Multiplication by eye and by ear for Chinese-speaking and English-speaking adults

How do distracting events influence children's arithmetic performance?

To understand how distraction influences children's arithmetic performance, we examined effects of irrelevant sounds on children's performance while they solve arithmetic problems. Third and fifth graders were asked to verify true/false, one-digit addition problems (e.g., 9 + 4 = 12. True? False?) under silence and sound conditions. The sounds began when the problems started to appear on the screen (Experiment 1; N = 76) or slightly after (Experiment 2; N = 92) and continued until participants responded. The results showed that (a) children solved arithmetic problems more quickly in the sound condition than in the silence condition when the sounds started with problem display (phasic arousal effects); (b) children were slower on the arithmetic problem verification task when the sounds was played slightly after the problems started to appear on the screen (distraction effects); (c) phasic arousal effects were found only in third graders, whereas distraction effects were found in both grades, although their magnitudes were smaller in fifth graders; (d) distraction effects increased with increasing latencies in third graders but did not change across the entire latency distribution in fifth graders; and (e) distraction effects on current trials were smaller after sound trials than after silence trials in both age groups (sequential modulations of distraction effects). These findings have important implications for furthering our understanding of effects of irrelevant sounds on arithmetic performance as well as cognitive processes involved in children's arithmetic.

Effects of presentation modality and duration on children's strategy use: A study in computational estimation

Two experiments were run to determine how presentation modality and duration influence children's arithmetic performance and strategy selection. Third and fourth graders were asked to find estimates for two-digit addition problems (e.g., 52 + 39). Children were tested in three conditions: (1) time-unlimited visual, (2) time-limited visual, or (3) time-limited auditory conditions. Moreover, we assessed children's working-memory updating and arithmetic fluency. Children were told which strategy to use on each problem to assess arithmetic performance while executing strategies, in Experiment 1, and were asked to choose the best strategy of three available strategies to assess strategy selection, in Experiment 2. Presentation modality influenced strategy execution (i.e., children were faster and more accurate in problems under visual than auditory conditions) but only in children with low updating abilities. In contrast, presentation modality had no effect on children's strategy selection. Presentation duration had an effect on both strategy execution and strategy selection with time-limited presentation leading to a decline in children's performance. Interestingly, specifically in children with low updating abilities, time-limited presentation led to poorer performance. Hence, efficient updating seemed to compensate for detrimental effects of auditory in comparison to visual and time-limited in comparison to time-unlimited presentation. These findings have important implications for determining conditions under which children execute strategies most efficiently and select the best strategy on each problem most often, as well as for understanding mechanisms underlying strategic behaviour.

Children's Strategy Choices on Complex Subtraction Problems: Individual Differences and Developmental Changes

We examined how children's strategy choices in solving complex subtraction problems are related to grade and to variations in problem complexity. In two studies, third- and fifth-grade children (N≈160 each study) solved multi-digit subtraction problems (e.g., 34–18) and described their solution strategies. In the first experiment, strategy selection was investigated by means of a free-choice paradigm, whereas in the second study a discrete-choice approach was implemented. In both experiments, analyses of strategy repertoire indicated that third-grade children were more likely to report less-efficient strategies (i.e., counting) and relied more on the right-to-left solution algorithm compared to fifth-grade children who more often used efficient memory-based retrieval and conceptually-based left-to-right (i.e., decomposition) strategies. Nevertheless, all strategies were reported or selected by both older and younger children and strategy use varied with problem complexity and presentation format for both age groups. These results supported the overlapping waves model of strategy development and provide detailed information about patterns of strategy choice on complex subtraction problems.

The role of early language abilities on math skills among Chinese children

BACKGROUND

The present study investigated the role of early language abilities in the development of math skills among Chinese K-3 students. About 2000 children in China, who were on average aged 6 years, were assessed for both informal math (e.g., basic number concepts such as counting objects) and formal math (calculations including addition and subtraction) skills, language abilities and nonverbal intelligence.

METHODOLOGY

Correlation analysis showed that language abilities were more strongly associated with informal than formal math skills, and regression analyses revealed that children's language abilities could uniquely predict both informal and formal math skills with age, gender, and nonverbal intelligence controlled. Mediation analyses demonstrated that the relationship between children's language abilities and formal math skills was partially mediated by informal math skills.

RESULTS

The current findings indicate 1) Children's language abilities are of strong predictive values for both informal and formal math skills; 2) Language abilities impacts formal math skills partially through the mediation of informal math skills.

The retrieval and selection of arithmetic facts in oral arithmetic

We examined the co-activation and the selection of arithmetic facts in oral arithmetic. In two experiments, participants had to verify whether simple additions were correct or not. In Experiment 1, additions were presented in the auditory-verbal format; in Experiment 2, additions were presented in the digit format but simulating the temporal sequence of auditory problems of Experiment 1. Results were similar in both experiments. Firstly, participants took the same time to respond when an addition was incorrect but the result was that of multiplying the operands (e.g., 2+4=8) relative to a control addition with unrelated result. Secondly, participants took more time to respond when the result of multiplying the operands of the first trial was presented again in a correct addition problem (e.g., 2+6=8) relative to a control addition. This pattern of results is discussed in terms of the temporal resolution to which auditory problems are resolved and the role of an inhibitory mechanism involved in the selection of arithmetic facts.

