| dc.description.abstract | This study addresses procedural learning gaps by repositioning practices from promoting procedural skill automatization to developing deep procedural understanding, improving the effectiveness of strategy selection. Productive practice is introduced as an effective type for nurturing deep procedural understanding. This study explores its potential by conducting the addition and subtraction productive practices on GeoGebra Classroom.
Deep procedural learning involves cognitive understanding of computational processes and flexible strategy selection. Claiming achievement is challenging since this cannot be simply defined by success rates or a single choice of strategy. It should be examined by delving into the mathematical thinking process and exploring the shifts of students’ strategy selection. Considering mathematical thinking as an important generative mechanism, this study adopts the view of critical realism to explore explanatory causality in the learning process and aims to understand the real structure beneath observable experiences and outcomes.
The study employed a design-based research approach across three iterative cycles. Both cycles involved participants from Hong Kong. Due to the COVID-19 pandemic, cycle 1 and cycle 2 were conducted in the form of online lessons via video conferencing, while cycle 3 was conducted in actual classrooms. Cycle 1 tested the initial design with five voluntary Grade 2 students. Cycle 2 included six Grade 2 students in School A, and the revised task design was implemented over 4 lessons with their own teacher serving as a facilitator. Cycle 3 involved 133 Grade 1 students in School B, and the similar design from cycle 2 was delivered without teacher’s guiding instruction. Data were collected according to the specific objectives of each cycle. In cycle 1, video recordings were made during lessons. In cycle 2, there were pre-test and post-test performances, video recordings during lessons, screen recordings and quantitative data from individual work on the GeoGebra platform, and video recordings of individual student interviews. In cycle 3, quantitative data from pre-test and post-test were collected.
The qualitative findings revealed that productive practices initiated mathematical thinking process and nurtured deep procedural understanding, with observed variations in mathematical thinking trajectories depending on how the causal powers intertwined. Internal causal powers (e.g., conjecturing styles, process of stabilising, seeking for transferability) were derived from students and directly shaped mathematical thinking trajectories. For instance, a student who adopted the inductive way of conjecturing tended to gather evidence from the empirical results, while those who thought deductively worked more on the initial structure of the context. Nonetheless, their ways of conjecturing were not unchanged, instead, they switched to a new conjecturing approach with different reasons. External causal powers (task design and digital platform features) indirectly influenced mathematical thinking by impacting students’ internal causal powers. For example, the features of instant feedback provided by GeoGebra Classroom allowed diverse ways of approaching the productive practice tasks. Students might take this advantage to exercise greater learning autonomy and shift their approach throughout the mathematical thinking process.
The quantitative findings from the pre-test and post-test showed a significant improvement in both overall scores and response time. The analysis of different categories of test items indicated a more frequent application of inverse relationship and related arithmetic laws to the test items. This is not only evidence of the positive impact on mechanical arithmetic skills (superficial procedural understanding), but also indicates the transferability of applying flexible strategies (deep procedural understanding) from productive practice contexts to routine arithmetic equations.
This study presents a holistic view of deep procedural learning with the designed learning environment of productive practices on GeoGebra Classroom, offering practical contributions to the instructional strategies through 1) constructing design principles for productive practices, 2) demonstrating the contribution of technology, and 3) acknowledging the potentials of various causal powers. Also, by identifying and explaining the interplay of causal powers within the mathematical thinking process, it offers a better overview of progress in deep procedural learning. From the view of theoretical implication, it provides a comprehensive way to explore deep procedural learning by introducing an integrated framework of mathematical thinking and instrumented activity situations model. | en_GB |
