SKEP The Psychology of learning and teaching mathematics
- Teaching and learning mathematics have many myths associated with them. Two of them are: mostly men can learn and enjoy mathematics, and math is a combination of rote computational activities and the logic of word problems – nothing more, nothing less. As is the case, understanding and using mathematics involves many more strategic operations than implied by memory and logic. Instead, math involves the application of a rather specific sequence of operations. Some are logical, others linguistic, and a few are social. In addition, many prove to be rather difficult for teachers and students to understand, teach and learn. However, recent research in cognitive psychology and the use of technology offer additional pedagogical assistance that may help to overcome this diversity of operations.
Introduction
The teaching and learning of math can seem very complicated, causing people both young and old to have a false estimate of their abilities in the subject. The truth is, math does not have to be as complicated as everyone claims it is. Math also does not have to be only computational problems that require brain busting logic. Math can be fun and creative, which leads to a better sense of understanding. There are strategies when using mathematics, and applications vary depending on the situation.
History of Math
Perhaps the study of the history of math is far less developed than the actual study of math. Ancient civilizations were using math to solve common problems in societies long before they even knew to define it as math. In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development. The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realization that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration. The major Greek progress in mathematics was from 300 BC to 200 AD. After this time, progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks. From about the 11th Century Islamic and Greek mathematics back knowledge into Europe. Major progress in mathematics in Europe began again at the beginning of the 16th Century with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionized the applications of mathematics to the study of the universe.O'Conner & Robertson
The 17th Century led mathematicians to greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century. Newton developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18th Century. The most important mathematical invention of the 18th Century was two new branches, namely the calculus of variations and differential geometry. Toward the end of the 18th Century, a rigorous theory of functions and of mechanics began to develop. The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. There was fundamental work produced on analytic geometry and synthetic geometry. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. He also contributed in a major way to astronomy and magnetism. The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century. (O'Conner & Robertson, http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/History_overview.html)
Myths About Math
There are many common myths about math. However, how many of these myths can be proven true, and how many are in fact just a myth? According to McGraw-Hill Book Company, there are twelve main myths about math:
1. MEN ARE BETTER IN MATH THAN WOMEN. Research has failed to show any difference between men and women in mathematical ability. Men are reluctant to admit they have problems so they express difficulty with math by saying, "I could do it if I tried." Women are often too ready to admit inadequacy and say, "I just can't do math."
2. MATH REQUIRES LOGIC, NOT INTUITION. Few people are aware that intuition is the cornerstone of doing math and solving problems. Mathematicians always think intuitively first. Everyone has mathematical intuition; they just have not learned to use or trust it. It is amazing how often the first idea you come up with turns out to be correct.
3. MATH IS NOT CREATIVE. Creativity is as central to mathematics as it is to art, literature, and music. The act of creation involves diametrical opposites--working intensely and relaxing, the frustration of failure and elation of discovery, satisfaction of seeing all the pieces fit together. It requires imagination, intellect, intuition, and aesthetic about the rightness of things.
4. YOU MUST ALWAYS KNOW HOW YOU GOT THE ANSWER. Getting the answer to a problem and knowing how the answer was derived are independent processes. If you are consistently right, then you know how to do the problem. There is no need to explain it.
5. THERE IS A BEST WAY TO DO MATH PROBLEMS. A math problem may be solved by a variety of methods which express individuality and originality-but there is no best way. New and interesting techniques for doing all levels of mathematics, from arithmetic to calculus, have been discovered by students. The way math is done is very individual and personal and the best method is the one which you feel most comfortable.
6. IT'S ALWAYS IMPORTANT TO GET THE ANSWER EXACTLY RIGHT. The ability to obtain approximate answer is often more important than getting exact answers. Feeling about the importance of the answer often are a reversion to early school years when arithmetic was taught as a feeling that you were "good" when you got the right answer and "bad" when you did not.
7. IT'S BAD TO COUNT ON YOUR FINGERS. There is nothing wrong with counting on fingers as an aid to doing arithmetic. Counting on fingers actually indicates an understanding of arithmetic-more understanding than if everything were memorized.
8. MATHEMATICIANS DO PROBLEMS QUICKLY, IN THEIR HEADS. Solving new problems or learning new material is always difficult and time consuming. The only problems mathematicians do quickly are those they have solved before. Speed is not a measure of ability. It is the result of experience and practice.
9. MATH REQUIRES A GOOD MEMORY. Knowing math means that concepts make sense to you and rules and formulas seem natural. This kind of knowledge cannot be gained through rote memorization.
10. MATH IS DONE BY WORKING INTENSELY UNTIL THE PROBLEM IS SOLVED. Solving problems requires both resting and working intensely. Going away from a problem and later returning to it allows your mind time to assimilate ideas and develop new ones. Often, upon coming back to a problem a new insight is experienced which unlocks the solution.
