How to Study Math
Learn the rules and concepts of math at each level (what does it mean, not just saying words and writing/drawing symbols)., Accept learning new mathematical definitions (vocabulary) so they become natural to you., Try to work ahead in the...
Step-by-Step Guide
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Step 1: Learn the rules and concepts of math at each level (what does it mean
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. -
Step 2: not just saying words and writing/drawing symbols).
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. , 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". , 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. ,, Sometimes the more difficult problems will involve the example plus other things you should already know from previous sections or chapters. , 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. , 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. , 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. , 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... , 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. , -
Step 3: Accept learning new mathematical definitions (vocabulary) so they become natural to you.
-
Step 4: Try to work ahead in the assignments of math problems by your teacher.
-
Step 5: Have a plan of attack for each worded (story) problem.
-
Step 6: Draw multiple representations when possible to understand the math in diagrams
-
Step 7: sketches
-
Step 8: graphs of parts of the problem to be solved.
-
Step 9: If there is a sample/example problem for the section in your book
-
Step 10: work through it yourself
-
Step 11: and use the sample in your book to guide you on the simpler problems.
-
Step 12: Check your work
-
Step 13: as you go.
-
Step 14: Don't shrug and say "Oh
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Step 15: well."
-
Step 16: Complete your work in an orderly manner.
-
Step 17: Form a study group
-
Step 18: to work quietly
-
Step 19: not to chat together.
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Step 20: Classify preventing errors into 3 categories -- common sense
-
Step 21: psychological
-
Step 22: and math tips.Use common sense.
-
Step 23: Find math review books such as at Half-Price Books
-
Step 24: 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.
Detailed Guide
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.
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. , 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". , 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. ,, Sometimes the more difficult problems will involve the example plus other things you should already know from previous sections or chapters. , 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. , 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. , 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. , 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... , 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. ,
About the Author
Natalie Rivera
Writer and educator with a focus on practical organization knowledge.
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