Traditional mathematics

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Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives.^[1] Several in times in history, traditional mathematics have been challenged by new methodology such as new math, now largely abandoned and discredited.

Standards-based mathematics is easily distinguised in curricula such as TERC in the deliberate ommission of instruction of some or most traditional mathematics methods, and emphasis on goals outside the traditional field of mathematics such as equity for women minorities, preserving the environment, and promoting multiculturalism where most algorithms and methods had come from European cultures and men. This comes from the belief system that such instruction promotes failure, discourages minorities and women, and is harmful to mathematical understanding in a world of new higher world class standards. The stated outcomes were to equip all students with mathematical power rather than tracking only a few students into advanced mathematics, and to make mathematics relevant to all groups, not only the successful. No such ambitious goals had ever been part of traditional mathematics programs which sought only to teach the same methods used by previous generations. Several such curricula were developed largely at federal expense. With the incentive of federal grants money, these were widely adopted by a consensus process in the United States by the 2000s as an integral part of standards-based education reform, another education initiative which had failed under its former name of outcomes-based education. It received bipartisan support by legislators and the public education bureacracy, as well as business interests such as the Business Roundtable.

Nevertheless, many adoptions of reform programs have been troublesome. Although many of these projects show research claiming they are effective, some of these students and parents would later complain that they entered college very poorly prepared for college mathematics.Reform math been the subject of harsh criticism from those knowledgeable about real life application of mathematics. Consequently, by the early 2000s, standards-based mathematics was also in a process of being abandoned and discredited as a massive failure in some school districts and states. Even the NCTM would abandon support of initiatives and texts which completely ommitted traditional arithmetic, and texts such as TERC which had consistently ommitted all traditional instruction^[2] would restore some traditional methods and textbooks which parents could use to the curriculum.

1 Traditional methods
2 Political motivation
3 Raising expectations
4 Replacement by reform mathematics
5 Criticism
6 Strategies
7 Homeschool
8 Traditional mathematics texts
9 Organizations promoting traditional mathematics
10 Which to choose
11 External links
12 Notes

[edit] Traditional methods

The topics and methods of traditional mathematics are well documented in books and open source articles of many nations and languages:

Most documents outside of standards-based mathematics texts only cover traditional methods. Most parents and those who work in professional that require mathematics such as accounting, science, or engineering were taught, and apply traditional mathematics. Most of mathematics notation and methods that are taught and used internationally in all languages have been standardized on traditional mathematics for thousands of years.

Open source encyclopedias entries like those listed above contain very little information on reform mathematics computation methods such as circling, reasoning, coloring in 100s or 10,000s charts, creating projects with glue and scissors, singing, working in groups, creating a fantasy lunch, writing an essay about rain forests, or explaining how to two 1-digit numbers, or the lattice method of multiplication.

The NCTM standards were proposed before curricula implementing the proposals had been created or tried, although they had been agreed upon by consensus by many reputable math educators. Instead, curricula based on the standards were created after they were proposed. Use of the term "standards" is somewhat misleading, as "standards" based methods generally eschew any "standard method" of arithmetic or mathematics. Reform methods are generally not well known, or were developed with government grant money quite recently, being tested with actual students in pilot projects. Teachers trained in one method such as TERC will not be prepared for Everyday Mathematics because the methods taught are completely different. Neither teach methods familiar to teachers, parents, professional mathematicians or engineers.

Elementary arithmetic generally covers content which was traditionally taught in elementary school, a level of education generally achieved even by immigrant populations with low rates of high school graduation. Most teachers and parents know enough math to be familiar with these concepts. Many reform text cover topics such as median, mode, bar-and-whisker charts, and concepts such similar abd obtuse triangles as early as 4th grade which may not have been taught to parents who have completed a masters degree. Thus, reform programs generally require extensive teacher re-training. Letters to parents are sent explaining that "math is not what is used to be", as parents find it difficult to even comprehend the homework sent home which may require cutting, pasting, finding newpaper articles, or interviewing members of the family or playing card games, with very little classic arithmetic.

[edit] Political motivation

Supporting traditional and opposing standards based mathematics has been called politically motivated. Yet traditional mathematics was largely concerned with arithmetic for its own sake. It was based on presenting content, practice, and testing for retention. It had no role for promoting any political or social agenda. Reform mathematics methods is motivated, as stated by the NCTM, to use mathematics to increase and create social equity between races, genders and classes, and promote awareness of issues such as the environment and multiculturalism. Many reform textbooks such as the Core-Plus Mathematics Project have a significant content of "context" such as rain forests, or diverse minority cultures, with "indexes of contexts", while traditional texts tend to be thinner, and are largely devoid of such content.

