Positive impact of sleep on recall of multiplication facts

1. Introduction

The beneficial effects of sleep on memory are well established. Sleep supports the consolidation of learnt facts, shown by improved recall of word pairs by both children and adults when learning occurred prior to sleep, compared with learning followed by a period of daytime waking [1,2]. There is a moderate beneficial effect size for recalling novel word pairs, where the second word in a pair is recalled when the first word of the pair is presented following sleep versus wake (Hedges g = 0.57) [3]. The vast majority of research investigating the impact of sleep on memory processes has been focused on word learning, which draws on declarative memory. However, sleep has also been shown to benefit spatial cognition [4] and procedural memory in young adults, such as memorization of a spatial sequence [5]. However, there are many other domains in which memory processes play an important role, for example, in mathematics.

It is recognized that learning and performing mathematics draws heavily on memory. For example, there is a close relationship between mathematical achievement and both short-term and working memory [6]. On the basis of this association, educational researchers have a growing interest in how memory management techniques can be applied within the classroom [7]. While memory processes are implicated in a wide range of mathematical skills [8], one evident role concerns the memorization and retrieval of number facts (e.g. number bonds, multiplication tables). Being able to correctly recall number facts from memory is an important component of mathematical achievement [9]. Fluent retrieval of number facts frees up cognitive resources that would otherwise be required to derive the answers to simple calculations, allowing individuals to focus on conceptual and strategic aspects of maths. This is recognized by the UK government which has recently introduced a new statutory national multiplication tables test to be taken by all children aged 8–9 years old. Given the critical role of memory in mathematics education, it is valuable to identify factors that can support learning and retrieval. No studies have yet explored whether the learning and recall of number facts, such as multiplication tables, are improved following sleep. This research is therefore timely.

While it might be plausible that the well-established beneficial effects of sleep on word learning could directly extend to learning number facts, there are some important differences between the domains. Multiplication fact learning is not a purely declarative process, as strategies such as decomposition, a standard algorithm or repeated addition can be used to solve complex multiplication problems [10]. However, the fact that the same strategies from simple multiplications can be used to solve complex multiplications demonstrates that learning complex multiplications is not without a familiar context. Learning stimuli that adhere to a familiar structure may particularly benefit from sleep-related memory consolidation. Indeed in the word-learning literature, it has been demonstrated that using nonsense words that follow familiar lexical patterns shows a greater sleep-related recall benefit (moderate effect size) than words in a foreign language that do not follow familiar patterns (small effect size) [3]. Where participants are given the opportunity to engage with the complex multiplication stimuli using familiar strategies, through the use of untimed trials in the first instance, it may enhance sleep-related benefit compared with no opportunity to consider the complex multiplication stimuli as a continuum of simple multiplication, in which only timed trials are used to constrain the strategy to rote memorization.

As discussed, initial engagement with complex multiplication stimuli is likely to involve problem solving using strategies rather than through simple memorization. However, when the response time is constrained to ensure that participants use retrieval rather than strategies to associate the multiplication problem and its answer [11] multiplication facts can be learnt the same way that word-pairs are presented in other studies of sleep-related memory consolidation. Indeed, research suggests that rote memorization techniques are frequently used by children to learn multiplication problems as part of a range of strategies [12]. This raises the potential of leveraging sleep-associated benefits to improve children's multiplication fact knowledge.

Evidence from brain imaging studies supports the potential benefit to number fact learning from sleep due to the wide network of brain regions involved. Brain imaging following the drill learning of answers to complex multiplications indicates greater activation in the left angular gyrus for learnt answers versus novel problems [13–15]. There is brain imaging evidence to suggest that solving arithmetic problems using procedural knowledge, which relies on working memory versus fact retrieval from long-term memory recruits different brain areas [14]. Other evidence indicates the involvement of hippocampus and the para-hippocampal and retrosplenial structures in learnt problems compared with novel problems, with many of the frontal clusters activated but no significant difference in the activation of the angular gyrus between these two conditions [16]. The activation of a wide network of brain regions, including both the hippocampus and frontal cortex, suggests that the learning of arithmetic facts might benefit from sleep-related consolidation. The active consolidation system hypothesis, in which declarative memories in the hippocampus are reactivated and transferred to the neocortex, driven by slow wave sleep oscillations, would suggest that the transfer of arithmetic facts to long-term memory in the neocortex is optimized when learning is followed by a period of sleep, compared with a period of wakefulness.

In the current study, we investigated whether benefits of learning prior to sleep compared with learning prior to wakefulness were observed when adult participants learnt complex multiplication facts (i.e. with either multiplier or multiplicand > 12). Recall of the newly learned items was tested 10.5 h later. We predicted that there would be a significant interaction between condition (learning prior to sleep versus learning prior to wakefulness) and session (learning versus recall), specifically that there would be significantly higher accuracy in the recall session of the sleep learning condition compared with the wake learning condition.

