Reform mathematics

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Standards-based mathematics is one name for a reform methods of mathematics instruction, usually based on recommendations published in 1989 by the National Council of Teachers of Mathematics [10] (NCTM). The original document was Principles and Standards for School Mathematics. It attempted to set forth a national vision for precollege mathematics education in the United States and Canada. The recommendations would be largely adopted by most education agencies from local to federal levels by the mid 2000s. Today, it still serves as the primary basis for most states' mathematics standards, many federally funded textbook projects, as well as an influence on standards in other nations.

[edit] Controversy

At the same time, this document and curricula based on it would later be fiercely opposed by many parents and mathematics professionals, and rejected by many states and school districts. As a major component of standards-based education reform, the NCTM standards would be the mathematics counter part to whole language, which was received with much opposition, science, and history curriculum standards largely aligned with constructivism. Parents would complain about replacing instruction in arithmetic with writing, coloring, counting, and inventing mathematics unrecognizable to any previous generation of mathematicians or educators. The term "standards-based mathematics" is somewhat misleading as the most ambitious implementations of the standards are distinguished by the complete expulsion of any standard methods of computation. Compared to traditional sequences, many innovators had determined that elementary arithmetic was too difficult and unproductive for many students in the age of calculators. At the same time, since only a few students advanced to topics such as algebra, statistics, and calculus, it became important to move such topics to all students as early as kindergarten. Since some of these topics exceed the level of math instruction of many parents and even teachers, it caused many problems from adults who attempted to help students, but were never familiar with this level of material, such as bar and whisker charts or mean, median and mode in fourth grade assessments such as WASL.

[edit] Standards

The purpose of the term standards, as is use by the greater standards-based education reform movement is rather than codifying traditional expectations, is to change the very nature of mathematics instruction. Instead of simply transmitting old methods to the next generation, mathematics education would become a tool of social restructuring to advance those who had been held back from advancement by old instruction methods.

A major goals of outcome-based education, and later standards-based reform was to insure that "all children will succeed" by setting "higher standards" of "what every child is expected to know and be able to do". This movement is based on the (some say unrealistic) belief that, through systemic change, and holding all participants in the system accountable, the expectation of a normal curve can be replaced by a system of standards and assessments. This system will insure that every student will be able to read and compute, and graduate at a world-class grade level. However, in states such as Washington, over half of students still failed to pass these new standards even after a decade of system alignment, and legislators were considering passing laws essentially discouraging new reform mathmatics. While the standards are available on the internet, full access by the public is only available by an expensive purchase or subscription.

[edit] 2006 Focal Points

In 2006, NCTM issued a document called "Focal Points" which presented a more concise set of goals and objectives on a grade by grade basis, for grades K through 8. The "Focal Points" were perceived by the press (notably the Wall Street Journal (Sept 12, 2006), the New York Times, the Chicago Tribune and other newspapers to be an admission that previous standards had permitted the creation of curricula such as "Investigations" which had almost completely omitted any instruction in traditional arithmetic methods. In response to a firestorm of criticism over the 1989 standards, the NCTM replaced its call for de-emphasis with a strong emphasis of direct instruction of basic skills.

While the PSSM was championed by education theorists and administrators as raising standards for all students, it was sharply attacked by mathematicians, parents, and even some teachers over the new teaching methods which inspired lampooned exercises such as Mathland's Fantasy Lunch, Rainforest Algebra, and academic papers finding that teaching arithmetic harmed mathematical understanding. Some officials were quoted as valuing understanding processes more than learning one correct way to get one correct answer. Although still widely adopted in the United States and abroad by the mid-2000s, some states such as California and many local districts such as Tacoma, Washington would reject the standards as a massively misguided mistake in favor of more traditional approaches such as Singapore Math and Saxon math.

[edit] Terminology

Mathematics in this style have also been called "standards-based" instruction or "standards-based mathematics,^[1] , or simply "reform mathematics"^[2].

Less favorable terminology which have appeared in press and web articles include fuzzy math, "Where's the math"^[3], "anti-math"^[4], "math for dummies"^[5], "no-math mathematics",^[6] , rainforest algebra ^[7], "Math for women and minorities, ^[8] and "new new math".^[9]

Traditional mathematics education has been called "Parrot Math" by critics. The direct instruction method has been criticized as "drill and kill".

