Strategic development of multiplication problem solving: Patterns of students' strategy choices

Low mathematics achievement is a persistent problem in the United States, and multiplication is a fundamental area in which many students manifest learning difficulties. This study examined the strategic developmental levels of multiplication problem solving among 121 elementary school students in Grades 3 through 5. A latent class analysis modeling was used to identify three valid groups representing different patterns of strategy choices for each of three types of multiplication problems. Findings indicated intra-group variability for problem-solving accuracy, for frequency of using different strategies, and for accuracy of executing direct retrieval/algorithm (DR/AG) strategies. Students demonstrated relative consistency in their strategy choices for solving the three problem types. Students who used DR/AG strategies most frequently showed the highest problem-solving accuracy and the highest accuracy of executing the DR/AG strategies. Students who most frequently relied on incorrect operations or who indicated they did not know how to solve problems demonstrated the lowest problem-solving accuracy among the three groups; the number of students in this group increased with problem difficulty levels. Implications are discussed in terms of identifying students' strategic developmental levels and providing differentiated instruction based on the identified levels.     

A comparison of strategic development for multiplication problem solving in low-, average-, and high-achieving students

The present study investigated the differences of strategy use between low-, average-, and high-achieving students when solving different multiplication problems. Nineteen high-, 48 average-, and 17 low-achieving students participated in this study. All participants were asked to complete three different multiplication tests and to explain how they solved these problems. Results suggested that low achievers used incorrect operation strategies more frequently, indicating a lack of conceptual understanding of multiplication. High-achieving students demonstrated greater flexibility in problem-solving and were more accurate in performing direct retrieval or math algorithm strategies. Results were discussed about improving low achievers’ use of advanced strategies, enhancing their flexibility in choosing strategies and improving students’ accuracy in using direct retrieval or math algorithms.     

Chinese and Singaporean sixth-grade students’ strategies for solving problems about speed

This study examined 361 Chinese and 345 Singaporean sixth-grade students’ performance and problem-solving strategies for solving 14 problems about speed. By focusing on students from two distinct high-performing countries in East Asia, we provide a useful perspective on the differences that exist in the preparation and problem-solving strategies of these groups of students. The strategy analysis indicates that the Chinese sample used algebraic strategies more frequently and more successfully than the Singaporean sample, although the Chinese sample used a limited variety of strategies. The Singaporean sample’s use of model-drawing produced a performance advantage on one problem by converting multiplication/division of fractions into multiplication/division of whole numbers. Several suggestions regarding teaching and learning of mathematical problem solving, algebra, and problems about speed and its related concepts of ratio and proportion are made.     

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.     

Comparing deaf and hearing college students' mental arithmetic calculations under two interference conditions

Deaf and hearing college students' mean reaction times (RTs) were compared on a mental calculation task in which they had to verify the accuracy of solutions to addition and multiplication problems. The deaf students were divided into higher and lower readers. Higher deaf readers and hearing students had similar RTs and accuracy on addition problems; their RTs were greater in the voicing interference mode than in the manual tapping interference mode. The lower deaf readers showed no RT differences between the two interference modes and had consistently lower RT performance and score accuracy across the verification tasks. On the verification task for multiplication problems, all participants showed a greater RT effect for manual tapping. The lower deaf readers were significantly less accurate on multiplication problems.     

The influence of self-efficacy and metacognitive prompting on math problem-solving efficiency

A regression design was used to test the unique and interactive effects of self-efficacy beliefs and metacognitive prompting on solving mental multiplication problems while controlling for mathematical background knowledge and problem complexity. Problem-solving accuracy, response time, and efficiency (i.e. the ratio of problems solved correctly to time) were measured. Students completed a mathematical background inventory and then assessed their self-efficacy for mental multiplication accuracy. Before solving a series of multiplication problems, participants were randomly assigned to either a prompting or control group. We tested the motivational efficiency hypothesis, which predicted that motivational beliefs, such as self-efficacy and attributions to metacognitive strategy use are related to more efficient problem solving. Findings suggested that self-efficacy and metacognitive prompting increased problem-solving performance and efficiency separately through activation of reflection and strategy knowledge. Educational implications and future research are suggested.     

