Reducing Interference Improves the Memorization of Multiplication Facts in Case of Hypersensitivity to Interference

Comparing DL With Control Participants

Statistical comparisons of DL’s performance to control groups were done using Crawford and Garthwaite's (2002) one-tailed t-test, with effect size defined as z_cc (CC stands for case-controls) – DL’s score normalized according to the control group distribution (Crawford, Garthwaite, & Porter, 2010). Control participants with outlier error rates (higher than the 75^th percentile by more than 150% the inter-quartile range) were excluded. When comparing DL’s performance versus the control group we typically report one-tail p values, reflecting the assumption that DL may have some cognitive deficit underlying her difficulty in multiplication, and such deficit should be manifested in poorer performance than the control group.

Comparing Single-Subject Data Between Two Conditions

One-tailed comparisons of DL’s performance between two conditions were done using a bootstrapping method. The effect we examine was indexed as the difference between the two conditions in the mean score across items:

One-tailed p values were obtained by comparing the observed ScoreDiff value versus the random distribution of ScoreDiff under the null hypothesis that the two conditions do not differ. This random distribution was generated by arbitrarily classifying all scores into two conditions 10,000 times, and computing the value of ScoreDiff for each such random classification. For a paired analysis, in which each tested item has one score in condition 1 and one score in condition 2, each random classification was created by arbitrarily labeling each item’s two scores as “condition 1” and “condition 2”. For an unpaired analysis, each random classification was created by reshuffling the scores of the two conditions into two sets arbitrarily labeled “condition 1” and “condition 2”, while maintaining the original number of items in each condition. When the total number of items was too small such that there were fewer than 10,000 different possible random classifications, we computed the random distribution based on all possible classifications. As effect size we report z – the observed ScoreDiff value, standardized to the mean and standard deviation of the random distribution.

Reading, Lexical Retrieval, and Symbolic Number Processing

DL performed well in reading words, nonwords, and word pairs (TILTAN battery, Friedmann & Gvion, 2003a), as well as in picture naming (SHEMESH test, Biran & Friedmann, 2004) (not significantly different from controls, Crawford and Garthwaite's one-tailed t-test; Table 2). She also performed well in processing symbolic numbers (digit strings and number words): reading aloud from paper a list of multi-digit numbers with 3-6 digits, repeating the same numbers, and writing numbers (3-5 digits) to dictation as digit strings. Thus, DL had good reading, lexical retrieval, and processing of symbolic numbers. Her difficulty in memorizing the multiplication facts did not originate in any of these processes.

Verbal Memory

Interim Summary

DL performed well in the tasks that examined reading, lexical retrieval, verbal memory, and symbolic number processing. Thus, her difficulty in memorizing multiplication facts does not originate in a general deficit of language, number processing, or memory. These results replicate previously-reported dissociations between good memory functions and impaired knowledge of arithmetic facts (Butterworth, Cipolotti, & Warrington, 1996; Kaufmann, 2002).

DL had difficulty in only one memory task – the 2-back task. We proposed that this difficulty actually originated in her hypersensitivity to interference, because the 2-back task may induce a high degree of interference.

Method

The task included a list of 12 fictitious person names (first name + surname) and a country in Africa or Asia where each person allegedly lived. Unknown to DL, the 12 names were the mixture of two lists with 6 names in each: in the low-similarity list, each first name and each surname appeared only once. In the high-similarity list, there were only 3 first names and 3 surnames, and each repeated twice to create 6 different combinations. DL memorized the 12-item list in 5 successive and identical learning stages, each of which was administered as follows. First, the experimenter said aloud each item (name-surname-country) and DL repeated it. After completing the list of 12 items, the experimenter said each name-surname (in random order), DL answered where that person lived, and the experimenter corrected her errors. The 5 learning stages were followed by a final test stage: DL was presented with 24 name-surname-country combinations, and judged whether each combination was correct or not. In this final test, each name-surname appeared twice – once with the correct country, and once with the country of one of the 5 other persons in the same list. Both during learning and during testing, two subsequent items never had the same first name, surname, or country. DL's performance in this task was compared with 24 age-matched control participants (mean age = 40;3, SD = 5;2, range = 31;6 to 48;5) with no reported cognitive deficits and with a mean digit span of 7.17 (SD = 1.19). One additional control participant was excluded due to outlier performance (chance level) in the final test. The detailed results of this task (DL and controls) are available in Supplementary Online Material.