Our learning sessions were designed following prior research of effective approaches. Evidence suggests that integrated training where both strategies are modelled and timed trials are used to teach the answers to simple multiplications for school children are effective in increasing and maintaining response accuracy [11]. Therefore, our learning sessions included blocks in which participants could use procedural strategies as well as those in which participants were required to retrieve the answer. Permitting participants to use untimed strategies such as decomposition, a standard algorithm or repeated addition (for definitions, see [10]) to solve the multiplication problem upon first presentation also reflects real-world learning, as less than 10% of participants used retrieval strategies on first presentation of complex multiplications [10]. We also employed an incremental rehearsal method (in which novel items are added one-by-one (see electronic supplementary material: https://osf.io/avy6d/files/osfstorage) which has shown a large effect size in the recall of simple multiplication problems relative to traditional drill, in which problems are repeated until all are correct [17].

2. Method

The design, sample size, exclusion criteria and analysis plan were pre-registered and can be found at https://aspredicted.org/3mb8d.pdf.

3. Results

Sixteen participants were assigned to the sleep condition first and 21 were assigned to the wake condition first. Learning scores did not differ between the sleep condition first and the wake condition first participants, all p's > 0.06. There was also no significant difference in any of the learning or recall scores between participants who completed the learning session with set A first (N = 17) or set B first (N = 20), all p's > 0.466.

3.1. Pre-registered analyses

A 2 (sleep, wake) × 2 (learning, recall) repeated measures ANOVA revealed significant effects of session (learning versus recall) F_{1, 36} = 21.90, p < 0.001 and condition (sleep versus wake) F_1,36 = 4.34, p = 0.044 on accuracy. Importantly, there was also a significant interaction between session and condition F_1,36 = 8.65, p = 0.006. Post hoc tests revealed significantly lower recall accuracy in the wake condition (M 18.46, s.d. 5.71) compared with the sleep condition (M 20.68, s.d. 4.22), t = 3.097, p = 0.004, significant after Bonferroni adjustment, Cohen's d = 0.51, moderate effect size [18] mean difference 2.22 (s.d. 4.35), equating to two additional facts recalled (see figures 1 and 2 and tables 1 and 2).

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Table 1. Descriptive statistics for N = 37.

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	sleep	wake	p-value
learning score (accuracy out of 28)	22.16 (3.56)	22.14 (3.55)	0.964
recall score (accuracy out of 28)	20.68 (4.22)	18.64 (5.71)	0.004
time between end of learning session and start of recall session (in minutes)	635.1 (61.74)	605 (69.88)	0.049
tiredness score after learning session (1 not at all tired to 5 extremely tired).	3.27 (1.12)	2.70 (1.08)	<0.001

Table 2. Proportions of participants who benefitted from the sleep learning condition above the wake learning condition.

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	met criterion (n = 37) % (n)	did not meet criterion (n = 33) % (n)
better recall in wake condition	29.7 (11)	45.5 (15)
the same recall in sleep and wake conditions	10.8 (4)	9.1 (3)
better recall in sleep condition	59.5 (22)	45.5 (15)

The interval between the end of the learning session and start of the delayed test ranged from 485 to 750 min in the sleep condition and 410 to 728 min in the wake condition. A paired sample T-test revealed a significantly shorter interval in the wake condition, t = 2.04, p = 0.049, with a mean of 10 h 35 min for the sleep condition and 10 h and 5 min for the wake condition; however, both intervals were close to the 10 h 30 min interval which was pre-registered. Following the sleep learning session, participants reported significantly greater tiredness (modal response: moderately tired) compared with the wake learning session (modal response: slightly tired). However, tiredness ratings did not explain a significant proportion of variance in the sleep learning score: standardized β = −0.275, t = −1.70, p = 0.099, R² = 0.076, nor the difference between the sleep learning score and the sleep recall score: standardized β = 0.098, t = 0.585, p = 0.562, R² = 0.01.

Following the sleep learning session, participants' sleep was an average of 8 h and 0 min (s.d. 1 h 16 min). Sleep duration did not explain any significant difference between the sleep learning score and the sleep recall score: standardized β = −0.071, t = −0.420, p = 0.677, R² = 0.005.

To establish whether participants followed the instructions to memorize the multiplication problems, at the end of the experiment they were asked whether they used non-retrieval strategies to provide the answer to the problem. The majority of participants never (32.4%) or occasionally (45.6%) used strategies to solve the problem, while 13.5% used strategies half of the time and 8.1% used strategies most of the time.

4. Discussion

Here we found, for the first time, a sleep-related benefit in recall for learning complex multiplication facts. The effect size for this benefit (0.33–0.51) is slightly smaller than that identified in a meta-analysis of studies using similar designs for word-pairs for reduced forgetting in the sleep versus wake condition (Hedges g = 0.57) [3]. This extends existing evidence of the benefits of sleep from word learning to mathematical facts. Below we first compare the process of encoding of complex multiplication problems relative to the encoding of word-pairs used in previous studies. We also discuss the implications that the benefit of learning prior to sleep was found regardless of initial levels of learning of the complex multiplication problems. Some limitations of the present study are also considered.