[edit] Raising expectations

Rather than teaching to current expectations and ranking students against each other, standards-based education reform is a movement in all areas of curriculum which builds up content-independent problem solving skills in place of learning specific facts and methods. It sets and demands HIGHER standards for all students, and numerical goals and timetables with sanctions such as denying diplomas, based on the (some say unrealistic) belief that all students can perform at world class levels, regardless of native ability. Thus, even students in the top tenth percentile could be classified as failures if they failed to show improvement compared to last year's students. Yet many parents and professionals are less concerned about demanding skills for 20 or more years in the future than if their children or students were at least as capable as was expected 20 years ago.

[edit] Replacement by reform mathematics

Since 1989, governments and education agencies have been in the process of replacing traditional mathematics with standards-based mathematics. However increasing numbers of schools districts are abandoning reform standards because of protests from parents, students, and mathematics experts and very low test scores, even from tests aligned with the new standards.

[edit] Criticism

Most standards-based mathematics methods are distinguished by either de-emphasizing, discouraging or completely ommitting any instruction in traditional methods listed above. Many widely cited research papers such as The Harmful Effects of Algorithms in Grades 1-4 take the position that instruction of such traditional methods is harmful to deep understanding of mathematics.

Because of the popular movement against standards, traditional mathematics education has been called "Parrot Math" by critics. The direct instruction method (teaching that 2 + 2 = 4 as a fact rather than an exploration) has also been disparagingly called "drill and kill". The battle of philosiphies has appeared so often on the front pages of national newspapers such as the Wall Street Journal and websites such as Mathematically Correct that the debate has been called the Math wars.

Traditional mathematics has been called a failure because only a few students achieve the highest levels of mathematics achievement such as calculus, and it is believed that not enough master even algebra, which is seen a gateway to high paying jobs such as computer programming. Because much of mathematics has roots in European culture, and developed by men, it is seen by some as not being appropriate for women, or people of color.

[edit] Strategies

Reform mathematics emphasizes understanding of concepts and exploration. A single method of arriving at a correct answer may not even be included in the student text, and traditional methods may be explicitly discouraged. Traditional presents a concept, usually a single method of solution, and then uses drill and practice.

Reform uses extra materials which must be refreshed each year, some don't use traditional textbooks, or break them up into several booklets. Traditional mathematics use a single textbook for the entire year which can be used over several years. For example, here is the list of student materials included in the 2002 Grade 5 package for TERC Investigations: 4 rolls of adding machine tape; 36 blank 5/8" cubes; 1,000 stickers for blank cubes; 200 1-cm cubes; 16 transparent blank spinners; 4 450-piece sets of power polygons; 4 buckets of square color tiles (400 per bucket); 1,000 Snap(TM) cubes; 1 set of elementary bar mass set-Ohaus; 4 graduated measuring prisms (2-cm x 5-cm x 21-cm); 4-liter measuring pitcher (calibrated 100 ml - 1,000 ml); 4 spectrum school balance (includes 7-piece mass set); 4 sets standard measuring pitchers (3 pitchers: quart, pint, cup per set); 10 measuring tapes; 12 meter/yard sticks. The total package for Grade 5 is listed at $1,388.42, and within that total the cost of the just mentioned student materials, for a class of 32, is $817.00^[3]

Reform mathematics omits or de-emphasizes traditional concepts, terms or skills such as regrouping, long division, or average. Traditional mathematics uses concepts and terms which have changed little since their invention hundreds or thousands of years ago. For example, Core-Plus introduces a NOW NEXT notation unfamiliar to most mathematics professors, while TERC instructs teachers to not use the traditional add-the-sum and divide by the number of data points method to compute the average.

Reform mathematics emphasizes multiculturalism. For example, TERC in one elementary unit asks the students to sing Happy Birthday, and as a variation sing it in one or more immigrant languages. Traditional textbooks such as Singapore Math reflect either no culture, or only one dominant culture.

Reform mathematics emphasizes problem solving, while traditional has tended to teach basic skills. However, in a comparison of a traditional Houghton-Mifflin curriculum against the Cord Applied Math, there were no advantage SAT total scores and SAT problem-solving scores, but the students with the traditional text showed a significant improvement in SAT procedure scores, or basic skills.

Reform mathematics is preferred by federal and state education agencies, as well as professional organizations such as the NCTM. Opposition to reform and preference for traditional mathematics has come from parents, college educators, and professionals in mathematics and science.

Traditional mathematics separates students into pre-college and vocational tracks. Not all students are expected to take or master algebra, geometry, statistics, or calculus. Reform programs like Core-Plus teach the same advanced content, including matrix mathematics to all high school students for 3 years.

Reform mathematics believes that mathematics classroom instruction must be changed to improve mathematics instruction and increase student achievement. Stronger connections must be made between mathematics and students' lives outside of the mathematics classroom. Traditional mathematics was abstract, and oriented towards college preparation. Traditional education expects different outcomes for different students rather than increasing achievement for all students every year.