Results

Notably, in the final test (Table 3) even the control group was affected by the similarity between list items: their performance in low-similarity items was marginally better than in high-similarity items, paired t(23) = 1.64, one-tailed p = .06, Cohen’s d = 0.41. This effect could also be observed during learning – e.g., in the last learning stage they recalled 4.21 out of 6 low-similarity items (SD = 1.72), but only 2.79 out of 6 high-similarity items (SD = 1.28; paired t(23) = 4.30, one-tailed p = .0001, d = 0.92). These results agree with previous findings of similarity-induced interference in normal population (Corman & Wickens, 1968; Hall, 1971; Mark-Zigdon & Katzoff, 2015; Oberauer, Lewandowsky, Farrell, Jarrold, & Greaves, 2012; Oppenheim, Dell, & Schwartz, 2010; Posner & Konick, 1966; Runquist, 1970, 1971).

As for DL, she showed a dramatic effect of similarity in the final test: she performed almost at ceiling on the low-similarity list, having only a single error – fewer errors than the control group, and not significantly more than zero errors (Fisher's one-tailed p = .50). Conversely, her performance was poor on high-similarity items – significantly worse than the control group, not significantly different from the chance level of 50% (Fisher’s one-tailed p = .50), and significantly worse than her own performance in the low-similarity items (using the paired bootstrap method described in General Method, one-tailed p = .01, z = 1.83). This pattern meets the criteria for classical dissociation (Crawford, Garthwaite, & Gray, 2003). Crucially, increasing the similarity level (low-similarity list versus high-similarity list) disrupted DL’s performance significantly more than it disrupted the control group’s performance (dissociation analysis of Crawford, Garthwaite, & Porter, 2010: t(23) = 2.15, one-tailed p = .02). These results clearly show that DL was sensitive to the level of item similarity significantly more than the control group. Namely, she had hypersensitivity to verbal interference.

Grouping the Trained Facts Into Sets

The 16 trained facts were grouped into 4 sets, with 4 facts per set. The sets differed from each other in the level of within-set similarity, which was computed using De Visscher and Noël’s (2014b) method: first, the similarity between each two multiplication facts was defined as the number of digit pairs that appeared in both facts, irrespectively of the digits’ position in the fact and of the relative order of the two digits. For example, the facts 8×7=56 and 8×3=24 have no common digit pair (only the digit 8 appears in both) so their similarity is 0. The facts 3×4=12 and 3×7=21 have three common digit pairs (1-2, 2-3, and 1-3) so their similarity is 3. Then, the similarity index for a set of 4 facts was computed by summing the pairwise similarities of all 6 fact pairs in the set. Below, in Appendix A, we consider alternative methods to compute similarity and their fit to the observed results.

Of the four sets of facts, one set had high similarity (7×4=28, 7×6=42, 8×4=32, 9×4=36, similarity=9). The three other sets had lower similarities (4×4=16, 8×3=24, 8×7=56, 5×3=15, similarity=0; 8×8=64, 9×7=63, 6×2=12, 8×6=48, similarity=3; 9×6=54, 6×5=30, 8×5=40, 7×5=35, similarity=4).

The Training Program

Each set of 4 facts was trained during one week, in four 5-minute sessions held in four separate days. Each week started with a session that tested DL’s knowledge of the facts that she learned during the previous weeks, and continued with 3 identical training sessions (Figure 1). Each session took about 5 minutes. The high-similarity set was trained on the second week.

Training sessions. The session started with a pretest: the experimenter said each fact and DL said the answer. After responding to all 4 facts, her errors were corrected. Next, 3 memorization-and-recall phases were done. In each phase, the experimenter said the 4 facts (exercise and result, e.g., "four times five, twenty"), presented in the same order in each session, and DL repeated each fact immediately after it was said; then DL recalled the 4 facts in free recall. Errors, “I don’t know” responses, and omissions of facts were corrected immediately. A post-test, administered like the pretest, completed the session. The order of presenting the facts in each session, and DL’s full responses, are available as Supplementary Online Material.

