Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification

The odd-even status of a sum depends on the odd-even status of its addends. A sum must be odd if an odd number of its addends are odd; else it must be even. A proposed sum that violates the required odd-even status of the sum—that is, deviates from the correct sum, whether odd or even, by an odd value (e.g., splits of ±1, ±3, ±5)—can be rejected immediately as false. Subjects in the present study did indeed use the odd-even rule in sum verification, because they were as fast and accurate in rejecting a split of ±1 as one of ±2, and a split of ±3 as one of ±4, even though a larger split generally is easier to reject (symbolic distance effect), and splits of ±3 and ±4 were rejected faster and more accurately than those of ±1 and ±2. Performance on separate odd-even tasks indicated that the odd-even properties of numbers and sums are readily available for use by adults, and that persons who do well on such tasks are especially likely to use the odd-even rule in sum verification.     

The odd-even effect in addition: an analysis per problem type

The odd-even effect is a well documented finding in the literature on mental arithmetic, at least for multiplication. It implies that false answers with the same parity as the correct answer are rejected more slowly than false answers with a different parity. For addition, this effect is not so well documented. The study by Krueger and Hallford (1984) is the only one that investigated odd-even effects for addition. However, they did not investigate odd-even effects per problem, even though there are indications that problem type can modulate odd-even effects for multiplication (Lemaire & Reder, 1999). Therefore, we wanted to get more insight into odd-even effects for addition by investigating odd-even effects per problem type. Our results extended the findings of Krueger and Hallford. First of all, we found an interaction between split and problem type. The most important and new result of present study, however, was a strong parity effect for E + E problems. We discuss our results in terms of two alternative explanations for odd-even effects, namely use of a parity rule on the one hand and familiarity with even outcomes on the other.     

The odd-even effect in Sudoku puzzles: effects of working memory, aging, and experience

The odd-even effect in numerical processing has been explained as the easier processing of even numbers compared with odd numbers. We investigated this effect in Sudoku puzzles, a reasoning problem that uses numbers but does not require arithmetic operations. Specifically, we asked whether the odd-even effect occurred with Sudoku puzzles and whether individual differences in working memory (WM), aging, and experience with Sudoku modulated this effect. We manipulated the presence of odd and even numbers in Sudoku puzzles, measured WM with the Wisconsin Card Sorting Test and backward digit span task, tested older and younger adults, and collected Sudoku experience frequency. Performance on Sudoku was more accurate for even puzzles than odd ones. Younger, experienced, and higher-WM participants were more accurate on Sudoku, but these individual difference variables did not interact with the odd-even effect. Odd numbers may impose more cognitive load than even numbers, but future research is needed to examine how age, experience, or WM may influence the odd-even effect.     

Why 2×2=5 looks so wrong: On the odd-even rule in product verification

Subjects judged whether the proposed product for two multipliers was true or false. Each equation could be judged as plausible or implausible because a product must be even if any of its multipliers is even; otherwise, it must be odd. A proposed product that violates the required odd-even status of the product—that is, deviates from the correct product, whether odd or even, by an odd value (e.g., splits of ±l, ±3, ±5)—can be rejected as false before normal processing is completed (i.e., before the correct product is retrieved and compared with the proposed product). Subjects were indeed faster and more accurate in rejecting a split of +1 or +3 than a split of +2 or +4, and this effect increased as the number of even multipliers in the pair increased. Subjects did not use the odd-even rule when either multiplier was 0 or 1 (Experiment 2), perhaps because other rules are available to bypass normal processing in those cases. A similar odd-even rule is used in sum verification (Krueger &Hallford), and use of the odd-even rules may help to explain why oddness and evenness are such salient features of numbers as abstract concepts (Shepard, Kilpatric, & Cunningham).     

When plausibility judgments supersede fact retrieval: The example of the odd-even effect on product verification

In the present study, we examined the conditions that favor the plausibility judgment strategy over the retrieval strategy when we verify some statements. In particular, we examined the effect of odd-even status of numbers on subjects' verification of single-digit arithmetic problems. In Experiment 1, we explored how factors such as problem difficulty and stimulus onset asynchrony (SOA) influence this effect in adults. In Experiment 2, we present evidence that this odd-even effect is also present in elementary school children, although it varies with the age of the children, the difficulty of the problems, and the SOA. We argue that the odd-even information is helpful in arithmetic verification tasks with difficult problems early in the verification processes and that the time course of these effects differs across ages. The present results are consistent with the view that the plausibility strategy is preferred over the retrieval strategy at an early stage of processing and with information that is not quickly accessible. Finally, we discuss the implications of the present experiments for understanding of single-digit arithmetic and for understanding the more general issue of how people coordinate use of multiple strategies.     

Optimum split-half reliabilities for the Rorschach: projective techniques are more reliable than we think

A new technique for optimizing split-half reliability estimates yielded substantial and comparable increases in indexes of internal reliability for some major Rorschach variables across two diverse samples. Practical suggestions for applying this method to other projective tests were advanced. The inadvisability of computing odd-even reliability coefficients without regard to split-half distributional anomalies is addressed.     

