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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.
Advocates of reform
The largest supporter of reform in the US has been the National Council of Teachers of Mathematics.
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 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.
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).
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 is criticized for not containing explicit instruction of the standard algorithms
- Core-Plus Mathematics Project, initially accused of placing students in remedial college math courses, a report that was later challenged.
- Connected Mathematics, 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 w
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Answers:Dunno what she means by justify either but just add the x's to get 22x-4 and common factor I guess if you want to go further to get 2(11x-2) solve for x maybe? 2(11x-2)=0 x=2/11
Answers:1) A 2) B 3) A 4) C 5) C 6) D
Answers:b d a c
Answers:Expand: (x - 4)(x + 8) x + 8x - 4x -42 Simplify: x + 4x - 32 I think FOIL is a confusing way to remember it. Look at the following picture: http://i35.tinypic.com/a49yrb.png See the curved lines? To me, they look like a crab claw. That is how I remember how to expand: just draw a crab claw.