Math wars
Encyclopedia
Math wars is the debate over modern mathematics education
, textbooks and curricula in the United States
that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics
(NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.
While the discussion about math skills has persisted for many decades, the term "math wars" was coined by commentators such as John A. Van de Walle and David Klein
. The debate is over traditional mathematics
and reform mathematics
philosophy and curricula, which differ significantly in approach and content.
.
One aspect of the debate is over how explicitly children must be taught skills based on formulas or algorithms (fixed, step-by-step procedures for solving math problems) versus a more inquiry-based approach in which students are exposed to real-world problems that help them develop fluency in number sense, reasoning, and problem-solving skills. In this latter approach, conceptual understanding is a primary goal and algorithmic fluency is expected to follow secondarily.
A considerable body of research by mathematics educators has generally supported reform mathematics and has shown that children who focus on developing a deep conceptual understanding (rather than spending most of their time drilling algorithms) develop both fluency in calculations and conceptual understanding. Advocates explain failures not because the method is at fault, but because these educational methods require a great deal of expertise and have not always been implemented well in actual classrooms.
A backlash which advocates call "poorly understood reform efforts" and critics call "a complete abandonment of instruction in basic mathematics" resulted in "math wars" between reform and traditional methods of mathematics education.
develop computational skills before they can understand concepts of mathematics. These
skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method. In this view, estimating answers is insufficient and, in fact, is considered to be dependent on strong foundational skills. Learning abstract concepts of mathematics is perceived to depend on a solid base of knowledge of the tools of the subject.
Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language. Student innovation is acceptable, even welcome, as long as it is mathematically valid. Calculator use can be appropriate after number sense has developed and basic skills have been mastered. Constructivist methods which are unfamiliar to many adults, and books which lack explanations of methods or solved examples make it difficult to help with homework. Compared to worksheets which can be completed in minutes, constructivist activities can be more time consuming. (Reform educators respond that more time is lost in reteaching poorly understood algorithms.) Emphasis on reading and writing also increases the language load for immigrant students and parents who may be unfamiliar with English.
Critics of reform point out that traditional methods are still universally and exclusively used in industry and academia. Reform educators respond that such methods are still the ultimate goal of reform mathematics, and that students need to learn flexible thinking in order to face problems they may not know a method for. Critics maintain that it is unreasonable to expect students to "discover" the standard methods through investigation, and that flexible thinking can only be developed after mastering foundational skills.
Some curricula incorporate research by Constance Kamii
and others that concluded that direct teaching of traditional algorithms is counterproductive to conceptual understanding of math. Critics have protested some of the consequences of this research. Traditional memorization methods are replaced with constructivist activities. Students who demonstrate proficiency in a standard method are asked to invent another method of arriving at the answer. Some teachers supplement such textbooks in order to teach standard methods more quickly. Some curricula do not teach long division. Critics believe the NCTM revised its standards to explicitly call for continuing instruction of standard methods, largely because of the negative response to some of these curricula (see below).
Critics of reform textbooks say that they present concepts in a haphazard way. Critics of the reform textbooks and curricula support traditional textbooks such as Singapore Math, which emphasizes direct instruction of basic mathematical concepts, and Saxon math, which emphasizes perpetual drill.
Reform educators have responded by pointing out that research tends to show that students achieve greater conceptual understanding from standards-based curricula than traditional curricula and that these gains do not come at the expense of basic skills. In fact students tend to achieve the same procedural skill level in both types of curricula as measured by traditional standardized tests. More research is needed, but the current state of research seems to show that reform textbooks work as well as or better than traditional textbooks in helping students achieve computational competence while promoting greater conceptual understanding than traditional approaches.
released the Principles and Standards for School Mathematics (PSSM)
, which was seen as more balanced than the original 1989 Standards. This led to some calming, but not an end to the dispute. Two recent reports have led to considerably more cooling of the Math Wars. In 2006, NCTM released its Curriculum Focal Points,, which was seen by many as a compromise position. In 2008, the National Mathematics Advisory Panel, created by George Bush, called for a halt to all extreme positions.
, which are being adopted by most of the United States.
. The National Math Panel examined and summarized the scientific evidence related to the teaching and learning of mathematics, concluding in their 2008 report, "All-encompassing recommendations that instruction should be entirely 'student centered' or 'teacher directed' are not supported by research. If such recommendations exist, they should be rescinded. If they are being considered, they should be avoided. High-quality research does not support the exclusive use of either approach." The Panel effectively called for an end to the Math Wars, concluding that research showed "conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided."
The Panel's final report met with significant criticism within the mathematics education community for, among other issues, the selection criteria used to determine "high-quality" research, their comparison of extreme forms of teaching, and the amount of focus placed on algebra.
Mathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
, textbooks and curricula in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics
National Council of Teachers of Mathematics
The National Council of Teachers of Mathematics was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally....
(NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.
While the discussion about math skills has persisted for many decades, the term "math wars" was coined by commentators such as John A. Van de Walle and David Klein
David Klein (California State University Northridge)
David Klein is a professor of Mathematics at California State University in Northridge. He is an advocate of increasingly rigorous treatment of mathematics in school curricula and a frequently cited opponent of reforms based on the NCTM standards...
. The debate is over traditional mathematics
Traditional mathematics
Traditional mathematics is a term used to describe the predominant methods of Mathematics education in the United States in the early-to-mid 20th century. The term is often used to contrast historically predominant methods with non-traditional approaches to math education...
and reform mathematics
Reform mathematics
Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics . The NCTM document, Curriculum and Evaluation Standards for School Mathematics, attempted to set forth a vision...
philosophy and curricula, which differ significantly in approach and content.
Advocates of reform
The largest supporter of reform in the US has been the National Council of Teachers of MathematicsNational Council of Teachers of Mathematics
The National Council of Teachers of Mathematics was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally....
.
One aspect of the debate is over how explicitly children must be taught skills based on formulas or algorithms (fixed, step-by-step procedures for solving math problems) versus a more inquiry-based approach in which students are exposed to real-world problems that help them develop fluency in number sense, reasoning, and problem-solving skills. In this latter approach, conceptual understanding is a primary goal and algorithmic fluency is expected to follow secondarily.
A considerable body of research by mathematics educators has generally supported reform mathematics and has shown that children who focus on developing a deep conceptual understanding (rather than spending most of their time drilling algorithms) develop both fluency in calculations and conceptual understanding. Advocates explain failures not because the method is at fault, but because these educational methods require a great deal of expertise and have not always been implemented well in actual classrooms.
A backlash which advocates call "poorly understood reform efforts" and critics call "a complete abandonment of instruction in basic mathematics" resulted in "math wars" between reform and traditional methods of mathematics education.
Critics of reform
Those who disagree with the inquiry-based philosophy maintain that students must firstdevelop computational skills before they can understand concepts of mathematics. These
skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method. In this view, estimating answers is insufficient and, in fact, is considered to be dependent on strong foundational skills. Learning abstract concepts of mathematics is perceived to depend on a solid base of knowledge of the tools of the subject.
Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language. Student innovation is acceptable, even welcome, as long as it is mathematically valid. Calculator use can be appropriate after number sense has developed and basic skills have been mastered. Constructivist methods which are unfamiliar to many adults, and books which lack explanations of methods or solved examples make it difficult to help with homework. Compared to worksheets which can be completed in minutes, constructivist activities can be more time consuming. (Reform educators respond that more time is lost in reteaching poorly understood algorithms.) Emphasis on reading and writing also increases the language load for immigrant students and parents who may be unfamiliar with English.
Critics of reform point out that traditional methods are still universally and exclusively used in industry and academia. Reform educators respond that such methods are still the ultimate goal of reform mathematics, and that students need to learn flexible thinking in order to face problems they may not know a method for. Critics maintain that it is unreasonable to expect students to "discover" the standard methods through investigation, and that flexible thinking can only be developed after mastering foundational skills.
Some curricula incorporate research by Constance Kamii
Constance Kamii
Dr. Constance Kamii is a Professor, Early Childhood Education ProgramDepartment of Curriculum and Instruction at the University of Alabama at Birmingham.-Overview:...
and others that concluded that direct teaching of traditional algorithms is counterproductive to conceptual understanding of math. Critics have protested some of the consequences of this research. Traditional memorization methods are replaced with constructivist activities. Students who demonstrate proficiency in a standard method are asked to invent another method of arriving at the answer. Some teachers supplement such textbooks in order to teach standard methods more quickly. Some curricula do not teach long division. Critics believe the NCTM revised its standards to explicitly call for continuing instruction of standard methods, largely because of the negative response to some of these curricula (see below).
Reform curricula
Examples of reform curricula introduced in response to the 1989 NCTM standards and the reasons for initial criticism:- MathlandMathlandMathLand was one of many controversial mathematics curricula that were designed around the 1989 NCTM standards. It was developed and published by Creative Publications and was initially adopted by the U.S. state of California and schools run by the US Department of Defense by the mid 1990s...
