(V) Implications of Research Items I, II, III, IV for School Curricula
"Changes in emphases require more than simple adjustments in the amount of time to be devoted to individual topics; they also will mean changes in emphases within topics. For example, although students should spend less time simplifying radicals and manipulating rational exponents, they should devote more time to exploring examples of exponential growth and decay that can be modeled using algebra."
- Taken from the National Council for Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (1989, p. 151)
The Standards document referred to in the quotation on the left-hand page has been the primary reference in all curricula revisions made in North America since its publication in 1989.
The flavor of the quotation on the left-hand page is characteristic of the whole document. Throughout the K-12 curricula, this document recommends an increased emphasis on problem solving and a decreased emphasis on numeric and symbolic manipulation. This is not necessarily a bad thing per se, but taken to the extreme, as indeed it has been in the implementation of the new curricula, it is disastrous.
The reason for the disastrous effect upon the learning of mathematics comes from cognitive psychology. The following excerpt from "Teaching the Rules of Exponents: A Resource-Based Approach" by May, Rabinowitz and Mantyka, accounts for this outcome.
Norman and Shallice (1986) provided a fairly simple framework to deal with mathematical problem solving and the relationship between reasoning and remembering. They suggested that the supervisory attentional mechanism is the limited resource and assume that deliberate attentional resources are employed in tasks that involve planning or decision making, troubleshooting, ill-learned or novel sequences, dangerous or technically difficult components, and the inhibition of strong habitual responses . . . . Perhaps all the attention-demanding characteristics are involved in solving difficult mathematical problems.
Norman and Shallice also suggested that the automatization of skills plays an important role in problem-solving tasks which require attentional resources. Within the context of resource theories, skill seems to represent any mental or physical response that is performed in completing a task. It is assumed that fewer resources are required as skills become more rapidly executed and that over-learned skills become automatic and no longer require resources. Automatic skill use is thought to be implicit, rather than explicit, and, therefore, less likely to be forgotten (see Berry & Dienes, 1993). One way this could happen is that over-learned skills could be directly retrieved from memory rather than constructed (Logan, 1988). For example, with sufficient practice most children know that 2 + 3 = 5 without needing to count and many high school students recognize that without multiplying the terms in the expression. At a more advanced level, a skilled high school student would automatically add the exponents, rather than first retrieving and then applying the addition rule when presented with . Automaticity is probably the most appealing resource-based concept from the perspective of a mathematics educator because over time, a child and later, a young adult, gets to practice a large number of skills repeatedly. Some of these highly practiced skills (e.g., adding, subtracting, multiplying, dividing, simplifying fractions, factoring) are useful in a large number of mathematical tasks and should be associated with developing expertise. The absence or slow execution of many of these basic skills in our remedial mathematics students is certainly an indicant that something went wrong. (May, Rabinowitz and Mantyka, 2001, pp. 68-69)
These paragraphs highlight two major flaws in teaching mathematics the way the Curriculum and Evaluation Standards for School Mathematics advocates.
Firstly, in curricula guided by the Standards document, new skills are taught in the context of problem solving. This directly contravenes the principles of cognitive psychology outlined in paragraph one of the quotation (on p. 67). Because new skills represent either novel or ill-learned sequences that also may be technically difficult, they place unusual demands on capacity. Thus, if a new skill is being taught, it is best to minimize other demands until that skill is mastered. For example, word problems should not be introduced until the relevant mathematical skill(s) is/are mastered.
Secondly, in de-emphasizing numeric and symbolic manipulation, skills are not being practiced enough to achieve automaticity. Demands on attentional resources are less if the level of relevant skills is higher. Therefore, skill enhancement should reduce demands on capacity and foster complex problem solving. Three ways in which relevant skills can be improved are: practice, the substitution of efficient skills for cumbersome skills, and the use of shortcuts to reduce steps. None of these is emphasized in modern curricula guided by the Curriculum and Evaluation Standards for School Mathematics.
For teachers and students alike, this is easily corrected. Teachers must require more practice of basic skills outside the context of problem solving, and students must be willing to do the practice.