The authors of this article declare no conflict of interest.

The present study explores the effect of two instructional methods for children with different levels of mathematical skills. One of these methods uses a conventional approach to learning multiplication and emphasizes the memorization of all arithmetic facts, whereas the other method is based on psychological principles and combines: a) the memorization of a small subset of problems aided by color cues and a portable time-table, with b) the use of single-step rules. One hundred and sixty second-grade children (aged 7-8) received instruction in one of these approaches – either the conventional method or the memory and rules method (M&R) – over the course of 6 months as part of their normal school education. Moderation analysis revealed that children with poor mathematical skills in the conventional group scored significantly better than their counterparts in the M&R group, whereas a significant advantage was observed in the M&R group for those children with strong mathematical skills.

El presente estudio explora el efecto de dos métodos de enseñanza de la multiplicación simple en alumnos de primaria con diferentes niveles de habilidades matemáticas. Un método se basa en el enfoque convencional para el aprendizaje de las multiplicaciones que enfatiza la memorización de todas ellas, mientras que el otro se basa en principios psicológicos y combina: a) la memorización de un pequeño subconjunto de multiplicaciones auxiliadas con claves de color y una tabla portátil con las multiplicaciones con b) el uso de reglas de un solo paso. Ciento sesenta niños y niñas de segundo de primaria (de 7 a 8 años) recibieron instrucción en uno de estos métodos, ya fuera el convencional o el método de memorización y reglas (M&R), durante 6 meses como parte de su educación escolar normal. El análisis de moderación reveló que los niños con habilidades matemáticas bajas en el grupo convencional obtuvieron puntuaciones significativamente mejores que sus pares en el grupo M&R, mientras que se observó una ventaja significativa en el grupo M&R para aquellos niños con altas habilidades matemáticas.

The mastery of single-digit multiplications (such as 3 x 4) is probably within the minimum targets of most educational systems around the world (e.g., see National Mathematics Advisory Panel [NMAP, 2008] in the USA;

Traditionally, it is assumed that curricular methods aimed at learning multiplication at school should provide children with: i) a conceptual understanding of the arithmetic operation and ii) fluency, that is, the skill of solving single-digit multiplications quickly and accurately. Understanding the meaning of multiplication is fundamental in our daily lives, but developing fluency is also needed, as this allows students to free-up cognitive resources that will be necessary when, in subsequent years of learning, more complex computations such as multi-digit multiplications or divisions are encountered (e.g.,

Different approaches to the teaching of single-digit multiplications are used in various parts of the world, but in that their objective is to attain a certain level of fluency they usually share the common feature that learning should rely mainly on memory. The focus on memorization is based on evidence that, as a means of solving single-digit multiplications, this is the most efficient, fastest, and least error-prone strategy (

Although rote verbal memory and repeated practice are the basis of most conventional methods aimed at teaching multiplication, such strategies are not without difficulties. First, many children simply struggle to learn multiplication using these strategies (e.g.,

The outlook in the area of designing new curricular methods based on scientific evidence is becoming more positive thanks to collaboration between researchers and practitioners. Cognitive and educational psychologists are currently offering new insights into the mechanisms involved in mathematical learning, and this knowledge is available to be employed in developing new educational methods (

The evidence reviewed above suggests that educational strategies aimed at reducing interference should facilitate the learning of multiplication. An easy way to diminish interference in multiplication retrieval is to reduce the set of problems to memorize (i.e., fewer problems, less competition), and rules can help here. By using rules for some tables (1 and 10), as well as the commutativity principle, the matrix of 10 x 10 problems can be reduced to 36 problems to be memorized. Relying on rules to learn some multiplications is not new. However, the recent literature has shown that “using single-step rules” allows children to rapidly obtain the solution to a problem without the effort of executing complex multi-steps procedures or memorizing facts by pure association (

Taking this as starting point, we designed and applied a new teaching method. With the aim of reducing interference, it combines the learning by “memory” of a reduced set of problems with the use of “rules” (from now onwards, M&R method). Moreover, to help with the memorization of the problems, along with fact rehearsal and guided practice, we added a portable time-tables. The rationale for this is that it may work in a similar way to flash cards, which in primary education have been shown to facilitate learning and to promote a sense of control over learning (e.g.,

