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.     

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.     

What affects strategy selection in arithmetic? The example of parity and five effects on product verification

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

Practice effects on strategy selection and strategy efficiency in simple mental arithmetic

Two experiments were conducted to investigate the effects of practice on strategy selection and strategy efficiency in mental arithmetic. Participants had to solve simple addition or multiplication problems, after having received 0, 3, or 6 practice sessions (Experiment 1), and before and after having received 3 practice sessions (Experiment 2). Strategy selection was measured by means of trial-by-trial strategy reports, whereas strategy efficiency was measured by means of response latencies. Results showed significant practice effects on retrieval frequency, procedural frequency, retrieval efficiency, and procedural efficiency. However, practice effects on strategy efficiency appeared to be both strategy-specific (i.e., only for procedural strategies) and operation-specific (i.e., only for multiplication problems). Implications of the present results for mathematic cognition and its modeling are discussed.     

Changing the focus of attention: The interacting effect of valence and arousal

We examined how valence and arousal of an image influence visual attention. “Spotlight of attention” theory suggests that positive affect broadens, and negative affect narrows, one's aperture of attention, whereas the arousal theory literature suggests that arousal level is what modulates attentional focus, with highly arousing affect capturing attention, regardless of valence. In two experiments, a digit parity task was used to index the influence of valence, and arousal, on visual attention. Positive or negative images were displayed centrally on each trial, with single digits presented more peripherally (Experiment 1) or more centrally (Experiment 2) to the image. In both Experiments participants were slower, and less accurate at making parity decisions (e.g., both digits odd or both even) when the image was negative relative to positive, and of high arousal. For low arousal images, positive, relative to negative, valence images led to greater impairment of the digit parity task. Findings suggest that arousal level of images modulates the influence of valence on distribution of visual attention. Highly negative emotional images may command or capture attention, but there are other factors that can lead to attention capture, even in low arousing positive stimuli.     

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     

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

English-speaking (n = 32) and Chinese-speaking adults (n = 32) solved single-digit multiplication problems. In one condition, problems were presented as visual digits (e.g., 8 x 9). In the other condition, problems were presented as auditory number words in the participant's first language (e.g., /eit/ /taimz/ /nain/). Chinese-speaking adults made proportionately more operand-intrusion errors (e.g., 4 x 8 = 24) than English-speaking adults. Both groups made more operand-intrusion errors with auditory than with visual presentation. These findings are similar to those found when participants solve problems presented as visual number words (e.g., eight x nine), suggesting that in both cases the activation of phonological codes interferes with processing.     

Simple arithmetic processing: surface form effects in a priming task

Models of numerical processing vary on whether they assume common or separate processing pathways for problems represented in different surface forms. The present study employed a priming procedure, with target naming task, in an investigation of surface form effects in simple addition and multiplication operations. Participants were presented with Arabic digit and number word problems in one of three prime-target relationships, including congruent (e.g., '2+3' and '5'), incongruent (e.g., '9+7' and '5') and neutral (e.g., 'X+Y' and '5') conditions. The results revealed significant facilitatory effects in response to congruent digit stimuli at SOAs of 300 and 1000ms, in both operations. In contrast, inhibitory effects were observed in response to incongruent word stimuli in both the addition and multiplication operations at 300ms, and in the addition operation at 1000ms. The overall priming effects observed in the digit condition were significantly greater than in the word condition at 1000ms in the multiplication operation and at 300ms in the addition operation. The results provide support to separate pathway accounts of simple arithmetic processing for problems represented in different surface forms. An explanation for variation in processing due to differences in access to visual and phonological representations is provided.     

Early learning shapes the memory networks for arithmetic: evidence from brain potentials in bilinguals

Language and math are intertwined during children's learning of arithmetic concepts, but the importance of language in adult arithmetic processing is less clear. To determine whether early learning plays a critical role in the math-language connection in adults, we tested retrieval of simple multiplication in adult bilinguals who learned arithmetic in only one language. We measured electrophysiological and behavioral responses during correctness judgments for problems presented as digits or as number words in Spanish or English. Problems presented in the language in which participants learned arithmetic elicited larger, more graded, and qualitatively different brain responses than did problems presented in participants' other language, and these responses more closely resembled responses for digits, even when participants' other language was more dominant. These findings suggest that the memory networks for simple multiplication are established when arithmetic concepts are first learned and are independent of language dominance in adulthood.     

