The references that follow are organized to match chapters and sections of Benchmarks, which in turn mostly match those of Science for All Americans. The list is very selective and includes only those references that met two criteria. One was relevance—some excellent papers were not included because they did not bear on one of the Benchmarks topics. The other criterion was quality—papers, however relevant, were bypassed if they were seen to have design flaws or their evidence or argument was weak. Even then, however, not all relevant and good papers are included. In many cases, a single paper has been used as representative of a number of similar reports.
It will immediately be clear that mathematics and the physical sciences have had the benefit of many more studies than have other fields. Perhaps that is because the subject matter lends itself to research more easily; in the next few years, though, perhaps the attention to cognitive research will increase in all fields.
The research on Habits of Mind has focused on the development of students' Computation and Estimation skills and Critical-Response skills. There is a rich and growing literature on the development of basic arithmetic skills, on the development of estimation skills, and on students' ability to solve problems that involve proportional reasoning. The development of skills for interpreting evidence has also received considerable research attention.
12a. Values and Attitudes
No applicable research findings.
12b. Computation and Estimation
Research suggests using word problems as a basis for teaching addition and subtraction concepts, rather than teaching computational skills first and then applying them to solve problems (Carpenter & Moser, 1983). Students should be exposed to a large variety of addition and subtraction situations (compare, combine, equalize, change add to, and change taken from) and given opportunities to consider different meanings for the +, -, and = marks. For example, when 9 - 3 = 6 represents the situation "John has 3 cars. Andy has 9 cars. How many more cars does Andy have?" the minus sign means compare rather than take away (Fuson, 1992).
Research has identified a developmental progression of concepts and skills that students use for addition and subtraction (Fuson, 1988; Fuson, 1992). There is some evidence that instruction based on this progression can help (Romberg & Carpenter, 1986). For example, after a year-long instruction based on this progression, 2nd-graders could solve almost all add or subtract problems with sums up to 18 (Fuson & Willis, 1989).
Students make a variety of errors in multi-digit addition and subtraction calculations (Brown & Van Lehn, 1982). Given traditional instruction, a substantial number of 4th- and 5th-graders are not able to subtract some whole numbers successfully (Fuson, 1992). Student errors suggest students interpret and treat multi-digit numbers as single-digit numbers placed adjacent to each other, rather than using place-value meanings for the digits in different positions (Fuson, 1992). With specially designed instruction, 2nd-graders are able to understand place value and to add and subtract four-digit numbers more accurately and meaningfully than 3rd-graders receiving traditional instruction (Fuson, 1992). Research also suggests students interpret multiplication of whole numbers mainly as repeated addition. This interpretation is inadequate for many multiplication problems and can lead to restrictive intuitive notions such as "multiplication always makes larger" (Greer, 1992).
Elementary- and middle-school students make several errors when they operate on decimals and fractions (Benander & Clement, 1985; Kouba et al., 1988; Peck & Jencks, 1981; Wearne & Hiebert, 1988). For example, many middle-school students cannot add 4 + 0.3 correctly or 7 1/6 + 3 1/2 (Kouba et al., 1988; Wearne & Hiebert, 1988). These errors are due in part to the fact that students lack essential concepts about decimals and fractions and have memorized procedures that they apply incorrectly. Interventions to improve concept knowledge can lead to increased ability by 5th-graders to add and subtract decimals correctly (Wearne & Hiebert, 1988).
Students of all ages misunderstand multiplication and division (Bell et al., 1984; Graeber & Tirosh, 1988; Greer, 1992). Commonly held misconceptions include "multiplication always makes larger," "division always makes smaller," "the divisor must always be smaller than the dividend." Students may correctly select multiplication as the operation needed to calculate the cost of gasoline when the amount and unit cost are integers, then select division for the same problem when the amount and unit cost are decimal numbers (Bell et al., 1981). Numerous suggestions have been made to improve student concepts of multiplication (Greer, 1992), but further research is needed to determine how effective these suggestions will be in the classroom.
Lower middle-school students may have difficulties understanding the relationship between fractions and decimal numbers (Markovits & Sowder, 1991). They may think that fractions and decimals can occur together in a single expression, like 0.5 + 1/2, or they might believe that they must not change from one representation to the other (from 1/2 to 0.5 and back) within a given problem. Instruction that focuses on the meaning of fractions and decimals forms a basis on which to build a good understanding of the relationship between fractions and decimals. Instruction that merely shows how to translate between the two forms does not provide a conceptual base for understanding the relationship (Markovits & Sowder, 1991).
Lower elementary students do not have procedures to compare the size of whole numbers. By 4th grade, students generally have no difficulty comparing the sizes of whole numbers up to four digits (Sowder, 1992). Students are less successful when the number of digits is much larger or when more than two numbers are to be compared. This might be due to increased memory requirements of working with more or larger numbers (Sowder, 1988). Upper elementary- and middle-school students taught traditionally cannot successfully compare decimal numbers (Sowder, 1988, 1992a). Rather they overgeneralize the features of the whole number system to the decimal numbers (Resnick et al., 1989). They apply a "more digits make bigger" rule (according to which .1814 > .385). After specially designed instruction which develops good meanings for decimal symbols, many students are able to compare decimal numbers with understanding by 5th grade (Wearne & Hiebert, 1988). Upper elementary- and middle-school students taught traditionally cannot compare fractions successfully (Sowder, 1988). Students' difficulties here indicate they treat the numerator and the denominator separately. Specially designed instruction to teach meanings for fractions can help to improve ordering fractions by as early as the end of the 5th grade (Behr et al., 1984).