On the other hand, to help with the learning and use of rules the M&R method promotes not only the mechanistic learning of single-step rules; it also encourages the understanding of these rules, because conceptual understanding constitutes greater achievement than simply heuristic learning (e.g.,

The M&R method was compared to a conventional one, that is, a method that, once the concept of multiplication had been explained, was based on the memory-based learning of the whole multiplication table. With this aim in mind, it uses fact rehearsal and guided practice as the main strategies (see

An additional aspect of the present study has to do with individual differences in multiplication fact learning. Educational systems inevitably have to deal with the issue of diversity in the student population; teaching methods that are valid for some students are not always useful for others (

From a cognitive perspective, differences in arithmetic learning have been related to differences in the mechanisms of working memory, either globally (e.g.,

Experiential factors have also been related to differences in mathematical attainment. For instance, children with low socio-economic status (SES) suffer from reduced exposure to situations involving numeracy, and this may affect the normal development of their mathematical skills, including multiplication (

Finally, and most importantly for the current study, the literature on interventions aimed at increasing fluency in arithmetic has also identified the importance of individual differences in levels of mathematical skill relating to how students take advantage of different interventions. Children with poor accuracy in multiplication tasks respond better to interventions that focus on modelling, and those who show acceptable accuracy but poor fluency respond better to interventions that focus on repeated practice (

In this study we address two research questions related to the effectiveness of curricular methods aimed at attaining certain levels of fluency in solving single-digit multiplications: a) does a method aimed to reduce interference, the M&R, which combines memory retrieval and single-step rules, lead to greater achievement than a conventional method based on memory retrieval of the whole time-table? b) to what extent is the effectiveness of these methods moderated by children’s levels of mathematical skills?

To answer these questions, we compared the effectiveness of the M&R method, which seeks to reduce interference through combining memory and rules, with a conventional method, but paying attention to the modulator effect of children’s mathematical skills. Notably, our study compared curricular methods, i.e., comprehensive methods aimed at teaching multiplication and attaining fluency with time-tables in the classroom and did not involve the comparison of small intervention methods simply aimed at achieving fluency in a small subset of multiplication facts or in a special population.

We employed moderation analysis (based on linear regression analysis; see

As we will note in the description of the M&R method (see below), there is theoretical and experimental support for: i) limiting the number of problems to memorize as a means of reducing interference; ii) the effectiveness of single-step rules in providing fast and accurate solutions to arithmetical problems; iii) the relevance of conceptual understanding in learning math; and iv) the benefits of using complimentary material, such as flash cards (equivalent to the portable time-table) and using color cues, in multiplication learning. So, it is hypothesized that students following this method should outperform those on the conventional method. Less certain is the question of the role of mathematical skills in terms of benefiting from the conventional and the M&R methods. Previous studies have suggested that children with poor skills benefit from strategies based on memory retrieval, such as conventional ones, which include practicing and modelling at the same time (

The respondents were 160 children (89 girls) aged 7-8 years, all of whom were second grade students in a charter school in Malaga (Spain). Children were from diverse socio-economic backgrounds, but most were from a medium socioeconomic level. An additional group of eighteen students diagnosed with developmental disabilities (i.e., dyscalculia, dyslexia, attentional problems) took part in the study, but were excluded from the analyses. Participants were drawn from 8 different classrooms. In four of these (80 children, 41 girls) the conventional procedure was followed during the 2014 academic year, and in the other four classrooms (80 children, 49 girls) the M&R method was followed in 2015. However, in both cases the methods were applied while children were in second grade. Informed consent for children was obtained thorough the school’s staff.

The two multiplication methods were followed for 6 months (January to June) as part of children’s second grade math classes. Both groups were evaluated at the end of the school year, after having finished the method.

A mathematical skills test (BERDE:

To assess multiplication fluency, the multiplication fluency test of the BERDE was used in a different session. Children were provided with a booklet including two sheets with single-digit multiplications in two columns, with 15 problems in each column, for a total of 60 problems. Ties problems (e.g., 3 x 3), smaller-operand first (e.g., 3 x 7), and larger-operand first problems (e.g., 7 x 3) were included in the set. Problems were semi-randomly ordered with the aim of avoiding the consecutive appearance of the same problem with a different order. It was explained to the children that they had two minutes to solve as many problems as they could. Instructions stressed that problems should be performed in columns and that no problems could be skipped. The score was calculated as the number of problems correctly solved minus those solved incorrectly.