Is it a letter? Is it a number? Processing of numbers within SMS shortcuts

For efficiency reasons, words in electronic messages are sometimes formed by combining letters with numbers, as in gr8 for “great.” The aim of this study was to investigate whether a digit incorporated into a letter-digit shortcut would retain its numerosity. A priming paradigm was used with letter-digit shortcuts (e.g., gr8) and matched pseudoshortcuts (e.g., qr8) as primes. The primes were presented simultaneously with sets of dots (targets) for which even/odd decisions were required, or they appeared 250 msec before target onset. When pseudoshortcuts were presented, decision latencies were shorter when the target and the digit in the prime were matched in parity than when they were mismatched. This main effect of match was not significant for shortcuts. The results suggest that the number concepts of digits combined with letters become activated but are quickly suppressed or deactivated when the digit is part of an existing shortcut.     

Automatic and directed search processes in solving simple semantic-memory problems

The cognitive processes involved in simple semantic-memory problems were investigated in four experiments. On each trial of Experiments 1 and 2, two stimulus words were presented, with the instructions to find a third word (i.e., the solution) that, when coupled with each of the stimuli, would yield two word pairs used in everyday language (e.g., surprise and birthday, for which the solution is party). The results of the two experiments indicated that informing the subject whether the solution constituted the first or the second element in the word pairs facilitated both likelihood and speed of solution attainment. In addition, solution attainment was relatively high for items based on frequently used word pairs (Experiment 1) and for items in which the stimuli appear, in everyday language, in a small number of word pairs (Experiment 2). In Experiment 3, the subjects were required to produce word pairs containing one of the two stimulus words from the items used in Experiment 2. Solution production was facilitated by rehearsing the second stimulus word of the specific item. The conclusion, supported by a post hoc analysis of the results of Experiments 2 and 3 (Experiment 4), was that indirect priming from one stimulus word may facilitate solution production from a searched word. These results are interpreted in terms of automatic and controlled processes, and their relevance to two different models for retrieval from semantic memory is discussed.     

The facilitative influence of phonological similarity and neighborhood frequency in speech production in younger and older adults

A tip-of-the-tongue (TOT) elicitation task and a picture-naming task were used to examine the role of neighborhood frequency as well as word frequency and neighborhood density in speech production. As predicted for the younger adults in Experiment 1, more TOT states were elicited for words with low word frequency and with sparse neighborhoods. Contrary to predictions, neighborhood frequency did not significantly influence retrieval of the target word. For the older adults in Experiment 2, however, more TOT states were elicited for words with low neighborhood frequency. Furthermore, in Experiment 3, pictures with high neighborhood frequency were named more quickly and accurately than pictures with low neighborhood frequency. These results show that the number of neighbors and the frequency of those neighbors influence lexical retrieval in speech production. The facilitative nature of these factors is more parsimoniously accounted for by an interactive model rather than by a strictly feedforward model of speech production.     

Simple arithmetic processing: Individual differences in automaticity

This study investigated individual differences in the ability to automatically access simple addition and multiplication facts from memory. It employed a target-naming task and a priming procedure similar to that utilised in the single word semantic-priming paradigm. In each trial, participants were first presented with a single digit arithmetic problem (e.g., 6+8) and were then presented with a target that was either congruent (e.g., 14) or incongruent (e.g., 17) with this prime. Response times for congruent and incongruent conditions were then compared to a neutral condition (e.g., X+Y, with target 14). For the high skilled group, significant facilitation in naming congruent multiplication and addition targets was found at SOAs of 300 and 1000?ms. In contrast, for the low skilled group, facilitation in naming congruent targets was only observed at 1000?ms. Significant inhibition in naming incongruent multiplication and addition targets at 300?ms, and addition targets at 1000?ms, was found for the high skilled group alone. This advantage in access to simple facts for the high skilled group was then further supported in a problem size analysis that revealed individual differences in access to small and large problems that varied by operation. These findings support the notion that individual differences in arithmetic skill stem from automaticity in solution retrieval and additionally, that they also derive from strategic access to multiplication solutions.     

Does number data entry rely on the phonological loop?

Two experiments investigated effects of articulatory processing on number data entry. Participants entered four‐digit numbers presented as either words or numerals on a keyboard, either under an articulatory condition or in silence. In Experiment 1, the articulatory condition was articulatory suppression; in Experiment 2, it was vocalisation. In Experiment 1, the articulatory suppression group typed initial digits faster than the silent group, but for subsequent digits, the opposite pattern occurred at least with word stimuli. In Experiment 2, the silent group typed initial digits faster but typed subsequent digits somewhat slower than the vocalisation group. Thus, articulation of numbers, which promotes entry into the phonological loop of working memory, retards processing of initial digits but enhances processing of subsequent digits.     

