Inverse reference in subtraction performance: An analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6?+?_?=?13) than in standard subtraction format (e.g., 13 – 6?=?_). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 – 6?=?7, “I know that 13 is 6?+?7”) because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

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

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.     

Calculation,culture, and the repeated operand effect

A basic phenomenon of cognitive arithmetic is that problems composed of a repeated operand, so-called "ties" (e.g. 6+6, 7 x 7), typically are solved more quickly and accurately than comparable non-tie problems (e.g. 6+5, 7 x 8). In Experiment 1, we present evidence that the tie effect is due to more efficient memory for ties than for non-ties, which participants reported solving more often using calculation strategies. The memory/strategy hypothesis accounts for differences in the tie effect as a function of culture (Asian Chinese vs. non-Asian Canadian university students), operation (addition, multiplication, subtraction, and division), and problem size (numerically small vs. large problems). Nonetheless, Blankenberger (Cognition 82 (2001) B15) eliminated the tie response time (RT) advantage by presenting problems in mixed formats (e.g. 4 x four), which suggests that the tie effect with homogenous formats (4 x 4 or four x four) is due to encoding. In Experiment 2, using simple multiplication problems, we replicated elimination of the tie effect with mixed formats, but also demonstrated an interference effect for mixed-format ties that slowed RTs and increased errors relative to non-tie problems. Additionally, practicing non-tie problems in both orders (e.g. 3 x 4 and 4 x 3) each time ties were tested once (cf. Cognition 82 (2001) B15) reduced the tie effect. The format-mismatch effect on ties, combined with a reduced tie advantage because of extra practice of non-ties, eliminated the tie effect. Rather than an encoding advantage, the results indicate that memory access for ties was better than for non-ties.     

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.     

Inverse reference in subtraction performance: an analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6 ± =?13) than in standard subtraction format (e.g., 13 - 6?=?). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 - 6?=?7, "I know that 13 is 6?+?7") because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

Operator and operand preview effects in simple addition and multiplication: A comparison of Canadian and Chinese adults

Using an arithmetic-based retrieval-induced forgetting (RIF) paradigm, researchers have found evidence that participants with very high arithmetic proficiency (Chinese adults), but not less-skilled participants (Canadian adults), solved some simple additions (e.g. 3 + 2) using fast procedural skills. Here we sought converging evidence for this using the operator-priming paradigm. Previous research testing simple addition and multiplication found that a 150-ms preview of the operator (+ or ×) facilitated only addition performance. This was taken as evidence that addition, but not multiplication, was solved by procedural algorithms that could be primed by presentation of the plus sign. In the present study, Chinese and Canadian adults (N = 144) were tested in the operator-priming paradigm but, in contrast to the RIF results, there was little evidence that operator-priming effects differed between the groups and robust operator priming was observed in both addition and multiplication. Thus, the operator preview results did not reinforce the results of previous research but the experiment revealed robust group differences in operand preview effects: For the Chinese, but not the Canadians, a preview of the numerical operands produced much greater facilitation for multiplication than addition. The fact that CN obtained a mean 103-ms gain for multiplication from the 150-ms preview of the operands strongly suggests that multiplication was their default operation in this paradigm. This result adds a potentially important new phenomenon to the behavioural distinctions between Chinese and North American adults' arithmetic skills.     

Spatial congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval

It is known that number and space representations are connected to one another in numerical and arithmetic abilities. Numbers are represented using the metaphor of a mental number line, oriented along horizontal and vertical space. This number line also seems to be linked to mental arithmetic, which is based partly on arithmetic fact retrieval. It seems that number representation and mental arithmetic are linked together. The present study tested the effect of spatial contextual congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval, using number-matching and addition verification tasks. For both tasks in Experiment 1, a contextual congruency effect was present horizontally (i.e., horizontal presentation of stimuli and horizontal response key alignments) but not vertically (i.e., vertical presentation of stimuli but horizontal response key alignments). In Experiment 2, both tasks showed a contextual congruency effect for both spatial conditions. Experiment 1 showed that the interference and distance effects were found in the horizontal condition, probably because of the spatial congruency between stimulus presentation and response key arrangements. This spatial congruency could be related to the activation of the horizontal number line. Experiment 2 showed similar interference and distance effects for both spatial conditions, suggesting that the congruency between stimulus presentation and response alignment could facilitate the retrieval of arithmetic facts. This facilitation could be related to the activation of both horizontal and vertical number lines. The results are discussed in light of the possible role of a mental number line in arithmetic fact retrieval.     