Testing during training. In the first session in each week (except the first week), we tested DL's knowledge of all the facts she learned since the beginning of the training program. Namely, 4 facts were tested in the beginning of the 2^nd week, and all 16 facts were tested in the beginning of the 5^th week. In each of these sessions, each fact was presented 3 times in a pseudo-random order, such that the same fact never appeared twice in a row, and the fact a×b was never followed by the facts a×(b±1) or (a±1)×b. Errors were not corrected, and there was no teaching during these test sessions. DL’s full list of responses in these tests is available as Supplementary Online Material.

Testing Before and After the Training

DL's knowledge of the multiplication facts was tested in 4 time points (hereby, “testing times”): before training, immediately after training, two months after the training ended, and 3 years after the study ended. Each testing time consisted of 3 testing sessions, administered in 3 days of a single week. In each of these testing sessions, DL was asked to solve the 55 multiplication facts. Reaction times were defined as the delay between the time when the experimenter finished asking the question and the time when DL started saying the result. This delay was measured by inspecting the recordings with an audio-processing software.

Three kinds of responses were classified as errors: (1) Incorrect responses, even if preceded or followed by a correct response. (2) “I don’t know” responses. (3) Extremely slow responses, which suggest that DL used calculation rather than retrieval: reaction times that exceeded the 75^th percentile of the correct-response trials by more than 150% the inter-quartile range. To compute the outlier threshold, items that were classified as errors by one of the two other criteria were not included. The outlier threshold was computed per testing time, but essentially the same results were obtained when using a common threshold for all testing times.

In each testing time, DL was tested 3 times on all facts, yielding an accuracy score between 0 and 3 for each fact. The 16 facts with lowest pre-training scores were selected for training (10, 3, and 3 facts with score = 0, 1, and 2, respectively).

On top of these four testing times, we also examined DL’s performance in the testing sessions that were administered during the training period, in the first session in each week. In each of these sessions, DL was tested on the facts that she already learned (i.e., on 4 facts in the beginning of the second week, and on 16 facts in the beginning of the fifth week). From each session we considered only the facts learned in the previous week. Below, we refer to this data as “testing during training”.ⁱⁱⁱ

Effectiveness of Training: Performance Over all Facts

In the rule-based facts (N×0 and N×1), DL had merely a single error in the pre-training test and in the 3-year follow-up, and no error in the other testing times. Table 4 shows her error rates in the non-rule facts. Outlier reaction times of the non-rule facts were classified as errors, as explained in the previous section (“Testing Before and After the Training”): responses slower than 2408 ms in the pre-training test, 3265 ms in the post-training test, 4010 ms in the 2-month follow-up test, and 2380 ms in the 3-year follow-up test. To examine the effect of training, we compared the pre-training performance with each of the other testing times using the paired bootstrapping method described in General Method, based on DL’s pre-training and post-training scores (a 0-3 score for each fact).

DL’s performance in the non-rule facts (Table 4) significantly improved from the pre-training test to the post-training test, i.e., the training was effective. As predicted, this overall improvement was driven by a significant improvement in the trained facts, with no significant improvement in the untrained facts^iv. Strikingly, DL’s relatively short training program has succeeded where years of schooling did not: it gave rise to an improvement in the trained facts. This improvement persisted even after 3 years during which she received no additional training (Table 4), and – according to her – did not use the multiplication table very often. Still, unsurprisingly, her knowledge of the trained facts after 3 years was not as good as at the end of the training program (comparing the two using the bootstrapping method described above, one-tailed p = .03, z = 1.69).

