What is 7 x 6?
Multiplication facts like the one above are a cornerstone of the mathematics curriculum around the world. Having a good grasp of basic mathematical skills like multiplication sets learners up for success in more complex mathematical domains. Children in primary education in the Netherlands are expected to achieve fluent mastery of the times tables up to 10 (Noteboom, Aartsen, & Lit, 2017). In this context, mastery is typically described as automation and memorization: learners should demonstrate the ability to solve multiplications quickly and accurately, by efficiently computing the answer or by retrieving it directly from memory. We built the Tafel Trainer application to help learners reach this goal. In this blog post, I will focus on the design of the adaptive algorithm that powers the application.
A complex skill
Learning to solve multiplication problems is a multifaceted process that researchers have been studying for decades. Learners typically start out using so-called computational strategies. For example, they might use repeated addition to solve a problem: 7 x 6 = 6 + 6 + 6 + 6 + 6 + 6 + 6. Such strategies are simple to perform, but they can be quite slow and error-prone. With practice, a learner progressively transitions towards using faster and more accurate solution methods (Lemaire & Siegler, 1995; van der Ven et al., 2012). Learners eventually end up knowing solutions by heart: rather than having to calculate anything, they can directly retrieve answers from memory. In this more advanced stage, learners may also still use computational methods from time to time (eg, when a direct memory retrieval is too difficult), but these will be much more efficient versions of the computations that they performed in earlier stages of learning. For instance, a learner solving 7 x 6 might know 6 x 6 = 36 by heart, and then add 6 to get to the answer. Even though this computation involves an additional step compared to a direct retrieval, it can still be done quite quickly.
Learning multiplication
Teachers and pupils use a variety of methods for multiplication learning, some of which are known to be more effective than others. Methods that rely on passive study, such as having learners look at a multiplication table written down or listen to a recording, often work less well than methods that require the learner to actively process the material, such as practicing with flashcards (Steel & Funnell, 2001; Ophuis-Cox, Catrysse, & Camp, 2023). As we have written about before, even if we know how we should be studying, it can be difficult to bring proven study methods into practice. Computer-based adaptive learning systems can play an important role in this regard. Such systems can guide learners towards using methods that are known to be effective, like spaced retrieval practice with feedback. Just as in other domains, there are clear individual differences: some multiplication problems are more difficult than others, and some learners more skilled at multiplication than others (van der Ven et al., 2015). An adaptive learning system, which can accommodate and adapt to such differences, can help make the learning experience appropriately challenging, but not too difficult, for each individual learner.
Mastering multiplication
We designed the Tafel Trainer application to support pupils in the process of achieving mastery of the multiplication tables. At its core, the system uses the same adaptive spacing algorithm on which the MemoryLab learning system is based. This algorithm uses a computational cognitive model of the learner's memory to adaptively schedule retrieval practice trials, responding to the learner's performance in real-time. The current context required making several alterations to the way in which the algorithm is employed.
Because multiplication problems can be solved successfully through different strategies, and because a learner's use of these strategies is expected to gradually shift over time from primarily computation to primarily retrieval, we divided the learning path for each multiplication table into three stages, or levels. While the basic task remains the same across levels—give the solutions to a sequence of retrieval problems—the learning goals, and consequently the realization of that task, are different in each level.
Fig. 1: Each multiplication table is mastered in three progressively more challenging levels that are designed to encourage a shift towards fluent automation and memorization.
Level 1: solving multiplications
The first level is intended for pupils to demonstrate that they are able to successfully solve each multiplication in the selected multiplication table. At this stage, we assume that pupils have already been introduced to the concept of multiplication by their teacher, but haven't yet had much practice. Multiplications are presented in a fixed order. The level is complete when the student has answered each problem correctly.