11. SOME PEOPLE HAVE A "MATH MIND" AND SOME DON'T. Belief in myths about how math is done leads to a complete lack of self-confidence. But it is self-confidence that is one of the most important determining factors in mathematical performance. We have yet to encounter anyone who could not attain his or her goals once the emotional blocks were removed.
12. THERE IS A MAGIC KEY TO DOING MATH. There is no formula, rule, or general guideline which will suddenly unlock the mysteries of math. If there is a key to doing math, it is in overcoming anxiety about the subject and in using the same skills you use to do everything else.
Source: "Mind Over Math," McGraw-Hill Book Company, pp. 30-43(http://swt.edu/slac/math/skills/12Myths.html)
Solving Math Problems
What is a math problem?
A mathematical problem, like any problem in life, is defined as a problem because it causes us much difficulty in attaining a solution. If the solution, or even the procedure for solving it, is obvious to you then it is no longer a problem but just an exercise. Much of our classroom mathematics is composed of repetitive exercises. (This teaching method does have a useful purpose but it should not be all that mathematics is about.)
A question is a problem if the procedure or method of solution is not immediately known to you but requires you to apply creativity and previous knowledge in new and unfamiliar situations. In a problem, you are not aware of any algorithm that will guarantee a solution.
"To have a problem means to search consciously for some action appropriate to attain some clearly conceived but not immediately attainable aim. To solve a problem means to find such an action." (George Polya)
(Crews, http://www.qerhs.k12.nf.ca/projects/math-problems/intro.html)
Algorithms vs. Heuristics
Algorithms are special methods specifically designed for solving a certain type of question. We have all learnt algorithms for specific situations, such as, the FOIL method for multiplication of binomials. This method has been developed for that specific situation.
Heuristics are general suggestions that may be applicable to all types of questions. They contain a series of tasks, each containing a series of decisions, that are loosely combined to form a model which can assist in problem solving.
(Crews, http://www.qerhs.k12.nf.ca/projects/math-problems/intro.html)
Process of Solving Problems
According to Richard Mayer, there is a distinct process when solving math problems. The first step is problem translation, followed by problem integration, then solution planning and monitoring, and finally, solution execution.
Problem Translation
This step is necessary to translate the problem into an internal representation for yourself. In order to solve the problem, some kind of representation or connection needs to be made in the mind. The problem solver will be restating the problem givens and restating the problem goals. Linguistic knowledge is used in this step, whcih involves some knowledge of the language the problem is given in, and some knowledge of the world around them. They will also need some element of factual knowledge, or previous knowledge.
Problem Integration
The next step is translating the statements in the problem into some kind of problem representation. This involves asking yourself what is actually being asked of the problem. The problem solver will be recognizing the problem types and relevant and irrelevant information. They will also have to determing the information that is needed for the solution and represent the problem as a diagram or picture. Schematic knowledge is necessary for this step in problem solving. This knowledge involves being able to categorize the problems and subproblems.
Solution Planning and Monitoring
This is where the problem solver would decide the method of solving the problem. This could be different for different people attempting to solve the problem at question. The solver will represent the problem as a number sentence, equation, or list of necesssary operations, etc. They will also establish subgoals, and draw conclusions. Strategic knowledge involves using actual pretaught strategies to solve the problem.
Solution Execution
This step involves using the basics of arithmetic to actually solve the problem. The solver will carry out single or a chain of calculations to achieve the end answer. In this step, the idea of a right or wrong answer is displayed. This step requires procedural knowledge, which is using already taught procedures to solve the problem. (Mayer, p. 149-152)
Strategies for Solving Problems
There are many different strategies for solving math problems. The ones listed below are just a few.
Draw a diagram
Look for a pattern
Simplify: solve a simplified problem
Solve for critical or extreme cases
Make an organized list or table
Estimation: guess and check
Work backwards
Use logical reasoning
Write equations or ratios
Act it out
Guess and check
Make a model
Solve a similar problem
Use objects
One approach to problem solving is to determine possible strategies, choose one to solve the problem, and decide whether this was an effective choice.
(Crews, http://www.qerhs.k12.nf.ca/projects/math-problems/intro.html)
(Miller, p. 2)
Understanding, Teaching and Learning Math
Understanding Math
The National Center for Improving Student Learning and Achievement in Mathematics and Science is very dedicated to student understanding of math and science. They feel as though understanding is the core of success in math because classrooms that promote understanding fundamentally respect student's dignity; understanding promotes student achievement, learning, and persistence in mathematics and science;understanding draws upon what students already know and provides a basis for students to use mathematics and science both in their everyday lives and as they study other subjects, and understanding of mathematics and science prepares students to live in a technological world where important personal and ethical decisions, career opportunities, and civic participation will require levels of scientific and mathematical literacy never before necessary in this society (Secada, 8).
