Learn mathematics well: Difference between revisions

Latest revision as of 22:37, 27 December 2022

Key takeaways

If you had sufficient time and resources, a case may be made that you should learn all subjects in the curriculum well. But that's not possible in the real world. We still strongly recommend that you consider learning mathematics well, in addition to the subjects that fascinate you.
In general, a number of people do not learn math as well as their potential. Not everybody can become a genius at mathematics, but most people (including you) can get a lot better than they currently are. A lot of people mistakenly believe, or affirm, statements of the form "I'm not a math person" and these prevent them from achieving their potential.
If the resources at your school are not helpful (for instance, you have a bad teacher or peer learning environment) there are still many ways for you to learn math well. Some strategies are described and linked on this page.
Learning mathematics conceptually is hardest, but it generates greater benefits than learning mathematical techniques through rote memorization and practice. However, even the latter may be quite helpful to you relative to not learning the subject at all. If you are thorough with the computational processes, it would make it easier for you to learn math at a conceptual level later when you have access to quality teachers and resources.

Why learn math?

Further information: Learning mathematics: benefits

The following are some general reasons for learning mathematics well.

Hierarchical: Later material in the curriculum depends heavily on earlier material (with a few important exceptions, such as many parts of geometry, plus advanced techniques within individual sections).
Unidirectional transfer to many other domains: Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra, is particularly important for understanding statistics rigorously.
Importance for college gateway examinations (such as SAT and ACT): A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.
General reasoning ability: Mathematics, if understood well, can help with general reasoning.

Psychological barriers to learning math well

Many people experience mathematical anxiety and believe that they're not well-suited to doing mathematics well, independently of their skill level. This belief can hinder their capacity to develop mathematical mastery.

It is certainly the case that some people have stronger cognitive skills and can learn mathematics faster. However, this does not mean that other people will be unable to learn math. There are many strategies for people to overcome limitations with working memory or processing speed in order to learn math well.
People often have the impression that others who seem to grasp a new idea conceptually and deeply somehow do so naturally and effortlessly. However, the people who have a strong mathematical intuition generally spend a lot of time thinking about math in general in order to develop that intuition. You may not want to reach that level yourself if math does not fascinate you enough to put in the effort. But you can reap good rewards by moving somewhat in the direction.

How can one learn math well?

There are some general good study habits that help with learning any subject well. There are also some general online mathematics learning resources. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.

Supplementing rote and immediate practice

In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject.

There are several problems with this type of learning:

The practice is being done immediately after reading the procedure and looking at the worked examples. This means that the student may be learning the material using his/her short term memory rather than long term memory. He/she may well forget the technique and be unable to do problems of the same type just a few days later.
The textbook from which they are obtaining the problems may have deliberately designed the worked problem, and other problems to match each other very well. So students who learn in this way may be thrown off by similar but slightly different problems that they encounter later .
The student may not obtain any conceptual understanding of the material, so he/she may be completely unable to use the ideas behind the procedure in different contexts, or reconstruct the procedure in the event that he/she forgets it.

The following approaches can help you avoid these pitfalls:

Maintain some spacing being doing a worked problem (where you follow indicated steps) and doing practice problems. In some cases, you may prefer to do one practice problem immediately after the worked problem, and a few more after a few days. This is a special case of the more general strategy of spaced repetition.
When you attempt practice problems a few days after doing the worked problem, pay particular attention to the parts that you are stuck at. These are probably the parts that you have conceptual trouble with. Do not just open up the worked problem again without making a serious effort.
Attempt practice problems that are framed somewhat differently from the original worked problem. Most books have large numbers of exercises to choose from, and good teachers generally give a varied mix of problems, but pay conscious attention to this issue if you are studying by yourself, or if your teacher isn't giving you a varied mix.
Wherever possible, try to acquire the conceptual intuition behind the steps. The texts or resources you use often do offer conceptual explanations, but these may not be highlighted because the authors expect that most students just want to master the algorithms and do not care about the explanations.

Online mathematics learning resources

Further information: Online mathematics learning resources

Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.

