MIDDLE SCHOOL STUDENTS' PERCEPTIONS, PERSISTENCE, AND PERFORMANCE IN MATHEMATICAL PROBLEM SOLVING.Abstract. The purpose of this study was to explore middle school students' (N= 54) perceptions of problem difficulty, persistence (1) In a CRT, the time a phosphor dot remains illuminated after being energized. Long-persistence phosphors reduce flicker, but generate ghost-like images that linger on screen for a fraction of a second. , and knowledge and use of problem-solving problem-solving n → resolución f de problemas; problem-solving skills → técnicas de resolución de problemas problem-solving n → strategies in solving mathematical word problems. Students identified as learning disabled, average achieving, or gifted were tested individually as they solved six word problems classified as 1-, 2-, or 3-step problems. After the examiner read each problem, the student rated the problem's difficulty on a 1-to-6 scale (very easy to very hard) and then solved the problem. Results indicated that students with learning disabilities rated problems as significantly more difficult and had a significantly lower total word problem score than both average and gifted students. In comparison, average students rated problems as significantly more difficult than gifted students but did not differ significantly on total word problem score. There was no significant difference between students with learning disabilities and average achievers in the length of time they spent solving problems, but both groups took significantly longer than the gifted students. Students with learning disabilities used significantly fewer problem-solving strategies on the two- and three-step problems than both the average and the gifted students, who did not differ. Findings suggest that although students with learning disabilities perceive problems as more difficult than do their more successful peers, they do not spend more time solving problems. Even with greater persistence, however, they would still be at a serious disadvantage compared with better problem solvers because they seem to lack important problem-solving strategies for effective and efficient mathematical problem Mathematical problem may mean two slightly different things, both closely related to mathematical games:
This study explored middle school students' perceptions of problem difficulty, persistence, and knowledge and use of problem-solving strategies in solving mathematical word problems. Previous research indicates that despite a positive attitude toward mathematics, students with learning disabilities (LD) are significantly poorer mathematical problem solvers than nondisabled students (e.g., Montague The name Montague can refer to the following: People Surnames
Applegate may also refer to:
Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. (Hutchinson Hutchinson, city (1990 pop. 39,308), seat of Reno co., S central Kans., on the Arkansas River; inc. 1872. It is a commercial and industrial center in a grain (especially wheat), livestock, and oil region. , 1993; Mayer, 1985; Montague, Marquard
Problem representation strategies are needed to process linguistic and numerical numerical expressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive. numerical nomenclature a numerical code is used to indicate the words, or other alphabetical signals, intended. information, comprehend and integrate the information, form internal representations in memory, and develop solution plans (Silver, 1985). These strategies facilitate translating and transforming problem information into problem structures or descriptions that are verbal, graphic, symbolic, and/or and/or conj. Used to indicate that either or both of the items connected by it are involved. Usage Note: And/or is widely used in legal and business writing. quantitative in nature (Heller & Hungate, 1985; Janvier Janvier may refer to:
, 1987; Mayer, 1985). These verbal and visual representations in turn assist in organizing and integrating problem information as the problem solver develops a logical solution plan. Specific problem representation strategies include (a) paraphrasing or restating problems in one's own words; Co) visualizing visualizing, v 1., holding an image in one's mind. 2., forming an image of a goal or destination in one's mind before undertaking it, so as to facilitate success. problems by drawing pictures, constructing diagrams or charts, and making mental images; and (c) hypothesizing or establishing goals and setting up a plan to solve the problem. Research has also suggested that academic performance may be influenced not only by cognitive factors Noun 1. cognitive factor - something immaterial (as a circumstance or influence) that contributes to producing a result cognition, knowledge, noesis - the psychological result of perception and learning and reasoning such as ability to represent problems, but also by noncognitive factors, for example, self-perceptions of ability or academic competence and perceptions of task difficulty (Heath heath, tract of open land heath, tract of open land characterized by a few scattered trees, abundant moss cover, and numerous low shrubs, principally of the heath family (see heath, in botany). , 1996; Montague, 1997b). Meltzer, Roditi, Houser, and Perlman Perl·man , Itzhak Born 1945. Israeli-born American violinist noted for his technical brilliance. (1998) found that, although students with LD rated their academic performance and organization as average to above average in seven of nine academic domains, their ratings were still significantly lower than the ratings of average achievers in all nine domains. Moreover, teachers rated the students with learning disabilities as weak in strategy use and below average in their performance on all nine academic domains. Other studies have shown that students with LD realize the importance of mathematical problem solving in today's society and value good problem solving, but have poor self-perceptions of their mathematical performance, especially mathematical problem solving (Montague, 1997b). Students' self-perceptions may directly influence how they approach a task and the amount of effort they put forth. Characteristically, students with LD have a tendency to attribute their failure to lack of ability and success to luck or other external factors. Their nondisabled peers, however, tend to attribute performance to the effort expended ex·pend tr.v. ex·pend·ed, ex·pend·ing, ex·pends 1. To lay out; spend: expending tax revenues on government operations. See Synonyms at spend. 2. in task preparation or engagement (Grimes Grimes is a surname, that is believed to be of a Scandinavian decent and may refer to
