Reading direction and spatial effects in parity and arithmetic tasks.

This study investigated the relationship between numerical and spatial processing and reading direction, conducting conceptual replications of the Shaki et al. (Psychonomic Bulletin & Review 16(2): 328-331, 2009) parity task and the Mathieu et al. (Cognition 146: 229-239, 2016, Experiment 1) simple addition (e.g., 3 + 2) and subtraction (e.g., 3 - 2) task. Twenty-four left-to-right readers (LTR) and 24 right-to-left readers (RTL) were tested. The response time (RT) analysis of the parity task presented a robust spatial-numerical association of response codes (SNARC) effect (left-side response advantage for smaller numbers and right-side advantage for larger numbers) for LTR but not RTL readers. In the arithmetic task, the three problem elements (e.g., 3 + 4) were presented sequentially with the second operand displaced slightly to the left or right of fixation. RTL but not LTR readers presented a RT advantage for subtraction relative to addition with a right-shifted second operand compared to it being left-shifted. This is consistent with a spatial bias linked to native reading direction. For both reading-direction groups, effects of the left vs. right side manipulation in the arithmetic or parity task did not correspond to parallel effects in the other task. The results imply that the parity-based SNARC effects and side-related effects in cognitive arithmetic are not equivalent measures of space-related processes in cognitive number processing and likely reflect distinct mechanisms.

Transfer of training in alphabet arithmetic.

In recent years, several researchers have proposed that skilled adults may solve single-digit addition problems (e.g., 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure, however, often leads to transfer of learning to unpracticed items; consequently, the fast counting theory was potentially challenged by subsequent studies that found no generalization of practice for simple addition. In two experiments reported here (Ns = 48), we examined generalization in an alphabet arithmetic task (e.g., B + 5 = C D E F G) to determine that counting-based procedures do produce generalization. Both experiments showed robust generalization (i.e., faster response times relative to control problems) when a test problem's letter augend and answer letter sequence overlapped with practiced problems (e.g., practice B + 5 = C D E F G, test B + 3 = C D E ). In Experiment 2, test items with an unpracticed letter but whose answer was in a practiced letter sequence (e.g., practice C + 3 = DEF, test D + 2 = E F) also displayed generalization. Reanalysis of previously published addition generalization experiments (combined n = 172) found no evidence of facilitation when problems were preceded by problems with a matching augend and counting sequence. The clear presence of generalization in counting-based alphabet arithmetic, and the absence of generalization of practice effects in genuine addition, represent a challenge to fast counting theories of skilled adults' simple addition.

Effects of pitch on auditory number comparisons.

Three experiments investigated interactions between auditory pitch and the numerical quantities represented by spoken English number words. In Experiment 1, participants heard a pair of sequential auditory numbers in the range zero to ten. They pressed a left-side or right-side key to indicate if the second number was lower or higher in numerical value. The vocal pitches of the two numbers either ascended or descended so that pitch change was congruent or incongruent with number change. The error rate was higher when pitch and number were incongruent relative to congruent trials. The distance effect on RT (i.e., slower responses for numerically near than far number pairs) occurred with pitch ascending but not descending. In Experiment 2, to determine if these effects depended on the left/right spatial mapping of responses, participants responded "yes" if the second number was higher and "no" if it was lower. Again, participants made more number comparison errors when number and pitch were incongruent, but there was no distance × pitch order effect. To pursue the latter, in Experiment 3, participants were tested with response buttons assigned left-smaller and right-larger ("normal" spatial mapping) or the reverse mapping. Participants who received normal mapping first presented a distance effect with pitch ascending but not descending as in Experiment 1, whereas participants who received reverse mapping first presented a distance effect with pitch descending but not ascending. We propose that the number and pitch dimensions of stimuli both activated spatial representations and that strategy shifts from quantity comparison to order processing were induced by spatial incongruities.

Retrieval-induced forgetting of multiplication facts and identity rule.

This research investigated retrieval-induced interference between counterpart multiplication (2 × 3 = 6) and addition facts (2 + 3 = 5). Adults (N =72) repeatedly solved either a set of simple addition (0 + 2, 1 + 5, 2 + 3) or multiplication problems (0 × 2, 1 × 5, 2 × 3) during a practice phase and then switched operations during a test phase that included counterparts to the practiced problems and control problems. The paradigm afforded measurement in response time both of inter-operation retrieval-induced forgetting (RIF) and generalization of practice across different problems within operations. The experiment demonstrated generalization of practice for the rule-based 0 + N = N problems (e.g., practicing 0 + 2 facilitated performance on 0 + 7) as well as for problems governed by the multiplicative identity principle (1 × N = N) and zero-product principle (0 × N = 0), but not the fact-based 1 + N problems. The experiment also demonstrated for the first time inter-operation RIF of fact-based multiplication, which was as large as the effect observed for fact-based addition. The 0 × N, 0 + N, and 1 + N problems did not present item-specific RIF from practice of cross-operation counterparts, but 1 × N problems did, despite the generalization-of-practice evidence that 1 × N problems were solved using an item-general procedure. The item-specific RIF for 1 × N = N must reflect item-specific interference rather than item-level competitor inhibition given that there is no item-level representation of 1 × N = N facts in long-term memory.