Effect of Within-Set Interference on the Post-Training Knowledge

Our main question was whether we can detect differences in the training effect between the facts that were trained together with highly-similar facts and the facts that were trained with dissimilar facts. Figure 2 presents DL’s detailed performance for each exercise in each of 4 testing times – before training, during training (on the testing sessions in the beginning of each week), after training, and in the 2-month follow-up^v. The table clearly shows an effect of similarity: both during training and in the post-training tests, accuracy was better in the sets with lower within-set similarity than in the high-similarity set. To examine whether this effect was significant we considered, for each testing time, the within-set similarity values corresponding with each of DL’s answer attempts. We classified these values into two groups according to her answer (correct or incorrect), and compared the two groups using Mann-Whitney U test (the effect size r was computed according to Fritz, Morris, & Richler, 2012). The correct answers had significantly lower similarity values than the incorrect answers, confirming that lower within-set similarity in the training period improved the post-training accuracy (during training: U = 94, one-tailed p = .006, r = .36; after training: U = 217.5, one-tailed p = .08, r = .20; two-month follow-up: U = 161.5, one-tailed p = .01, r = .33; i.e., a medium effect size during training and in the 2-month follow-up). To eliminate any possible effect of prior knowledge, we also ran the same analysis only on the 9 exercises with pre-training score = 0. The results were essentially the same: for the tests administered during training, this analysis showed a slightly stronger effect than in the previous analysis (U = 35.5, one-tailed p = .01, r = .43). For the tests administered after training, the analysis now showed a slightly weaker effect than before, which was marginally significant (immediately after training: U = 58, one-tailed p = .08, r = .27; two-month follow-up: U = 64, one-tailed p = .09, r = .26).

Click to enlarge

Figure 2

DL's performance before, during, and after training. During training, DL’s score in the low-similarity set was better than in the high-similarity set. This difference persisted when she was tested 2 months later. The within-set similarity index (S) was computed as explained in the "Grouping the Trained Facts Into Sets" section. The “Score” columns show the score for each trained fact in each testing time (scale: 0-3 per fact, 0-12 per set; red/green shading indicates below/above 50% correct). The “Answers During Training” columns show DL's accuracy in each answer attempt during the training sessions.

These results were not an artifact of problem size (better knowledge of facts with smaller operands, Zbrodoff & Logan, 2005). First, a fact’s average operand size did not correlate with the within-set similarity level (r = .20, p = .46). Second, when directly comparing operand size and similarity as explanations to DL’s performance during and after training, similarity stands out as a genuine predictor of accuracy. This was examined by submitting DL’s accuracy scores to a logistic regression with two predictors: the average operand size and the within-set similarity level – 1, 2, or 3 (weeks 3 and 4, whose similarities were nearly identical, were both assigned similarity level 2). Both during training and in the 2-month follow-up test, the within-set similarity had a significant effect on accuracy (during training: z = 2.59, one-tailed p = .005; follow-up: z = 2.23, one-tailed p = .01), whereas the problem size had a smaller or non-significant effect (during training: z = 0.95, one-tailed p = .17; follow-up: z = 1.73, one-tailed p = .04). Namely, the effect of similarity could not be reduced to a problem size effect. In the post-training test, however, the regression showed a significant effect only for problem size (z = 2.46, one-tailed p = .007), with no significant effect of similarity (z = 0.76, one-tailed p = .22).

A second analysis ignored the specific similarity values, and just compared the per-fact accuracy after training (correct or incorrect for each answer attempt) between the high-similarity set (H) and the three lower-similarity sets (L). The comparison used the unpaired bootstrapping method described in General Method, based on the 0-3 score of each fact (the random distribution for H₀ was computed by considering all $\left(\begin{array}{c}16\\ 4\end{array}\right)$ possible classifications of the 16 facts into arbitrary “H” and “L” groups containing 4 and 12 facts respectively). The effect of similarity was significant in the tests administered during training (H: 50%, L: 89%, one-tailed p = .003, z = 1.85), in the post-training test (H: 42%, L: 61%, p = .03, z = 0.81), and in the two-month follow-up test (H: 17%, L: 81%, p < .001, z = 2.81). These results survived the exclusion of the three facts with the highest pre-training scores (during training: one-tailed p = .003, z = 1.47; post-training: p = .02, z = 0.55; two-month follow-up: p < .001, z = 2.42).

A third analysis examined how quickly DL learned each fact until reaching ceiling performance. Each fact’s learning duration was defined as the last answer attempt, out of the fact’s 12 answer attempts (right columns in Figure 2), in which DL made an error (here we did not code slow answers as errors, because most answer attempts during training were made as free recall, in which the response time is undefined). The 4 facts in the high-similarity set were the slowest to be learned – they had the longest learning durations (p = $1/\left(\begin{array}{c}16\\ 4\end{array}\right)$ < .001).