Reform mathematics is expensive to implement. Teachers must be trained and supplementary kits must be purchased. Traditional mathematics does not require teacher retraining, or extra consumable materials. The study of the Cord text concluded that that if a reform mathematics curriculum did not promote an increase in student achievement, such extra expense was not worthwhile.

Reform mathematics is often funded by federal money from organizations such as the Department of Education and the National Science Foundation. School districts are not offered extra grant money to buy or promote traditional mathematics textbooks, which must be funded from local education dollars.

Traditional math textbooks are more easily understood by parents and teaches the same methods parents learned, so they can help their children with homework. Some standards-based curricula do not have textbooks to bring home^[4], or do not contain explanations of correct methods of solving problems directly. For example, a Core-Plus Mathematics Project unit introducing matrices asks to graph and interpet data on charts, but the text only provides information on Nike and Reebok sales. In another exercise, students are asked to draw a fantasy fractal picture, but are not given a definition of what a fractal picture is.

Traditional math text problems typically have only one correct answer. In a Core-Plus matrix problem, the student is asked to choose the best chart format for data, but multiple correct answers are possible.

[edit] Homeschool

Homeschool students and parents often favor traditional texts. Since the parents are teachers, they are more familiar with standard methods that they were taught. Traditional texts also tend to be much simpler since they take generally only a few minutes to explain a skill, and a few more minutes to practice rather than long group-oriented projects that require mathematical communication, music or art projects, and nightly homework which requires a half to an hour of work on the part of the student and the parent, or the rest of the family in the case of interviews. They are generally based on inexpensive, thin books which contain only mathematical information rather than contexts on rain forests or running shoes or a tour of the United States, and do not require expensive manipulatives such as number cubes or blocks. Standards-based methods usually require a significant investment and specialized teacher training, often with federal grant money to be adopted by a large school district, and are not suitable for use in home school environment.

[edit] Traditional mathematics texts

In general, any math textbook which contains instruction in standard arithmetic methods can be categorized as a traditional math textbook. Any math textbook which de-emphasizes or omits significant standard methods can be considered to be standards-based. The following current texts are often cited as good for those wishing for a traditional approach, often also favored by homeschoolers.

[edit] Organizations promoting traditional mathematics

Most of these organizations are critical of standards-based mathematics and have given poor reviews to textbooks such as TERC, Mathland and Core-Plus Mathematics Project

Mathematically Correct a website which supports traditional mathematics
NYC HOLD a New York-based organization of teachers, professional mathematicians, parents and others which has been extremely active in recent years in working for adoption of mastery-based, traditional math programs
Illinois Loop - extensive web coverage of math issues and specific math programs
Where's The Math - a website which supports traditional mathematics and more focused standards for the state of Washington

[edit] Which to choose

Mathematics texts are generally chosen by local school districts, or promoted by state education agencies, often with the support of business organizations such as the Business Roundtable by a consensus process. A blue ribbon panel is created to determine that current math standards are inadequate and unacceptable, and draw up a plan to implement texts which are compliant with standards-based education reform, of which mathematics is only one field of curricula. At no time is there a place for public debate of whether such new standards should be adopted. Instead, the public is invited to be informed of the new standards, and are invited to provide input, and break up into small groups on how they can help adopt and embrace the new standards.

Similar standards-based curricula also exist for science, social studies, and language arts, also with their own controversies. Many districts apply for the federal grants which are generally given only to districts which adopt standards-based curricula. In general, only a small number of parents or mathematics experts are even aware of such an adoption process, and a few parents and citizens will be recruited to support such an adoption. Once adopted, citizens often find it difficult or impossible to persuade education officials to discard such curricula, despite a wide body of information about the ineffectiveness of such curricula, but citizens in some states and districts have succeeded in some cases such as California, and Tacoma, Washington.

Standards-based curricula have the advantage of promising world class mathematical power for students of all cultures, races, incomes and genders, and promoting social issues such as multiculturalism, the environment, and careers.

Traditional curricula are more suitable if the community is more interested in accurate computation, easy to understand homework, and instruction in time-tested math methods used by all nations.

[edit] External links

Mathematically Correct, a website which supports traditional mathematics
NYC HOLD, a New York-based organization of teachers, professional mathematicians, parents and others which has been extremely active in recent years in working for adoption of mastery-based, traditional math programs
Illinois Loop - extensive web coverage of math issues and specific math programs
Where's The Math, a website which supports traditional mathematics and more focused standards for the state of Washington

[edit] Notes

^ [1]A comparison of traditional and reform mathematics curricula in an eighth-grade classroom Education, Summer 2003 by Alsup, John K., Sprigler, Mark J.
^ [2] Quirk "TERC Omits All Standard Computational Methods"
^ [3] Reviews of TERC
^ [4] Back to basics on kids’ math Alarmed by low scores, Tacoma school officials OK added textbook DEBBY ABE; The News Tribune (Tacoma, WA) "parents need easier-to-understand math textbooks so they can help their children with homework."