Concepts in Mathematics
By David Alderoty © 2015
Over 2540 Words
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Studying and Learning Mathematics, and Related Concepts
Studying and Learning Mathematics
There are four important strategies for understanding, studying, and learning mathematics, which are as follows:
1) Using the proper strategy for Reading and Comprehending Mathematics
2) Adequate Preparation for Mathematics Courses
3) Mathematics is a skill that requires Practice, Practice, and More Practice
4) Developing an understanding that Mathematics is NOT a Set of Rules, it is a Set of Definitions, Postulates, Theorems, Formulas, and Techniques, Based on LOGIC
Without the four strategies mentioned above, most students will have difficulties with mathematics courses, especially at the college and graduate level. These strategies are discussed in the following four subsections.
Reading and Comprehending Mathematics
Mathematical material cannot be read, comprehended, memorized, or mastered in the same way that other subjects can be learned. One page of mathematics may require more time and effort to master than one or two chapters of conventional course work, such as sociology, history, or English literature. With most course material, you are not practicing an actual skill that requires complex sets of logical steps, unless the course is based on mathematics.
When students initially fail to master a topic in other subjects, they can usually learn the material by reading a textbook. However, this is not the case with mathematics. Most textbooks on mathematics are written with the assumption that the student will learn the material step-by-step from an instructor.
The only practical way of learning and comprehending mathematics is with extensive practice, COUPLED with the assistance of instructors, tutors, and/or videos. This can be reinforced with other techniques, such as computerized tutorials, and trial and error learning, and studying text-based material on mathematics. Ideally, all of the above should be incorporated into your personal study plans, to optimize your success in mathematics.
When students do not understand the ideas presented above, they may erroneously conclude that they do not have an inborn ability to master mathematics. This notion might be easily reinforced if the student failed a mathematics course, or even if he or she quickly read a couple of pages from a math textbook. As a result, they may change their major, to an area that does not involve mathematics.
Adequate Preparation for Mathematics Courses
Most mathematics courses start with the assumption that the student has achieved a specific level of mathematical knowledge and skills. For example, a college-level course on algebra might be created for students that already have substantial skills and knowledge in algebra and trigonometry from their studies in high school. If this is not the case, the student is likely to fail the course, or have an excessively difficult time obtaining a passing grade.
A solution to the problem described above, is to evaluate the content of a mathematics course before registering. If you have not acquired the knowledge and skills to do well in the course, take a simpler course first. This can sometimes involve a preparation course in mathematics, which many colleges provide. Some students might be able to prepare themselves for a mathematics course, with self-study based on mathematics videos. This is especially useful for people that already have a background in mathematics, or for individuals that are planning to take a course for reviewing a topic in mathematics.
Practice, Practice, and More Practice
A major part of learning mathematics is practice, and without practice, it is unlikely that most students could pass a mathematics course. There are two very important reasons why practice is necessary with mathematics. First, mathematics is a skill, and to develop any skill requires practice.
The second reason is it is difficult to convey mathematical techniques and concepts in words. As a result, mathematics usually requires at least some trial-and-error learning, which takes place when students try to solve math problems. This can indicate if you fully understand the techniques and concepts that were explained to you by your instructor. The trial and error learning frequently results in developing optimized problem-solving strategies in mathematics.
Many students, instructors, and some sources, describe mathematics as a set of rules. To solve a specific type of problem, you follow a specific set of rules, and related steps. This is an oversimplified way of learning mathematics, and it can be dysfunctional, especially for students that are majoring in subjects that involve mathematics.
To master mathematics, it is necessary to understand the difference between rules, and the logical definitions, postulates, theorems, formulas, and techniques that comprise mathematics. This is explained in the following two paragraphs.
Rules can be based on values, practical utility, and/or on arbitrary beliefs or traditions. Rules cannot be evaluated to determine if they are right or wrong, but they can be evaluated in terms of good or bad, or in terms of functionality or dysfunctionality. You cannot use logic to create your own rules, to solve your problems, unless the rules relate to your household or personal space.
The definitions, postulates, theorems, formulas, algorithms, and techniques that comprise mathematics, cannot be evaluated in terms of our values, or in terms of good or bad. They can be evaluated in terms of true or false, or in terms of their utility for solving specific types of problems. Unlike rules, you can use logic to derive your own theorems, formulas, and techniques to solve math problems. In mathematics, if you do not know how to solve a problem, you can probably devise your own technique to obtain a solution.
If you think of axioms and theorems as rules, you lose the versatility, and the creativity that mathematics offers. Instead of a system of logic that can be used to solve many problems, mathematics becomes a meaningless and uninteresting game, with a complex set of tedious rules.