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.     

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.     

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.     

Language-specific memory for everyday arithmetic facts in Chinese-English bilinguals

The role of language in memory for arithmetic facts remains controversial. Here, we examined transfer of memory training for evidence that bilinguals may acquire language-specific memory stores for everyday arithmetic facts. Chinese-English bilingual adults (n = 32) were trained on different subsets of simple addition and multiplication problems. Each operation was trained in one language or the other. The subsequent test phase included all problems with addition and multiplication alternating across trials in two blocks, one in each language. Averaging over training language, the response time (RT) gains for trained problems relative to untrained problems were greater in the trained language than in the untrained language. Subsequent analysis showed that English training produced larger RT gains for trained problems relative to untrained problems in English at test relative to the untrained Chinese language. In contrast, there was no evidence with Chinese training that problem-specific RT gains differed between Chinese and the untrained English language. We propose that training in Chinese promoted a translation strategy for English arithmetic (particularly multiplication) that produced strong cross-language generalization of practice, whereas training in English strengthened relatively weak, English-language arithmetic memories and produced little generalization to Chinese (i.e., English training did not induce an English translation strategy for Chinese language trials). The results support the existence of language-specific strengthening of memory for everyday arithmetic facts.     

Continuity in Representation Between Children and Adults: Arithmetic Knowledge Hinders Undergraduates' Algebraic Problem Solving

This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184) solved addition facts or participated in one of several control conditions. Those who solved addition facts were less likely to solve prealgebra equations (e.g., 6 + 8 +4 = 7 + __) correctly under speeded conditions. In a fourth experiment, the negative effects of solving arithmetic problems extended to undergraduates (N = 74) solving algebra problems with no time pressure. Taken together, results suggest that arithmetic activates knowledge that hinders performance on algebra problems. Thus, an operational view of equations, which is prevalent in children, does not seem to be revised or abandoned, even after years of experience with algebra.     

Retrieval-induced forgetting in recall and recognition of thematically related and unrelated sentences

In three experiments, we assessed the effects of type of relation and memory test on retrieval-induced forgetting of facts. In Experiments 1 and 2, eight sets of four shared-subject sentences were presented for study. They were constructed so that half were thematically related and half were unrelated. A retrieval practice phase required participants to recall a subset of the studied sentences. In the final test, the participants were prompted to recall all the sentences (character cued in Experiment 1 and character plus stem cued in Experiment 2). The results showed that the retrieval-induced forgetting (RIF) effect was similar for thematically related and unrelated sentences, indicating that the presence of episodic relations among the sentences was sufficient to produce the effect. In Experiment 3, a recognition task was introduced and the RIF effect emerged in accuracy as well as in latency measures. The presence of this effect with item-specific cues is difficult to accommodate for noninhibitory theories of retrieval.     

Negative numbers in simple arithmetic

Are negative numbers processed differently from positive numbers in arithmetic problems? In two experiments, adults (N?=?66) solved standard addition and subtraction problems such as 3?+?4 and 7 – 4 and recasted versions that included explicit negative signs—that is, 3 – (–4), 7?+?(–4), and (–4)?+?7. Solution times on the recasted problems were slower than those on standard problems, but the effect was much larger for addition than subtraction. The negative sign may prime subtraction in both kinds of recasted problem. Problem size effects were the same or smaller in recasted than in standard problems, suggesting that the recasted formats did not interfere with mental calculation. These results suggest that the underlying conceptual structure of the problem (i.e., addition vs. subtraction) is more important for solution processes than the presence of negative numbers.     