The N3 is sensitive to odd-even congruency information in arithmetic fact retrieval.

This study investigated the behavioral and electrophysiological effects elicited by adults' simple addition verification when false answers agree or disagree with the odd-even status of the correct sum (parity congruency vs. parity incongruency), while they are near or far from correct (small vs. large splits). Event-related brain potentials were recorded from 18 students using a first-answer-then-problem paradigm. The results showed that odd-even congruency had a significant effect on the N3 latency with a small, but not a large split. Specifically, odd-even congruent answers with a small split elicited an N3 with a longer latency. Analyses of RT similarly indicated a bigger parity-congruency effect with small-split answers compared with large-split answers. This pattern parallels the corresponding effects on N3 and confirms that the N3 is sensitive to odd-even information in arithmetic fact retrieval and that there are clear links between the event-related brain potential pattern and behavioral effects.

Semantic alignment and number comparison.

Are the quantity representations activated by Arabic digits influenced by semantic context? We developed a novel paradigm to examine semantic alignment effects (e.g., Bassok et al. in J Exp Psychol Learn Mem Cogn 34:343-352, 2008) in number comparison. A horizontal word pair (either less more or few many) appeared for 480 ms to prime either relative magnitude (less more) or quantity (few many). Then a horizontal pair of single digits that were either successors (near) or differed by at least four (far) appeared above the word pair. Participants indicated verbally whether or not the word and digit pairs were congruent with respect to left-to-right ascending or descending relative magnitude. The RT advantage for far number pairs compared to near pairs (the distance effect) was greater with magnitude primes (81 ms) than quantity primes (17 ms), demonstrating a semantic alignment effect. This effect disappeared in Experiment 2 in which participants received identical stimuli but named the larger of the two digits and were free to ignore the primes. Nonetheless, mean RT in Experiment 2 was faster with prime and target pairs both ascending or both descending, but only with quantity primes. This prime-dependent order-congruity effect suggests that semantic alignment with respect to numerical order affected number comparison in Experiment 2. The results thereby demonstrate that number comparison exhibits task-dependent semantic alignment effects and recruits distinct numerical representations as a function of semantic context (e.g., Cohen Kadosh and Walsh in Behav Brain Sci 32:313-373, 2009).

MorePower 6.0 for ANOVA with relational confidence intervals and Bayesian analysis.

MorePower 6.0 is a flexible freeware statistical calculator that computes sample size, effect size, and power statistics for factorial ANOVA designs. It also calculates relational confidence intervals for ANOVA effects based on formulas from Jarmasz and Hollands (Canadian Journal of Experimental Psychology 63:124-138, 2009), as well as Bayesian posterior probabilities for the null and alternative hypotheses based on formulas in Masson (Behavior Research Methods 43:679-690, 2011). The program is unique in affording direct comparison of these three approaches to the interpretation of ANOVA tests. Its high numerical precision and ability to work with complex ANOVA designs could facilitate researchers' attention to issues of statistical power, Bayesian analysis, and the use of confidence intervals for data interpretation. MorePower 6.0 is available at https://wiki.usask.ca/pages/viewpageattachments.action?pageId=420413544 .

Neighborhood consistency and memory for number facts.

Verguts and Fias (Memory & Cognition 33:1-16, 2005a) proposed a new model of memory for simple multiplication facts (2 x 3 = 6; 8 x 7 =56) in which learning and performance is governed by the consistency of a problem's correct product with neighboring products in the times table. In the present study, to directly investigate effects of neighborhood consistency, participants memorized a set of 16 novel "pound" arithmetic equations. The pound arithmetic table included eight tie equations with repeated operands (e.g., 4 # 4 = 29) and eight nontie equations (e.g., 5 # 4 = 39). In the consistent problem set, tie and nontie answers in adjacent columns and rows shared a common decade or unit value. In the inconsistent problem set, neighboring tie and nontie problems did not share a common decade or unit. Across 14 study-test blocks, memorization of the pound arithmetic table presented a robust effect of neighborhood consistency, with the rate of learning nearly doubling that of the inconsistent condition. An analysis of error types showed that consistency fostered the development of a categorical structure based on problem operands and that tie problems were encoded as a distinct subcategory of problems. There was also a substantial learning advantage for tie problems relative to nonties both with consistent and inconsistent neighbors. The results indicate that neighborhood consistency can have a major impact on memory for number facts.