v. cor·re·lat·ed, cor·re·lat·ing, cor·re·lates v.tr. 1. To put or bring into causal, complementary, parallel, or reciprocal relation. 2. with problem-solving success, may be a contributor to poor problem solving for these students (Sternberg Stern·berg , George Miller 1838-1915. American army physician who was US surgeon general (1893-1902) and organized (1900) the Yellow Fever Commission. , 1985). Thus, both cognitive and noncognitive factors such as mathematical ability, problem-solving strategies, attributions, general self-concept self-concept n. An individual's assessment of his or her status on a single trait or on many human dimensions using societal or personal norms as criteria. , academic self-perceptions, and perceptions of tasks may play a role in academic performance. In the present study, we investigated several variables that may influence the mathematical problem solving of middle school students. These included perceptions of problem difficulty, problem-solving accuracy, persistence as measured by time taken to solve the problems, use of problem-solving strategies, and problem-solving method (silent problem solving versus thinking aloud). We hypothesized that students with LD, compared with average-achieving and gifted peers, would perceive the mathematical problems as more difficult, take less time solving them, get more wrong answers, and use fewer problem-representation strategies. These hypotheses were based on a previous study, which suggested that task difficulty has a direct influence on persistence and the number and type of cognitive strategies and processes students use (Montague & Applegate, 1993a). In that study, unsuccessful problem solvers took more time and made more cognitive and metacognitive verbalizations in their attempt to solve a one-step one-step n. 1. A ballroom dance consisting of a series of unbroken rapid steps in 2/4 time. 2. A piece of music for this dance. intr.v. mathematical problem compared to successful problem solvers. This suggests that if problems were perceived as easy, students persisted longer in their attempt to solve them even though they were unsuccessful. As problems increased in difficulty level, students with LD took less time, suggesting that they may have perceived the problems as being beyond their capability. For students with LD, less time spent on a task may be related to perceptions of ability relative to perceptions of problem difficulty. However, equally important, they made fewer cognitive and metacognitive verbalizations than their average-achieving and gifted peers, which suggests that they lacked a cognitive approach for solving problems and had few resources (i.e., problem-solving strategies). In contrast, as problems became more difficult (i.e., more cognitively demanding), the average and gifted problem solvers persisted longer and articulated ar·tic·u·la·ted adj. Characterized by or having articulations; jointed. more cognitive and metacognitive strategies. On the most difficult problems, the gifted students articulated significantly more metacognitive strategies than the average students, suggesting that more able students have more sophisticated strategic repertoires. This may not be surprising given that metacognitive abilities mature during adolescence adolescence, time of life from onset of puberty to full adulthood. The exact period of adolescence, which varies from person to person, falls approximately between the ages 12 and 20 and encompasses both physiological and psychological changes. (Berk, 1997). Thus, the gifted students' metacognitive abilities may be more developed, and, therefore, more accessible than the average students'. With this in mind, we predicted that ability level would have an impact on performance and that all groups would differ in strategy use. We also were interested in validation See validate. validation - The stage in the software life-cycle at the end of the development process where software is evaluated to ensure that it complies with the requirements. of the six word problems as a measure of mathematical problem solving for middle school students. In previous studies, both concurrent and discriminate dis·crim·i·nate v. dis·crim·i·nat·ed, dis·crim·i·nat·ing, dis·crim·i·nates v.intr. 1. a. validity of the problems was demonstrated (Montague & Applegate, 1993b; Montague, Applegate, & Marquard, 1993). Finally, we investigated whether thinking out loud while solving problems improves performance, as espoused by Whimbey (1990). Thinking aloud or overt Public; open; manifest. The term overt is used in Criminal Law in reference to conduct that moves more directly toward the commission of an offense than do acts of planning and preparation that may ultimately lead to such conduct. OVERT. Open. verbalization is a technique associated with concurrent assessment of cognitive processing during task performance (Ericcson & Simon, 1980; Garner & Alexander, 1989; Montague & Applegate, 1993b) and with cognitive behavior modification Cognitive Behavior Modification (CBM) is a therapeutic technique in which clients challenge their internal beliefs and assumptions regarding matters that are upsetting them. The objective is to eliminate debilitating cognitions and replace them with productive ones. , an intervention A procedure used in a lawsuit by which the court allows a third person who was not originally a party to the suit to become a party, by joining with either the plaintiff or the defendant. that promotes self-regulation The term self-regulation can signify
