NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS:
The National Council of Teachers of Mathematics is an educational subject matter group for mathematics teachers in elementary, junior high, and high school. Unfortunately it has tried to be more than what it is capable of being, in my opinion.
Several subject matter organizations in the 1980's and 1990's put forth "Standards" using that word to suggest that there was some kind of verifiable, vetted program that would improve the teaching of a subject.
Nothing could be further from the truth!
"Standards" were not standards at all! They were merely a list of cockamamie "education school" concepts that were developed and based on hokum science. That anyone took them seriously is a testament to the gullibility of the people residing in the fields of education.
But believe them they did. And why shouldn't they? Most of the people making decisions on such matters are themselves products of non-academic college schools of education!
Say something enough times, and it becomes real. Say it even more and the public will buy into it too!
Here are the suggestions put forth by the NCTM in the 1980's:
The National Council of Teachers of Mathematics recommends that:
Problem solving be the focus of school mathematics in the 1980s;
Basic skills in mathematics be defined to encompass more than computational facility;
Mathematics programs take full advantage of the power of calculators and computers at all grade levels;
Stringent standards of both effectiveness and efficiency be applied to the teaching of mathematics;
The success of mathematics programs and student learning be evaluated by a wider range of measures than conventional testing;
More mathematics study be required for all students and a flexible curriculum with a greater range of options be designed to accommodate the diverse needs of the student population;
Mathematics teachers demand of themselves and their colleagues a high level of professionalism;
Public support for mathematics instruction be raised to a level commensurate with the importance of mathematical understanding to individuals and society.
The development of problem-solving ability should direct the efforts of mathematics educators through the next decade. Performance in problem solving will measure the effectiveness of our personal and national possession of mathematical competence.
Problem solving encompasses a multitude of routine and commonplace as well as nonroutine functions considered to be essential to the day-to-day living of every citizen. But it must also prepare individuals to deal with the special problems they will face in their individual careers.
Problem solving involves applying mathematics to the real world, serving the theory and practice of current and emerging sciences, and resolving issues that extend the frontiers of the mathematical sciences themselves.
This recommendation should not be interpreted to mean that the mathematics to be taught is solely a function of the particular mathematics needed at a given time to solve a given problem. Structural unity and the interrelationships of the whole should not be sacrificed.
True problem-solving power requires a wide repertoire of knowledge, not only of particular skills and concepts but also of the relationships among them and the fundamental principle that unify them. Each problem cannot be treated as an isolated example. This recommendation looks toward the need to solve problems in an uncertain future as well as here and now. As such, mathematics needs to be taught as mathematics, not as an adjunct to its fields of application. This demands a continuing attention to its internal cohesiveness and organizing principles as well as to its uses.
Recommended Actions
1.1 The mathematics curriculum should be organized around problem solving.
The current organization of the curriculum emphasizes component computational skills apart from their application. These skills are necessary tools but should not determine the scope and sequence of the curriculum. The need of the student to deal with the personal, professional, and daily experiences of life requires a curriculum that emphasizes the selection and use of the skills in unexpected, unplanned settings.
Mathematics programs of the 1980s must be designed to equip students with the mathematical methods that support the full range of problem solving, including:
the traditional concepts and techniques of computation and applications of mathematics to solve real-world problems, the rational and real number systems, the notion of function, the use of mathematical symbolism to describe real-world relationships, the use of deductive and inductive reasoning to draw conclusions about such relationships, and the geometrical notions so useful in representing them;
methods of gathering, organizing, and interpreting information, drawing and testing inferences from data, and communicating results;
the use of the problem-solving capacities of computers to extend traditional problem-solving approaches and to implement new strategies of interaction and simulation;
the use of imagery, visualization, and spatial concepts.
Mathematics programs should give students experience in the application of mathematics, in selecting and matching strategies to the situation at hand. Students must learn to:
formulate key questions;
analyze and conceptualize problems;
define the problem and the goal;
discover patterns and similarities;
seek out appropriate data;
experiment;
transfer skills and strategies to new situations;
draw on background knowledge to apply mathematics.
Fundamental to the development of problem-solving ability is an open mind, an attitude of curiosity and exploration, the willingness to probe, to try, to make intelligent guesses.
The curriculum should maintain a balance between attention to the applications of mathematics and to fundamental concepts.
1.2 The definition and language of problem solving in mathematics should be developed and expanded to include a broad range of strategies, processes, and modes of presentation that encompass the full potential of mathematical applications.
Computational activities in isolation from a context of application should not be labeled "problem solving."
The definition of problem solving should not be limited to the conventional "word problem" mode.
As new technology makes it possible, problems should be presented in more natural settings or in simulations of realistic conditions.
Educators should give priority to the identification and analysis of specific problem-solving strategies.