When is an odd number not odd? Influence of task rule on the MARC effect for numeric classification

When classifying numbers as odd or even with left-right keypresses, performance is better with the mapping even-right/odd-left than with the opposite mapping. This linguistic markedness association of response codes (MARC) effect has been attributed to compatibility between the linguistic markedness of stimulus and response codes. In 2 experiments participants made keypresses to the Arabic numerals or number words 3, 4, 8, and 9 using the odd-even parity rule or a multiple-of-3 rule, which yield the same keypress response for each stimulus. For both stimulus modes, the MARC effect was obtained with the odd-even rule, but tended to reverse with the multiple-of-3 rule. The reversal was complete for the right response, but task rule had little influence on the left response. The results are consistent with the view that the MARC effect and its reversal are caused by correspondence of the stimulus code designated as positive by the task rule with the positive-polarity right response code.     

阿拉伯数字与汉字大写数字认知的大脑功能一侧化实验研究

用半视野速示术检查阿拉伯数字和汉字大写数字在三种任务加工条件（认读、简单相加、奇偶概念异同判断）下进行认知的大脑两半球机能活动一侧化效应。结果表明：以被试正确反应的百分数为指标,两类数字的三种任务加工均没有表现出明显的视野（大脑半球）优势。以被试正确反应的反应时为指标,两类数字的认读和简单相加也未出现视野（大脑半球）的明显差别,而两类数字的奇偶概念异同判断却出现了差异显著的右视野（左半球）优势。     

Reliability of the Ego-Grasping Scale

Research using Knoblauch and Falconer's Ego-Grasping Scale is reviewed. Using a sample of 695 undergraduate students, the scale had moderate reliability (Cronbach alpha, odd-even numbered items, and test-retest), but a principal-components analysis with a varimax rotation identified five components, indicating heterogeneity in the content of the items. Lower Ego-Grasping scores appear to be associated with better psychological health. The scale has been translated and used with Korean, Kuwaiti, and Turkish students, indicating that the scale can be useful in cross-cultural studies.     

Optimized Split-Half Reliability for the Bender Visual Motor Gestalt Test: Further Evidence for the Use of the Maximization Procedure

Bender-Gestalt protocols from a clinical sample (N = 100) were scored according to the Pascal-Suttell method and γ?11? derived for all possible split-half combinations (n = 35). The distribution of these 35 reliability coefficients proved to be negatively skewed, justifying the use of the maximum split-half reliability rather than the usual odd-even or alpha coefficients. The maximized γ?11?, using only eight designs, was .72, which was higher than has heretofore been reported. Implications of low ns and negative skews for correlational distributions are discussed, and the maximization procedure is recommended even for psychometric tests when the number of items is low.     

情境对序数的空间表征之影响

通过设置垂直维度上不同的情境,本研究采用“奇偶判断任务”探讨了情境对序数空间表征的影响。结果发现,只有序数的情况下,被试对小数的上键反应或下键反应、对大数的上键反应或下键反应都没有显著差异;在楼层情境下,被试对小数的下键反应更快,对大数的上键反应更快;在家谱情境下,被试对小数的上键反应更快,对大数的下键反应更快。以上结果表明,垂直维度上序数的空间表征受到情境的影响,这说明在垂直维度上数字的空间表征具有动态性,且受到具体和情境的调节。     

An Examination of the Selection of Criterion Scores for the Study of Motor Learning and Retention

150 Ss were given 50 practice trials on the stabilometer over a period of 4 days (20 trials on Day 1 followed by a 1-day rest, 10 trials on Day 2, followed by a 7-day rest, 10 trials on Day 3 followed by a 14-day rest, and 10 trials on Day 4). A series of learning scores reflecting improvement in performance over the 4 days and retention scores reflecting loss of performance between days were obtained. The reliability of individual differences in these scores were calculated using the odd-even method and by progressively increasing the number of trials included in the measure of initial and final performance level. On the basis of the findings obtained it was concluded that using 4 – 6 trials in initial and final ability levels results in a difference (criterion) score which combines relatively high reliability with a relatively high estimate of the amount of change in performance.     

The odd-even effect in multiplication: Parity rule or familiarity with even numbers?

This study questions the evidence that a parity rule is used during the verification of multiplication. Previous studies reported that products are rejected faster when they violate the expected parity, which was attributed to the use of a rule (Krueger, 1986; Lemaire & Fayol, 1995). This experiment tested an alternative explanation of this effect: the familiarity hypothesis. Fifty subjects participated in a verification task with contrasting types of problems (even x even, odd x odd, mixed). Some aspects of our results constitute evidence against the use of the parity rule: False even answers were rejected slowly, even when the two operands were odd. We suggest that the odd-even effect in verification of multiplication could not be due to the use of the parity rule, but rather to a familiarity with even numbers (three quarters of products are indeed even).     

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.     