Resource	Link to website	Videos explaining the content?	Text explanations of concepts?	Practice problems sorted by area?	Dependencies between topics explicitly codified?
ALEKS	www.aleks.com	No	No	Yes	Yes, done at a considerable level of detail using knowledge spaces.
Khan Academy	khanacademy.org	Yes; comprehensive coverage but not high quality	No	Yes	Yes, but this is a new feature and may not be very thorough.
Brilliant	brilliant.org	No	No (with a few exceptions)	Yes	No (though they may be doing this soon)
PatrickJMT	patrickjmt.com	Yes	No	No	No
Edia	edia.app	Yes (as of January 2021; the videos are sourced from third parties)	Yes (text explanation of each problem)	Yes	No (as of January 2021)

Math reading recommendations

See our math reading recommendations page.

@@ Line 7: / Line 7: @@
 ==Why learn math?==
+{{further|[[Learning mathematics: benefits]]}}
 The following are some general reasons for learning mathematics well.
 # '''Hierarchical''': Later material in the curriculum depends heavily on earlier material (with a few important exceptions, such as many parts of geometry, plus advanced techniques within individual sections).
-# '''Unidirectional transfer to many other domains''': Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra , is particularly important for understanding statistics rigorously.
+# '''Unidirectional transfer to many other domains''': Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra, is particularly important for understanding statistics rigorously.
 # '''Importance for college gateway examinations (such as SAT and ACT)''': A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.
 # '''General reasoning ability''': Mathematics, if understood well, can help with general reasoning.
-===Hierarchical nature of the curriculum===
+==Psychological barriers to learning math well==
-Later material in the mathematics curriculum relies heavily on earlier material, both at a computational and a conceptual level:
+Many people experience [http://en.wikipedia.org/wiki/Mathematical_anxiety mathematical anxiety] and believe that they're not well-suited to doing mathematics well, independently of their skill level. This belief can hinder their capacity to develop mathematical mastery.
-* At a computational level, techniques for solving problems for later parts of the curriculum typically involve reducing those problems to problem types seen earlier in the curriculum. For instance, techniques for solving equations in high school algebra rely on basic algebraic manipulation and arithmetic computation. Techniques in calculus (such as finding local maxima and minima) rely on solving algebraic equations.
+* It is certainly the case that some people have stronger cognitive skills and can learn mathematics faster. However, this does not mean that other people will be unable to learn math. There are many strategies for people to overcome limitations with working memory or processing speed in order to learn math well.
-* At a conceptual level, it is hard to understand ideas in later parts of the curriculum if one does not understand earlier ideas well. For instance, it is difficult to understand how differentiation without understanding limits. It is difficult to understand any of calculus without understanding functions and graphs of functions. It is difficult to understand functions without understanding basic algebraic notation. It is difficult to understand basic algebraic notation without familiarity with arithmetic.
+* People often have the impression that others who seem to grasp a new idea conceptually and deeply somehow do so naturally and effortlessly. However, the people who have a strong mathematical intuition generally spend a lot of time thinking about math in general in order to develop that intuition. You may not want to reach that level yourself if math does not fascinate you enough to put in the effort. But you can reap good rewards by moving somewhat in the direction.
-* Even the techniques in a part of a mathematics curriculum that do ''not'' get used later may be helpful for [[overlearning]] important techniques that are a part of the curriculum and get used a lot in later parts of the curriculum. For instance, the long division algorithm helps review key skills of subtraction, multiplication, number sense, estimation, and place value, and may be useful to master even if you never do long division by hand.
-One exception to the generally hierarchical nature of the curriculum is high school geometry. A bare minimum of geometry, including lines, circles, and angles, is essential for understanding coordinate geometry. However, many aspects of geometry, such as geometric facts about triangles and circles, are not used in the rest of mathematics. Geometry may be worthwhile as a way of improving one's general reasoning and proof discovery skills, but it is not the only way to develop such skills, and it is not ''specifically'' important to master material about triangles and circles. That said, many people may deeply enjoy learning geometry and it may ignite their passion for mathematics, so this should not be construed as a recommendation against learning geometry. The point here is simply that the general arguments for learning math well do not apply with the same force to geometry.
