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Weaknesses in Our Current Math Education SystemSix General Principles Specified by the NCTMTen General Standards Specified by the NCTMA Mathematics Expertise Scale ApproachOther Topics That Might be Added to This PageWeaknesses in Our Current Math Education SystemMuch of the weakness in our current Math Education system is historical in nature and can be discerned by carefully thinking about the following diagram. It is a simplified 4-step model of using mathematics to solve a problem. Historically, Math Education systems focused on helping students to learn to carry out a number of different types of "step 2" using some combination of mental and written knowledge and skills. It takes a typical students hundreds of hours of study and practice to develop a reasonable level of speed and accuracy in performing addition, subtraction, multiplication, and division on integers, decimal fractions, and fractions. Even this amount of instructional time and practice -- spread out over years of schooling -- tends to produce modest results. Speed and accuracy decline relatively rapidly without continued practice of the skills. During the past 5,000 years there has been a steady increasing body of knowledge in mathematics, science, and engineering. The industrial age and our more recent information age has lead to a steady increase in the use of "higher" math in many different disciplines and on the job. Our Education System has moved steadily toward the idea that the basic computational numeracy described above is insufficient. Students also need to know basic algebra, geometry, statistics, probability, and other higher math topics. As these topics began to be introduced into the general curriculum, a gap developed between the math that students were learning in school and the math that most people used in their everyday lives. More and more, Math Education focused on learning math topics in a self-contained environment where what was being learned had little immediate use in the lives of the students and little use in the lives of their parents. A pattern of Math Education curriculum developed in which one of the main reasons for learning the material in a particular course was to be prepared to take the next course. Students developed little skill at transferring their math knowledge and skills into non-math disciplines or into problems that they encountered outside of school. Only a modest number of adults maintain the math knowledge and skills that they initially developed while studying algebra, geometry, and other topics beyond basic arithmetic. That brings us up to current times. Many high schools require students to take three years of math (during their four years of high school) in order to graduate. There is considerable pressure to have all students take an algebra course. The nature of the instruction and the learning in many of these math courses follows the "80-percent on step 2" that has been noted above. Students are now learning the underlying concepts, or how to make use of the math in other courses or outside of a formal school setting. As a final note in this subsection, the 4-step diagram represents only part of the field of math. For example, it does not include math as a human endeavor with its long and rich history. ------------------------------------------------------ Here, and perhaps elsewhere in this Website, we want to emphasize that computers can carry out step 2 both accurately and rapidly. Thus, our current Math Education system is spending about 80-percent of its time teaching students to compete with machines in a domain where the machine is far far superior to even the best of humans. ------------------------------------------------------ Note to self: The February 1999 issue of the Phi Delta Kappan is a theme issue focussing on The Mathematical Miseducation of America's Youth. It is an excellent resource. Phi Delta Kappan. Select articles are available online. Accessed 8/15/01: http://www.pdkintl.org/kappan/karticle.htm. See specifically the following three articles from the February 1999 issue: The Mathematical Miseducation of America's Youth: Ignoring Research and Scientific Study in Education, by Michael T. BattistaParrot Math, by Thomas C. O'Brien ReferenceGlazer, Evan (Date?). Technology Enhanced Learning Environments that are Conducive to Critical Thinking in Mathematics: Implications for Research about Critical Thinking on the World Wide Web [Online]. Accessed 1/14/02:http://www.arches.uga.edu/~eglazer/EDIT6400.html. Quoting the Abstract from this draft paper:Mathematics education reform has emphasized a need for increased experience with technology and critical thinking skills in order to better prepare students for a modern society that is dependent on access to and use of information. However, the complexity and diversity of these skills leaves some uncertainty about their successful implementation in the mathematics classroom. The goal of this paper is to reveal a model that promotes critical thinking in a technology-enhanced mathematics classroom. Moreover, this paper will also examine critical thinking opportunities that are unique to technology-enhanced environments by examining discrepancies from critical thinking in non-technology enhanced environments.A domain specific definition of critical thinking in mathematics will be formed in order to identify research and literature that supports this model. A review of literature suggests that environmental factors, such as the nature of a critical thinking task, the students' learning responsibilities, and the teacher's role to foster learning, influence and enhance critical thinking opportunities. The nature of each of these elements in technology and non-technology-enhanced environments will be elaborated through the model. The paper will conclude with implications for research about critical thinking using resources from the World Wide Web.This Evan Glazer article includes a lengthy discussion of what is meant by critical thinking in general, and in mathematics. Suppose one decides that one of the goals of mathematics education is to develop critical thinking. Then, what does progress in Brain Science, ICT, and SoTL do to help us here? Six General Principles Specified by the NCTMThe National Council of Teachers of Mathematics (NCTM) it the leading professional society for Pre K-12 Mathematics Education in the United States. Much of the contents of this section are direct quotes from NCTM [Online] Accessed 7/28/01:http://www.nctm.org/standards/overview.htm.NCTM's Math Standards are based on six general principles:
The Six General Principles are carefully thought out, powerful statements that NCTM feels should underlie mathematics in curriculum, instruction, and assessment. Here are a few observations that relate some of these principles to this workshop and Website. |
Tuesday, December 20, 2011
goals in math education
maths as a language
Foundational InformationHome Page |