The center claims that understanding is relating new knowledge to existing knowledge and that understanding math involves understanding the mathematical ideas, relating to those ideas and making connections within those ideas to the outside world. This association feels as though understanding develops as students construct new relationships among ideas, as they strengthen existing relationships among those ideas, and as they reorganize their ideas. Center researchers Thomas Carpenter and Richard Lehrer have proposed that understanding develops as a result of five mental activities: (1) constructing relationships among ideas, (2) extending and applying what one knows in new situations, (3) reflecting on one's own and othersÕ experiences, (4) communicating what one knows, and (5) actively trying to acquire knowledge(Secada, 9).
Teaching and Learning Math
Some possible strategies for teaching and learning math that go beyond the traditional approach to math include, but are not limited to, journal writing, cooperative learning, the constructivist approach, student centered learning.
The National Council of Teachers of Mathematics and the American Mathematical Association of Two Year Colleges emphasize that students should be able to communicate mathematically, both in written and oral forms, using mathematical and vocabulary notations. NCTM notes that writing in math allows for students to express their thinking. One possible strategy for teaching math is through journal writing. This strategy allows the teacher to see how the student is progressing in the subject, as well as the students likes and dislikes with math. Journals provide great feedback for the teaching and learning of math. Hari Koirala directed a study at North Eastern University that allowed students to use journals in math. By the end of the five year study, over 1800 journals had been written by over 200 students. Conclusion of the study gave the teacher valuable insight on his teaching of mathematics to students. The journals proved that students did have logical understanding of the concepts, and were more freely able to discuss their thoughts and feelings about math. (Kiorala, p. 2)
Another strategy in teaching math is cooperative learning. Jaqueline Bernero, of Saint Xavier University conducted a study with second grade students involving cooperative learning. The study showed the use of cooperative learning in math fostered more interest in math and made the subject more enjoyable for the students and teacher. (Bernero, p. 1)
Cooperative learning is defined as the instructional use of small groups so that students work together to maximize their own and each other's learning. Within cooperative learning groups or cohorts, students are given two responisibilities, to learn the assigned material and to make sure all the other members of the group do likewise. In cooperative learning situations, students perceive that they can reach their learning goals only if the other students in the group do so. Students discuss the material to be learned with each other, help and assist each other to understand it, and encourage each other to work hard.(Johnson and Johnson, 1992, p. 174 as cited by Taylor, 1999, p. 3)
The constructivist approach to teaching mathematics involves sitting back and letting the students be the problem solvers on their own. Teachers can start with an experiment or problem for the students to solve and then it is the job of the teacher to sit back and let the students explore. This allows the students to travel down mathematical paths they may not have ever tried with computational math. Although this approach is very difficult for students to catch on to due to traditional methods of learning math, it can be very rewarding.
Student centered learning is where the teacher lets the students work in both groups and individually to explore problems and become active learners rather than passive recipients. Instead of the teacher being the sole provider of instruction, the students are allowed to explore and teach themselves and each other the information. This in turn leads them to learn how to learn through discovery, inquiry, and problem solving.
National Council of Teachers of Mathematics (NCTM)
The National Council of Teachers of Mathematics feels as though all people encounter math within their everyday life, whether it be knowing how many cookies to divide among the group, or deciding how much of a pay check will go to taxes. NCTM views math as a very important subject that all students, even with varying abilities and goals, has the opportunity to master.
The NCTM believes that all students should have a future in which everyone has access to rigorous, high-quality mathematics instruction, including four years of high school mathematics. The council feels as though there should be knowledgeable math teachers that have adequate support and ongoing access to professional development. The curriculum of NCTM is mathematically rich, providing students with opportunities to learn important mathematical concepts and procedures with understanding, and students have access to technologies that broaden and deepen their understanding of mathematics.
Currently, NCTM feels that not all students have access to adequate math programs in schools for one reason or another. The goal of NCTM is to bring all districts, schools, and classrooms to the same level of mathematics, where enriched and meaningful learning is taking place.
Source: http://www.nctm.org/standards/overview.htm
Principles for School Mathematics
The National Council of Teachers of Mathematics has outlined six principles that will encourage the beneficial learning of all students.
The Equity Principle--Excellence in mathematics education requires equity--high expectations and strong support for all students.
The Curriculum Principle--A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.
The Teaching Principle--Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
The Learning Principle--Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
The Assessment Principle--Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.
The Technology Principle--Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.
Source: http://www.nctm.org/standards/principles.htm
Standards for School Mathematics
NCTM describes the standards that students should acquire for mathematical understanding, knowledge, and skills in grades Pre K-12.