n. Control of one's emotions, desires, or actions by one's own will. (Meichenbaum, 1977). Thinking aloud is also an important part of cognitive strategy instruction for students with LD. As part of the instructional routine, teachers model verbalization of cognitive and metacognitive processes and strategies as they complete problem-solving activities. Students then practice solving problems while thinking aloud as their teachers did. As they become more familiar with the process and their performance improves, students progress to covert COVERT, BARON. A wife; so called, from her being under the cover or protection of her husband, baron or lord. instructions and internal monitoring. Although research suggests that thinking aloud is important during strategy acquisition, there is no evidence that students in general are better problem solvers when they think aloud (Montague, 1997a). METHOD Subjects Students in an urban South Florida Florida, state, United States Florida (flôr`ĭdə, flŏr`–), state in the extreme SE United States. A long, low peninsula between the Atlantic Ocean (E) and the Gulf of Mexico (W), Florida is bordered by Georgia and middle school who met both district and research eligibility criteria participated in the study. Students (N = 54) represented three levels of problem-solving ability (i.e., learning disabled, average-achieving, and gifted) and two levels of grade placement (i.e., seventh and eighth grade). Appendix A contains the district eligibility criteria for placement in the learning disabilities and gifted programs. In addition to meeting district eligibility criteria, students with learning disabilities had to have a full scale score of 85 or more on the Wechsler Wechsler is a German word meaning "exchanger" (from '', "(ex)change"). Wechsler (or Wexler) may refer to:
n a unit consisting of one-ninth of the total range of the standard scores (SDs) of a normal distribution. The term is a condensation of “standard nine. of 3 on the district-administered group achievement test. Gifted students had a score of at least 130 on either the verbal or performance scale of the WISC-R. Finally, average-achieving students had an estimated IQ equivalent between 85 and 115 on the short form of the WISC-R (Kaufman, 1976). School records were screened to identify students who met the IQ and reading criteria. Permission forms were distributed to the students with learning disabilities and gifted students who qualified for participation and to 60 average achievers. The first 18 students in each group who returned a signed permission form participated. All students were individually administered the Woodcock-Johnson Psychoeducational psychoeducational (sīˈ·kō·ed·j Battery (WJ) passage comprehension comprehension Act of or capacity for grasping with the intellect. The term is most often used in connection with tests of reading skills and language abilities, though other abilities (e.g., mathematical reasoning) may also be examined. subtest and the mathematics subtests (calculation, applied problems, and quantitative concepts) by a trained graduate student (Woodcock woodcock: see snipe. woodcock Any of five species (family Scolopacidae) of plump, sharp-billed migratory birds of damp, dense woodlands in North America, Europe, and Asia. & Johnson, 1977). ANOVAs were conducted to evaluate if group differences existed on age, IQ and WJ achievement variables. Table 1 presents the subject demographic data. No significant difference was found for age, F(2,51) = 0.74, p = 48.20. However, significant group differences were found for IQ and the achievement variables: WISC-R full scale, F(2,48) = 94.50, p = .0001; WJ passage comp-rehension, F(2,51) = 19.76, p = .0001; WJ calculation, F(2,51) = 37.47, p = .0001; applied problems, F(2,51) = 31.60, p = .0001; quantitative concepts, F(2,51) = 28.42, p = .0001; broad math, F(2,51) = 46.36, p = .0001; and basic math, F(2,51) = 39.01, p = .0001. Post hoc post hoc adv. & adj. In or of the form of an argument in which one event is asserted to be the cause of a later event simply by virtue of having happened earlier: analyses indicated that the gifted students had significantly higher scores on the IQ and reading variables than the other groups (p [is less than] .05). However, there were no significant differences on these variables between the students with learning disabilities and the average students (p [is greater than] .05). On the math variables, gifted students scored significantly higher than the average students, who scored significantly higher than the students with learning disabilities (ps [is less than] .05).
Table 1
Subject Demographic Data
Variable Group
LD AA G
(N = 18) (N = 18) (N = 18)
Grade
Seven 9 7 14
Eight 9 11 4
Age
M (mos.) 164.94 161.50 160.89
SD 9.85 12.72 9.47
Gender
Male 12 10 9
Female 6 8 9
Ethnicity
White 7 7 10
Hispanic 10 9 8
African-American 1 1 0
Asian 0 1 0
WISC-R(*) Full Scale Score
M 96.06 100.76 130.78
SD 9.79 7.85 6.45
WJ Subtests
Passage Comprehension
M 96.89 103.00 124.61
SD 9.00 10.68 19.61
Math Calculation
M 86.94 108.56 129.28
SD 10.79 17.13 15.35
Applied Problems
M 94.78 107.28 125.11
SD 9.42 11.36 13.39
Quantitative Concepts
M 85.22 101.22 120.06
SD 7.31 17.12 15.19
Broad Math
M 88.28 107.06 129.22
SD 10.09 14.40 13.42
Basic Math
M 83.28 103.78 127.44
SD 8.88 18.42 16.06
(*) Estimated full scale score based on short form for average achievers (Kaufman, 1974). Procedures All testing was done individually in a private room by a graduate assistant who had been trained by the researcher (first author) in task administration procedures. The measures were administered during one 55-minute class period. Students solved three of the six word problems silently and three aloud. Problem type (number of steps) was counterbalanced coun·ter·bal·ance n. 1. A force or influence equally counteracting another. 2. A weight that acts to balance another; a counterpoise or counterweight. tr.v. within solution method (silent, aloud), which was then counterbalanced to control for order effects. Before solving the think-aloud problems, students received training in which the examiner first modeled thinking aloud while solving a verbal reasoning Verbal reasoning is understanding and reasoning using concepts framed in words. It aims at evaluating ability to think constructively, rather than at simple fluency or vocabulary recognition. problem and then had students think aloud while solving a similar problem. During the training and testing, students were cued at 5-second intervals if they were not verbalizing .and told to "say everything you are thinking and doing." Additionally, before solving each mathematical problem, the examiner read the problem to the student who was then asked to rate the problem difficulty on a 1-to-6 scale (very easy to very hard). Figure 1 presents the scale and the six word problems used. Figure 1. Word problems. Directions: Read the problem. Think about solving the problem. Circle the number of the problem that you think describes how easy or hard the problem is.
