Acquisition of Mental Multiplication Skill: Evidence for the Transition Between Counting and Retrieval Strategies

Strategies used by third and fourth graders to perform simple multiplication problems were examined. Children participated in an interview and then completed a timed production task in which they solved single-digit multiplication problems. Analyses of children's verbal reports and of solution latency data were found to support the view that the acquisition of mental multiplication begins with the use of counting strategies. By the fourth grade, however, there was a marked transition toward the use of a retrieval strategy. Children in both grade levels were found to apply rules to solve problems that involved multiplication by 1 or 0; however, not all students reported using these rules, and those students who did report the use of these rules were not consistent in the application of the rules. Comparisons between groups' and individual subjects' performance revealed that some important individual differences were obscured when the group served as the unit of analysis. Although several discrepancies were noted between the analyses of verbal reports and the analyses of chronometric data, correlations between the two methods were moderately high.     

Calculation latency: The μ of memory and the τ of transformation

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N=47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into μ (reflecting shift) and t (reflecting skew) confirmed that retrieval trials contributed predominantly to μ, whereas procedure trials contributed predominantly to τ. The format × problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems) was associated entirely with μ and not with τ. Reported use of procedures presented a corresponding format × size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.     

“2 x 3” primes naming “6”: Evidence from masked priming

It is a common assumption for multiplication-solving models that single-digit multiplications are automatically retrieved. However, the experimental evidence for this is based on paradigms under suspicion. In this research, we employed a new procedure with the aim of assessing the automatic retrieval of multiplication more directly. In two experiments, multiplication automatism was studied using briefly presented primes (stimulus onset asynchrony = 48 msec) in a number-naming task. In Experiment 1, in the congruent conditions, the target and the prime were the same numbers (e.g., prime, 6; target, 6) or the target was the solution to the multiplication prime (e.g., prime, 2×3=; target, 6). In the incongruent conditions, no relationship existed between the primes and the targets (e.g., prime, 32; target, 6; or prime, 4×8=; target, 6). Experiment 2 explored the relevance of the equal sign for the multiplication-priming effect. Data showed that naming was faster when the solution of the multiplication prime matched the target, as compared with the incongruent condition (multiplication-priming effect), and that these effects were found irrespective of the presence of the equal sign. The fact that this priming effect was found even though the participants were unaware of the presentation of the primes supports the automatic character of single-digit multiplication. We conclude by arguing that this procedure is highly valuable for exploring the mechanisms involved in simple arithmetic solving.     

Do people combine the parity- and five-rule checking strategies in product verification?

The basic question of the present experiment was whether people use a combination of arithmetic problem solving strategies to reject false products to multiplication problems or whether they simply use the single most efficient strategy. People had to verify true and false, five and non-five arithmetic problems. Compared with no-rule violation problems, people were faster with (a) five problems that violated the five rule (i.e., N×5=number with 5 or 0 as the final digit; e.g., 15 × 4=62), (b) problems that violated the parity rule (i.e., to be true, a product must be even if either or both of its multipliers is even; otherwise, it must be odd; 4 × 38=149), and (c) problems that violated both the parity and five rules (e.g., 29 × 5=142). Finally, people were equally fast and accurate when they solved two-rule violation problems than when they solved five-rule violation problems, and faster for those two types of problems than for parity-rule violation problems. Clearly, people use the single most efficient strategy when they reject false product to multiplication problems. This result has implications for our understanding of strategy selection in both arithmetic in particular and human cognition in general. Received: 18 October 1999 / Accepted: 27 January 2000     

Retrieval-induced forgetting of arithmetic facts but not rules

Retrieval of a multiplication fact (2×6 =12) can disrupt retrieval of its addition counterpart (2+6=8). We investigated whether this retrieval-induced forgetting effect applies to rule-governed arithmetic facts (i.e., 0×N=0, 1×N=N). Participants (n=40) practised rule-governed multiplication problems (e.g., 1×4, 0×5) and multiplication facts (e.g., 2×3, 4×5) for four blocks and then were tested on the addition counterparts (e.g., 1+4, 0+5, 2+3, 4+5) and control additions. Increased addition response times and errors relative to controls occurred only for problems corresponding to multiplication facts, with no problem-specific effects on addition counterparts of rule-governed multiplications. In contrast, the rule-governed 0+N problems provided evidence of generalisation of practice across items, whereas the fact-based 1+N problems did not. These findings support the theory that elementary arithmetic rules and facts involve distinct memory processes, and confirmed that previous, seemly inconsistent findings of RIF in arithmetic owed to the inclusion or exclusion of rule-governed problems.     