First, we note that both methods described here emphasize the practice of fact retrieval as a key active component, and both use the same textbook (^{nd} grade. Connect with Pupi], SM Editorial. However, whereas in the M&R method this was used as complementary material (it was mainly used as an activities database), in the conventional method it was the main guide for learning. This difference involved that rules, including the commutative principle, and time-tables, were presented in a different temporal sequence and with different purposes in each method (see below for more details on this). In both cases the children practiced retrieval with the exercises in the textbook and with additional work-sheets, and these materials were always adjusted to the lesson in question. The textbook includes sections which explain the concept of multiplication as repeated addition, then the tables, and finally the commutative rule is briefly explained. The main differences between methods are described in the following sections.

The basis of this method is memorizing the tables from 1 x 1 to 10 x 10 through the use of rote verbal learning and repeated practice with problems. Although during the learning process some conceptual knowledge is presented, and it is usually pointed out that some tables can be learned by rules, the time-tables with all the problems are ultimately learned by memorization.

Three phases were included in this method. During the first phase the concept of multiplication was explained to children using verbal and visual examples. They were also asked to convert repeated additions into multiplications (e.g., 3 + 3 + 3 + 3 + 3 = 5 x 3). In the second phase the rote verbal learning of tables from 1 x 1 to 10 x 10 was stressed by reciting the sequences in the classroom and by practicing problems. Following the textbook, the 2 and 4 time-tables were learned first, then 1 and 10, followed by 5, and finally 3, 6, 7, 8, 9, and 0. For the learning of each time-table, the procedure in the classroom was as follows: initially the table was presented and was recited by all children several times. They were then provided with time to recite the table themselves, to practice problems, and to study it at home. During the following schooldays they were asked the table in full and also in individual problems presented verbally or in a booklet. After a variable delay, according to what the teacher considered appropriate, the learning of a new table was presented. Finally, in the third phase, once all the tables had been studied, the children were presented periodically with single-digit problems to be solved as part of math class activities, and then the commutative rule is explained.

With some minor variations, this is the method followed by the majority of schools in Spain

This method combines the use of memory learning and single-step rules with the aim of learning to solve single-digit multiplications. It shares with the conventional method the fact that learning by memory provides an efficient way of attaining fluency but emphasizes conceptual understanding and promotes the use of single-step rules in order to reduce the number of problems to memorize, and then the interference associated with this process. Additionally, the method complements the memory learning process by cueing multiplication tables with color in a portable time-table. In this way it seeks to reinforce the association between problems and their solutions (the materials of the method, in Spanish, can be downloaded from www.ladiscalculia.es).

The method is based on educational strategies applied in certain countries (e.g., China) and in response to suggestions in previous research (e.g.,

In the M&R method, rules were used to learn the 0, 1, and 10 tables, since their solutions involve a single-step procedure: a x 0 is always 0, a x 1 is a, and a x 10 is a0 ( ‘a’ being any natural number). Additionally, this method used another single-step rule, the commutativity principle: a x b = b x a (a and b being any natural numbers, and a ≤ b), so that only problems in one direction (larger x smaller or smaller x larger) needed to be learned, in this case a x b problems. Instruction was also provided for children, so when faced with problems of the type b x a, they were asked to apply a change of order. By using these single-step rules the method reduces the number of problems to be memorized from the original 100 problems (from 1 x 1 to 10 x 10), to 36 (see

After the application of single-steps rules, children needed to learn a small subset of problems by memory. To help in this process we employed two strategies:

a) A portable time-table was provided for the children (see

b) Problems in the portable time-table were cued with colors. The aim of this was to provide implicit information that is stored in memory and can help children to organize a network of problems. According to some models, during the learning process a network of operator and result nodes is built (e.g.,

The application of the M&R method had the following phases: first, children were presented with the concept of multiplication and its understanding was encouraged. As part of this understanding, multiplications by 0 and by 1 were presented. Subsequently, the commutative property was introduced. The last rule presented was the multiplication by 10 rule. Each of these steps was accompanied by the presentation of the time-table and the consequence for memory learning of using this rule. For instance, after understanding the rule for multiplication by 1, these problems were shaded in the portable time-table. After learning the rules of the multiplication tables, the table with shaded problems was presented and children were encouraged to learn by memory those 36 problems that were not shaded, starting with the 2 time-table and moving successively to the 9 time-table. Emphasis was placed on starting each table with the ties (e.g., 2 x 2; 3 x 3, etc.) and using the commutative principle to retrieve the smaller x larger problems (e.g., 2 x 3).