+==How can one learn math well?==
-Note that these arguments do ''not'' apply with the same force to other subjects such as physics, chemistry, economics, history, biology, or English literature. In most other subjects, there is a relatively small base of foundational knowledge and skills, and a large variety of other topics within the subject that build on those. The foundational base is important to understand, but dependencies between later topics are minimal.
+There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online mathematics learning resources]]. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.
-===Unidirectional transfer to many other domains===
+===Supplementing rote and immediate practice===
-Concepts as well as computational skills learned in math classes are used in other subjects. The usefulness of math for other subjects is greater than the usefulness of other subjects for each other, or the usefulness of other subjects for math.
+In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject.
-* '''Usefulness for Advanced Placement and college prep coursework''': Advanced Placement classes (or equivalents) in physics, chemistry, and economics rely on basic knowledge of the precalculus math curriculum. Some parts of these classes benefit from a knowledge of calculus. It is ''possible'' to do well in these classes without knowing the math, but doing so may require considerably more rote memorization. Math knowledge can also help with SAT subject tests in these subjects.
+There are several problems with this type of learning:
-* '''Usefulness for undergraduate coursework''': The mathematics curriculum leading up to calculus, as well as multivariable calculus and linear algebra, is useful for physics, chemistry, biology, economics, and other natural and social sciences at the beginning and intermediate undergraduate level. More advanced mathematics is useful for these subjects at the graduate level. This means that whether you want to major in the natural or the social sciences, taking mathematics is useful. Even humanities majors do benefit by mastering some college-level social science material, and knowledge of mathematics can help with such mastery.
-* '''Usefulness via statistics''': The mathematics curriculum leading up to calculus, as well as multivariable calculus and linear algebra, is crucial for a deep understanding of statistics, which in turn is widely used in the natural and social sciences.
-* '''Usefulness in programming''': Although advanced mathematical concepts are not directly used in programming, mathematical thinking is closely related to the type of thinking needed for programming. Some mathematical concepts that have closely related concepts in programming include basic algebra (the use of variables for unknowns), functions, piecewise definitions of functions, indexed summations, and logic (use of ''and'', ''or'', and quantifiers). Learning the concepts in mathematics is neither necessary nor optimal for learning them in the programming context, but this does count as an additional advantage.
-* '''Usefulness for graduate coursework''': Graduate coursework in most of physics and in some parts of chemistry, economics, and biology require even more math than the above. In particular, parts of mathematics that are widely used in other subjects include differential equations, differential geometry, linear programming and optimization, etc.
-* '''Usefulness in the real world''': Functions and their graphical representation are crucial for understanding real-world data. Computational formulas are rarely used, but a conceptual understanding of derivatives and integrals can occasionally be useful in data analysis. The main utility of mathematics in the real world may arise from its role in giving a number sense and basic intuition for statistics.
-Another related point is that topics in mathematics are often taught in a manner that is more general than equivalent topics in other subjects, so it is easier for students to transfer learning from mathematics to other subjects. There is some [http://reasoninglab.psych.ucla.edu/KH%20pdfs/Bassok_Holyoak%5B1%5D.1989.pdf research] that demonstrates greater transfer of learning from the algebra of arithmetic progressions to the physics of constant acceleration than in the opposite direction.
+* The practice is being done immediately after reading the procedure and looking at the worked examples. This means that the student may be learning the material using his/her short term memory rather than long term memory. He/she may well forget the technique and be unable to do problems of the same type just a few days later.
+* The textbook from which they are obtaining the problems may have deliberately designed the worked problem, and other problems to match each other very well. So students who learn in this way may be thrown off by similar but slightly different problems that they encounter later .
+* The student may not obtain any conceptual understanding of the material, so he/she may be completely unable to use the ideas behind the procedure in different contexts, or reconstruct the procedure in the event that he/she forgets it.
-==How can one learn math well?==
+The following approaches can help you avoid these pitfalls:
-There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online learning resources]] that help with all subjects. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.
+* Maintain some spacing being doing a worked problem (where you follow indicated steps) and doing practice problems. In some cases, you may prefer to do one practice problem immediately after the worked problem, and a few more after a few days. This is a special case of the more general strategy of [[spaced repetition]].
+* When you attempt practice problems a few days after doing the worked problem, pay particular attention to the parts that you are stuck at. These are probably the parts that you have conceptual trouble with. ''Do not'' just open up the worked problem again without making a serious effort.
+* Attempt practice problems that are framed somewhat differently from the original worked problem. Most books have large numbers of exercises to choose from, and good teachers generally give a varied mix of problems, but pay conscious attention to this issue if you are studying by yourself, or if your teacher isn't giving you a varied mix.
+* Wherever possible, try to acquire the conceptual intuition behind the steps. The texts or resources you use often do offer conceptual explanations, but these may not be highlighted because the authors expect that most students just want to master the algorithms and do not care about the explanations.
-===Rote and immediate practice===
+===Online mathematics learning resources===
-In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject. "Boring, but straightforward."
+{{further|[[Online mathematics learning resources]]}}
-There are several problems with this type of learning:
+Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.
-* The practice is being done immediately after reading the procedure and looking at the worked examples. This means that the student may be learning the material using his/her short term memory rather than long term memory. He/she may well forget the technique a few days later and blank out on equivalent problems.
+{| class="sortable" border="1"
-* The textbook or other source from which they are obtaining the problems may have deliberately designed the worked problem, and other problems to match each other very well. Students may be thrown off by problems that are even somewhat different numerically, even if the same algorithm works in principle.
+! Resource !! Link to website !! Videos explaining the content? !! Text explanations of concepts? || Practice problems sorted by area? !! Dependencies between topics explicitly codified?
-* The student may not obtain any conceptual understanding of the material, so he/she may be completely unable to modify the algorithm for a slightly different situation, or even to fill in gaps in the algorithm in case he/she forgets it.
+|-
+| [[ALEKS]] || [http://www.aleks.com www.aleks.com] || No || No || Yes || Yes, done at a considerable level of detail using knowledge spaces.
+|-
+| [[Khan Academy]] || [http://www.khanacademy.org khanacademy.org] || Yes; comprehensive coverage but not high quality || No || Yes || Yes, but this is a new feature and may not be very thorough.
+|-
+| [[Brilliant]] || [http://www.brilliant.org brilliant.org] || No || No (with a few exceptions) || Yes || No (though they may be doing this soon)
+|-
+| [[PatrickJMT]] || [http://www.patrickjmt.com patrickjmt.com] || Yes || No || No || No
+|-
+| [[Edia]] || [https://edia.app/ edia.app] || Yes (as of January 2021; the videos are sourced from third parties) || Yes (text explanation of each problem) || Yes || No (as of January 2021)
+|}
-The following approaches help tackle these problems:
+===Math reading recommendations===
-* Maintain some spacing being doing a worked problem (where you follow indicated steps) and doing practice problems. In some cases, you may prefer to do one practice problem immediately after the worked problem, and a few more after a few days. This is a special case of the more general strategy of [[spaced repetition]].
+See our [[math reading recommendations]] page.
-* When you attempt practice problems a few days after doing the worked problem, pay particular attention to the parts that you are stuck at. These are probably the parts that you have conceptual trouble with. ''Do not'' just open up the worked problem again without making a serious effort.
-* Attempt practice problems that are framed somewhat differently from the original worked problem. Most books have large numbers of exercises to choose from, and good teachers generally give a varied mix of problems, but pay conscious attention to this issue if you are studying by yourself, or if your teacher isn't giving you a varied mix.
-* Wherever possible, try to acquire the conceptual intuition behind the steps. This does not necessarily mean obtaining a ''proof'' for why the algorithm works (though a proof would certainly be nice, it may not be worth the effort to master). There may be other ways, such as trying variations of the algorithm and understanding where they fail. The texts or resources you use often do have such explanations, but they may not be highlighted because the authors expect that most people just want to master the algorithms and do not care about the explanations.