Taken together, these results form a clear picture: DL’s learning of the multiplication facts was strongly affected by similarity. The effect of similarity was clearly observed during the 4-week training period, and critically – also in the test administered two months after DL’s training has ended. In the test administered immediately after the 4-week training period, the effect of similarity was observed only in one of the two analyses. The weak similarity effect immediately after the 4-week training period can be explained by the fact that in this test, the between-fact variance in the time elapsed since each fact was learned was the highest: DL had learned the facts of set #4 only few days before the post-training test, but she had learned the facts of set #1 almost 4 weeks before the post-training test. The idea of time-elapsed-since-learning as a confounding factor is supported by comparing the results of the immediately-after-training test with those of the two other post-training tests (during training, 2-month follow-up): in the immediately-after-training test, DL performed more poorly for facts that were learned more recently. Whether this effect is random or has a genuine cognitive origin remains an open question, as the amount of data in the present study is insufficient to evaluate the effect reliably.

The effect of similarity did not go unnoticed by DL herself: during the 2^nd week of training, when she learned the high-similarity set, she commented more than once that "it is hard for me to learn these exercises because of all these 4's that repeat over and over again" – an accurate description of her hypersensitivity to interference.

Pre-Training Scores

If DL’s learning is affected by the similarity between facts, this should also be reflected in her performance in the pre-training test, because even before we started the training program, DL’s knowledge of specific arithmetic facts was presumably affected by similarity-induced interference (De Visscher & Noël, 2014b). In particular, we predicted that she would show lower pre-training knowledge for multiplication facts that have higher similarity with the rest of the multiplication table.

To examine this prediction, we considered each of DL’s answer attempts in the pre-training test for all 36 non-rule facts. For each answer attempt, we computed the fact’s similarity with the rest of the multiplication table (hereby denoted Sim) as the sum of that fact’s similarities with all other multiplication facts between 2×2 and 9×9 (the similarity between each pair of facts was defined as explained in the “Grouping the Trained Facts Into Sets” section). We predicted that DL’s pre-training knowledge would be better for facts with lower fact-table similarity (lower Sim). Namely, the Sim values should be larger for DL’s incorrect pre-training answers than for her correct answers. To assess whether this was the case, we used the unpaired bootstrapping method described in General Method. The difference between Sim of correct answers (15.25) and that of incorrect answers (18.19) was significant (one-tailed p = .03, z = 1.83). Essentially the same results were obtained when the fact-table similarity computation excluded the tie problems (Sim = 13.78 versus 16.21, one-tailed p < .05, z = 1.72). Namely, as predicted, DL’s pre-training knowledge was better for multiplication facts that were less similar to the rest of the multiplication table.

The Processing Stage Sensitive to Interference

According to Campbell’s network interference model, interference arises because one multiplication fact is associated with another, so they activate overlapping representations (Campbell, 1987, 1995; Oberauer, 2009). When the facts are similar to each other, irrelevant associations between them are strengthened. Furthermore, processing different facts in temporal proximity may strengthen the irrelevant associations between these facts. In our experiment, manipulating similarity and temporal proximity during learning affected how well DL learned the facts. Importantly, the effect of similarity could reliably be attributed to processes that occur during learning time, because similarity and temporal proximity were not manipulated when DL’s knowledge was tested (but note that the learning sessions included not only encoding and storage but also retrieval). Additionally, Campbell (1987) showed that arithmetic performance was affected by manipulating the degree of interference during retrieval, for arithmetic facts that the participants already knew beforehand. It therefore seems that interference affects both learning-time and retrieval-time processes. Indeed, other studies too showed that high levels of interference may take an effect in different processing stages (Bartko, Cowell, Winters, Bussey, & Saksida, 2010; Farrell, 2006; Fernandes & Moscovitch, 2000; Kaufmann, Lochy, Drexler, & Semenza, 2004; Lochy, Domahs, & Delazer, 2004; Van Dyke & McElree, 2006; Wixted, 2004). Interference may disrupt the encoding and storage of data in memory while learning the facts (Farrell & Lewandowsky, 2002; Lewandowsky & Farrell, 2008), and it may disrupt the retrieval stage (Burgess & Hitch, 1999; Campbell, 1987; Henson, 1996, 1998).