[edit] Origins

Based on a consensus process that involved classroom teachers, mathematicians, and educational researchers from across the country, but later criticized by many people who actually used advanced mathematics for a living, the document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe high-quality mathematics programs. Mathematical equity was a principle previously unseen in the field of mathematics, reflecting the influence of the politics of race, gender, and class of the 1960s and 1970s. Ten general strands or standards of mathematics content and processes were defined that cut across the school mathematics curriculum.

Specific expectations for student learning derived from beliefs of outcome-based education are described for ranges of grades (preschool to 2, 3 to 5, 6 to 8, and 9 to 12). The draft standards and the final standards make explicit goals that all students should learn higher level mathematics, particularly under-served groups such as minorities and women. These standards were made an integral part of nearly all outcome-based education and later standards-based education reform programs that were widely adopted by consensus across the United States by the 2000s.

Previously, de-facto standards had been set by textbook publishers. Mathematics texts were largely devoid of goals such as social justice and race and gender issues (equity). The new mathematics would reflect thinking in education since the activism of the 1960s and 1970s. Across education came the new context of the rise of multiculturalism and affirmative action as the primary goals of education rather than just academic content. This document built on several earlier standards documents produced by the National Council of Teachers of Mathematics [11] — including the Curriculum and Evaluation Standards for School Mathematics (1989), the Professional Standards for Teaching Mathematics (1991), and the Assessment Standards for School Mathematics (1995).

[edit] Dumbing down or raising the bar?

Critics decried the dumbing "down" of mathematics, and called for giving minorities the same standards and instruction which had served previous generations of mathematics and engineering professionals. Reformers pointed to the "basics" as being the dumbed down alternative to teaching representation, relating and communicating higher order thinking skills. The standards introduced new terminology such as mathematical power, which should be given to all students, not merely the successful few who were tracked into technical college majors, and number sense, which would go far beyond memorizing a few traditional computing methods.

As parents and math / science professionals revolted against curricula which in the case of Mathland and Investigations in Number, Data, and Space dispensed with instruction of traditional arithmetic as obsoleted by calculators in favor of writing, cutting, pasting, singing and coloring, the New York Times and Wall Street Journals made "Math Wars" a new headline story. While the standards were widely and nearly universally adopted by the mid-2000s, at the same time many schools, school districts and even states such as California effectively rejected the standards, instead adopting rigorous traditional content and skill based standards and supplementing or replacing standards based curricula with Saxon math and Singapore Math which resulted in much higher test scores. Even the 2006 revision to NCTM guidelines lauded Singapore Math, though they would downplay headlines that that the standards had retreated back towards basic skills.

[edit] Emphasis on finding "correct answers" reduced

Traditional mathematics is often perceived as teaching a single path that leads to a single correct answer. This approach is de-emphasized in the new, higher standards. ^[10]

The NCTM recommends "decreased attention" for "finding exact forms of answers". (5.8.O)

"Although written tests structured around a single correct answer can be reliable measures of performance, they offer little evidence of the kinds of thinking and understanding advocated in the Curriculum Standards." (EVAL.2)

"Students might like mathematics but not display the kinds of attitudes and thoughts identified by this standard. For example, students might like mathematics yet believe that problem solving is always finding one correct answer using the right way. These beliefs, in turn, influence their actions when they are faced with solving a problem. Although such students have a positive attitude toward mathematics, they are not exhibiting essential aspects of what we have termed mathematical disposition." (EVAL.2)

The "decreased attention" statement above is one of many reasons for a bitter conflict between the self-described "traditionalists" and the reformers. The reformers point out that they do not oppose correct answers but would prefer to focus students' attention on the process leading to the answer rather than the answer itself. In a formal evaluation, it is difficult to credit the work of a student who approaches a problem informally and exhibits substantially correct analysis while failing to use, for example, proper terminology or exact computations. The PSSM-supported approach would be to encourage the student to develop his or her arguments and to formalize them in appropriate mathematical language. In more traditional instruction, such student's answer would often be simply marked incorrect or insufficient. It is important to note that PSSM offers only guidelines, along with some exemplary practices in supporting materials. However, the practical implementations of the guidelines sometimes fail in achieving the balance. The programs created in response to the reform have been criticized for overreaching in decreasing attention to some topics to the point of failing to teach them. In contrast, although the more traditional textbook series always claim adherence to and compliance with the Standards, in practice, this amounts to lip service. The situation is complicated further by the fact that any textbook that hopes to be sold nationally must comply not only with the national standards, but also with a host of state standards--often contradictory in both content coverage and pedagogical intent.

