Math wars

The largest supporter of reform in the US has been the National Council of Teachers of Mathematics.[4]

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.[1] Some parents and other stakeholders blame educators saying that failures occur 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.

Those who disagree with the inquiry-based philosophy maintain that students must first 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.[1]

Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language.[5] 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[6] 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 that 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.[7] Commentators have argued that there is philosophical support for the notion that "algorithmic fluency" requires the very types of cognitive activity whose promotion reform advocates often claim is their approaches' unique virtue.[8] However, such arguments assume that reformers do not want to teach the standard algorithms, which is a common misunderstanding of the reform position.

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 parents have accused reform math advocates of deliberately slowing down students with greater ability in order to "paper-over" the inequalities of the American school system. 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). College professors and employers have sometimes claimed that students that have been taught using reform curricula do not possess basic mathematical skills. One study found that, although first-grade students in 1999 with an average or above-average aptitude for math did equally well with either teacher-directed or student-centered instruction, first-grade students with mathematical difficulties did better with teacher-directed instruction.[9]

Examples of reform curricula introduced in response to the 1989 NCTM standards and the reasons for initial criticism:

• Mathland (no longer offered)
• Investigations in Numbers, Data, and Space, criticized for not containing explicit instruction of the standard algorithms
• Core-Plus Mathematics Project, criticized for failing "to convey critical mathematical concepts and ideas that should and can be within reach for high school students",[10] downplaying "algebraic structure and skills" and inability to build geometry up from foundations in a mathematically sound and coherent way".[11]
• Connected Mathematics, criticized for not explicitly teaching children standard algorithms, formulas or solved examples
• Everyday Math, also known as "Fuzzy Math",[12] criticized for putting emphasis on non-traditional arithmetic methods.

Critics of reform textbooks say that they present concepts in a haphazard way.[13] Critics of the reform textbooks and curricula support methods such as Singapore math, which emphasizes direct instruction of basic mathematical concepts, and Saxon math, which emphasizes frequent cumulative review.

Reform educators have responded by pointing out that research[14][15][16] 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.

In 2000 the National Council of Teachers of Mathematics (NCTM) 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,[17] which was seen by many as a compromise position. In 2008, the National Mathematics Advisory Panel, created by George W. Bush, called for a halt to all extreme positions.

• Anti-racist mathematics, efforts to reduce racial bias in mathematics pedagogy
• David Klein
• Education in the United States
• Jo Boaler
• Marian Small
• New Math – a reform in mathematics education in western countries during the late 1950s to the early 1970s
• Three-part lesson