Very Easy Easy Somewhat Easy Somewhat Hard Hard Very Hard
1 2 3 4 5 6
(1-Step: Problem 1) Janice has a paper route. She delivers papers to 88 customers every day. Janice must also collect money from her customers each week. On Saturday Saturday: see week; Sabbath. , she collected money from 43 customers. How many customers must Janice collect money from on Sunday Sunday: see Sabbath; week. ? (2-Step: Problem 2) Four friends have decided they want to go to the movies on Saturday. Tickets are $2.75 for students. Altogether they have $8.40. How much more do they need? (3-Step: Problem 3) Chain sells for $1.23 a foot. How much will Farmer Jones have to spend for chain in order to enclose en·close also in·close tr.v. en·closed, en·clos·ing, en·clos·es 1. To surround on all sides; close in. 2. To fence in so as to prevent common use: enclosed the pasture. a 70 foot by 30 foot patch of ground, leaving a 4 foot entrance in the middle of each of the 30 foot sides? (1-Step: Problem 4) Bill and Shirley Shir·ley , William 1694-1771. British colonial administrator who was governor of Massachusetts (1741-1749 and 1753-1756) and commanded British forces in the French and Indian War. need to arrange the chairs for a play that the class is having. They took 252 chairs from the storeroom. Their teacher told them to make rows of 12 chairs each. How many rows will they have? (2-Step: Problem 5) A group bought 52 tickets. Each ticket was $26 less than the $280 regular price ticket. How much did the group spend for the tickets? (3-Step: Problem 6) A store sells shirts for $13.50 each. On Saturday, it sold 93 shirts. This was 26 more than it had sold on Friday Friday: see Sabbath; week. Friday young Indian rescued by Crusoe and kept as servant and companion. [Br. Lit.: Robinson Crusoe] See : Servant . How much did the store charge for all the shirts sold on both days? Design and Analyses The dependent variables in the study were (a) total word problem score, (b) computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. accuracy, (c) problem solution time (in seconds), (d) problem difficulty ratings, (e) total problem-solving strategies, (f) total problem representation strategies, and (g) problem-solving method (silent vs. aloud). To analyze differences in computation accuracy, the number of operations completed and the number of computational Having to do with calculations. Something that is "highly computational" requires a large number of calculations. errors per problem were determined. Then, accuracy per problem was calculated by dividing the number of errors by number of operations and subtracting from one. Differences in these variables were examined as a function of group (learning disabled, average, gifted) and problem type (1-step, 2-step, 3-step). The primary design was a 3 (Group) by 3 (Problem Type) factorial factorial For any whole number, the product of all the counting numbers up to and including itself. It is indicated with an exclamation point: 4! (read “four factorial”) is 1 × 2 × 3 × 4 = 24. ANOVA anova see analysis of variance. ANOVA Analysis of variance, see there . For these analyses, Tukey's post hoc procedure was used to determine the nature of the differences. RESULTS Word Problem Score To analyze differences among groups on total word problem score, a one-way one-way adj. 1. Moving or permitting movement in one direction only: a one-way street. 2. Providing for travel in one direction only: a one-way ticket. ANOVA was performed. The analysis resulted in a significant effect for group, F(2,51) = 10.61, p = .0001. Tukey's post hoc procedure showed no significant difference between the means of the average achievers and gifted students. However, the means of both of these groups were significantly higher than the mean of the students with learning disabilities (LD, M = 1.61, SD = 1.14; AA, M = 2.89, SD = 1.23; G, M = 3.56, SD = 1.46). Computation Accuracy First, a 3 (Group) by 3 (Problem Type) factorial ANOVA was performed for number of operations, resulting in a significant group by problem type interaction, F(4,102) = 14.61, p = .0001. Because the interaction was significant, simple main effects were examined. Table 2 presents the means and standard deviations In statistics, the average amount a number varies from the average number in a series of numbers. (statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. by group and problem type. On the 1-step problem, no significant group difference was found for the number of operations used. On the 2-step problem, gifted students used significantly more operations than the students with LD, F(2,51) = 3.20, p = .0490. Finally, on the 3-step problem, both gifted and average achievers used significantly more operations than the students with LD, F(2,51) = 17.29, p = .0001. Students with LD did not differ significantly in the number of operations across problem type. However, both average achievers, F(2,34) = 28.47, p = .000, and gifted students, F(2,34) = 94.16, p = .0001, used significantly more operations as problems increased in difficulty (3-step [is greater than] 2-step [is greater than] 1-step). Table 2 Means and Standard Deviations for Number of Operations by Group and Problem Type
Group
LD AA G
(N = 18) (N = 18) (N = 18)
Problem Type
1-Step
M 2.44 2.11 2.00
SD 1.15 .58 .34
2-Step
M 3.22 3.67 3.94
SD .94 1.02 .54
3-Step
M 3.17 5.67 6.78
SD 1.38 2.38 1.77
Second, to analyze differences in computational errors, a 3 (Group) by 3 (Problem Type) factorial ANOVA was performed, resulting only in a significant main effect for problem type, F(2,102) = 23.29, p = .0001. Tukey post hoc tests revealed that students made significantly more computational errors as the problems became harder (3-step, Least Squares Mean [LSM LSM Linux Software Map LSM Louisiana State Museum LSM Linux Security Module LSM Living Stream Ministry LSM Laser Scanning Microscopy LSM Legato Storage Manager LSM Land-Surface Model LSM Lutheran Student Movement LSM Logical Storage Manager ] = 1.19, SD = 1.08 [is greater than] 2-step, LSM = .57, SD = .72 [is greater than] 1-step, LSM = .24, SD = .51). Third, to analyze differences in computational accuracy, a 3 (Group) by 3 (Problem Type) factorial ANOVA was performed, resulting only in a significant main effect for problem type, F(2,102) = 7.26, p = .0011. Tukey post hoc tests revealed that students made significantly more computational errors on the 1-step and 2-step problems than on the 3-step problems ([1-step, LSM = .90, SD = .22] = [2-step, LSM = .83, SD = .22] [is greater than] [3-step, LSM = .75, SD = .26]). Solution Time To analyze differences in solution time, a 3 (Group) by 3 (Problem Type) factorial ANOVA was performed, resulting in significant main effects for group, F(2,51) = 4.48, p = .0162, and problem type, F(2,102) = 80.87, p = .0001. No