Age-related differences in sequential modulations of problem-size and rule-violation effects during arithmetic problem verification tasks

Young and older adults were asked to verify true (e.g., 5 × 61 = 305) and false (5 × 61 = 315) arithmetic problems. Half the problems were small (e.g., 5 × 17 = 85) and half were large problems (e.g., 5 × 93 = 465). Half the false problems respected the five rule (i.e., the product of an operand multiplied by five ends with either 5 or 0), and half violated this rule (e.g., 21 × 5 = 115 vs. 21 × 5 = 113). Both young and older adults showed problem-size effects (i.e., they verified small problems more quickly than large problems) and five-rule violation effects (i.e., they verified problem violating five rule more quickly than problems respecting five rule). Moreover, we found sequential modulations of these problem-size and five-rule effects. Problem-size effects were larger on current problems following large problems than after small problems, and five-rule violation effects were larger after problems violating the five rule than after no-rule violation problems. Finally, sequential modulations of problem-size effects were larger in older adults than in young adults, and there were no age-related differences in sequential modulations of five-rule violation effects. These findings speak to the determiners of arithmetic performance, as to how well arithmetic calculation and non-calculation strategies are executed and selected on current problems depends on strategies used with preceding problems.

The relation between language and arithmetic in bilinguals: insights from different stages of language acquisition

Solving arithmetic problems is a cognitive task that heavily relies on language processing. One might thus wonder whether this language-reliance leads to qualitative differences (e.g., greater difficulties, error types, etc.) in arithmetic for bilingual individuals who frequently have to solve arithmetic problems in more than one language. The present study investigated how proficiency in two languages interacts with arithmetic problem solving throughout language acquisition in adolescents and young adults. Additionally, we examined whether the number word structure that is specific to a given language plays a role in number processing over and above bilingual proficiency. We addressed these issues in a German-French educational bilingual setting, where there is a progressive transition from German to French as teaching language. Importantly, German and French number naming structures differ clearly, as two-digit number names follow a unit-ten order in German, but a ten-unit order in French. We implemented a transversal developmental design in which bilingual pupils from grades 7, 8, 10, 11, and young adults were asked to solve simple and complex additions in both languages. The results confirmed that language proficiency is crucial especially for complex addition computation. Simple additions in contrast can be retrieved equally well in both languages after extended language practice. Additional analyses revealed that over and above language proficiency, language-specific number word structures (e.g., unit-ten vs. ten-unit) also induced significant modulations of bilinguals' arithmetic performances. Taken together, these findings support the view of a strong relation between language and arithmetic in bilinguals.

Mathematics development and difficulties: the role of visual-spatial perception and other cognitive skills

Several neurocognitive abilities, including visual-spatial and language-based processes, attention, and fine motor/finger skills, are thought to play important roles in mathematical development and disability. Evidence for relations of specific neurocognitive skills and mathematical development and disability is presented, with a particular emphasis on findings from longitudinal studies. Why these particular neurocognitive skills are related to math is also discussed. We suggest that mathematics learning in children with congenital and acquired neurodevelopmental disorders, including children treated for cancer, is particularly vulnerable to disruption because these disorders often affect one or more of the neurocognitive systems that support math learning and performance. Implications for assessment of and interventions for math difficulties are discussed. The article ends with implications for mathematical functioning in children treated for acute lymphoblastic leukemia and brain tumors.

Separate pathway characteristics of numerical surface form processing: evidence from operand-related error effects

Two perspectives compete to explain how the surface form of digits affects cognitive processing of numerical magnitude; one argues for a common pathway and the other for separate pathways. This study examined the operand-related error effect in simple multiplication operations using different combinations of visually presented Arabic digits and auditorily presented Mandarin number words. The study suggested two conclusions, both consistent with the separate pathway perspective. First, the numerical surface form (Arabic digits, spoken Mandarin number words) affected retrieval. That is, surface properties were maintained as specific codes throughout processing. Second, the phonological code activated by spoken Mandarin number words interfered with activation of answers during retrieval.

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.

The operand-order effect in single-digit multiplication: An ERP study of Chinese adults

Unlike those used in the West, a typical Chinese multiplication table includes only smaller-operand-first entries (e.g., 3 x 7=21, but not 7 x 3=21). Due to this unique feature, multiplication for Chinese subjects has been found to show an operand-order effect. The present study aims to investigate the neural bases of the operand-order effect. Subjects were 20 Mainland Chinese subjects who learned as children the half multiplication table (i.e., smaller-operand-first entries only) and 20 Hong Kong and Macao Chinese subjects who learned as children the whole multiplication table (i.e., both smaller- and larger-operand-first entries) under the British and Portuguese educational systems, respectively. ERP data showed that, for those who learned the half table (Mainland Chinese), but not for those who learned the whole table (Hong Kong and Macao Chinese), the larger-operand-first problems elicited greater negative potentials across representative electrodes of the whole scalp, emerging at about 120 ms after the onset of the second operand and lasting until around 750 ms. These results suggest that the particular experience of acquiring multiplication facts had pronounced impact on their representations in the brain.

Interactive Effects of Numerical Surface Form and Operand Parity in Cognitive Arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.

Phonological and visual working memory in mental addition

The goal of the present research was to examine the role of working memory in mental arithmetic. Adults (n = 96) solved multidigit arithmetic problems (e.g., 52 + 3; 3 + 52) alone and in combination with either a phonological memory load (i.e., nonwords, such as gup) or a visual memory load (i.e., random pattern of asterisks). The participants solved problems presented in a vertical format significantly faster than problems presented in a horizontal format. They also solved double digit first problems (e.g., 52 + 3) more quickly than the reverse (e.g., 3 + 52), but only when the problems were presented horizontally. Performance was worse in the phonological load condition than in the visual load condition for the participants who solved problems presented horizontally, whereas performance was worse in the visual load condition than in the phonological load condition when problems were presented vertically. The present research provides evidence that both phonological and visual aspects of working memory are involved in mental arithmetic but that the role of each working memory component will depend on such factors as presentation format.