(no longer offered) - Investigations in Numbers, Data, and SpaceInvestigations in Numbers, Data, and SpaceInvestigations in Number, Data, and Space is a K-5 mathematics curriculum, developed at in Cambridge, Massachusetts, USA. The curriculum is often referred to as Investigations or simply TERC. Patterned after the NCTM standards for mathematics, it is among the most widely used of the new reform...
is criticized for not containing explicit instruction of the standard algorithms - Core-Plus Mathematics ProjectCore-Plus Mathematics ProjectThe Core-Plus Mathematics Project is one of the five NCTM-standards-based high school mathematics curriculum development projects funded by the National Science Foundation. The project has developed, tested, and published a four-year comprehensive high school mathematics textbook series...
, initially accused of placing students in remedial college math courses, a report that was later challenged. - Connected MathematicsConnected MathematicsConnected Mathematics is a comprehensive, problem-centered curriculum designed for all students in grades 6-8 based on the NCTM standards. The curriculum was developed by the at Michigan State University and funded by the National Science Foundation....
, criticized for not explicitly teaching children standard algorithms, formulas or solved examples - Everyday Math , criticized for putting emphasis on non-traditional arithmetic methods.
Critics of reform textbooks say that they present concepts in a haphazard way. Critics of the reform textbooks and curricula support traditional textbooks such as Singapore Math, which emphasizes direct instruction of basic mathematical concepts, and Saxon math, which emphasizes perpetual drill.
Reform educators have responded by pointing out that research tends to show that students achieve greater conceptual understanding from standards-based curricula than traditional curricula and that these gains do not come at the expense of basic skills. In fact students tend to achieve the same procedural skill level in both types of curricula as measured by traditional standardized tests. More research is needed, but the current state of research seems to show that reform textbooks work as well as or better than traditional textbooks in helping students achieve computational competence while promoting greater conceptual understanding than traditional approaches.
Recent developments
In 2000 the National Council of Teachers of Mathematics (NCTM)National Council of Teachers of Mathematics
The National Council of Teachers of Mathematics was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally....
released the Principles and Standards for School Mathematics (PSSM)
Principles and Standards for School Mathematics
Principles and Standards for School Mathematics are guidelines produced by the National Council of Teachers of Mathematics in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada...
, which was seen as more balanced than the original 1989 Standards. This led to some calming, but not an end to the dispute. Two recent reports have led to considerably more cooling of the Math Wars. In 2006, NCTM released its Curriculum Focal Points,, which was seen by many as a compromise position. In 2008, the National Mathematics Advisory Panel, created by George Bush, called for a halt to all extreme positions.
NCTM 2006 recommendations
In 2006, the NCTM released Curriculum Focal Points, a report on the topics considered central for mathematics in pre-kindergarten through eighth grade. Its inclusion of standard algorithms led editorials in newspapers like the Chicago Sun Times to state that the "NCTM council has admitted, more or less, that it goofed," and that 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." NCTM responded by insisting that it considers "Focal Points" a step in the implementation of the Standards, not a reversal of its position on teaching students to learn foundational topics with conceptual understanding. Francis Fennell, president of the NCTM, stated that there had been no change of direction or policy in the new report and said that he resented talk of “math wars”. The Focal Points were one of the documents consulted to create the new national Common Core StandardsCommon core state standards initiative
The Common Core State Standards Initiative is a U.S. education initiative that seeks to bring diverse state curricula into alignment with each other by following the principles of standards-based education reform. The initiative is sponsored by the National Governors Association and the Council...
, which are being adopted by most of the United States.
National Mathematics Advisory Panel
On April 18, 2006, President Bush created the National Mathematics Advisory Panel, which was modeled after the influential National Reading PanelNational Reading Panel
The National Reading Panel was a United States government body. Formed in 1997 at the request of Congress, it was a national panel with the stated aim of assessing the effectiveness of different approaches used to teach children to read....
. The National Math Panel examined and summarized the scientific evidence related to the teaching and learning of mathematics, concluding in their 2008 report, "All-encompassing recommendations that instruction should be entirely 'student centered' or 'teacher directed' are not supported by research. If such recommendations exist, they should be rescinded. If they are being considered, they should be avoided. High-quality research does not support the exclusive use of either approach." The Panel effectively called for an end to the Math Wars, concluding that research showed "conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided."
The Panel's final report met with significant criticism within the mathematics education community for, among other issues, the selection criteria used to determine "high-quality" research, their comparison of extreme forms of teaching, and the amount of focus placed on algebra.
External links
- An A-Maze-ing Approach To Math by BARRY GARELICK A parent outlines how "the National Science Foundation (NSF) promoted the NCTM standards beginning in 1991 and awarded millions of dollars in grant money for the writing of math texts that embraced them and to state boards of education whose math standards aligned with them"