The surface form x problem size interaction in cognitive arithmetic: evidence against an encoding locus.

Sixty-four university students received simple addition problems with operands presented as arabic digits (e.g. 2 + 3, 8 + 6) or as English number words (two + three, eight + six). Operands either were displayed simultaneously or sequentially with the left operand appearing 800 ms before the right operand. Consistent with previous findings, word problems were slower than digit problems and this word-format cost was larger for large- than small-number problems. The central question addressed by this experiment concerned whether this Format x Size interaction arises during problem encoding processes or in subsequent retrieval or production processes. In the simultaneous condition, both operands would contribute to format-related differences in encoding, whereas in the sequential condition encoding differences would arise only in connection with the second operand. Critically, however, the Format x Size interaction did not differ between the sequential and simultaneous conditions, although the experiment had ample power to detect such an effect. The results argue that the Format x Size interaction does not arise during encoding, but instead arises during calculation or production processes.     

I 5683 you: dialing phone numbers on cell phones activates key-concordant concepts

When people perform actions, effects associated with the actions are activated mentally, even if those effects are not apparent. This study tested whether sequences of simulations of virtual action effects can be integrated into a meaning of their own. Cell phones were used to test this hypothesis because pressing a key on a phone is habitually associated with both digits (dialing numbers) and letters (typing text messages). In Experiment 1, dialing digit sequences induced the meaning of words that share the same key sequence (e.g., 5683, LOVE). This occurred even though the letters were not labeled on the keypad, and participants were not aware of the digit-letter correspondences. In Experiment 2, subjects preferred dialing numbers implying positive words (e.g., 37326, DREAM) over dialing numbers implying negative words (e.g., 75463, SLIME). In Experiment 3, subjects preferred companies with phone numbers implying a company-related word (e.g., LOVE for a dating agency, CORPSE for a mortician) compared with companies with phone numbers implying a company-unrelated word.     

The effect of divided attention on memory for items and their context.

Memory for context, in comparison to memory for items, is a more demanding task and requires more attentional resources. We examined differences between item and context memory using divided attention at encoding and retrieval. Participants were presented with word lists and were instructed to learn the items (i.e., words), the intrinsic context (i.e., the colour of the cards on which each word was presented), and the extrinsic context (i.e., the temporal order of the words). Among 72 young adults, in comparison to conditions of full attention, divided attention applied at encoding only or retrieval only resulted in equally lower performance on all memory tasks; in contrast, divided attention applied at both encoding and retrieval resulted in lower performance only on memory for temporal order. The findings support the idea that memory for temporal order requires greater attentional resources and strategic processing than memory for items.     

Retrieval-induced forgetting of arithmetic facts

Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 × 3 = 6). In both experiments, robust RIF expressed in response times occurred only for high-strength small-number addition facts with sums ≤ 10, indicating that RIF from multiplication practice was interference dependent. RIF of addition-fact memory was produced by multiplication retrieval (2 × 3 = ?) but not multiplication study (2 × 3 = 6), supporting an inhibitory mechanism of RIF in arithmetic memory. Finally, RIF occurred with multiplication practiced in word format (three × four) and addition tested later in digit format (3 + 4), which provides evidence that digit and written-word formats for arithmetic accessed a common semantic retrieval network. The results support the view that addition and multiplication facts are stored in an interrelated semantic network and that RIF of competing addition facts is an intrinsic process of multiplication fact retrieval.     

Adults’ strategy choices for simple addition: Effects of retrieval interference

Simple addition (e.g., 3 + 2, 7 + 9) may be performed by direct memory retrieval or by such procedures as counting or transformation. The distribution of associations (DOA) model of strategy choice (Siegler, 1988) predicts that procedure use should increase as retrieval interference increases. To test this, 100 undergraduates performed simple addition problems, either after blocks of simple multiplication (high-interference context) or after blocks of simple division problems (low-interference context). Addition took longer and was more error prone after multiplication; in particular, there were more multiplication confusion errors on the relatively easy, small-number addition problems (e.g., 3 + 2 = 6, 4 + 3 = 12), but not on the more difficult, large-number additions. Consistent with the DOA, participants reported greater use of procedures for addition after multiplication, but more so for small addition problems. The findings demonstrate that adults’ use of procedural strategies for simple addition is substantially influenced by retrieval interference.     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.