interaction effect was found, F(4,102) = 2.45, p = .0506. Tukey post hoc tests revealed that the students with LD did not differ significantly from average achievers in the amount of problem-solving time (average number of seconds for all six problems); however, both of these groups took significantly more time than the gifted students (LD, M = 160.41, SD = 85.94; AA, M = 164.23, SD = 104.12; G, M = 114.04, SD = 65.43). The post hoc analyses also revealed that all problem type means differed significantly from one another (1-step, M = 93.45, SD = 76.45; 2-step, M = 130.02, SD = 62.15; 3-step, M = 215.20, SD = 80.25). Problem Difficulty Ratings To analyze differences in problem difficulty ratings, a 3 (Group) by 6 (Problem) factorial ANOVA was performed. Significant main effects were found for group, F(2,51) = 11.62, p = .0001, and for problem, F(5,255) = 41.68, p = .0001. No interaction effect was found, F(10,255) = 1.63, p = .0983. Tukey post hoc tests showed significant differences in problem ratings among all groups. Specifically, the students with LD rated the problems as significantly more difficult than the average achievers, who rated the problems as significantly more difficult than the gifted students. The post hoc tests also detected significant differences in problem ratings among the problems (problem 3 [is greater than] problems 6 and 5 [is greater than] problems 2, 4, and 1). The descending descending /des·cend·ing/ (de-send´ing) extending inferiorly. order of problem difficulty was 3-step, 2-step, and then 1-step. Table 3 presents the means and standard deviations for the groups and problem difficulty ratings. Table 3 Means and Standard Deviations for Problem Difficulty Ratings by Group and Problem
Problem Difficulty Ratings
M SD
Group
LD 3.41 1.28
AA 2.60 1.25
G 2.31 .94
Problem
1-Step - Problem 1 2.09 1.23
1-Step - Problem 4 2.33 1.08
2-Step - Problem 2 2.35 .93
2-Step - Problem 5 2.96 1.16
3-Step - Problem 6 2.96 1.06
3-Step - Problem 3 3.94 1.07
Problem-Solving Strategies To examine use of problem-solving strategies, audiotapes of the think-aloud problems were transcribed. Of the 54 student recordings, only 33 could be used due to the quality of the audio, LD (n = 12), AA (n = 11), G (n = 10). Therefore, only 33 transcriptions were used in the analyses. To determine the number and type of problem-solving strategies applied, a previous study's coding procedures were applied (Montague & Applegate, 1993). Appendix B presents the cognitive categories, codes, and operational definitions. The researcher coded all protocols, and an independent rater rat·er n. 1. One that rates, especially one that establishes a rating. 2. One having an indicated rank or rating. Often used in combination: a third-rater; a first-rater. who was trained in coding procedures scored 33% of the protocols (n = 11). Interrater agreement, determined by dividing the number of agreements by the number of disagreements and multiplying mul·ti·ply 1 v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies v.tr. 1. To increase the amount, number, or degree of. 2. Mathematics To perform multiplication on. by 100 (Kazdin, 1982), was 94%. To analyze differences in strategy use, 3 (Group) by 3 (Problem Type) factorial ANOVAs were performed for (a) total number of problem-solving strategies and (b) total number of problem representation strategies. The analysis for the total number of problem-solving strategies yielded a significant group-by-problem-type interaction, F(4,60) = 4.64, p = .0025; therefore, only simple main effects were analyzed an·a·lyze tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es 1. To examine methodically by separating into parts and studying their interrelations. 2. Chemistry To make a chemical analysis of. 3. (see Figure 2 for a graph of the interaction). Table 4 presents the means and standard deviations for group and problem type. The findings of these analyses indicated no significant difference among problem types for the students with LD, F(2 22) = .56, p = .5783. However, significant differences were detected among problem types for the total number of problem-solving strategies used by the average achievers, F(2,20) = 9.19, p = .0015 (3-step [is greater than] 2-step = 1-step), and by the gifted students, F(2,18) = 16.79, p = .0007 (3-step = 2-step [is greater than] 1-step). Findings also indicated no significant difference among groups in the total number of problem-solving strategies for the 1-step problems, F(2,30) = 1.06, p = .3606, or the 2-step problems, F(2,30) = 2.07, p = .1444. However, a significant difference was detected among groups for the 3-step problems, F(2,30) = 11.24, p = .0002 (G = AA [is greater than] LD). Table 4 Means and Standard Deviations for Total Problem-Solving Strategies by Group and Problem Type
Group
LD AA G
Problem Type (n = 12) (n = 11) (n = 10)
1-Step
M 4.92 6.36 5.00
SD 2.27 3.53 1.70
2-Step
M 5.92 10.00 9.50
SD 4.38 6.83 4.12
3-Step
M 5.50 14.55 12.30
SD 2.84 7.51 2.58
The analysis for the total number of problem representation strategies also yielded a significant group-by-problem-type interaction, F(4,60) = 2.62, p = .0496. Again, only simple main effects were analyzed (see Figure 3 for a graph of the interaction). Table 5 presents the means and standard deviations for group and problem type. The findings of these analyses indicated no significant difference among problem type in the total number of representation strategies for the students with LD, F(2,22) = .83, p = .4128, or the average achievers, F(2,22) = 1.31, p = .2891. However, a significant difference was detected among problem types for the gifted students, F(2,30) = 11.24, p = .0002 (3-step [is greater than] 1-step). Findings also indicated no significant difference among groups in the total number of representation strategies for the 1-step problems, F(2,30) = 2.28, p = .1199. However, significant differences were detected among groups for the 2-step problems, F(2,30) = 12.82, p = .0001 (G = AA [is greater than] LD), and the 3-step problems, F(2,30) = 7.73, p = .0020 (G = AA [is greater than] LD). Table 5 Means and Standard Deviations for Total Problem Representation Strategies by Group and Problem Type
Group
LD AA G
Problem Type (n = 12) (n = 11) (n = 10)
1-Step
M 1.17 1.82 1.90
SD .94 .87 .88
2-Step
M .75 2.36 2.50
SD 1.62 1.12 0.97
3-Step
M .83 3.00 4.00
SD 1.19 2.61 1.83
Problem-Solving Method To analyze differences in accuracy as a function of problem-solving method (think-aloud, silent), a 3 (Group) by 2 (Method) factorial ANOVA was performed, which resulted in a significant main effect for group only, replicating the results in the section on word problem score. No significant main effect was detected for method, F(2,51) = .02, p = .8833, and there was no interaction, F(2,51) = .94, p = .3988 (see Table 6 for the means and standard deviations for problem type accuracy by group and method). Table 6 Means and Standard Deviations for Problem-Solving Accuracy by Group and Problem Method