The reliability of speeded tests

Some methods are presented for estimating the reliability of a partially speeded test without the use of a parallel form. The effect of these formulas on some test data is illustrated. Whenever an odd-even reliability is computed it is probably desirable to use one of the formulas noted in Section 2 of this paper in addition to the usual Spearman-Brown correction. Since the formulas given here involve the mean and the standard deviation of the “number unattempted score,” a method is given in Section 4 for computing this mean and standard deviation from item analysis data. If the item analysis data are available, this method will save considerable time as compared with rescoring answer sheets.     

Task switching and response correspondence in the psychological refractory period paradigm

Three experiments examined the effects of task switching and response correspondence in a psychological refractory period paradigm. A letter task (vowel-consonant) and a digit task (odd-even) were combined to form 4 possible dual-task pairs in each trial: letter-letter, letter-digit, digit-digit, and digit-letter. Foreknowledge of task transition (repeat or switch) and task identity (letter or digit) was varied across experiments: no foreknowledge in Experiment 1, partial foreknowledge (task transition only) in Experiment 2, and full foreknowledge in Experiment 3. For all experiments, the switch cost for Task 2 was additive with stimulus onset asynchrony, and the response-correspondence effect for Task 2 was numerically smaller in the switch condition than in the repeat condition. These outcomes suggest that reconfiguration for Task 2 takes place after the central processing of Task 1 and that the crosstalk correspondence effect is due to response activation by way of stimulus-response associations.     

Visual field differences in the processing of numerical stimuli

Twenty-four right-handed subjects received random presentations of the numbers 1-6 in the form of words, digits, and dot patterns, to the left and right visual fields. Accuracy and reaction time were recorded for an odd-even judgment requiring a manual response. A significant stimulus type of visual field interaction was obtained, with words showing a left-hemisphere advantage and digits and dot patterns showing a right-hemisphere advantage. This pattern supports Coltheart's (1980, Deep dyslexia: A right hemisphere hypothesis, In M. Coltheart, K. Patterson, & J.C. Marshall (Eds.), Deep dyslexia, London: Routledge & Kegan Paul) right hemisphere reading hypothesis, which suggests that the left hemisphere's general advantage in processing linguistic material may be specific to stimuli which involve phonological processing. When phonological processing is not possible (e.g., for arabic digits and other ideographic orthographies), the right hemisphere may have an advantage because of its superior visuospatial processing capabilities.     

GLOBAL PROCESSING-TIME COEFFICIENTS CHARACTERIZE INDIVIDUAL AND GROUP DIFFERENCES IN COGNITIVE SPEED

Abstract— Forty participants performed seven different information processing tasks (choice reaction time, letter classification, visual search, abstract matching, line-length discrimination, mental rotation, and mental paper-folding) Slow (top quartile) and fast (bottom quartile) processors were selected based on their mean z scores Response times (RTs) of the slow and fast groups in the 21 conditions of the seven tasks were linear functions of the mean RTs of the entire group (both r²s = 99) In addition, individuals' RTs were well described by linear functions (median r²= 93) When tasks were ranked in order of complexity, the odd-even reliability of the ratio of an individual's RT to the average RT was 80, indicating that such ratios remain relatively stable across tasks Taken together, these findings indicate that the performance of an individual on diverse tasks can be predicted on the basis of a single processing-time coefficient Such coefficients may provide useful indices of the efficiency with which different individuals process information     

Spatial memory in rats: A cross validation study

Two experiments were performed to investigate whether two measures of spatial memory, working memory (WM) and reference memory (RM), can be generalized over learning situations. In the first experiment, outbred WU rats were used; in the second four inbred strains. In both experiments rats were given 50 training trials in both a holeboard and a high-sided radial maze: 4 out of 16 holes and 4 out of 8 arms were baited. The number of WM (revisits to baited holes/arms) and RM (visits and revisits to never baited holes/arms) errors were calculated for 5 blocks of 10 trials; odd-even reliabilities were determined for the last 20 trials and were shown to be satisfactory for both measures in both types of apparatus. Error-decrements were analyzed by trend components. For both WM and RM, simple and canonical correlations were determined between means and significant trends in maze and holeboard as a test for construct validity; in both experiments r's and R_c's were very low. It was concluded that the concepts WM and RM are not validly generalizable over tests.     

The cuing and priming of cognitive operations

The effect on performance of advance information about the specific cognitive operations to be performed on a stimulus was investigated in two experiments using cues (information useful and necessary for performance) and primes (information useful but not necessary for performance). In the first experiment, a cue presented prior to a digit stimulus indicated whether the digit was to be classified as odd or even, or low (less than 6) or high (greater than 5). Results showed that performance improved with increasing time between cue and digit and with practice. A stroop-like asymmetric interference of the low-high operation on the odd-even operation was also observed. In the second experiment, a prime that matched the cue, mismatched it, or was neutral was presented before the cue. Results showed facilitatory and inhibitory priming effects, as well as a distance effect based on the position of a digit relative to the boundary between 5 and 6. The results of both experiments were discussed in terms of a model based on relative processing speeds of the two relevant properties of the digits.