The emphasis on analysis rather than the answer is common in professional education, e.g. law school, but many question whether such thinking is appropriate for children in kindergarten who must first memorize their basic facts. Legal problems are often presented without a possibility of a single correct answer but rather encouraging multiple, deep analyses of the factors that might contribute to an answer. As a result, finding a single answer generally results in a lower grade on law school exams. Although this parallel is not perfect, many professionals in mathematics and in education continue to argue that this is a valid approach.

[edit] High school freshman calculus and elementary algebra

According to the 1989 standards: ^[11]

It is "essential that in grades 5-8, students explore algebraic concepts in an informal way." (5-8.9)

In grades 9-12, the mathematics curriculum should include the informal exploration of calculus concept" (9-12.13)

[edit] Traditional content

K-12 math should no longer necessarily cover the same content that has been traditionally taught under the headings "arithmetic", "algebra", and "geometry". According to the NCTM Standards: Introduction:

Calculators and computers have "changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them".

"quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence."

"For many non mathematicians, arithmetic operations, algebraic manipulations, and geometric terms and theorems constitute the elements of the discipline to be taught in grades K-12. This may reflect the mathematics they studied in school or college rather than a clear insight into the discipline itself."

[edit] Technology

The standards put emphasis on using computers and calculators to render much manual calculation, and therefore instruction of such methods obsolete.

"Contrary to the fears of many, the availability of calculators and computers has expanded students' capability of performing calculations. There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations." (Intro)

"Calculators must be accepted at the K-4 level as valuable tools for learning mathematics." (K-4.O)

"Calculators enable children to compute to solve problems beyond their paper-and-pencil skills." (K-4.8)

"The calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses." (5-8.O)

"By assigning computational algorithms to calculator or computer processing, this curriculum seeks not only to move students forward but to capture their interest." (9-12.O)

[edit] Math appreciation and culture

According to the introduction:

The first goal for all students is "they learn to value mathematics". ( Intro) ** See Quotes

"Students should have numerous and varied experiences related to the cultural, historical, and scientific evolution of mathematics so that they can appreciate the role of mathematics in the development of our contemporary society." (Intro)

[edit] Problem Solving

General Problem-Solving Skills is seen as a way to solve problems without requiring teaching or memorizing specific math knowledge, such as common denominators or using a formula to compute an average.

"Problem solving should be the central focus of the mathematics curriculum" (K-4.1)

"Mathematical problem solving, in its broadest sense, is nearly synonymous with doing mathematics." (9-12.1)

"A vital component of problem-solving instruction is having children formulate problems themselves." (K-4.1)

The problem-solving strategies identified by the NCTM for the K-4 level are "using manipulative materials, using trial and error, making an organized list or table, drawing a diagram, looking for a pattern, and acting out a problem." At the 5-8 level the NCTM adds "guess and check". (K-4.1) and (5-8.1)

Critics cite "trial and error" and other content-independent "problem solving" skills as inappropriate ways of applying the fundamental strategy of "discovery learning" to math.

[edit] Mathematical communications

"The communication standard (Standard 2) calls for the integration of language arts as children write and discuss their experiences in mathematics." (K-4.4)

"Students should be encouraged to explain their reasoning in their own words." (5-8.3)

"In mathematics, just as with a building, all students can develop an understanding and appreciation of its underlying structure independent of a knowledge of the corresponding technical vocabulary and symbolism." (9-12.14)

Critics agree communication skills may be even more important than math skills, but they're not math. Some parents worry about when their children writing, cutting, pasting and drawing more than computing. Many standards based exercises such as drawing pie charts and finding patterns in data appear to be Powerpoint rather than mathematics skills.