Group
LD AA G
(n = 18) (n = 18) (n = 18)
Problem Method
Aloud M .72 1.39 1.89
SD .75 1.50 1.67
Silent
M .89 .78 .90
SD .68 .79 .84
Validity of the Six Word Problems To further examine the validity of the six word problems used in this study (see Montague & Applegate, 1993a), a 3 (Group) by 6 (Individual Problems, pooled over group) factorial ANOVA was performed for problem solution time. Significant main effects were found for group, F(2,51) = 4.48, p = .0162, and for problem, F(5,255) = 28.76, p = .0001. No interaction effect was found, F(10,255) = 1.35, p = .2055. Tukey post hoc tests revealed that the students with LD and the average achievers, who did not significantly differ in problem-solving time (average number of seconds for all six problems), took significantly less time to solve the problems than the gifted students (see the results of the solution time analysis in this section for means and standard deviations). Solution times for the six problems increased as the problems became more difficult. Specifically, times for the 3-step problems were significantly greater than for the 2-step and 1-step problems, and the solution time for problem 5 (2-step problem) was significantly greater than for problem 1 (1-step problem). Although the solution time for problem 2 (2-step problem) was greater than for problem 4 (1-step problem), they were not significantly different. See Table 7 for means and standard deviations for each problem. Table 7 Means and Standard Deviations for Solution Time for the Six Word Problems
Solution Time (in seconds)
M SD
Problem
1-Step - Problem 1 81.85 58.48
1-Step - Problem 4 105.06 128.33
2-Step - Problem 2 118.30 71.82
2-Step - Problem 5 141.75 85.70
3-Step - Problem 6 211.44 95.38
3-Step - Problem 3 218.95 102.88
DISCUSSION The validity of the six word problems as a measure of mathematical problem solving was further investigated. The systematic variation in problem solution time and difficulty ratings of the ability groups suggested increasing difficulty and information-processing demands for the problem types. However, even though the means of the student ratings for the problems suggested increasing difficulty as the number of steps increased, no significant differences in student ratings were found between problem 6 (3-step) and problem 5 (2-step) or problem 2 (2-step) and the 1-step problems. Problem 6 may have been perceived by the students as easier than it actually is, especially when compared with problem 3 (also a 3-step but rather complex geometric problem.) However, these problems did not differ in the length of time students spent solving them, indicating similar task requirements and difficulty level. Problem 2, on the other hand, may not be sufficiently different from the 1-step problems, as determined by student ratings of difficulty and problem-solving time, to warrant its inclusion in the six-problem set. The good problem solvers used more strategies overall and more problem representation strategies specifically as problems became more difficult. Thus, the problems were differentiated with respect to a need for more and different strategies, which is evidence of their construct validity construct validity, n the degree to which an experimentally-determined definition matches the theoretical definition. . An informal, but valid measure of mathematical problem solving for middle school students could be very useful to classroom teachers, particularly for screening or initial assessment of students' strengths and weaknesses, to facilitate instructional planning (see also, Montague, 1997a). But further validation studies of these six problems is necessary if we want teachers to have confidence in using them as a measure of mathematical problem solving. For example, problem 2 may need to be replaced by another problem that is more clearly differentiated in difficulty level from the other problem types. Several conclusions regarding the performance of these middle school students can be drawn. First, as expected, the students with LD performed more poorly on the six mathematical problems than their average and gifted peers. Because there were no significant differences in scores on the reading achievement, cognitive ability, and computational accuracy measures between students with LD and the average achievers, we can assume that their poor performance was due to factors other than poor reading, low ability, or computation errors. These other factors may include an inability to judge the difficulty level of the problems, poor perceptions of their ability to solve the problems, lack of persistence, and strategy deficits. This study corroborates previous research suggesting that students with LD lack critical problem-solving strategies, most notably strategies that facilitate problem representation (Montague, Marquard, & LeBlanc, 1993). The students with LD perceived problems as more difficult than the average achievers and gifted students, but their persistence (as measured by solution time) was not significantly different from that of the average achievers. Theoretically, though, if they perceived the problems as more difficult than the other students, they should have exerted more effort and persisted longer in their attempt to solve them. These results support the notion that some students "shut down" cognitively when they perceive problems as difficult or when information-processing demands seem excessive, and simply do not attempt the task because they "cannot do it" (Ericcson & Simon, 1980). Another related and, perhaps, more important reason for their lack of fortitude Fortitude See also Bravery. Fratricide (See MURDER.) Asia despite torture, refuses to deny Moses. [Islam: Walsh Classical, 35] Calantha fulfills wifely and queenly duties despite losses. [Br. Lit. is that they may not know how to approach the task (the "what do I do?" stage) and have limited knowledge of essential problem-solving strategies (the "how do I do it?" stage). In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke" put differently , they do not have the cognitive tools to perform the task and consequently spend their time using ineffective trial-and-error strategies. Indeed, there was virtually no change in the number of strategies used by students with LD across the problem types. Thus, these students' poor performance most likely can be attributed to both cognitive and noncognitive variables including strategy deficits, low academic self-perceptions, lack of confidence in their ability, and lack of effort. As a result, they may become easily frustrated frus·trate tr.v. frus·trat·ed, frus·trat·ing, frus·trates 