To sum up, the M&R method, as with the conventional method, emphasizes memory in learning the association between problems and solutions. However, it does so on a reduced subset of problems and uses color as an additional cue for recall, whereas for the rest of the problems the method relies on single-step rules. An additional difference between the methods is related to the accessibility of the tables. The portable time-table makes it easier to consult the solution when doubts arise and also makes studying it easier. Moreover, its use as a novel resource probably has the effect of showing children the relevance of learning the tables, and the potential motivational role of the portable time-table as a didactic tool cannot be dismissed.

Descriptive statistics were conducted separately for the conventional group and the M&R group. The exploratory analysis indicated that distributions departed from normality, and that homoscedasticity was not attained, so non-parametric statistics were preferred for data analyses.

Non-parametric correlations between mathematical skills and multiplication fluency were developed separately for each group; we also compared the mathematical skills and the multiplication fluency between groups.

A moderation analysis was run to evaluate our hypotheses. This analysis is an alternative to ANOVA for exploring the interaction between the method factor and the level of mathematical skills in predicting multiplication fluency. The moderation analysis was conducted with the Process module by

_{1}quantifies the effect of X (Curricular method) on Y (Multiplication fluency), b

_{2}quantifies the effect of M (Mathematical skills) on Y (Multiplication fluency), and b

_{3}quantifies whether the effect of X (Curricular method) on Y is moderated by M (Mathematical skills).

Descriptive statistics for the dependent variable and the moderator for each method group are shown in

Overall, the correlation between multiplication fluency and the moderator, mathematical skills, was significant rho = .42 (

^{2} = .25. No effects of type of curricular method on multiplication fluency (b_{1} = -0.196, _{2} = 0.259, _{3} = -0.229,

Using a Pick-a-Point approach, the interaction was analyzed by testing the effects of type of method at three levels of mathematical skills: one standard deviation below the mean, at the mean and one standard deviation above the mean (e.g.,

A deeper analysis of this interaction with the Johnson-Neyman technique, that is, without categorizing a priori children in terms of mathematical skill levels, showed a difference between methods in favor of the conventional group for those participants with values of mathematical skills lower than -19.60 (15.6% of the sample). On the other hand, for participants with values of mathematical skills higher than +16.47 (22.5% of the sample), those in the M&R method showed higher multiplication fluency than those in the conventional method. No differences were found between methods for children with values of mathematical skills in the range -19.60 to 16.47 (see

The present study has sought to compare the effectiveness of two curricular methods for learning multiplications. The M&R method, based on evidence and principles of cognitive and educational psychology, combines the use of memory and rules, the other, a conventional method, uses memory-based learning. An additional question was the role of individual differences in mathematical skills in terms of benefitting from these methods. Each method was followed in four classrooms over a six-month period as part of the math curriculum of second grade students. At the end of the school year, children were evaluated. Results indicate that the relationship between method and multiplication fluency was moderated by mathematical skills: a) among children with strong mathematical skills, those following the M&R method had achieved greater multiplication fluency; b) no difference was seen in children with intermediate levels of mathematical skills; and c) among children with poor mathematical skills, those following the conventional method scored better. In what follows we will discuss these results.