Retrieval-induced forgetting of arithmetic facts across cultures

Retrieving a single-digit multiplication fact (3×4 =12) can slow response time (RT) for the corresponding addition fact (3+4=7). The present experiment investigated effects of problem type (i.e., tie addition problems such as 3+3 vs. non-ties such as 3+4) and cultural background on this retrieval-induced forgetting (RIF) phenomenon in young adults. Canadians answering in English (n=36), Chinese adults answering in English (n=36), and Chinese answering in Chinese (n=36) received four blocks of multiplication practice and then two blocks of the addition counterparts and control additions. Tie addition problems presented a robust RIF effect that did not differ between groups, but only the Canadian group showed RIF for non-ties and only for small non-ties with sum≤10 (3+4). The Chinese groups' RIF effect for addition ties, but not small non-ties, converges with recent evidence that ties are solved by direct memory retrieval whereas small non-ties may be solved by highly efficient procedural processes in skilled performers.     

高校教师教学水平评价概化理论预算限制下最佳样本量估计

使用“高校教师教学水平评价问卷”,要求566名学生对19名教师进行评价,对收集到的数据作不同的概化设计,包括t×i、（s：t）×i、（s：t）×（i：v）和（s：t）×（i：v）×o四种设计。基于概化理论,结合预算限制,统一LaGrange乘法公式,自行推导不同设计的最佳样本量公式,联合估计的方差分量,计算出不同设计的最佳样本量。结果表明：（1）LaGrange乘法统一公式表现出较强的通用性,能够适用于预算限制下各种概化设计;（2）评价场合是影响高校教师教学水平评价一个相当重要的因素;（3）（s：t）×（i：v）×o是高校教师教学水平评价概化理论预算限制下最优概化设计;（4）高校教师教学水平评价概化理论预算限制下,每位教师最佳评价学生人数为20人,每个维度最佳评价题目数为3题。     

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college‐educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse primes for the equation that immediately followed it (e.g., 4 × 3/4 = 3 followed by 3 × 8/6 = 4). Students with relatively high math ability showed relational priming (speeded solution times to the second of two successive relationally related fraction equations) both with and without high perceptual similarity (Experiment 2). Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers (e.g., 4 × 3/4 = 3 followed by 3 × 4/3 = 4). Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers.     

Mental Arithmetic: A Componential Analysis of Speed-of-Processing Across Monolingual,Weak Bilingual,and Strong Bilingual Adults

Two experiments compared rates of solving simple and complex addition and multiplication problems in groups of speakers of French or English in Experiment 1 (n = 35) and Spanish or English in Experiment 2 (n = 84). Subjects were divided into groups of English unilinguals, weak bilinguals, and strong bilinguals according to their performance on a naming task. In both experiments, simple problems consisted of two single-digit numbers. At least three single-digit numbers were used for complex problems in Experiment 1 and double-digit numbers in Experiment 2. Mean solution times, particularly for complex problems, were lowest for the monolingual group, followed in turn by the weak bilingual and strong bilingual groups, but these differences were not statistically reliable in either experiment. In Experiment 2, however, componential analyses of solution times indicated that strong bilingual subjects were slower at executing the carry operation when solving complex problems, relative to the two remaining groups. Results were interpreted in terms of the relationship between bilingualism and the representation and processing of numerical information.     

