Principles and Standards for School Mathematics

Principles and Standards for School Mathematics is a document produced in 1989 by the National Council of Teachers of Mathematics [10] (NCTM) to set forth a national vision for precollege mathematics education in the US and Canada. The controversial document would be largely adopted by most education agencies from local to federal levels by the mid 2000s and also serve as a basis for most states' mathematics standards, as well as an influence on standards in other nations. It would later be fiercely opposed by many parents and mathematics professionals, and rejected by many states and school districts who complained about replacing instruction in arithmetic with writing, coloring, counting, and inventing mathematics unrecognizable to any previous generation of mathematicians or educators.

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.

Mathematics in this style have also been called "standards-based" instruction or "standards-based mathematics, , or simply "reform mathematics".

Less favorable terminology which have appeared in press and web articles include fuzzy math, "Where's the math", "anti-math", "math for dummies", "no-math mathematics", , rainforest algebra , "Math for women and minorities, and "new new math".

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

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).

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 hould 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 curriculums 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.

Traditional computation instruction

Although the standards are said to not have called for actual elimination of instruction of traditional arithmetic, the NCTM published a paper, The Harmful Effects of Algorithms in Grades 1-4 by Constance Kamii which was widely cited by other researchers and papers which advocated that such instruction was actually harmful. It's Time To Abandon Computational Algorithms is another paper that called for abolishing of the teaching of traditional arithmetic as harmful. Some of the curricula inspired by the NCTM standards such as Investigations in Numbers, Data, and Space and Mathland which have received the most severe reviews from Mathematically Correct are consistent with this view in that they contain very little in the way of direct instruction of traditional computing methods. They may direct students to invent their own methods such as using scissors and paste, skip counting on a 100s chart or using colored pencils on a 10,000s chart, which is said to increase student understanding and proficiency.

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.

• 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. Nevertheless, many standards-based curricula such as Investigations not only omit any instruction in traditional arithmetic, but instruct teachers to discourage use of traditional methods even if the child is already familiar and proficient in such methods. Many studies show that students who do not memorize basic facts and arithmetic do even worse when asked to construct their own computation methods based on weak strategies such as counting and coloring.

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, the original PSSM guidelines would appear to require 2nd graders to show the thinking skills of polished professional mathematicians, yet many professionals in mathematics and in education continue to argue that this is a valid approach.

High school freshman calculus and elementary algebra

According to the 1989 standards:

• 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)

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."

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)

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)

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

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.

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."

"Reform" textbooks teach concepts which used to be reserved for advanced students in higher grades, while de-emphasizing procedural skills such as long division. 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.

By contrast, "traditional" textbooks emphasize procedural mathematics, such as arithmetic calculation. They provide step-by-step examples with skill exercises, and devote little or no space to real-life contexts such as running shoe companies or geography. 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.

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?

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.

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. 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.

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.

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." 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."

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)

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 3 is even or odd. Show your work.

External links

• [12] Standards online, 120-day free access, otherwise payment required to purchase or view.
• [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

See also

• Education in the United States
• Mathematically Correct