Recommendations for the Teaching of Mathematics with an Emphasis on "Catching Up"
We assume that the intervention principles
(provision of a broad spectrum of services, flexibility, the
development of mutual respect and trust, and provision of services
that are coherent and easy to use), must be primary in the
implementation of these programs. Below, the cognitive principles,
abstracted from the literature review, which we believe should be
implemented in any mathematics curriculum are enumerated.
The student must understand the skill being taught in order to transfer that skill to new contexts.
The teacher must recognize that more than one skill is required to solve most problems. Although the solution to some problems (e.g., 2 + 3) can be retrieved directly from memory, a minimum of three skills is usually necessary: the relevant data must be recorded either on paper or in memory; at least one rule must either be retrieved from memory or constructed; and at least one rule must be applied to the data.
Capacity is used in and limits performance. In general, the more information and/or skills that are required to solve a problem, the more likely that capacity will be exhausted and performance will deteriorate. Klein and Bisanz (1999) recently reported an experiment which illustrates this principle. They found that 4-year old children's error rates on non-verbal arithmetic problems were closely related to the maximum number of units that needed to be held in memory to solve each of the problems (r2 = .88).
Because new skills represent either novel or ill-learned sequences which 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.
Because capacity varies across individuals, there will be individual differences in learning rates. Thus, whenever possible, individualized instructional programs are preferable. If a student is having difficulty learning a new mathematical skill, it is particularly important to analyze the skill and break it into components.
Capacity also varies across groups. For example, capacity is positively correlated with age during childhood (Rabinowitz et al., under review) and, in mathematical contexts, negatively correlated with mathematical anxiety (Ashcraft & Kirk, in press). Therefore, different instructional programs should be used for younger and older children, and for students with low and high mathematical anxiety.
Skills are generally used in a fixed order in attempting to solve a problem. Capacity is more likely to be exhausted and, therefore, errors will occur more frequently with skills implemented later in the sequence. Thus, skills may appear to be error prone in some sequences, but effective in others.
If the reasoning/remembering tradeoff observed with class inclusion is also characteristic of mathematical problem solving, then the most likely problem solving sequence is: remember the data, either retrieve or construct the relevant rules, apply the rules. If this assumption is correct, it is particularly important to teach new skills using minimal data sets.
Demands on capacity are negatively correlated with the level of relevant skills. 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. It is assumed that practice is associated with faster and more accurate skill use and that sufficient practice produces automatization (i.e., skill use does not require capacity). Since practice is often undervalued by contemporary curriculum developers, demonstrations that differential amounts of practice over more than a decade facilitates arithmetic skills (Campbell and Xue, in press) and concentrated practice facilitates complex problem solving (Haverty, 1999) are important to our arguments. Campbell and Xue presented 72 students registered in undergraduate and graduate programs 360 arithmetic questions. Participants were required to add three 1- or 2-digit numbers, divide 2- or 3-digit numbers by single-digit numbers, subtract 2-digit numbers from 2-digit numbers, and multiple 2-digit numbers by 1-digit numbers. The number of these problems participants successfully completed in 15 minutes was predicted (R2 = .59, p < .001) by reported calculator use before entering university (β = 1.29, p = .005) and mean reaction time to answer simple arithmetic (e.g., 4 x 3) questions (β = 170, p < .001). Presumably early practice in single-digit arithmetic lead to faster access to basic facts which transferred to more complex arithmetic while access to calculators reduced the number of opportunities to practice more complex arithmetic. Haverty trained seventh graders to generate either the 17 or 19 multiplication tables, but not both the 17 and 19 tables. Following their mastery of these facts, the children were presented inductive reasoning problems at a variety of difficulty levels by providing them with tables, each of which contained six x, y number pairs, and requiring them to describe the mathematical relationship between the variables represented by the numbers in each table. The prior learning of the relevant multiplication facts facilitated solving the most difficult problems [e.g., y = 17 (x + 1)]. Haverty's finding is consistent with the assertion that mathematical discovery is dependent on prior skill level. We believe that skill development usually is a consequence of practice over extended time periods that results in both the availability of and speedy access to facts and procedures.
An excerpt from Teaching Remedial
Mathematics at the University: Rationale, Principles, Procedures,
and Outcomes (full text of