Effects of problem format on division and multiplication performance: division facts are mediated via multiplication-based representations

In 2 experiments participants solved division problems presented in multiplication-based formats (e.g., 8 x _ = 72) more quickly than division problems presented in division-based formats (e.g., 72 / 8 = _). In contrast, participants solved multiplication problems presented in a division-based format (e.g., _ / 8 = 9) slowly and made many errors. In both experiments, the advantage for multiplication-based formats on division problems was found only for large problems (i.e., those with products or dividends greater than 25). These findings provide support for the view that large single-digit division facts are mediated via multiplication-based representations and that multiplication is the primary mode of representation for both division and multiplication facts.     

Student Preference for Explicit Timing and Interspersal Procedures as a Function of Math Problem Completion Rates: Testing the Discrete Task Completion Hypothesis

A within-subjects design was used to compare explicit timing and interspersal with college students. Students were given 3 minutes to complete problems on the explicit timing assignment (25 problems, 3 digits –3 digits) and the interspersal assignment (25 similar problems and 10 problems, 1 digit –1 digit). Results indicated that: (a) students completed more total problems during interspersal for both trials, (b) students completed more target problems during explicit timing for the second trial, and (c) students only preferred interspersal for the first trial. The data from trial one fit the discrete task completion hypothesis and matching law, yet the data from trial two do not match as closely (Skinner, 2002). Discussion focuses on continued need for more research on academic interventions, comparing academic interventions, the discrete task completion hypothesis, and the matching law.     

Effects of Interspersing Rate on Student Preferences for Mathematics Assignments

This study investigated the extent to which interspersing effects are consistent with the effects of reinforcement on predicting students preferences for mathematics assignments. Students were exposed to 4 pairs of assignments. Each assignment pair contained a control assignment with 15 problems requiring multiplication of a three digit number by a two digit number, and an experimental assignment consisting of 15 similar multiplication problems plus additional brief one-digit by one-digit multiplication problems interspersed at four different rates (i.e., no interspersing, every other, every third, or every fifth problem) across assignment pairs. Performance data were collected for accuracy, total problem completion rate and target problem completion rate. In addition to performance data, students were asked to rate each assignment with regard to relative difficulty, time, effort to complete, and preference between assignments for homework. Results suggest that although interspersing rates do not affect accuracy, they do affect problem completion rate, and student preferences for academic assignments. Discussion focuses on interspersing rate and schedules of reinforcement with emphasis on both applied and theoretical implications.     

Target strength and retrieval-induced forgetting in semantic recall

Retrieval-induced forgetting (RIF) occurs when practice of a memory item impairs retrieval of related, unpracticed items. Here, we demonstrated that RIF in semantic memory is retrieval dependent. University students either studied (7 × 8 = 56) or retrieved (7 × 8 = ?) the answers to a set of multiplication problems for 40 blocks and then were tested on their addition counterparts (7 + 8 = ?). For the retrieval practice group, but not the study practice group, response time for the multiplication-practiced addition facts was about 100 msec slower, relative to control addition problems, in the first of five postpractice addition blocks. Subsequent blocks of addition were interleaved with retrieval blocks of all the multiplication counterparts, which permitted measurement of RIF for the control addition problems after only a single retrieval of their multiplication counterparts. The control problems presented RIF in excess of 200 msec, much larger than the RIF observed after massive practice. This is consistent with the hypothesis that inhibition of competitors should be weaker when target strength is high than when target strength is only moderate (Anderson, 2003; Norman, Newman, &; Detre, 2007). The evidence that RIF in semantic retrieval is both retrieval dependent and weaker following massive target practice than following moderate target practice provides strong support for inhibition-based theories of RIF.     

Enhancing Multiplication Performance in Students with Moderate Intellectual Disabilities Using Pegword Mnemonics Paired with a Picture Fading Technique

The present study examined the effectiveness of an instructional package that included an adapted version of pegword mnemonics paired with a picture fading technique in teaching two students with moderate intellectual disabilities to recall 28 single-digit multiplication facts between 2 and 9. The instructional package was assessed using a multiple baseline design across students. Results indicated that instruction produced substantial and immediate effects in that both students increased their accuracy at recalling basic multiplication facts, maintained the newly acquired skill, and were able to generalize it across material, format, and another trainer. These results have implications for teaching students with intellectual disabilities basic math facts that are considered important for gaining access to the general curriculum.     

