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.
Students' understanding of the Mathematical World has been extensively researched. The Handbook of Research in Mathematics Teaching and Learning. (Grouws, 1992) as well as the papers presented at the "research agenda conferences" on number concepts in the middle grades (Hiebert & Behr, 1988) and on the learning and teaching of algebra (Wagner & Kieran, 1989), reveal that there is a rich and growing research base related to Numbers, Symbolic Relationships, Shapes, and Uncertainty. There is still little research related to Reasoning, however. As in other domains, research has focused on what students understand about mathematical concepts at isolated points in time or on how this understanding evolves naturally in students. Research on instructional interventions that improve student understanding has received less attention.
During preschool and elementary-school years, children develop meanings for number words in which sequence, count, and cardinal meanings of number words become increasingly integrated (Fuson et. al., 1982; Fuson, 1988). Students' own meanings for number words determine to some extent their strategies for adding and subtracting and the complexity of problems they can solve. Elementary- and middle-school students may have limited ability with place value (Sowder, 1992a). Sowder reports that middle-school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (for example, students have trouble determining how many boxes of 100 candy bars could be packed from 48,638 candy bars).
Upper elementary- and middle-school students often do not understand that decimal fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). For example, students have trouble writing decimals for shaded parts of rectangular regions divided into 10 or 100 equal parts (Hiebert & Wearne, 1986). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary-school students can establish rich meanings for decimal symbols and do a variety of decimal tasks well after specially designed instruction using base-10 blocks (Wearne & Hiebert, 1988, 1989).
Upper elementary- and middle-school students may exhibit limited understanding of the meaning of fractional number (Kieren, 1992). For example, many 7th-graders do not recognize that 5 1/4 is the same as 5 + 1/4 (Kouba et al., 1988). In addition, elementary-school students may have difficulties perceiving a frac-tion as a single quantity (Sowder, 1988), but rather see it as a pair of whole numbers. An intuitive basis for developing the concept of fractional number is provided by partitioning (Kieren, 1992) and by seeing fractions as multiples of basic units—for example, 3/4 is 1/4 and 1/4 and 1/4 rather than 3 of 4 parts (Behr et. al., 1983).
Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).
There is very little research into student understanding of number symbols as arbitrary conventions. It does indicate that not until 11 years of age do most children consider that correct counting with nonstandard symbols is as adequate as correct counting with standard symbols (Saxe et. al, 1989).
9b. Symbolic Relationships
Research on Symbolic Relationships examines student understanding of the concepts of variable and algebraic equality, their ability to construct and interpret graphs, and their ability to solve algebraic equations. Several reviews on the literature base in this area are available (Herscovics, 1989; Kieran, 1989, 1992; Leinhardt et al., 1990).
Students have difficulty understanding how symbols are used in algebra (Kieran, 1992). They are often unaware of the arbitrariness of the letters chosen to represent variables in equations (Wagner, 1981). Middle-school and high-school students may regard the letters as shorthand for single objects, or as specific but unknown numbers, or as generalized numbers before they understand them as representations of variables (Kieran, 1992). These difficulties tend to persist even after instruction in algebra (Carpenter et al., 1981) and are evident even in college students (Clement, 1982). Long-term experience (3 years) in elementary computer programming has been shown to help middle-school students overcome these difficulties, although short-term experiences (less than 6 months) are less successful (Kieran, 1992; Sutherland, 1987).
Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of the situations (Leinhardt, Zaslavsky, & Stein, 1990; McDermott, Rosenquist, & van Zee, 1987). Many students interpret distance/time graphs as the paths of actual journeys (Kerslake, 1981). In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate (McDermott et al., 1987; Clement, 1989). When constructing graphs, middle-school and high-school students have difficulties with the notions of interval scale and coordinates even after traditional instruction in algebra (Kerslake, 1981; Leinhardt et al., 1990; Vergnaud & Errecalde, 1980; Wavering, 1985). For example, some students think it is legitimate to construct different scales for the positive and the negative parts of the axes. Alternatively, students think that the scales on the X and Y axes must be identical, even if that obscures the relationship. When interpreting graphs, middle-school students do not understand the effect that a scale change would have on the appearance of the graph (Kerslake, 1981). Finally, students read graphs point-by-point and ignore their global features. This has been attributed to algebra lessons where students are given questions that they could easily answer from a table of ordered pairs. They are rarely asked questions about maximum and minimum values; intervals over which a function increases, decreases or levels off; or rates of change (Herscovics, 1989).
Students have difficulty translating between graphical and algebraic representations, especially moving from a graph into an equation (Leinhardt et al., 1990). Results from the second study of the National Assessment for Educational Progress showed, for instance, that given a line with indicated intercepts, only 5% of 17-year-olds could generate the equation (Carpenter et al., 1981).
Little is known about how graphic skills are learned and how graph production is related to graph interpretation. Microcomputer-based Laboratories (MBLs) are known to improve the development of students' abilities to interpret graphs. For instance, MBLs can help middle-school students learn that a graph is not a picture and overcome the height/slope confusion mentioned above (Mokros & Tinker, 1987).
Students of all ages often do not view the equality sign of equations as a symbol of the equivalence between the left and the right side of the equation, but rather interpret it as a sign to begin calculating (Kieran, 1992). For example, middle-school students may not accept statements like 3x + 4 = x + 8 as legitimate because they think the right side should indicate the answer. Introducing the equal sign from the beginning as a symbol indicating "equivalence" between arithmetic equalities can ameliorate this difficulty (Kieran, 1981).