First, the analysis indicates simple effects of mathematical skills on multiplication fluency and no effects of method, although these results were qualified by an interaction between both variables, i.e., moderation. This moderation analysis showed the relevance of individual differences in terms of the advantages of one instructional method or the other, the relative benefits of the two methods being modulated by children’s mathematical skills. Against our predictions, children with poor mathematical skills attained greater fluency with the conventional method, whereas children with strong mathematical skills progressed more with the M&R method. In understanding these results, it may be useful to bear in mind that the M&R makes more demands on numerical and arithmetical understanding than the conventional method, which relies more on the memorization of the tables. It is perhaps not surprising, then, that students with strong mathematical skills benefitted more from a method based on conceptual understanding of rules and on memory than from one based almost exclusively on memory. Their knowledge of mathematics facilitated not only the understanding of the concepts and the single-step rules on which the method is based, but also of how to use these rules. Additionally, the time saved in learning problems through the use of rules can be employed in learning by memorization the smaller subset of multiplication facts that cannot be learned by rules. Our results here coincide with other research in the area of multiplication fluency intervention, which has found greater benefits for interventions based on understanding than for those simply based on memory. For instance,

One of the aims of the present study was to compare the relative effectiveness of the conventional and the M&R methods. There is no single finding here since, as we have noted, benefits depend on children’s mathematical skills. The M&R method was designed in light of earlier studies (e.g.,

A direct implication of our results is that more time should be provided for understanding the rules when using the M&R method with children who have medium and poor mathematical skills. The worst observed effectiveness of this method with children with poor mathematical skills does not seem to be a consequence of incorrect design, but of incorrect implementation. A minimum level of mathematical skills is required in order to understand the rules, and indeed is a pre-requisite for running the method. The lack of specific mathematical skills may lead to a failure in understanding and in assuming the relevance of the rules and the logic behind the use of each strategy. So, when working with children with poor mathematical skills, the focus should be on increasing their mathematical knowledge prior to implementing methods that are based more on understanding than simply on memorization.

An important implication of our findings is the need to individualize instruction for students who present different mathematical skill levels, as suggested in previous studies (e.g.,

Certain limitations of this study might usefully be pointed out. First, as noted above, participants’ mathematical skill level used in the analysis was taken at the same time as the multiplication fluency task. Although it is expected that the measure of mathematical skills at the end of the application of the method would show a high correlation with the level of mathematical skill before starting the multiplication methods, a previous measure would allow us to establish a more causal relationship between mathematical skills and the effectiveness of the methods. In fact, the measure of mathematical skills in the M&R group taken one year before, that is, at the end of their first grade, showed a high correlation with the mathematical skills taken at the end of the application of the M&R method, rho =.516 (

Additionally, it should be noted that since the M&R method is based on different psychological principles, our study cannot disentangle the relative role of each of these in the success of the intervention. As indicated in the Introduction, there are good reasons to expect that both reducing the number of facts to memorize and using color cues to code the problems can help students in encoding and retrieving the problems more effectively in their long-term-memory (e.g.,

Finally, although the groups involved in the methods all belonged to the same school, it may be that the presence of different teachers and the non-random assignment of the students to the methods might have introduced some differences, and thus that students on the different methods may have differed in terms of mathematical knowledge, multiplication skills, or even in cognitive processes relevant for this learning (e.g., executive functions, working memory, phonology). Although this cannot be discounted, it should be borne in mind that no between-group differences were found in the measures of mathematical skills (see

Several other issues in our study might also be highlighted with regard to previous work in the area. First, we implemented and compared teaching within curricular methods, not interventions. So, the methods were implemented as part of math lessons given by the children’s math teachers during the school term. In this sense our study is more ecological and might serve to guide teachers’ daily school activity (

The present study has shown the differential benefits of two different methods of learning multiplication. A conventional memory-based method seems to be more efficient for children with poor mathematic skills, whereas a method based on understanding and using rules, and which limits the number of facts to be learned, seems more effective for children with stronger mathematical skills. Although these results should be taken with caution given the limitations of the current study, it seems that taking individual differences in mathematical skills into account appears to be a fundamental prerequisite of instructional methods as applied in the classroom for the learning of multiplication.

Math skills were collected at the start and at the end of the program for both groups. Unfortunately, an error in data coding led to those for the conventional group being unavailable, so in the analysis we used those data taken once the curricular methods were finished, what can be considered a proxy of earlier math skills.

Cite this article as: García-Orza, J. , Álvarez-Montesinos, J. A. , Luque, M. L. , & Matas, A. (2021). The moderating role of mathematical skill level when using curricular methods to learn multiplication tables.

^{rd}Ed.) (pp. 257-268). Routledge.