Arabic digit naming speed: task context and redundancy gain

There is evidence for both semantic and asemantic routes for naming Arabic digits, but neuropsychological dissociations suggest that number-fact retrieval (2x3=6) can inhibit the semantic route for digit naming. Here, we tested the hypothesis that such inhibition should slow digit naming, based on the principle that reduced access to multiple routes would counteract redundancy gain (the response time advantage expected from parallel retrieval pathways). Participants named two single digit numbers and then performed simple addition or magnitude comparison (Experiment 1), multiplication or magnitude comparison (Experiment 2), and multiplication or subtraction (Experiment 3) on the same or on a different pair of digits. Addition and multiplication were expected to inhibit the semantic route, whereas comparison and subtraction should enable the semantic route. Digit naming time was approximately 15ms slower when participants subsequently performed addition or multiplication relative to comparison or subtraction, regardless of whether or not the same digit pair was involved. A letter naming control condition in Experiment 3 demonstrated that the effect was specific to digit naming. Number fact retrieval apparently can inhibit Arabic digit naming processes.     

A Preliminary Investigation of the Relationship Between Fluency and Application for Multiplication

Research suggests component skill performance has a strong positive relationship with composite skill performance. This study examined the association between accuracy and fluency for the component-composite relationship within multiplication. One hundred and fifty-seven fifth-graders did one-minute assessments for single-digit, and multi-digit multiplication problems. The results demonstrated the students achieved high levels of accuracy but low levels of fluency. Strong correlations between the component-composite skill fluency suggest that fluent component skills may have a significant role in composite skill performance. Moderate/low correlations between component and composite skill accuracy indicate that more than one skill component may contribute to composite skill acquisition.     

Single-digit multiplication: is there an age-of-acquisition effect?

Evidence for 37 12-yr-old students is presented which suggests that an age-of-acquisition effect may be present in the performance of single-digit multiplication facts.     

A neuropsychological dimension for anchoring effects

Previous research has shown that strength of handedness predicts differences in sensory illusions, Stroop interference, episodic memory, and beliefs about body image and the origin of species. Recent evidence also suggests handedness differences in the susceptibility to information framing and persuasion. The present paper extends this line of work to decision anchoring effects. In Experiment 1, 131 introductory psychology students responded to 12 real‐world knowledge questions after being given random, uninformative high or low anchors. Results indicated that “strong‐handers” showed larger anchoring effects than “mixed‐handers.” In Experiment 2, 89 introductory psychology students responded to 6 real‐world knowledge questions in a modified, two‐step anchoring task in which participants were given a credible source for the anchored information and asked to give pre‐ and post‐anchor estimates. In contrast to Experiment 1, results revealed that mixed‐ and strong‐handers were affected similarly by anchoring. In Experiment 3, 158 students were asked to estimate the answer to one of two versions of 8! (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 or 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8)—a multiplication problem in which the high and low anchors are inherently informative. Here, mixed‐handers showed larger anchoring effects than strong‐handers. A theory centered around the notion of hemispheric specialization and the communication between the two halves of the brain as well as arguments about the informativeness of anchors, metacognition, and recent theorizing in the anchoring literature are used to account for these data. Copyright © 2005 John Wiley & Sons, Ltd.     

EFFECTS OF EXPLICIT TIMING ON MATHEMATICS PROBLEM COMPLETION RATES IN AFRICAN-AMERICAN THIRD-GRADE ELEMENTARY STUDENTS

A multiple baseline design was used to evaluate the effects of Van Houten and Thompson's (1976) explicit timing procedure on problem completion rates and accuracy levels in African-American third-grade students. During the explicit timing phase, students were told that they were being timed and were instructed to circle the last problem completed at each 1-min interval. Results showed that the explicit timing procedure increased problem completion rates. A decreasing trend in percentage of problems correct also occurred. Exploratory data analysis suggested that decreases in accuracy were not caused by the explicit timing procedure and did not occur in students who had attained high levels of preintervention accuracy. Discussion focuses on recommendations for educators who wish to use timing procedures to increase students' rates of accurate responding.