A more specific question concerns the locus of DL’s deficit. Her hypersensitivity to interference may have resulted from an encoding/storage deficit in creating the network of associations in long-term memory – e.g., she may have been creating too strong irrelevant associations. Alternatively, her deficit may be impaired retrieval processes, which fail retrieving the arithmetic facts from an intact storage. Sadly, it seems that our data cannot arbitrate between these possibilities. Within the framework of the network interference model, retrieving the correct answer to a multiplication problem requires that the representation of the fact is distinctive enough from other facts. When this is not met, incorrect responses occur. This can happen because the storage of facts is corrupted such that the facts are not distinguishable sufficiently; but it can also happen because the retrieval process is impaired and requires higher distinctiveness for successful retrieval. Under either assumption, increasing the distinctiveness of facts (as our manipulation probably did) would increase the activation of the correct solution relative to incorrect solutions, and would therefore help achieving the required threshold of distinctiveness. In short, both possibilities (storage deficit and retrieval deficit) make similar predictions about DL’s performance. Note, however, that from a clinical/intervention point of view, the picture emerging from our study is clear: a learning-time intervention can help overcoming similarity-induced interference.

Interference and Spacing

De Visscher and Noël (2014b) described interference as an effect of previously-learned items on the ability to learn a new item. Consequently, they defined an “interference parameter” for each multiplication fact as its degree of similarity versus all previously-learned facts. This formulation accords with the definition of proactive interference in working memory (Bennett, 1975). However, in the present study, it is impossible to define a clear order of learning the facts, because several multiplication facts were learned simultaneously during each given week (this is probably the case also in real-life situations of learning the multiplication table). The interference here does not arise from the similarity with previously-learned items, but from the similarity with simultaneously-learned items.

Our data suggest that the critical methodological factor is to create a sufficient temporal gap between the learning of similar multiplication facts. Future studies may examine the size of the temporal gap required to avoid interference. This idea of temporal separation accords with other studies showing that learning is improved by increasing the temporal delay between learning sessions that contain potentially-interfering items (Friedman & Korman, 2016).

The Type of Information Sensitive to Interference

The effect of similarity on the size of interference accords with the view of interference as arising from the amount of overlapping features between the items to be remembered (Oberauer & Lange, 2008). But what are these features? The representations sensitive to interference could be phonological (Baddeley, 1966, 1968; Farrell, 2006; Nelson et al., 1974; Runquist, 1970), semantic (Baddeley, 1966; Oppenheim et al., 2010), number-specific, or another representation.

In line with the possibility of phonological sensitivity-to-interference, the speed and accuracy of addition fact retrieval was shown to be affected by phonological similarity (Noël, Désert, Aubrun, & Seron, 2001). Further support to the role of phonological interference comes from studies of non-number words, which show that word memorization is affected by their phonological similarity to each other (Nelson et al., 1974; Pajak et al., 2016; Runquist, 1970). However, interpreting these findings as an explanation for difficulties in memorizing multiplication facts should be done with caution, because at least some phonological mechanisms treat words and numbers differently: e.g., the speech mechanisms handle words as sequences of phonemes, but numbers words as whole building blocks – in speech production (Bencini et al., 2011; Cohen, Verstichel, & Dehaene, 1997; Dotan & Friedmann, 2015, 2019; Shalev, Ophir, Gvion, Gil, & Friedmann, 2014), and apparently also in speech comprehension (Fischer-Baum, Mis, & Dial, 2018). Furthermore, the representation of multiplication facts in memory is apparently not purely phonological (Whalen, McCloskey, Lindemann, & Bouton, 2002).

In order to fully understand sensitivity to interference, and the effect of similarity, one must use a detailed cognitive model of arithmetic fact representation. Only a detailed model can specify the precise amount of representational overlap between some given facts, and the amount of interference between them. Consequently, each such model can predict how easy it should be to learn a given set of multiplication facts, so the model can be evaluated by comparing these predictions against the actual learnability of facts. In Appendix A we describe 7 detailed cognitive models of arithmetic fact representation, and we evaluate them based on their ability to predict several aspects of DL’s performance. This analysis lends support to the models that assume symbolic representations (digits / number words) of arithmetic facts, rather than magnitude representation – in accord with several studies indicating that multiplication facts are stored in verbal format (Dehaene, 1992; Dehaene & Cohen, 1995; Dehaene et al., 2003, 1999). Future studies may elaborate further on the precise representation of arithmetic facts.