[edit] Math Wars

The NCTM Standards have led to changes in mathematics textbooks. Perhaps because of the abruptness of these changes, debate over textbooks has sometimes been polarized. This debate is known as the "Math Wars." ^[12]

"Reform" textbooks teach concepts which used to be reserved for advanced students in higher grades, while de-emphasizing procedural skills such as long division. They often require the purchase of, and are dependent on graphic calculators which cost over US $100 since students are no longer expected to be proficient in manual arithmetic. Reform texts favor problem-solving in new contexts over template word problems with corresponding examples. Reform texts emphasize written and verbal communication, working in a group, connections between concepts, connections between representations, activities such as cutting, pasting, and in the case of "Investigations" singing that were once reserved for kindergarten. Some devote so much space in print on "contexts" that the Core-Plus Mathematics Project includes a separate index of contexts with topics ranging from the board-game Monopoly to Nike and rain forests.

The emphasis introducing so many topics so early has been criticized, even by the NCTM, as a curriculum that is a Mile wide and an inch deep. Some experts believe that many topics are introduced too early, though the 1989 standards call for bringing the introduction to algebra as early as elementary school, and calculus in early high school. Core-Plus introduces linear algebra and matrices, once taught in freshman college calculus, as early as junior high school in some districts. Teaching advanced mathematics to all students rather than only the students on the most advanced track may appear to promote equity, but may not be appropriate for students who have not even mastered basic arithmetic. Some integrated math texts have been criticized as covering too many topics in a haphazard sequence, while spending only brief time on topics such as solving linear equations which a traditional algebra class might devote months to deep understanding of a few important single topics.

By contrast, "traditional" textbooks emphasize procedural mathematics, such as arithmetic calculation. They provide step-by-step examples with skill exercises. Unlike texts which have been called Rainforest algebra, they have far fewer pages, and they devote little or no space to real-life contexts such as running shoe companies or geography. The entirety of the first page on matrices in Core-Plus is devoted to information on running shoe companies and their stores, and contains no content about what matrices are. Unlike the standards, texts adapted from Japan or Singapore include students and examples from only a single culture. They do not include historical figures to enhance cultural or gender identity diversity, make no reference to "mathematical power" and contain little or no content with regard to social justice or the equity sought by the standards. However, current traditional textbooks usually include some projects and exercises meant to address the NCTM Standards. Most of the parent and mathematics professional objections in the math wars have been in regard to the dearth or poor quality of mathematical content. In contrast, the lack of diversity, context, or equity laid out by the 1989 standards has mainly objected to by the administrators and officials who have promoted standards based mathematics, and have opposed adoption of more traditional texts such as Saxon math and Singapore math.

One of the subtexts to the debate over the Standards is the rapid development of technology. Has the invention of the calculator made some of the traditional mathematics curriculum obsolete? In the Information Age, are problem solving and communication more valuable than symbolic algebra?

[edit] Innovative curricula

Mathland asks 2nd graders to cut out and paste a Fantasy Lunch.
Investigations in Number, Data and Space does not contain instruction on any traditional computation methods, except to mention that they are to be discouraged. These include regrouping, the standard formulas for computing an average and volume, the standard notation for longhand division, and using "R" to indicate a remainder.

It has no student textbook. It uses 100 charts and skip counting, but not mulitiplication tables to teach multiplication. A study shows that a second grader who used his knowledge of the properties of negative numbers got more accurate results than another who used a traditional borrowing method. The second grade book includes sheet music to the song "happy birthday" which is meant to be sung in several languages although words to other languages are not provided. Decimal math is taught using colored pencils and 10,000 grid chart. Converting to a common denominator to add fractions is not taught.

The introductory chapter for Matrices in the Core-Plus Mathematics Project spends a page explaining about Nike and the running shoe industry. It contains no information on how to solve any of the charting or data problems which follow. A matrix is used as a place to put a data table at the high school level. It has a separate "index of contexts" to non-mathematics topics such as "Monopoly", "Nike" or "global warming".

By contrast, math texts such as Singapore Math and Saxon math contain very little content or methods outside the field of traditional mathematics, and they make little use of sophisticated graphic calculators, or pictures of many diverse cultural or disabled groups.

[edit] NCTM 2006 Recommendations

In 2006, the NCTM released “Curriculum Focal Points,” a report urging that math teaching in kindergarten through eighth grade focus on a few basic skills, largely reversing the controversial stand taken in the landmark 1989 standards document which launched the math wars of the 1990s and 2000s.^[13] Francis Fennell, president of the council played down the degree of change the new report, and said that he resented talk of “math wars.” Interviews of many who were committed to the standards said that, like the 2000 standards, these merely refined and focused rather than renounced the original 1989 recommendations.