Instructional programs from prekindergarten through grade 12 should enable all students to--
Number and Operations--Understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; compute fluently and make reasonable estimates.
Algebra--Understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; analyze change in various contexts.
Geometry--Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; apply transformations and use symmetry to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems.
Measurement--Understand measurable attributes of objects and the units, systems, and processes of measurement; apply appropriate techniques, tools, and formulas to determine measurements.
Data Analysis and Probability--Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them; select and use appropriate statistical methods to analyze data; develop and evaluate inferences and predictions that are based on data; understand and apply basic concepts of probability.
Problem Solving--Build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.
Reasoning and Proof--Recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof.
Communication--Organize and consolidate their mathematical thinking though communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely.
Connections--Recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.
Representation--Create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; use representations to model and interpret physical, social, and mathematical phenomena.
Source: http://www.nctm.org/standards/standards.htm
International Mathematics Education vs. United States Mathematics Education
The Trends in International Mathematics and Science Study (TIMSS) 2003 is the third comparison of mathematics and science achievement carried out since 1995 by the International Association for the Evaluation of Educational Achievement (IEA), an international organization of national research institutions and governmental research agencies. In 2003, some 46 countries participated in TIMSS, at either the fourth- or eighth-grade level, or both. Comparisons of the mathematics and science achievement of fourth-graders in 2003 are made among the 25 participating countries. In 2003, U.S. fourth-grade students exceeded the international averages in both mathematics and science. In mathematics, U.S. fourth-graders outperformed their peers in 13 of the other 24 participating countries, and, in science, outperformed their peers in 16 countries. In 2003, fourth-graders in three countries-Chinese Taipei, Japan, and Singapore-outperformed U.S. fourth-graders in both mathematics and science, while students in 13 countries turned in lower average mathematics and science scores than U.S. students. Among the 13 countries in which students were outperformed by U.S. fourth-grade students, five countries are members of the OECD (Australia, Italy, New Zealand, Norway and Scotland), and three are English-speaking countries (Australia, New Zealand and Scotland). Comparisons of the mathematics and science achievement of eighth-graders in 2003 are made among the 45 participating countries. In 2003, U.S. eighth-graders exceeded the international average in mathematics and science. U.S. eighth-graders outperformed their peers in 25 countries in mathematics and 32 countries in science. Eighth-graders in the five Asian countries that outperformed U.S. eighth-graders in mathematics in 2003-Chinese Taipei, Hong Kong SAR, Japan, Korea, and Singapore-also outperformed U.S. eighth-graders in science in 2003, with eighth-graders in Estonia and Hungary performing better than U.S. students in mathematics and science as well. (http://nces.ed.gov/timss/Results03.asp)
Another study in 2004 found that compared with peers in Europe, Asia and elsewhere, U.S. 15-year-olds are below average when it comes to applying math skills to real-life tasks, new test scores show. The U.S. students were behind most other countries in overall math literacy and in every specific area tested in 2003, from geometry and algebra to statistics and computation. The international test is not a measure of grade-level curriculum, but rather a gauge of the skills of 15-year-olds and how well students can apply them to problems they may face in life. Among 29 industrialized countries, the United States scored below 20 nations and above five in math. The U.S. performance was about the same as Poland, Hungary and Spain. When compared with all 39 nations that produced scores, the United States was below 23 countries, above 11 and about the same as four others, with Latvia joining the middle group. The test is run by the Organization for Economic Cooperation and Development, a Paris-based intergovernmental group of industrialized countries. The top math performers included Finland, Korea, the Netherlands, Japan, Canada, Belgium, Switzerland and New Zealand. Compared with peers from the OECD countries, even the highest U.S. achievers – those in the top percent of U.S. students – were outperformed. U.S. scores held steady from 2000 to 2003 in the two math subject areas tested in both years. But both times, about two-thirds of the major industrialized countries did better. (Feller, http://www.signonsandiego.com/news/education/20041206-2335-mathskills.html)
Conclusions
Math can be more than just the traditional worksheet, pencil, and calculator approach to learning. Studies have shown that students do better in math when different teaching strategies are applied. Problem solving in math is one area that students are able to use strategies in order to sharpen their skills. There are many people that are dedicated to the successful understanding, teaching, and learning of mathematics in order for students to achieve maximum potential. Hopefully, with time, these mathematical myths will be dispelled, in order for all students to have a positive and rewarding attitude towards math.
As a math teacher for 18 years, with 6 years in USA, I feel really bad for the students who failed math classes, but they are enrolled in the next level or semester of math. It may be because some financial reason for those schools, hoping that the student would make it up in evening or summer school, but this is totally giving up on the student. Honestly, I do not believe that somebody who did not pass the first semester of e.g. Geometry can understand what this is about in the second semester, because math is really a different language and the teacher is also struggling in this, too. A. Rosu
References
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