If you learn mathematics in terms of a blind set of rules, you are likely to forget what you learned. If you learn mathematics in terms of postulates, theorems, and logic, any material you forget, you can probably re-create it, based on logical concepts.
It is not necessary to learn the name of every postulate, and the derivation of every theorem. However, it is necessary to develop an understanding of the logic behind every step you use to solve a math problem. You should develop an understanding of what you are doing in terms of logic that relates to each problem you solve. All of this ultimate leads to an intuitive insight, based on mathematical logic.
If you can devise an alternative set of steps to solve a math problem, you have developed the ability discussed above. For example, how many ways can you solve the following division problem:
There are at least three ways to solve the above problem.
Internet Resources For Learning Mathematics
This section provides general information, about web-based resources for learning mathematics. This includes many links to websites and search engines to access resources. This should give you a good idea of what material is available on the web. However, you should carry out your own web-based searches to find the specific resources you need to study mathematics.
Six Categories of Web-Based Resources for Studying, Practicing, and Learning Mathematics
Many websites provide resources to help students learn mathematics. Most, but not all of these resources are free. The resources available on the web can be divided into the following six categories. (Note the blue underlined words are links to Internet searches or to specific websites):
1) VIDEOS ON MATHEMATICS: See also Mathematics Courses, and Advanced Mathematics Courses
2) MATHEMATICS PROBLEMS TO SOLVE FOR PRACTICE, see also Solving math problems for practice, and Mathematics problems and solutions
3) ARTICLES, AND TEXT BASED INSTRUCTIONS ON MATHEMATICS, see also International Journal for Mathematics Teaching and Learning
4) ONLINE CALCULATION DEVICES, see also “Online Calculators ”
5) MATHEMATICS SOFTWARE DOWNLOADS, see also free mathematics software downloads and “Microsoft Mathematics” and Microsoft Mathematics Add-In for Word and OneNote
6) MATHEMATICS TUTORIAL SOFTWARE, see also Mathematics tutorial software download Programmed learning for mathematics
There may be some overlap between the above categories, which depends on the design of the specific web resource. There are many additional links to these resources in the following subsections.
The best web-based resources to learn mathematics are VIDEOS, combined with, MATHEMATICS PROBLEMS TO SOLVE FOR PRACTICE. This is presented in the following subsections.
Videos for Learning Mathematics
It is not feasible for most people to learn mathematics solely by reading instructional material. Most students need an instructor to learn mathematics. Many math videos provide video-based mathematics instructors, without charge. This is an excellent way to learn mathematics. Unlike conventional instructors, a video can be watched at your convenience, and repeated if necessary. A video can be stopped at any point in time, such as to solve a problem that the instructor just explained. If you do not understand a concept presented by an instructor, you can consult another video.
Of course, videos do not provide all of the advantages of working with a live instructor. However, it eliminates all the disadvantages associated with conventional mathematics instructors. Some students might prefer to include a personal tutor, along with the videos, which is probably the ideal. This can be expensive, unless there are free math tutors available at the school you are attending. However, most students can probably master most mathematical material solely with videos, coupled with determination and practice.
The following subsection provides important instructions for using videos optimally, for learning mathematics.
How to Use Video Optimally to Learn Mathematics
To use videos optimally, you should learn how to stop the video at any point, such as to practice what the instructor presented, to be sure that you understand the material, or to take a break. To do this with most videos, you simply left click on the video, such as in the center of the screen. To restart the video, simply left click again in the center of the screen.
You should also learn how to repeat a section of a video, without replaying all of it. This can usually be done by first stopping the video, by left clicking on it. Then, on the bottom of the screen, there is usually a bar with a knob. In the screenshot below, the bar is red, and the knob is white. Place the mouse cruiser on the white circle, and left click, and hold down the mouse key, and move the knob to the left. (If you let go of the mouse key, the knob will not move.) If you want to turn back a video a minute or less, the required knob movement, will be relatively small. For longer time intervals, the movement will be further to the left. You also can move the video forward using the technique described above, except you move the white knob to the right.
When studying math with videos, you must practice solving problems. The more problems you solve, the more successful you will be. The problems you solve should be primarily related to the material you studied on the video. If you do NOT practice, by solving problems, you will probably learn very little from videos, or even from a live instructor, or personal tutor.
Below there is a subsection, with links to videos for studying mathematics. A few paragraphs below this, you will find a subsection with web links to obtain mathematics problems for practice.
How to Find Videos for Mathematics Instruction
There are many good search engines for finding videos with mathematics instruction. This includes Google videos (www.google.com/videohp), YouTube (www.youtube.com), Khan Academy (www.khanacademy.org), and Mashpedia (www.Mashpedia.com). (Technically speaking, YouTube, and Khan Academy, are not really search engines.)