Level 2: power
From the second level on, pupils build familiarity with individual multiplication problems through repeated spaced practice. The repetition serves to strengthen the memory representation of each multiplication, increasing the odds that the pupil can directly retrieve the answer from memory when asked. Repeated practice also allows pupils to develop more efficient computational strategies for when direct retrieval fails. Both of these developments should lead to faster, more accurate performance as learners gain experience.
In level 2, we expect learners to achieve consistent accuracy, even if they are not always quick to respond. At this stage, learners are likely using a mix of computation- and retrieval-based response strategies, which means that a response time may not necessarily be indicative of the duration of a memory retrieval. The regular MemoryLab algorithm uses response times to infer the activation of a piece of knowledge in memory: the longer it takes to retrieve the foreign-language translation of a word, for example, the weaker the memory for that word is assumed to be. Here, response times are less straightforwardly interpretable. For that reason, the adaptive algorithm in level 2 is sensitive to the accuracy, but not the speed, of a learner's responses. This means that problems on which a learner makes mistakes will be repeated more frequently, and that slow but correct responses are not penalized for being slow. The level is complete once the cognitive model that runs in the background makes the assessment that all multiplication facts have been mastered to a sufficient level.
Level 3: power and speed
Once a learner reaches level 3, both accuracy and speed become important. Level 3 tests whether a learner has the goal of automation or memorization: they should be able to respond to each problem within a limited time. By introducing time pressure, we encourage direct retrieval (or failing that, quick computation) of answers, since inefficient computational procedures will be too slow. Learners who successfully complete the third level demonstrate that they have achieved mastery, by being able to consistently respond accurately and correctly. In this final level, the adaptive algorithm is sensitive to both accuracy and response time, which means that a learner making a correct but slow response causes the problem to be repeated sooner, similar to when the response is incorrect. As in level 2, completing level 3 requires reaching a sufficient level of mastery on each multiplication fact.
Having completed level 3 does not mean that the learner is completely done, of course. Practice makes perfect! To maintain fluency, learners should revisit multiplication tables that they have previously mastered from time to time.
Summary
Learning multiplication facts usually involves transitioning through different solution strategies, going from primarily slow and error-prone computational methods to fast and accurate direct retrieval. We designed the Tafel Trainer adaptive learning system with the aim of supporting learners on this path. The three levels break the learning task down into achievable chunks. They are designed to encourage a shift towards more efficient strategies, helping learners achieve fluent mastery through adaptive spaced practice.
References
Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124(1), 83–97. https://doi.org/10.1037/0096-3445.124.1.83
Noteboom, A., Aartsen, A., & Lit, S. (2017). Intermediate objectives in arithmetic and mathematics for primary education. Elaborations of arithmetic goals for groups 2 to 8 on the way to target level 1S. SLO. https://www.slo.nl/@4587/tussendoelen-rekenen/
Ophuis‐Cox, FHA, Catrysse, L., & Camp, G. (2023). The effect of retrieval practice on fluently retrieving multiplication facts in an authentic elementary school setting. Applied Cognitive Psychology, acp.4141. https://doi.org/10.1002/acp.4141
Steel, S., & Funnell, E. (2001). Learning Multiplication Facts: A Study of Children Taught by Discovery Methods in England. Journal of Experimental Child Psychology, 79(1), 37–55. https://doi.org/10.1006/jecp.2000.2579
van der Ven, SHG, Boom, J., Kroesbergen, EH, & Leseman, PPM (2012). Microgenetic patterns of children's multiplication learning: Confirming the overlapping waves model by latent growth modeling. Journal of Experimental Child Psychology, 113(1), 1–19. https://doi.org/10.1016/j.jecp.2012.02.001
van der Ven, SHG, Straatemeier, M., Jansen, BRJ, Klinkenberg, S., & van der Maas, HLJ (2015). Learning multiplication: An integrated analysis of the multiplication ability of primary school children and the difficulty of single digit and multidigit multiplication problems. Learning and Individual Differences, 43, 48–62. https://doi.org/10.1016/j.lindif.2015.08.013