Cross-Reactive SARS-CoV-2 Neutralizing Antibodies From Deep Mining of Early Patient Responses.

Passive immunization using monoclonal antibodies will play a vital role in the fight against COVID-19. The recent emergence of viral variants with reduced sensitivity to some current antibodies and vaccines highlights the importance of broad cross-reactivity. This study describes deep-mining of the antibody repertoires of hospitalized COVID-19 patients using phage display technology and B cell receptor (BCR) repertoire sequencing to isolate neutralizing antibodies and gain insights into the early antibody response. This comprehensive discovery approach has yielded a panel of potent neutralizing antibodies which bind distinct viral epitopes including epitopes conserved in SARS-CoV-1. Structural determination of a non-ACE2 receptor blocking antibody reveals a previously undescribed binding epitope, which is unlikely to be affected by the mutations in any of the recently reported major viral variants including B.1.1.7 (from the UK), B.1.351 (from South Africa) and B.1.1.28 (from Brazil). Finally, by combining sequences of the RBD binding and neutralizing antibodies with the B cell receptor repertoire sequencing, we also describe a highly convergent early antibody response. Similar IgM-derived sequences occur within this study group and also within patient responses described by multiple independent studies published previously.

Switch costs and the operand-recognition paradigm.

Experimental research in cognitive arithmetic frequently relies on participants' self-reports to discriminate solutions based on direct memory retrieval from use of procedural strategies. Given concerns about the validity and reliability of strategy reports, Thevenot et al. in Mem Cogn 35:1344-1352, (2007) developed the operand-recognition paradigm as an objective measure of arithmetic strategies. Participants performed addition or number comparison on two sequentially presented operands followed by a speeded operand-recognition task. Recognition times increased with problem size following addition but not comparison. Thevenot et al. argued that the complexity of addition strategies increases with problem size. A corresponding increase in operand-recognition time occurs because, as problem size increases, working memory contains more numerical distracters. However, because addition is substantially more difficult than comparison, and difficulty increases with problem size for addition but not comparison, their findings could be due to difficulty-related task-switching costs. We repeated Thevenot et al. (Experiment 1) but added a control condition wherein participants performed a parity (odd or even) task instead of operand recognition. We replicated their findings for operand recognition but found robust, albeit smaller, effects of addition problem size on parity judgements. The results indicate that effects of strategy complexity in the operand-recognition paradigm are confounded with task-switching effects, which complicates its application as a precise measure of strategy complexity in arithmetic.

What is learned in procedural learning? The case of alphabet arithmetic.

This research pursued a fine-grained analysis of the acquisition of a procedural skill. In two experiments (n = 29 and n = 27), adults practiced 12 alphabet arithmetic problems (e.g., C + 3 = C D E F) in two sessions with 20 practice blocks in each. If learning reflected speed up of a counting algorithm, response time (RT) speed up should be proportional to the number of counting steps (+ 1, + 2, or + 3). Instead, we found about 50% of RT gains occurred in the first six blocks of practice during which speed up was parallel for + 1, + 2, and + 3 problems. In both experiments, RT initially was a linear function of addend size, reflecting a letter counting strategy. Mean RT for + 3 problems was eventually equal to + 2 problems, which suggests that speed up reflected a gradual shift to associative fact retrieval. Trial by trial strategy self-reports in Experiment 2 revealed that the proportion of trials reported as memory retrieval as opposed to counting predicted 96% of the variance in RT as a function of addend size and practice block. As such, the results provided no evidence for speed up of a counting algorithm and indicated that skill acquisition for this task entailed speed up of task-general processes independent of addend size and rapid transition from counting to fact retrieval. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Operation-specific effects of numerical surface form on arithmetic strategy.

Educated adults solve simple addition problems primarily by direct memory retrieval, as opposed to by counting or other procedural strategies, but they report using retrieval substantially less often with problems in written-word format (four + eight) compared with digit format (4 + 8). It was hypothesized that retrieval efficiency is relatively low with word operands compared with digits and that this promotes a shift to procedural backup strategies. Consistent with this hypothesis, Experiment 1 demonstrated greater word-format costs on retrieval usage for addition than subtraction, which was due to increased counting for addition but not subtraction. Experiment 2 demonstrated greater word-format costs on retrieval for division than multiplication, which was due to increased use of multiplication-fact reference to solve division problems. Format-related strategy shifts away from retrieval reflected both the efficiency of retrieval for a given operation and the availability of viable alternative strategies. The results demonstrate that calculation processes are not abstracted away from problem surface form. The authors propose that retrieval efficiency for arithmetic connects diverse performance and strategy-related effects across key arithmetic factors, including arithmetic operation, numerical size, and numeral format.