Fostering the acquisition of subtraction strategies with interleaved practice: An intervention study with German third graders

Interleaved practice is a promising approach to foster the adaptive use of subtraction strategies. By intermixing strategies, comparison processes are evoked, which prompt more task-based strategy use. However, the effectiveness of interleaved practice in primary school mathematics has not been investigated yet. In the current study, 236 German third graders were randomly assigned to either an interleaved or a blocked condition. Both groups were instructed in using number-based strategies and the standard written algorithm for solving subtraction problems over 14 lessons. The students in the interleaved condition were prompted to compare strategies (between-comparison), while the students in the blocked condition compared the adaptivity of one strategy for different tasks (within-comparison). Our findings show that the students in the interleaved condition solved subtraction tasks with greater adaptiveness and accuracy. The effect on correctness was mediated by greater adaptive strategy use in the interleaved condition.     

Effects of achievement standards,choice, sex,and educational level on mathematical performance

The study assessed the effects of achievement standards, and choice of such standards, on mathematics performance in conditions where no tangible rewards were presented for reaching such standards. Elementary, junior high, and high school students performed a multiplication task in conditions where they chose standards, had standards imposed by an experimenter, or had no standards. Results demonstrated that high school students worked more multiplication problems when they chose standards than when identical standards were imposed by an experimenter or when no standards were present. Moreover, male students who chose standards worked more problems than did male students who had no standards; this effect was not found for female students.     

EFFECT OF THE PRESENCE OF EXTERNAL REPRESENTATIONS ON ACCURACY AND REACTION TIME IN SOLVING MATHEMATICAL DOUBLE-CHOICE PROBLEMS BY STUDENTS OF DIFFERENT LEVELS OF INSTRUCTION

This study explores the effects of the presence of external representations of a mathematical object (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Performance was evaluated according to the reaction time (RT) required for solving the problem and the accuracy of the answer. Thirty high school students studying at high and regular levels of instruction in mathematics (HL and RL) were asked to solve half of the problems with ERs and half of the problems without ERs. Each task was solved by half of the students with ERs and by half of the students without ERs. We found main effects of the representation mode with particular effect on the RT and the main effects of the level of mathematical instruction and mathematical subject with particular influence on the accuracy of students’ responses. We explain our findings using the cognitive load theory and hypothesize that these findings are associated with the different cognitive processes related to geometry and algebra.     

Transition from concrete to abstract representations: the distributive property in a Chinese textbook series

Through examining a representative Chinese textbook series’ presentation of the distributive property, this study explores how mathematics curriculum may structure representations in ways that facilitate the transition from concrete to abstract so as to support students’ learning of mathematical principles. A total of 319 instances of the distributive property were identified. The representational transition among these instances was analyzed at three tiers: within one worked example, from the worked example to practice problems within one topic, and across multiple topics over grades. Findings revealed four features that facilitate the transition process in the Chinese textbook series. First, it situates initial learning in a word problem context, which serves as a starting point of the transition process. Second, it sets up abstract representations as an ultimate goal of the multi-tier transition process. Third, it incorporates problem variations with connections in carefully designed tasks that embody the same targeted principles. Fourth, it engages students in constant sense making of the transition process through various pedagogical supports. Implementations and future research directions are also discussed.     

Strategy use,listening problems,and motivation of high- and low-proficiency Chinese listeners

Building on previous listening strategy research, the author aimed to explore the differences between Chinese high-proficiency listeners (HLs) and low-proficiency listeners (LLs) on their strategy use, problems, and motivation in native language (L1) listening. It involved 1,290 Grade 7 and 1,515 Grade 9 students. Both quantitative and qualitative methods, including a listening comprehension test, questionnaires, and interviews, were adopted. The findings indicated that HLs possessed more types of strategies and used strategies more frequently and effectively than LLs. HLs not only reported fewer listening problems but also had a better awareness of listening problems and use of problem-solving strategies than LLs. Both HLs and LLs agreed with the importance of listening but showed little interest in doing listening tasks. The similarities and differences between the findings of this study and those of second-language listening research and implications for planning effective instruction to enhance native language listening proficiency are discussed.     