We have found that the benefit of learning before sleep extends beyond word learning to multiplication facts. One of the differences between encoding the association of a complex multiplication problem and its product compared with word-pair associations is the greater opportunity for error due to interference effects and problem size [19]. To ensure that the stimuli learnt were novel to participants, it was necessary to use stimuli with large problem sizes, i.e. a large maximum operand. Consequently, few participants met the 60% criterion typically used in studies of the benefits of sleep versus wake on cued recall of word pairs [20]. There is little semantic information attached to the operands and the product of a complex multiplication problem. The studies using word pairs with less semantic information to link the pairs, i.e. unrelated pairs, show less accuracy and a greater number of trials to reach criterion compared with related word pairs [20,21]. In one study, sleep following learning relative to wake following learning demonstrated a greater benefit for the unrelated pairs compared with the related pairs [20], although in a smaller study the learning prior to sleep condition evidenced beneficial recall scores compared to the wake condition for both the related and unrelated word pairs [21]. Therefore, like studies with little semantic information to link the two associations, the recall of complex multiplication stimuli does benefit from learning prior to sleep relative to a period of wakefulness.

A second difference between learning multiplication facts and word-pair associations is that non-retrieval, procedural strategies (such as decomposition or repeated addition) can be used to solve multiplication problems. We investigated the potential role of these procedural strategies. Unlike a previous study that found that accuracy using non-retrieval strategies was associated with accuracy using retrieval strategies of complex multiplication problems [10], there was no association between accuracy on the untimed trials, that facilitated the use of non-retrieval strategies, and the accuracy on the learning session test. However, participants who had lower accuracy on the untimed trials experienced greater benefit of the sleep condition compared with the wake condition for recall, albeit a small association. It is possible that sleep-related benefits for learning multiplication problems could particularly benefit learners who struggle with using procedural strategies to solve problems, although further work is needed to identify what strategies are used when presented with untimed multiplication problems and allowing participants to use a pen and paper for working out problems, which was not encouraged in the present study.

The present study indicated that learning prior to sleep benefits learners at all levels of initial learning, and that prior knowledge of simple multiplication problems is not associated with the extent of the sleep-related benefit for recall. This suggests that targeting learning of multiplication problems prior to sleep could benefit learners with varying memory capacities and prior knowledge of multiplication stimuli. There is mixed previous evidence for whether participants with weaker encoding of word-pairs experience greater sleep-related benefit than individuals with higher encoding [22–24]. Further research is needed to apply these findings to a real-world educational context. Future studies could evaluate whether primary school children with heterogeneous memory skills would benefit from learning novel simple multiplication problems in the evening compared with during the day.

One limitation of this study was that participants completed this study online in unsupervised conditions, and as the data indicate, not all participants were able to follow the instructions to answer the timed multiplication problems using retrieval. However, given the longitudinal design of this study and its conduct during the COVID-19 pandemic, the online data collection was necessary to ensure a sufficient sample size. Pre-registered exclusion criteria to maximize the quality of the data (taking over 1 h 45 min to complete the learning session, completing the experiment at the incorrect time, participants reporting that their sleep on the night of the sleep learning condition was greater than 2 h shorter than usual) mitigated some of the concerns over fraudulent behaviour or low effort that could occur due to lack of supervision. The 6-s time limit for responses was identified through piloting to ensure sufficient time to type a three-digit number but insufficient time to use alternative strategies. Another study has demonstrated that the use of timed questions reduced self-reported cheating compared with other strategies [25]. The rate of non-compliance with instructions to not use strategies for 50% or more of the trials was 22%, which is slightly lower than the 24–41% of self-reported cheating in student samples [25]. An additional limitation is the self-reported sleep duration, which is overestimated compared with polysomnography-assessed sleep duration [26]. In addition, night waking was not accounted for in the calculation of sleep duration, due to poor recall for night waking. Despite this limitation, another study has also demonstrated a lack of association between sleep duration and recall performance in a sleep restriction study [27]. A further limitation is the lack of other comparative stimuli with a similar complexity of encoding with the same number of presentations to conclusively demonstrate the specificity of sleep-related benefit on the recall of complex multiplication problems. However, in this study design the learning session for the complex multiplications took approximately 1 h, so the addition of further stimuli, increasing the learning sessions to 2 h would have inordinately increased participant burden and fatigue effects. Overall, further research is needed to evaluate the effect of learning multiplication problems prior to sleep on long-term recall.

In conclusion, learning complex multiplication problems prior to sleep conferred additional benefit on recall compared with learning problems during the daytime. This effect was found across all levels of initial encoding. This effect could be evaluated in an educational context for the learning of novel simple multiplication problems.

Ethics

This study was approved by the Loughborough University Ethics Review committee and all participants gave informed consent.

Data accessibility

The data are available at doi:10.25392/leicester.data.22148921 [28] and doi:10.25392/leicester.data.22148909 [29] and the materials are available via Gorilla Open Materials. The experiment was pre-registered: https://aspredicted.org/3mb8d.pdf [30].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors' contributions

J.S.: conceptualization, formal analysis, funding acquisition, methodology, project administration, writing—original draft; G.C.: conceptualization, funding acquisition, methodology, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

The authors have no relevant financial or non-financial interests to disclose.

Funding

This study was funded by a British Academy-Leverhulme Small grant and the ESRC Centre for Early Mathematics Learning (grant no. ES/W002914/1).