Educators should develop and disseminate examples of "good problems" and strategies and suggest the scope of problem-solving activities for each school level.
1.3 Mathematics teachers should create classroom environments in which problem solving can flourish.
Students should be encouraged to question, experiment, estimate, explore, and suggest explanations. Problem solving, which is essentially a creative activity, cannot be built exclusively on routines, recipes, and formulas.
The mathematics teacher should assist the student to read and understand problems presented in written form, to hear and understand problems presented orally, and to communicate about problems in a variety of modes and media.
The mathematics curriculum should provide opportunities for the student to confront problem situations in a greater variety of forms than the traditional verbal formats alone; for example, presentation through activities, graphic models, observation of phenomena, schematic diagrams, simulation of realistic situations, and interaction with computer programs.
1.4 Appropriate curricular materials to teach problem solving should be developed for all grade levels.
Most current materials strongly emphasize an algorithmic approach to the learning of mathematics, and as such they are inadequate to support or implement fully a problem-solving approach. Present textbook problems tend to be easily categorized and stylized and often bear little resemblance to highly diversified, real-life problems. They do not permit the full range of strategies and abilities actually demanded in realistic problem contexts.
The potential of computing technology for increasing problem-solving ability should be explored and exploited by the development of creative and imaginative software.
1.5 Mathematics programs of the 1980s should involve students in problem solving by presenting applications at all grade levels.
Applications should be presented that use the student's growing and changing repertoire of basic skills to solve a multitude of routine and commonplace problems essential to the day-to-day living of every citizen.
Applications of mathematics to other disciplines such as the social sciences, business, engineering, and the natural sciences should be presented.
The enormous versatility of mathematics should be illustrated by presenting as diversified a collection of applications as possible at the given grade level
At the college level, courses in mathematics and the mathematical sciences should give prospective teachers experiences that develop their capacities in modeling and problem solving.
1.6 Researchers and funding agencies should give priority in the 1980s to investigations into the nature of problem solving and to effective ways to develop problem solvers.
Support should be provided for:
the analysis of effective strategies;
the identification of effective techniques for teaching;
new programs aimed at preparing teachers for teaching problem-solving skills;
investigations of attitudes related to problem-solving skills;
the development of good prototype material for teaching the skills of problem solving, using all media.
You can go here to read the rest of their nonsense: http://www.nctm.org/standards/content.aspx?id=17279
If the reader has waded through this compilation of baloney, he will see the problem. Anyone with an academic background has to be nonplussed at some of these items!
These suggestions do not describe an intelligent way to teach mathematics!
The problem is that teachers of mathematics may never have taken a real, true college level mathematics course--ever! And for some of them, if they took such a course, they may not have even passed it. They have instead, taken "math education." Trust me--"math education" has not much to do with real mathematics. And it is easily "passed."
If one has the gumption, he should ask each of his children's mathematics teachers what college courses they had in mathematics. Ask too if they took a real college mathematics course what their grade was or if they even passed it! You could ask that question, by the way, of any public school administrator you happen to run into, also. You won't be surprised at the answer to that question--it will be just as I've described.
Having pointed out the inherent problem with the teaching of mathematics in our public schools, one must decide what to do about the situation.
After my suggestions elsewhere that parents either teach their children themselves all that they need to know the year before they are supposed to learn material in public schools, or that parents simply homeschool their children, or that they send them to a qualified private school, I would suggest that parents band together and raise Cain about the situation in our schools. When people run for the school board, ask them what they know about the NCTM Standards. If they say they don't know anything, educate them. If they say those Standards are fine, don't vote for them! Just for fun, you might want to ask them if THEY took a real college level math course and how they did!
But whatever you do, make certain that YOUR child knows his tables (addition, subtraction, multiplication and division) through 12's, that he can multiply and divide using fractions, that he can work with a decimal point, that he knows the square roots of the obvious numbers, and that he can long divide--and all without the help of a calculator. That's all he needs to know by the end of the fifth grade.
If he has to perform 33 steps to add 27 and 92, be sure that he knows there is a more commonsensical way to do it!
When you child gets to Algebra, make certain that he learns Algebra and that he is not being taught "rainforest Algebra."
(If you don't know what "rainforest Algebra is," Google it.)
If you want your child to be a scientist, an engineer, a physician, a physicist, a chemist, or a mathematician (or anything related to those subjects), you might want to think very seriously about hiring a tutor (if you cannot teach him yourself), or enrolling your child in a private school.
Remember that the thrust of public education these days is to dumb down 85% of the students so they may satisfy the requirements and needs of corporate America, and if you want your child to be in the 15% that are not a part of that thrust, you need to be pro-active and learn what this scheme by our government is all about. (The NCTM is part of the scheme whether they know about it or like it.)
Visit the topic "Mathematics" under the "Education" heading on this web site for more information about the teaching of mathematics.