Beginning algebra students use various intuitive methods for solving algebraic equations (Kieran, 1992). Some of these methods may help their understanding of equations and equation solving. Students who are encouraged initially to use trial-and-error substitution develop a better notion of the equivalence of the two sides of the equation and are more successful in applying more formal methods later on (Kieran, 1988). By contrast, students who are taught to solve equations only by formal methods may not understand what they are doing. Students who are taught to use the method of "transposing" are found to only mechanically apply the change side/change sign rule (Kieran, 1988, 1989).
Students of all ages can often solve algebraic equations without a deeper understanding of what a solution is. For example, middle- and high-school students do not realize that an incorrect solution, when substituted into the equation, will yield different values for the two sides of the equation (Greeno, 1982; Kieran, 1984). More research is needed to identify how students can come to understand what a solution means and why anyone would want to find it.
Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks like a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction. At the fourth level, students can produce a short sequence of statements to logically justify a conclusion and can understand that deduction is the method of establishing geometric truth.
Progress from one of Van Hiele's levels to the next is more dependent upon instruction than age. Given traditional instruction, middle-school students perform at levels one or two (Clements & Battista, 1992). Despite that, almost 40% of high-school graduates finish high-school geometry below level two (Burger & Shaughnessy, 1986; Clements & Battista, 1992; Suydam, 1985). Further research will help identify what levels of geometric thinking students can attain at different grades given effective instruction that takes account of their difficulties in learning geometry. Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976) and can understand the relations that connect the properties of shapes or make simple deductions by 8th or 9th grade (Clements & Battista, 1992).
Research on students' development of the ability to construct proofs reflects somewhat conflicting views (Clements & Battista, 1992). Piagetian research suggests that students can reason deductively from any assumptions once they reach the formal operational stage (roughly age 12 and beyond). Other research, however, suggests that the ability to construct proofs depends on the amount and organization of particular knowledge they have. For example, this research indicates that students are not likely to understand and construct geometric proofs before they can see the relationships between classes of figures (Senk, 1989). Still other research suggests that students may need to understand the nature of proof and how it differs from everyday argumentation before they are able to con-struct proofs (Clements & Battista, 1992). Clearly, further research is needed to identify how students can come to understand what it means to prove something in geometry and what such a proof entails.
Students' conceptions about uncertainty and students' probabilistic reasoning have been extensively researched, and there are several literature reviews on the topic (Garfield & Ahlgren, 1988; Hawkins & Kapadia, 1984; Shaughnessy, 1992). The research on summarizing data, which focuses on students' understanding of different measures of central tendency and dispersion, is less extensive.
Research presents somewhat contradictory results on elementary children's understanding of probability. Piagetian research says lower elementary children have no conception of probability (Piaget & Inhelder, 1975; Shayer & Adey, 1981), but other studies indicate that even lower elementary-school children have probabilistic intuitions upon which probability instruction can build. Falk et al. (1980) presented elementary-school students with two sets, each containing blue and yellow elements. Each time, one color was pointed out as the payoff color. The students had to choose the set from which they would draw at random a "payoff element" to be rewarded. From the age of six, children began to select the more probable set systematically. The ability to choose correctly precedes the ability to explain these choices.
Upper elementary students can give correct examples for certain, possible, and impossible events, but cannot calculate the probability of independent and dependent events even after instruction on the procedure (Fischbein & Gazit, 1984). That is partly because students at this age tend to create "part to part" rather than "part to whole" comparisons (e.g., 9 men and 11 women rather than 15% of men and 10% of women). By the end of 8th grade, students can use ratios to calculate probabilities in independent events, after adequate instruction (Fischbein & Gazit, 1984).
Upper elementary students begin to understand that there is an increase in regularity of a sample distribution with an increase in the sample size, but they can apply this idea only to relatively small numbers. It is postulated that to deal with large numbers, children must first cope with notions of ratio and proportion and that their failure to understand these notions creates "a law of small large numbers" (Bliss, 1978).
Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found even among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that it is "typical." For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up. Another common error is estimating the likelihood of events based on how easily instances of it can be brought to mind.
The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. Several difficulties have been documented in the literature: Students of all ages can talk about the algorithm for computing the mean and relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima, & Well, 1981); upper elementary- and middle-school students believe that the mean of a particular data set is not one precise numerical value but an approximation that can have one of several values (Mokros & Russell, 1992); some middle-school students cannot use the mean to compare two different-sized sets of data (Gal et al., 1990); high-school students may believe the mean is the usual or typical value (Garfield & Ahlgren, 1988); students (or adults) may think that the sum of the data values below the mean is equivalent to the sum above the mean (rather than that the total of the deviations below the mean is equal to the total above) (Mokros & Russell, 1992). Research suggests that a good notion of representa-tiveness may be a prerequisite to grasping the defin-itions for measures of location like mean, median, or mode. Students can acquire notions of representa-tiveness after they start seeing data sets as entities to be described and summarized rather than as "unconnected" individual values. This occurs typically around 4th grade (Mokros & Russell, 1992).
Research suggests students should be introduced first to location measures that connect with their emerging concept of the "middle," such as the median, and later in the middle-school grades, to the mean. Premature introduction of the algorithm for computing the mean divorced from a meaningful context may block students from understanding what averages are for (Mokros & Russell, 1992; Pollatsek et al., 1981).
No applicable research findings.
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