Do it this way! Metacognitive strategies in collaborative mathematical problem solving

Recent years have seen increasing interest in the role of metacognition in mathematical problem solving, and in the use of small group work in classroom settings. However, little is known about the nature of secondary students' metacognitive strategy use, and how these strategies are applied when students work together on problems. The study described in this paper investigated the monitoring behaviour of a pair of senior secondary school students as they worked collaboratively on problems in applied mathematics. Analysis of verbal protocols from think aloud problem solving sessions showed that, although the students generally benefited from adopting complementary metacognitive roles, unhelpful social interactions sometimes impeded progress. The findings shed some light on the nature of individual and interactive metacognitive strategy use during collaborative activity.     

The relative difficulty of signed arithmetic story problems for primary level deaf and hard-of-hearing students

This study determines the relative difficulty and associated strategy use of arithmetic (addition and subtraction) story problems when presented in American Sign Language to primary level (K-3) deaf and hard-of-hearing students. Results showed that deaf and hard-of-hearing students may consider and respond to arithmetic story problems differently than their hearing peers, with the critical dimension in problem difficulty being based on the operation typically used to solve the problem, not the story within the problem. The types of strategies used by the students supported the order of problem difficulty. The visual-spatial nature of the problem presentation appeared not to assist the deaf and hard-of-hearing students in solving the problems. Factors that may have contributed to this pattern of problem difficulty are discussed so that educators can better align mathematics instruction to the thinking of the deaf child.     

Effective strategies for mental and written arithmetic calculation from the third to the fifth grade

The strategies used to solve mental and written multidigit arithmetical addition, subtraction, multiplication and division were observed in 200 third, fourth and fifth grade children. A strategy was classified as effective if it resulted in the correct solution at least 75% of the time. For mental addition and subtraction, primitive strategies such as counting on fingers and counting on (mental counting from a specific point), and the more sophisticated strategy 1010 (solution of the calculation problem using tens and units separately) were more effective than the strategies learned at school. In written addition, subtraction and multiplication there was a shift from the CAR+to the CAR- strategy (tabulating with, or without, a carried amount) from the third to the later grades. Results show that typical strategies taught at school progressively substitute every other strategy both in mental and written calculation, but without reaching the criterion of effectiveness. The implications for maths curricula are discussed.     

Student and school SES,gender, strategy use,and achievement

A multilevel mediated regression model was fit to Programme for International Student Assessment achievement, strategy use, gender, and family‐ and school‐level socioeconomic status (SES). Two metacognitive strategies (i.e., understanding and summarizing) and one learning strategy (i.e., control strategies) were found to relate significantly and positively to achievement. These strategies were used more by females and students attending higher SES schools. In contrast, males and students attending lower SES schools tended to use a greater number of learning strategies that did not relate to achievement, including memorization and elaboration. In addition, the strategies that did not relate to achievement were used more frequently by students from higher SES families. The findings suggest that schools, as opposed to families, may be the primary vehicle for developing effective strategy use practices for students and thus, targeted interventions may be particularly useful for male students attending low SES schools.     

A Study of the Influence of Some Factors in Style of Composition on the Interest,Comprehension, and Rate of Reading of Fourth-Grade Pupils

The research hypothesis was that memory strategy deficits can occur because of students' failure to understand the task and to evaluate their own performance. The effect of performance feedback on memory strategy use, performance evaluation, and recall was assessed with students with and without mild mental retardation with a mental age of approximately 8.5 years. For students with retardation, feedback resulted in more accurate performance evaluation and recall, but memory strategy use did not increase. By contrast, for students without retardation, feedback resulted in more accurate performance evaluation and greater recall accuracy and memory strategy use. We concluded that students with retardation may not have the appropriate memory strategies in their repertoire and, therefore, feedback does not result in strategy use. It seems likely that memory strategy use in this population would be increased by a training package that includes strategy instruction as well as feedback.     