Our math education system pays some attention to the idea that math is a language. For example, many math teachers have their students do journaling on the math learning experiences and their math use experiences. Some math teachers make use of cooperative learning--an environment that encourages students to communicate mathematical ideas. Some math assessment instruments require that students explain what it is they are doing as they solve the math problems in the assessment. There has been a great deal of research on the teaching and learning of reading and writing in one's first (natural) language. In addition, there has been a great deal of research on the learning of a second language. It seems likely that some of the research findings and practical implementations of these findings would be applicable to teaching and learning of mathematics. In the early days of computer programming, there was quite a bit of research done how to identify people who might be good at computer programming. It turned out that music ability and math ability correlated well with computer programming ability. This is interesting from the point of view that in some sense music is a language, and computer programming requires learning programming languages and then solving problems using the languages. ======================================= The following article provides some research on the value of directly teaching language skills in various disciplines, including math: Marzano, Robert J. (September 2005) Preliminary Report on the 2004–05 Evaluation Study of the ASCD Program for Building Academic Vocabulary. Accessed 11/30/05:www.ascd.org/ASCD/pdf/ Building%20The following email from Garry Taylor is a valuable resource in exploring mathematics as a language. From: consultay@npgcable.com ReferencesMusic as a language. Quoting Howard Gardner:“It may well be easier to remember a list if one sings it (or dances to it). However, these uses of the ‘materials’ of an intelligence are essentially trivial. What is not trivial is the capacity to think musically.” (Howard Gardner)English as a Second or Other Language (ESOL) Research on Learning Computer Programming and Software EngineeringMathematics as a LanguageCrannell, Annalisa. Writing in Mathematics [Online]. Accessed 1/26/02: http://www.fandm.edu/Departments/Mathematics/writing_in_math/writing_index.html. Crannell gives writing assignments in the calculus classes she teaches at a university level. Her Website includes a 1994 booklet A Guide to Writing in Mathematics Classes. Quoting from the first part of that booklet:Language and the Learning of Mathematics [Online]. Accessed 1/26/02: http://www.mathematicallycorrect.com/allen4.htm. A speech delivered at the NCTM Annual Meeting Chicago, April 1988 by Frank B. Allen, Emeritus Professor of Mathematics Elmhurst College. Quoting from the paper:For most of your life so far, the only kind of writing you've done in math classes has been on homeworks and tests, and for most of your life you've explained your work to people that know more mathematics than you do (that is, to your teachers). But soon, this will change.Now that you are taking Calculus, you know far more mathematics than the average American has ever learned - indeed, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel like the mathematics you can do is simple and obvious (doesn't everybody know what a function is?), but you can be sure that other people find it bewilderingly complex. It becomes increasingly important, therefore, that you can explain what you're doing to others that might be interested: your parents, your boss, the media. This brings me to my major thesis that natural language, gradually expanded to include symbolism and logic, is the key to both the learning of mathematics and its effective application to problem situations. And above all, the use of appropriate language is the key to making mathematics intelligible. Indeed, in a very real sense, mathematics is a language. Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof.Mathematics as a Language [Online]. Accessed 1/26/02:http://www.cut-the-knot.com/language/. Quoting from the Website: However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that ambiguities exist in plain English (the number's number is one of them) and in other sciences as well. A a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.The Language of Mathematics [Online]. Accessed 1/26/02:http://www.math.montana.edu/~umsfwest/. This Website is based on a book by Warren Esty and a course at Montana State University by the same name. The first quote given below is from the Website, and the second is from the Warren Esty book.The Language of Mathematics [Online]. Accessed 1/26/02:http://www.chemistrycoach.com/language.htm.Jointly with Anne Teppo, Warren Esty published an article in the Mathematics Teacher (Nov. 1992, 616-618) entitled "Grade assignment based on progressive improvement" which was reprinted in the NCTM's Emphasis on Assessment. and posted on the web by the Eisenhower National Clearinghouse for Mathematics and Science Education. In a language course, you can expect continual improvement. This article discusses why grading should not be based on averages of unit-exam scores and how a course like "The Language of Mathematics" can be graded. This Website contains a number of quotations that relate to the topic of mathematics as a language. Here are two examples:Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell, (1872-1970) The Scientific Outlook, 1931. |
sureshmathematics
dear user you know what is mathematics?
What is Mathematics?Home Page |
A Tidbit of HistoryMathematics as a formal area of teaching and learning was developed about 5,000 years ago by the Sumerians. They did this at the same time as they developed reading and writing. However, the roots of mathematics go back much more than 5,000 years.Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone (see ahttp://www.math.buffalo.edu/mad/ Ancient-Africa/ishango.html andhttp://www.naturalsciences.be/expo/ishango/ en/ishango/riddle.html) is a bone tool handle approximately 20,000 years old.  Figure 1 The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago (seehttp://www.sumerian.org/tokens.htm). Such clay tokens were a predecessor to reading, writing, and mathematics. Figure 2 The development of reading, writing, and formal mathematics 5,000 years ago allowed the codification of math knowledge, formal instruction in mathematics, and began a steady accumulation of mathematical knowledge.Mathematics as a DisciplineA discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):
Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines? Beauty in MathematicsRelatively few K-12 teachers study enough mathematics so that they understand and appreciate the breadth, depth, complexity, andbeauty of the discipline. Mathematicians often talk about the beauty of a particular proof or mathematical result. Do you remember any of your K-12 math teachers ever talking about the beauty of mathematics?G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.
Problem SolvingThe following diagram can be used to discuss representing and solving applied math problems at the K-12 level. This diagram is especially useful in discussions of the current K-12 mathematics curriculum. Figure 3 The six steps illustrated are 1) Problem posing; 2) Mathematical modeling; 3) Using a computational or algorithmic procedure to solve a computational or algorithmic math problem; 4) Mathematical "unmodeling"; 5) Thinking about the results to see if the Clearly-defined Problem has been solved,; and 6) Thinking about whether the original Problem Situation has been resolved. Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by the process or attempting to solve the original Clearly-defined Problem or resolve the original Problem Situation. Click here for more information about problem solving.Final RemarksHere are four very important points that emerge from consideration of the diagram in Figure 3 and earlier material presented in this section:
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