In the following paragraphs, I provide links for accessing over a dozen math-based subjects, with information on how to do your own video searches.
Mathematics Videos from Mashpedia
The best search engine to find videos for studying mathematics, or for learning any subject, is Mashpedia (www.Mashpedia.com). A simple technique that I use to find videos with Mashpedia is to place a / after the.com, followed by the subject I am searching for, such as the following examples:
www.Mashpedia.com/matrix-algebra
The following Mashpedia links provide OVER 100 videos, per link. (I have sometimes found over 800 relevant videos per link.) Each link opens a Mashpedia webpage, which usually contain approximately 50 videos per page. To go from one webpage to another on Mashpedia, scroll down to the bottom of the page, and left click on the word NEXT>>
Mashpedia Web Links (www.Mashpedia.com)
www.Mashpedia.com/advanced-mathematics
www.Mashpedia.com/basic-mathematics
www.Mashpedia.com/differential-calculus
www.Mashpedia.com/Hyperbolic-Functions
www.Mashpedia.com/integral-calculus
www.Mashpedia.com/linear-algebra
www.Mashpedia.com/matrix-algebra
www.Mashpedia.com/quadratic-equations
www.Mashpedia.com/solid-geometry
www.Mashpedia.com/trigonometry
The Khan Academy (www.khanacademy.org)
Khan Academy, was specifically created as a learning resource, for a variety of subjects, including mathematics. The Khan Academy has its own online library, with a search engine for all types of courses, including mathematics. See the following www.khanacademy.org/library. Khan Academy is a nonprofit organization, and all the services they offer are free.
Khan Academy has two major advantages over the search engines presented in this section. The most important is the material is organized into video courses focused on a specific subject, such as algebra, or calculus. The other advantage is each video course also includes some problems to evaluate what you need to learn, or for practice.
The Khan Academy has one major disadvantage over the video search engines, which is a limited number of courses. If you want to study an unusual or highly specialized branch of mathematics, you will probably not find it on the Khan Academy. Specialized fields of mathematics can usually be found easily with Google videos, Mashpedia, or YouTube.
Below there are a number of links to Khan Academy webpages, where you can access videos on the subjects indicated in the links.
Khan Academy Web Links (www.khanacademy.org)
www.khanacademy.org/math/algebra
www.khanacademy.org/math/arithmetic
www.khanacademy.org/math/calculus
www.khanacademy.org/math/differential-calculus
www.khanacademy.org/math/geometry
www.khanacademy.org/math/integral-calculus
www.khanacademy.org/math/linear-algebra
www.khanacademy.org/math/physics
www.khanacademy.org/math/precalculus
www.khanacademy.org/math/trigonometry
From YouTube: Hyperbolic functions
From YouTube: Differential calculus
From YouTube: Integral calculus
Mathematics Problems to Solve for Practice
It is important to obtain a good source of math problems to help you master the material you are studying. These problems should relate to the mathematics course that you are studying, and they should have answers that are easily accessible to check your calculations. Ideally, there should be a step-by-step explanation on how to solve each problem, along with the answers.
If you cannot find the ideal list of problems with answers online, it is best to purchase a mathematics book that has the specific type of math problems you are studying. In some cases, software-based tutorials can provide some or even all of the problems you need for practicing. Listed below there are a few resources that provide mathematics problems that are useful for practice.
Sample Problems From Intermediate Algebra
Practice Solving Percent Problems
Practice with Solving Quadratic Equations
Solving Factorable Quadratic Equations
Using Algebra Tiles to Help Solve Equations
THE CALCULUS PAGE PROBLEMS LIST Problems and Solutions Developed by :D. A. Kouba
THE CALCULUS PAGE PROBLEMS LIST Problems and Solutions Developed by :D. A. Kouba
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To Access Additional Information with Hyperlinks
Studying and Learning Mathematics, and Related Concepts
Studying and Learning Mathematics.
Reading and Comprehending Mathematics
Adequate Preparation for Mathematics Courses
Practice, Practice, and More Practice
Internet Resources For Learning Mathematics
Six Categories of Web-Based Resources for Studying, Practicing, and Learning Mathematics
Videos for Learning Mathematics
How to Use Video Optimally to Learn Mathematics
How to Find Videos for Mathematics Instruction
Mathematics Videos from Mashpedia.
Mashpedia Web Links (www.Mashpedia.com)
The Khan Academy (www.khanacademy.org)
Khan Academy Web Links (www.khanacademy.org)
Mathematics Problems to Solve for Practice
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