Effects of interspersing rates on students performance on and preferences for mathematics assignments: Testing the discrete task completion hypothesis

The current study investigated the discreet task completion hypothesis presented by C. H. Skinner (2002) by investigating how the rate of interspersing affects performance on and preferences for academic assignments. Specifically, 70 sixth‐, seventh‐, and eighth‐grade students were presented with four assignment pairs of multiplication problems. Each pair consisted of a control assignment (i.e., no interspersing) and an experimental assignment (i.e., interspersing) that interspersed at one of four rates (i.e., no interspersing, every other problem, every third problem, or every fifth problem). After working on each assignment pair, assignment acceptability was measured. Results indicated that although students completed the same number of target problems with the same level of accuracy within assignment pairs, total problem completion rates were affected by the rate of interspersing. In addition, students' acceptability of the assignments was strongly related to the discrepancy in total problems completed across assignment types within assignment pairs. Discussion focuses on predicting students' preferences for academic assignments, implications for practitioners, and directions for future research. © 2007 Wiley Periodicals, Inc.     

Learning to spell: variability, choice, and change in children's strategy use

We examined whether the overlapping waves model, originally developed to account for strategy choices in arithmetic, could also account for strategy choices in spelling. The contrast was of particular interest because arithmetic is an algorithmic domain (a domain that includes strategies that always yield correct answers if executed properly), whereas spelling is not. Thirty first-grade students spelled words under 2 conditions, and 23 of these students were retested in second grade. Trial-by-trial analysis of strategy use was used to identify which strategies first and second graders used, how adaptively they chose among them, how effective the strategies were, and what changes occurred from first to second grade along each dimension. The model proved useful for understanding the development of spelling, despite the fact that explicit use of backup strategies had a minimal impact on accuracy. Implications for understanding adaptive strategy choices in algorithmic and nonalgorithmic domains are discussed.     

Thinking Strategies: Its Effectiveness in the Teaching of Multiplication Facts to Low Achievers of Mathematics

Many students have found the learning of multiplication a difficult task. Without mastering multiplication facts, a student will always resort to fundamental strategies which may be undesirable in problem-solving. Many teachers and educators agree that learning basic facts in the four operations are the fundamental steps that precede learning harder facts. Referring to the aspect of computational errors, Knifong and Holton (1975) indicated that computational errors accounted for 49% of the total errors made by 35 students who sat for the Metropolitan. Achievement Test. Suydam (1975) and NCTM (1977) stressed the importance of basic facts even though calculators and lately computers have been introduced into the school curriculum. The purpose of this study was to investigate the methodology of teaching basic facts which would alleviate the problems of teaching these facts to low achievers of Mathematics. A review of studies on thinking strategies shows that very few were carried out in learning the thinking strategies, particularly in multiplication using low achievers as the target. The success of using thinking strategies to teach multiplication number facts to most students does not imply its success in teaching all students. One of the intentions of this study is to investigate what strategies are useful in helping the low achievers in learning mathematics.     

Tower of Hanoi disk-transfer task: Influences of strategy knowledge and learning on performance

Tower of Hanoi has become a popular tool in cognitive and neuropsychology to assess a set of behaviors collectively referred to as executive functions. Substantial variability in performance on the Tower of Hanoi (TOH) disk-transfer task among normally functioning young adults, and potential contributions to these individual differences, were examined. In this expanded 60-problem version of the four-disk TOH, the degree to which problem administration (blocked vs. random) and strategy knowledge influenced overall performance and changes in accuracy across problems was examined. Eighty-seven college students were randomly assigned to a Blocked Group (problems given in ascending order of move-length) and a Random Group (problems given in a random order). After administration of the TOH task, participants described their problem solving and these verbal protocols were analyzed with regard to four elements of a strategic approach to problem solving. Problem administration order demonstrated no effect on task performance or on expressed strategy knowledge; however, strategy knowledge did predict performance on the TOH. An expected decrease in performance across trials was observed in the Blocked Group, and an increase in accuracy in the Random group indicated a learning effect. Strategy knowledge did not interact with these changes in performance across the items. These results suggest that external cues do not influence performance on the TOH to the same extent as individual differences in strategy induction relatively early in the problem solving process.