1. a. To prevent from accomplishing a purpose or fulfilling a desire; thwart: and quit prematurely. In contrast, the average achievers rated the problems as significantly more difficult and persisted significantly longer than the gifted students but did not differ significantly in problem-solving accuracy. They also behaved more like their gifted peers in their use of problem-solving strategies, increasing the number of strategies as problems became more difficult and using significantly more problem representation strategies on the 3-step problems. For average achievers, adequate knowledge and use of problem-solving strategies combined with greater persistence may have been responsible for better performance. The average achievers, like their gifted peers, seemed to have the cognitive resources necessary for solving the problems, and they also put forth appropriate effort. Moreover, they may be able to accurately perceive their ability to solve problems, judge the information-processing demands of a task, and access more strategies when a task demands are greater. As a consequence, these students may persevere per·se·vere intr.v. per·se·vered, per·se·ver·ing, per·se·veres To persist in or remain constant to a purpose, idea, or task in the face of obstacles or discouragement. more in their efforts to solve problems and be rewarded more for their effort. The gifted students clearly were both fast and accurate problem solvers. Thus, they were able to solve problems in significantly less time than the average achievers and students with LD but used as many strategies as the average students. Additionally, as problems became harder, they used more strategies, indicating that they have a repertoire Repertoire may mean Repertory but may also refer to:
Results also suggest that problem-solving method does not seem to affect accuracy. Thinking aloud, as a technique for improving performance advocated by Whimbey (1990), may not facilitate problem solving. Rather, cognitive modeling The term cognitive model can have basically two meanings. In cognitive psychology, a model is a simplified representation of reality. The essential quality of such a model is to help deciding the appropriate actions, i.e. and overt verbalization of the problem-solving process seem to be necessary parts of cognitive strategy instruction for students with LD when they are first learning and practicing strategies for solving mathematical problems (Montague, 1992; Montague, Applegate, & Marquard, 1993). Verbalization may not be as important for students who are good problem solvers and have a relatively well-developed well-developed adj [arm, muscle etc] → bien desarrollado; [sense] → agudo, fino well-developed adj [girl repertoire of problem-solving strategies. Indeed, several gifted students in previous studies reported that thinking aloud was "too slow" and interfered with their problem solving (Montague, 1991; Montague & Applegate, 1993b). The most disturbing, although not surprising, finding in the present study was that students with learning disabilities performed significantly less well than better performing peers on the standard mathematical tests that were administered. Their calculation skills, mathematical vocabulary and concepts, and application skills were clearly inferior INFERIOR. One who in relation to another has less power and is below him; one who is bound to obey another. He who makes the law is the superior; he who is bound to obey it, the inferior. 1 Bouv. Inst. n. 8. to those of better problem solvers. This finding further illustrates the need for more and better instruction in mathematics, generally, and explicit instruction in strategies for mathematical problem solving, specifically, for these students. Standardized standardized pertaining to data that have been submitted to standardization procedures. standardized morbidity rate see morbidity rate. standardized mortality rate see mortality rate. mathematical tests are characterized char·ac·ter·ize tr.v. character·ized, character·iz·ing, character·iz·es 1. To describe the qualities or peculiarities of: characterized the warden as ruthless. 2. by few items at any one level that increase in difficulty quickly. Most students with LD have not had the opportunity to learn advanced mathematical skills and, thus, are precluded from success on higher level items on these tests. Consequently, they display inferior achievement compared with peers who have been exposed to higher level mathematics. In conclusion, this study corroborates findings that most normally developing children acquire problem-solving strategies and are able to use them appropriately (Flavell, Miller, & Miller, 1993). It also underscores the need for students with LD to receive explicit cognitive strategy instruction to become better problem solvers (Montague & Applegate, 1993b). However, cognitive strategy instruction, although an effective academic intervention for students with LD (Swanson, Carson Carson, city (1990 pop. 83,995), Los Angeles co., S Calif., an industrial and residential suburb of Los Angeles; inc. 1968. Oil refining is the major industry; fabricated metals, paper, and other products are manufactured. The California State Univ. Dominguez Hills is there. , & Sachse-Lee, 1996; Swanson, O'Shaughnessy This article is about the Irish surname; for other versions, see Shaughnessy (disambiguation). O'Shaughnessy is family surname of Irish origin. The name is found primarily in County Galway and County Limerick. , McMahon McMahon is the family name of the following persons:
REFERENCES Berk, I.E. (1997). Child development (4th ed.). Boston Boston, town, England Boston, town (1991 pop. 26,495), E central England, on the Witham River. Boston's fame as a port dates from the 13th cent., when it was a Hanseatic port trading wool and wine. Having recovered from a decline in the 18th and 19th cent. : Allyn & Bacon. Ericcson, K.A., & Simon, H.A. (1980). Verbal reports as data. Psychological Review, 87, 215-251. Flavell, J.H., Miller, P.H., & Miller, S.A. (1993). Cognitive development (3rd ed.). Englewood Cliffs, NJ: Prentice-Hall. Garner, R., & Alexander, P.A. (1989). Metcognition: Answered and unanswered questions. Educational Psychologist psy·chol·o·gist n. A person trained and educated to perform psychological research, testing, and therapy. psychologist , 24, 143-158. Grimes, L. (1981). Learned helplessness learned helplessness In psychology, a mental state in which a laboratory subject forced to bear aversive stimuli becomes unable or unwilling to avoid subsequent applications, even if they are “escapable,” presumably through having learned that situational and attribution theory Attribution theory is a social psychology theory developed by Fritz Heider, Harold Kelley, Edward E. Jones, and Lee Ross. The theory is