Mental addition in third,fourth, and sixth graders

Research on mental arithmetic has suggested that young children use a counting algorithm for simple mental addition, but that adults use memory retrieval from an organized representation of addition facts. To determine the age at which performance shifts from counting to retrieval, children in grades 3, 4, and 6 were tested in a true/false verification task. Reaction time patterns suggested that third grade is a transitional age with respect to memory structure for addition—half of these children seemed to be counting and half retrieving from memory. Results from fourth and sixth graders implicated retrieval quite strongly, as their results resembled adult RTs very closely. Fourth graders' processing, however, was easily disrupted when false problems were presented. The third graders' difficulties are not due to an inability to form mental representations of number; all three grades demonstrated a strong split effect, indicative of a simpler mental representation of numerical information than is necessary for addition. The results were discussed in the context of memory retrieval versus counting models of mental arithmetic, and the increase across age in automaticity of retrieval processes.     

Retrieval-induced forgetting of performed and observed bizarre and familiar actions

To investigate whether people show retrieval-induced forgetting (RIF) for bizarre and familiar actions that they performed or observed, three experiments were conducted. In Experiment 1, participants performed bizarre and familiar actions with different objects during learning (e.g., pencil: balance the pencil across the cup, sharpen the pencil). They repeatedly performed a set of the bizarre or familiar actions during retrieval practice. After a distracter task, participants' cued recall was tested. Participants showed RIF for both bizarre and familiar actions. In Experiment 2, half of the participants performed the bizarre and familiar actions themselves; the other half observed the experimenter performing the actions. Replicating the results of Experiment 1, participants who performed the actions showed RIF for bizarre and familiar actions. In contrast, participants who observed the actions did not show RIF for either action type. Experiment 3 examined whether this lack of RIF for observed actions occurred due to a lack of active recall during retrieval practice; it did. Overall, the three experiments demonstrated RIF for both bizarre and familiar performed and observed actions. A distinctiveness account of the results is provided.     

Alzheimer's disease disrupts arithmetic fact retrieval processes but not arithmetic strategy selection

Three groups of healthy younger adults, healthy older adults, and probable AD patients, performed an addition/number comparison task. They compared 128 couples of additions and numbers (e.g., 4 + 9 15) and had to identify the largest item for each problem by pressing one of two buttons located under each item. Manipulations of problem characteristics (i.e., problem difficulty and splits between correct sums and proposed numbers) enabled us to examine strategy selection and specific arithmetic fact retrieval processes. Results showed that arithmetic facts retrieval processes, which were spared with aging, were impaired in AD patients. However, AD patients were able to switch between strategies across trials according to problem characteristics as well as healthy older adults, and less systematically than healthy younger adults. We discuss implications of these findings for further understanding AD-related differences in arithmetic in particular, and problem solving in general.     

Cognitive arithmetic across cultures.

Canadian university students either of Chinese origin (CC) or non-Asian origin (NAC) and Chinese university students educated in Asia (AC) solved simple-arithmetic problems in the 4 basic operations (e.g., 3 + 4, 7 - 3, 3 x 4, 12 divided by 3) and reported their solution strategies. They also completed a standardized test of more complex multistep arithmetic. For complex arithmetic, ACs outperformed both CCs and NACs. For simple arithmetic, however, ACs and CCs were equal and both performed better than NACs. The superior simple-arithmetic skills of CCs relative to NACs implies that extracurricular culture-specific factors rather than differences in formal education explain the simple-arithmetic advantage for Chinese relative to non-Asian North American adults. NAC's relatively poor simple-arithmetic performance resulted both from less efficient retrieval skills and greater use of procedural strategies. Nonetheless, all 3 groups reported using procedures for the larger simple subtraction and division problems, confirming the importance of procedural knowledge in skilled adults' performance of elementary mathematics.     

Retrieval or nonretrieval strategies in mental arithmetic? An operand recognition paradigm

According to LeFevre, Sadesky, and Bisanz, averaging solution latencies in order to study individuals' arithmetic strategies can result in misleading conclusions. Therefore, in addition to classical chronometric data, they collected verbal reports and challenged the assumption that adults rely systematically on retrieval of arithmetic facts from memory to solve simple addition problems. However, Kirk and Ashcraft questioned the validity of such a methodology and concluded that a more appropriate method has to be found. Thus, we developed an operand recognition paradigm that does not rely on verbal reports or on solution latencies. In accordance with LeFevre et al., we show in a first experiment that adults resort to nonretrieval strategies to solve addition problems involving medium numbers. However, in a second experiment, we show that high-skilled individuals can solve the same problems using a retrieval strategy. The benefits of our paradigm to the study of arithmetic strategies are discussed.