Evidence for a retrieval over a sum counting theory of adults' simple addition.

Researchers have recently proposed that educated adults solve the simplest addition problems (e.g., 3 + 2) by an automatic counting procedure, challenging the long-held view that educated adults solve small additions by associative memory retrieval. We tested predications of a sum-counting model that assumes a procedure in which the 2 quantities represented by the operands are encoded and counted sequentially. Here, we presented the 2 operands sequentially (e.g., "3 +" first and then "2") and manipulated the preview time for the first operand (O1) and operator across 2 experiments (both n = 36); the O1 preview times were 1000 ms and 500 ms in Experiments 1 and 2, respectively. We measured response time (RT) from the presentation of the second operand (O2) and compared it with RT when both operands appeared simultaneously. Contrary to the sum-counting model, with sequential presentation, problems with the same O2 sizes (e.g., 3 + 2, 4 + 2) demonstrated significant RT differences across levels of O1, and the sum of the operands was a better RT predictor than O2 with both sequential and simultaneous displays. These results challenge a sum-counting model of the present data but are consistent with a memory retrieval theory. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Does the min-counting strategy for simple addition become automatized in educated adults? A behavioural and ERP study of the size congruency effect.

Recent studies have proposed that the sum-counting strategy for simple addition (i.e., count up of the summed value of the two operands one by one) used at early age becomes automatized in adults, challenging the long held view that skilled adults solve simple addition problems by fact retrieval. As arithmetic skill develops, however, the sum-counting strategy usually is replaced by a more advanced and efficient min-counting strategy (i.e., start counting at the value of the larger addend and count up by ones equal to the smaller or "min" addend). Thus, one would expect the min strategy, rather than the sum strategy, to become automatized if we assume automatic counting procedures exist. The present study sought evidence of the min-strategy in adults by investigating the size congruency effect (SCE) through behavioural and event related brain potential (ERP) experiments. The SCE is observed in number comparison tasks (e.g., identify the larger of two numbers), where RT is slower when the physical and numerical size of the numbers are incongruent compared to when they are congruent. The min-counting strategy inherently requires a number comparison stage, because the min and max number must be determined before the counting begins. Experiment 1 tested 72 participants on addition and number comparison tasks. The results showed a robust behavioural SCE for number comparison but not for simple addition. Experiment 2 tested 20 participants with a large number of addition and number comparison problems and recorded ERP. The behavioural results replicated the findings of Experiment 1. The ERP results revealed brain signatures in line with previous studies and the current behavioural findings. No SCE indicated the absence of a number comparison stage for addition; thus, the present findings ruled out the possibility of a fast min-counting strategy, or more generally a min strategy, for adults' simple addition.

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.

Bidirectional associations in multiplication memory: conditions of negative and positive transfer.

A variety of experimental evidence indicates that the memory representation for multiplication facts (e.g., 6 x 9 = 54) incorporates bidirectional links with a forward association from factors to product and a reverse association from product to factors. Surprisingly, the authors did not find evidence in Experiment 1 of facilitative transfer-of-practice from multiplication (6 x 9 = ?) to factoring (54 = ? x ?); in fact, multiplication practice produced item-specific interference with factoring. Similarly, the authors found no evidence in Experiment 2 that repetition of specific factoring problems (54 = ? x ?) facilitated performance of corresponding multiplication problems (6 x 9 = ?). In Experiment 3, participants practiced both multiplication and factoring and presented facilitative transfer in both directions. Thus, bidirectional facilitation occurred if both operations were practiced, but interference occurred when only one operation was practiced. We propose that this seemingly paradoxical behavior occurs because it is adaptive for the bidirectional retrieval structure to retain operational flexibility in the context of practicing both operations, whereas it is adaptive to specialize the memory representation for the practiced operation (i.e., factoring or multiplication) when only one operation is practiced.

Spoken numbers versus Arabic numerals: differential effects on adults' multiplication and addition.

