as you go. Identify your errors early. Finding an error after finishing may seem discouraging. Still, discovering your own errors is the best way not to repeat them.
  8. Don't shrug and say "Oh, well." Don't just go on in your course or book without continuing to learn the material thoroughly, completely. Math builds upon itself.
    • A math course or book is like a novel, because it doesn't make sense unless you start at the beginning.
  9. Complete your work in an orderly manner. Endeavor to make all of your writing, numbers and symbols look the same way every time. The more complex the math, the more neatness counts.
  10. Form a study group, to work quietly, not to chat together. Be deep. When one person in the group has a problem, others can help. But, don't depend on others too much. Do your in-depth studying and thinking...
  11. Classify preventing errors into 3 categories -- common sense, psychological, and math tips.[1]
    • Use common sense. Avoid writing your "5 or 8" like a “6" or "3”. Do them clearly. For Algebra, write letter "l" (such as l for length) in a cursive form, to prevent confusion with 1 or i. Also, many people write z and 7 with a line in the middle, like "Ƶ" to avoid confusion with "2", or "7" to show it is not a "1" .
      • Leave time for checking. Speed requires practice and confidence, but leaving some time, about 5-10 minutes to check the paper is a good strategy. Your checking may spot obvious errors, like leaving out an entire question.
    • Think of psychology in scoring: If the question is worth 10 points, but your solution is very easy and short, something may have been done wrongly. Or if the question is worth only 1 point, but it took a long time to solve -- that may set off your alarm.
      • See if a final answer should be a “nice number“. If questions that are about whole objects, like the number of books, the answer should be a whole number, even if by rounding off. Then questions that require answers in 3 significant figures, may or may not have such “nice numbers”.
    • Try mathematical methods of verifying whether your answer seems "reasonable" or fit the kind of problem, to be sure of understandings:
      • Substitute back your final answer into the equations. For example, when solving "simultaneous equations" like x+y=3, x+2y=4, after getting the solution x=2, y=1, you should substitute back into the original two equations to check it. So, do x=2 and y=1 work for both equations. We substitute 2 for x and 1 for y. So then, x+y=3 gives us 2+1=3, true, and then x+2y=4 gives 2+2(1)=?; we get 2+2=4, true (here 2(1) is used to mean 2 times 1).
      • Substitute in nice possible values in an algebraic expression. For example, after finding your solution, you could substitute in a certain value for x, like x= 1, 2 or 10 or -1 or -2, etc. Then check whether both the left-hand (LHS) and right-hand sides (RHS) give the same value. (LHS=1/3, RHS=1/2-1/6=1/3). This indicates a high chance that your work is likely to be correct.
  12. Find math review books such as at Half-Price Books, or Amazon and such as suggestions online for math books for your or your child's exam reviews -- and look at the comments from actual customers on Amazon books for example about standardized testing as the Singapore Math Exams.

Video

Tips

  • Do not think it is so difficult. Take it easy. If you are stuck on a problem, get up and go play then do the problem again. Get active, unless you are school, then you ask your teacher.
  • Don't worry too much, if you feel like you don't understand something immediately. This might be a logical thing, as you now must stretch your thinking. You might have to "live with a problem", or a step in a proof, or even a definition, for a few days or a week before your mind absorbs it and grasps it. If it is confusing, stop working on it temporarily; do something else, and go back to it later. "Stay with it", but in timed intervals. Look at other descriptions or similar problems, perhaps in other books for ideas.
  • The best thing you can do when studying unfamiliar math is to leave a paper trail in steps. Take the extra time to make your handwriting look readable and orderly, and when you're having trouble, try not to skip steps. If you really want your math to look well done, try using a ruler (straight edge), for your straight lines and fraction bars. It's much easier to study, read or retrace your own work when it is written out clearly enough for you and others to easy read it.
  • Make songs or rap-poems about your math in your mind. Sing them. If your songs have tunes, then your math will have tunes. Some people love to remember such songs, and never forget them. Rap has rhythms and can be memorable in a similar manner.
  • If you're learning an odd theorem or property, make up and write down some examples that satisfy the conditions, and see how you can find the results yourself. This will allow you to get the properties "under your fingers." This is especially useful in abstract algebra and number theory.
  • Math is not just on paper. Use all of your senses and sensations to learn math and fix it in your mind. Write things out (uses visual brain/"photographic memory", and kinesthetic brain/muscle memory). Reading and saying definitions and theorems out loud or slowly and clearly in your mind, uses your auditory brain/memory of sounds.
  • If you are fuzzy about something that the book expects you to know before you read it, and the book does not review it, or review it in enough depth so that you are not fuzzy, you should look that up, and learn it first.

Warnings

  • Don't be afraid to ask for help from someone who has been doing math for a long time or is further along.
  • Do not push yourself too far, to fast (take breaks, look around, speak to someone), on math; if you do not relax some, your mind will be confused.
  • Don't be too hard on yourself. Realize that many have struggled with the same areas you are learning. Some people just take longer to understand math. Eventually, with enough, perseverance, you can succeed in math.
  • Do not let anyone in the group be lazy as to COPY OR CHEAT! Copying will be discovered eventually, one way or another. Besides, why would you wish someone else to get credit for your work?

Things You'll Need

  • Notebooks
  • Pencils
  • Math book
  • Math reviews -- optional (supplementary books)
  • Blue and black pen
  • Ruler
  • Eraser
  • Protractor and compass (for geometry)

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Sources and Citations

  1. ↑ *"Avoid Careless Mistakes in Math"
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