Nevertheless, many newspapers like the Chicago Sun Times reported that the "NCTM council has admitted, more or less, that it goofed". The new report cited "inconsistency in the grade placement of mathematics topics as well as in how they are defined and what students are expected to learn." The new recommendations are that students are to be taught the basics, including the fundamentals of geometry and algebra, and memorizing multiplication tables. ^[14]

Many school districts and states are committed to curricula and frameworks based on the now-obsolete mathematics standards which many parents and citizens claim robbed their children of an education in basic arithemetic skills.

[edit] Compared to international standards

American Institutes for Research (AIR) February 7, 2005: "Because topics are mapped out in such a general way, the NCTM requirements risk exposing students to unrealistically advanced mathematics content in the early grades"....Students are exposed to these complicated mathematical topics in kindergarten and first grade at the same time they are learning basic addition and subtraction. Singapore, in contrast, considers algebraic concepts to be advanced rather than introductory mathematics, so algebra is not introduced until the sixth grade. The specificity and logic in Singapore’s spiral approach offer a more effective, better sequenced framework for a mathematical curriculum."^[15] The California mathematics framework is modeled on Singaporean and Japanese frameworks. It is similar to the Singapore framework in that it is organized around a varying set of mathematical topics appropriate to the grades in which they are taught."

[edit] List of those who believe the standards may be harmful to mathematics teaching

Mathematically Correct rated several curricula based on the standards to be unacceptable.
Weapons of Math Destruction
Tacoma, Washington has adopted Saxon Math to supplement / replace standards.
American Institutes for Research (above) found Singapore Math to be superior and that the standards may introduce math too difficult for the grade level.
Prof David Klein (California State University Northridge)

[edit] Humour and culture

Weapons of math destruction is a website of standards-based mathematics humour.

Examples of some cartoons on NCTM math standards and standards-based curricula:

Worker uses a drill to drive a nail. "Should we teach him about the hammer? No, he's got to discover it for himself"

Agent to president "Mr. President, in 24 hours, this school district will be teaching Core-Plus Mathematics Project to thousands of children". Get Jack Bauer Now!

Mom, I can't finish the math homework. We're out of glue.

Teacher has fainted over 6+6+6 = 666. Police "This is why it's called Investigations. It was completely avoidable"

As long as we send home A's, they won't complain about the math.

Investigations in Numbers, Time and Space: Use 3 different methods to tell if 2 is even or odd. Show your work.

[edit] Notes

^ [1] Standards-Based Mathematics Curriculum Materials: A Phrase in Search of a Definition. By Paul R. Trafton, Barbara J. Reys, and Deanna G. Wasman
^ Reform Mathematics vs. the Basics
^ San Francisco Chronicle: Where's the Math?
^ The State's Invisible Math Standards: "With Zacarias' anti-math policies in force..."
^ [http://www.dehnbase.org/hold/ Math Framework in California NCTM "A State Dummies Down", editorial, The Business Journal (Sacramento), 10 April 1995]
^ Joanne Jacobs: Eth-no-mathematics "there's not much math in the new mathematics"
^ [2] TEXAS ADOPTS TEXTBOOK REJECTED BY NATION Adoption of "Rainforest Algebra" appears to contradict this logic
^ [3] David Klein: "This misguided view of women and minorities..."
^ [4] New, New Math = Controversy CBS News 5/28/2000
^ [5] The NCTM Calls it "Math"
^ [6] The NCTM Calls it "Math"
^ [7] Reform Mathematics vs. The Basics: Understanding the Conflict and Dealing with It John A. Van de Walle Virginia Commonwealth University: "Debate has degenerated to 'math wars'"
^ [8] Report Urges Changes in the Teaching of Math in U.S. Schools by TAMAR LEWIN New York Times September 13, 2006
^ [9] Chicago Sun Times "Fuzzy teaching ideas never added up" September 13, 2006
^ http://www.air.org/news/documents/Singapore%20Report%20(Bookmark%20Version).pdf What the United States Can Learn From Singapore’s World-Class Mathematics System

[edit] External links

[12] Standards online. 120-day free access, otherwise the public is required to pay to purchase or view the standards.
[13] Log in for full access to Principles and Standards online
[14]Original 1989 Curriculum and Evaluation Standards
[15] 1991 Professional Standards
[16] 1995 Assessment Standards