J.-A. LeFevre, Q. Lei, B. L. Smith-Chant, and D. B. Mullins (2001) examined effects of auditory versus Arabic visual presentation formats on performance of simple multiplication. They observed a smaller problem-size effect (response time [RT] increases with numerical size) with auditory stimuli compared with Arabic stimuli. If this arises during problem encoding, as opposed to during subsequent calculation processes, the authors would expect comparable Format x Problem Size interactions for both multiplication and addition. For multiplication, the authors replicated the finding of a smaller problem-size effect for auditory stimuli than for Arabic stimuli, but found the opposite pattern for addition whereby the problem-size effect was larger with auditory stimuli than with Arabic stimuli. Decomposition of mean RT into its ex-Gaussian components, mu and tau, demonstrated that the triple interaction arose entirely in connection with tau. This suggests that the effects of auditory versus Arabic format on RT substantially reflected format-related shifts in the use of procedural strategies.

Transfer of training in simple addition.

In recent years, several researchers have proposed that skilled adults may solve single-digit addition problems (e.g., 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure often leads to transfer of learning and faster performance of unpracticed items. Such transfer has been demonstrated using a counting-based alphabet arithmetic task (e.g., B + 4 = C D E F) that indicated robust generalization of practice (i.e., response time [RT] gains) when untrained transfer problems at test had been implicitly practiced (e.g., practice B + 3, test B + 2 or B + 1). Here, we constructed analogous simple addition problems (practice 4 + 3, test 4 + 2 or 4 + 1). In each of three experiments (total n = 108), participants received six practice blocks followed by two test blocks of new problems to examine generalization effects. Practice of addition identity rule problems (i.e., 0 + N = N) showed complete transfer of RT gains made during practice to unpracticed items at test. In contrast, the addition ties (2 + 2, 3 + 3, etc.) presented large RT costs for unpracticed problems at test, but sped up substantially in the second test block. This pattern is consistent with item-specific strengthening of associative memory. The critical items were small non-tie additions (sum ≤ 10) for which the test problems would be implicitly practiced if counting was employed during practice. In all three experiments (and collectively), there was no evidence of generalization for these items in the first test block, but there was robust speed up when the items were repeated in the second test block. Thus, there was no evidence of the generalization of practice that would be expected if counting procedures mediated our participants' performance on small non-tie addition problems.

"Compacted" procedures for adults' simple addition: A review and critique of the evidence.

We review recent empirical findings and arguments proffered as evidence that educated adults solve elementary addition problems (3 + 2, 4 + 1) using so-called compacted procedures (e.g., unconscious, automatic counting); a conclusion that could have significant pedagogical implications. We begin with the large-sample experiment reported by Uittenhove, Thevenot and Barrouillet (2016, Cognition, 146, 289-303), which tested 90 adults on the 81 single-digit addition problems from 1 + 1 to 9 + 9. They identified the 12 very-small addition problems with different operands both ≤ 4 (e.g., 4 + 3) as a distinct subgroup of problems solved by unconscious, automatic counting: These items yielded a near-perfectly linear increase in answer response time (RT) yoked to the sum of the operands. Using the data reported in the article, however, we show that there are clear violations of the sum-counting model's predictions among the very-small addition problems, and that there is no real RT boundary associated with addends ≤4. Furthermore, we show that a well-known associative retrieval model of addition facts-the network interference theory (Campbell, 1995)-predicts the results observed for these problems with high precision. We also review the other types of evidence adduced for the compacted procedure theory of simple addition and conclude that these findings are unconvincing in their own right and only distantly consistent with automatic counting. We conclude that the cumulative evidence for fast compacted procedures for adults' simple addition does not justify revision of the long-standing assumption that direct memory retrieval is ultimately the most efficient process of simple addition for nonzero problems, let alone sufficient to recommend significant changes to basic addition pedagogy.

Numerical order and quantity processing in number comparison.

We investigated processing of numerical order information and its relation to mechanisms of numerical quantity processing. In two experiments, performance on a quantity-comparison task (e.g. 2 5; which is larger?) was compared with performance on a relative-order judgment task (e.g. 2 5; ascending or descending order?). The comparison task consistently produced the standard distance effect (faster judgments for far relative to close number pairs), but the distance effect was smaller for ascending (e.g. 2 5) compared to descending pairs (e.g. 5 2). The order task produced a pair-order effect (faster judgments for ascending pairs) and a reverse distance effect for consecutive pairs in ascending order. The reverse effect implies an order-specific process, such as serial search or direct recognition of order for successive numbers. Thus, numerical quantity and order judgments recruited different cognitive mechanisms. Nonetheless, the reduced distance effect for ascending pairs in the quantity task implies involvement of order-related processes in magnitude comparison. Accordingly, distance effects in the quantity-comparison task are not necessarily a process-pure measure of magnitude representation.