The use of calculators in K–12 mathematics does not hinder the development of basic computation skills and frequently improves concept development and paper-and-pencil skills, both in basic operations and in problem solving (Hembree & Dessart, 1986; Kaput, 1992). The use of calculators in testing produces higher scores than paper-and-pencil efforts in problem solving as well as in basic operations (Hembree & Dessart, 1986).
Good estimators use a variety of estimating tactics and switch easily between them. They have a good understanding of place value and the meaning of operations, and they are skilled in mental computation. Poor estimators rely on algorithms that are more likely to yield the exact answer. They lack an understanding of the notion and value of estimation and often describe it as "guessing" (Sowder, 1992b). Before 6th grade, students develop very few estimation skills from their natural experiences (Case & Sowder, 1990; Sowder, 1992b). As a result, some researchers caution that teaching estimation to young children may have as its single effect that they master specific procedures in a superficial manner (Sowder, 1992b).
Early adolescents and also many adults have difficulty with proportional reasoning (Behr, 1987; Hart, 1988). Difficulty is influenced by the problem format, the particular numbers in the problem, the types of ratios used, and the problem situation (Heller et al., 1989; Karplus et al., 1983; Tournaire & Pulos, 1985; Vergnaud, 1988). Middle-school students can solve problems in proportions that involve simple numbers and simple wordings (Vergnaud, 1988), but troubles appear with more difficult numerical values or problem contexts. Problems using 2:1 ratios are easier than problems using n:1 ratios, and can be solved by elementary-school children (Shayer & Adey, 1981). Problems using n:1 ratios are easier than problems using other integer ratios (e.g., 6/2) which in turn are easier than problems using non-integer ratios (e.g., 6/4) (Tournaire & Pulos, 1985). Different ratio types (e.g., speed, exchange, mixture) appear to give more or less difficulty. For example, speed problems appear to be more difficult than exchange problems (Heller et al., 1989; Vergnaud, 1988). And these difficulties compound one another. Unfamiliarity with the problem situation causes even more difficulty when it occurs with a difficult ratio type (Heller et al., 1989).
12c. Manipulation and Observation
Upper elementary- and middle-school students who can use measuring instruments and procedures when asked to do so often do not use this ability while performing an investigation. Typically a student asked to undertake an investigation and given a set of equipment that includes measuring instruments will make a qualitative comparison even though she might be competent to use the instruments in a different context (Black, 1990). It appears students often know how to take measurements but not what or when.
12d. Communication Skills
No applicable research findings.
12e. Critical-Response Skills
Upper elementary-school students can reject a proposed experimental test where a factor whose effect is intuitively obvious is uncontrolled, at the level of "that's not fair" (Shayer & Adey, 1981). "Fairness" develops as an intuitive principle as early as 7 to 8 years of age and provides a sound basis for understanding experimental design. This intuition does not, however, develop spontaneously into a clear, generally applicable procedure for planning experiments (Wollman, 1977a, 1977b; Wollman & Lawson, 1977). Although young children have a sense of what it means to run a fair test, they frequently cannot identify all of the important variables, and they are more likely to control those variables that they believe will affect the result. Accordingly, student familiarity with the topic of the given experiment influences the likelihood that they will control variables (Linn & Swiney, 1981; Linn, et al., 1983). After specially designed instruction, students in 8th grade are able to call attention to inadequate data resulting from lack of controls (see for example Rowell & Dawson, 1984; Ross, 1988).
Middle-school students tend to invoke personal experiences as evidence to justify a particular hypothesis. They seem to think of evidence as selected from what is already known or from personal experience or second-hand sources, not as information produced by experiment (Roseberry et al., 1992). Most 6th-graders can judge whether evidence is related to a theory, although they do not always evaluate this evidence correctly (Kuhn et al., 1988). When asked to use evidence to judge a theory, students of all ages may make only theory-based responses with no reference made to the presented evidence. Sometimes this appears to be because the available evidence conflicts with the students' beliefs (Kuhn et al., 1988).
Students of all ages show a tendency to uncritically infer cause from correlation (Kuhn et al., 1988). Some students think even a single co-occurrence of antecedent and outcome is always sufficient to infer causality. Rarely do middle-school students realize the indeterminacy of single instances, although high-school students may readily realize it. Despite that, as covariant data accumulate, even high-school students will infer a causal relation based on correlations. Further, students of all ages will make a causal inference even when no variation occurs in one of the variables. For example, if students are told that light-colored balls are used successfully in a game, they seem willing to infer that the color of the balls will make some difference in the outcome even without any evidence about dark-colored balls (Kuhn et al., 1988).
Faced with no correlation of antecedent and outcome, 6th-graders only rarely conclude that the variable has no effect on the outcome. Ninth-graders draw such conclusions more often. A basic problem appears to be understanding the distinction between a variable making no difference and a variable that is correlated with the outcome in the opposite way than the students initially conceived (Kuhn et al., 1988).
Most high-school students will accept arguments based on inadequate sample size, accept causality from contiguous events, and accept conclusions based on statistically insignificant differences (Jungwirth & Dreyfus, 1990, 1992; Jungwirth, 1987). More students can recognize these inadequacies in arguments after prompting (for example, after being told that the conclusions drawn from the data were invalid and asked to state why) (Jungwirth & Dreyfus, 1992; Jungwirth, 1987).
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