concerned with the ways in which people explain (or attribute) the behavior of others, or themselves (self-attribution), with something : Redefining children's learning problems. Learning Disability Quarterly, 4, 88-99. Heath, N. (1996). The emotional domain: Self-concept and depression in children with learning disabilities. In T.E. Scruggs & M.A. Mastropieri (Eds.), Advances in learning and behavioral behavioral pertaining to behavior. behavioral disorders see vice. behavioral seizure see psychomotor seizure. disabilities (Vol. 10, Part A, pp. 47-76). Greenwich, CT: JAI JAI Java Advanced Imaging JAI Justice et Affaires Interiéures (French: Justice and Home Affairs) JAI Journal of ASTM International JAI Just An Idea JAI Jazz Alliance International JAI Joint Africa Institute Press. Heller, J., & Hungate, H. (1985). Implications for mathematics instruction of research in mathematical thinking. In E.A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 83-112). Hillsdale, NJ: Erlbaum. Hutchinson, N. (1993). Effects of cognitive strategy instruction on algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as problem solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 34-63. Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum. Kaufman, A.S. (1976). A four-test short form of the WISC-R. Contemporary Educational Psychology, 1, 180-196. Kazdin, A.E. (1982). Single-case research designs: Methods for clinical and applied settings. New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : Oxford University Press. Mayer, R.E. (1985). Mathematical ability. In R.J. Sternberg (Ed.), Human abilities: Information processing information processing: see data processing. information processing Acquisition, recording, organization, retrieval, display, and dissemination of information. Today the term usually refers to computer-based operations. approach (pp. 127-150). San Francisco San Francisco (săn frănsĭs`kō), city (1990 pop. 723,959), coextensive with San Francisco co., W Calif., on the tip of a peninsula between the Pacific Ocean and San Francisco Bay, which are connected by the strait known as the Golden , CA: Freeman Freeman can mean:
Meichenbaum, D. (1977). Cognitive behavior modification: An integrative approach. New York: Plenum In a building, the space between the real ceiling and the dropped ceiling, which is often used as an air duct for heating and air conditioning. It is also filled with electrical, telephone and network wires. See plenum cable. Press. Meltzer, L.J., Roditi, B., Houser, R.F., & Perlman, M. (1998). Perceptions of academic strategies and competence in students with learning disabilities. Journal of Learning Disabilities, 31, 437-451. Montague, M. (1991). Gifted and gifted-learning disabled students' knowledge of mathematical problem solving. Journal for the Education of the Gifted, 14, 393-411. Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248. Montague, M. (1997a). 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Scruggs & M.A. Mastropieri (Eds.), Advances in learning and behavioral disabilities (Vol. 12, pp. 1-42). Greenwich, CT: JAI Press. Wechsler, D. (1974). Wechsler Intelligence Scale for Children-Revised. New York: Psychological Corporation. Whimbey, A. (1990). Thinking through math word problems: Strategies for intermediate elementary school elementary school: see school. students, teacher's manual. Hillsdale, NJ: Lawrence Erlbaum Associates. Woodcock, R., & Johnson, W. (1977). Woodcock-Johnson PsychoEducational Battery. Boston: Teaching Corp. Zawaiza, T., & Gerber, M. (1993). Effects of explicit instruction on math word problem solving by community college students with learning disabilities. Learning Disability Quarterly, 16, 64-79. NOTES We wish to thank the teachers and students of the Dade County Dade County can refer to the following places:
Miami is a major city in southeastern Florida, in the United States. It is the county seat of Miami-Dade County. Miami is a gamma world city with an estimated population of 404,048. , for their cooperation and participation in this study. Correspondence concerning this article should be addressed to: Marjorie Montague, School of Education, University of Miami This article is about the university in Coral Gables, Florida. For the university in Oxford, Ohio, see Miami University. The University of Miami (also known as Miami of Florida,[2] UM,[3] or just The U , P. O. Box 248065, Coral Gables Coral Gables, city (1990 pop. 40,091), Miami-Dade co., SE Fla., SW of Miami; inc. 1925. Founded at the height of the Florida land boom, Coral Gables is a noted planned city, with tree-lined boulevards and Mediterranean-style buildings. , FL 33124; e-mail: mmontague@umiami.edu APPENDIX A District Eligibility Criteria for Placement in the Learning Disabilities and Gifted Programs Learning Disabilities (a) a disorder in one or more of the basic psychological processes including visual, auditory auditory /au·di·to·ry/ (aw´di-tor?e) 1. aural or otic; pertaining to the ear. 2. pertaining to hearing. au·di·to·ry adj. , or language processes, (b) academic achievement significantly below the student's level of intellectual functioning, (c) learning problems that are not due primarily to other disabling dis·a·ble tr.v. dis·a·bled, dis·a·bling, dis·a·bles 1. To deprive of capability or effectiveness, especially to impair the physical abilities of. 2. Law To render legally disqualified. conditions, and (d) ineffectiveness in·ef·fec·tive adj. 1. Not producing an intended effect; ineffectual: an ineffective plea. 2. Inadequate; incompetent: an ineffective teacher. of general educational alternatives in meeting the student's educational needs. Gifted (a) superior intellectual development as indicated by an IQ of two standard deviations or more above the mean on an individually administered standardized test A standardized test is a test administered and scored in a standard manner. The tests are designed in such a way that the "questions, conditions for administering, scoring procedures, and interpretations are consistent" [1] of intelligence with consideration for the standard error of measurement in individual cases, (b) evidence of a majority of characteristics of gifted children according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. a standard scale or checklist, and (c) documentation of the need for a special program. MARJORIE MONTAGUE, Ph.D., is professor, Department of Special Education, University of Miami. BROOKS APPLEGATE, Ph.D., is associate professor, Department of Educational Studies, Western Michigan University Western Michigan University, at Kalamazoo, Mich.; coeducational; founded in 1903 as Western State Normal School, became accredited in 1927 as a college, gained university status in 1957. . |
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