This study fulfils one cycle of design research that seeks to build a didactical model for the instruction of prime numbers and related mathematical ideas in accordance with the theoretical principles of Realistic Mathematics Education. Taking the context of online information security as a realistic starting point, the Primes in Context using Technology (PiC-T) instructional sequence was developed that supports learning trajectories connecting a series of fundamental mathematical ideas with the support of technological tools. In a two-week classroom teaching experiment, a PiC-T hypothetical learning trajectory was implemented using a mobile computer station with a class of 17 high-achieving middle school students in their classroom.
Data sources include interviews, student-generated artifacts, field notes, video-taped classroom sessions, student online activities, and daily debriefings. Data analysis under the emergent perspective on classroom research led to five themes that ran across the teaching experiment. First, students reported that they were experiencing meaningful mathematics that was useful, connected, and centered around problem solving. Second, students felt that they were well supported by contextual models, technology, and whole class interactions. Third, students felt that they were constantly distracted by a variety of factors in the learning environment, including the problem contexts, the computers, and fellow students. Fourth, studentsí experience along the PiC-T trajectory was deeply intertwined with their personal beliefs about the use of technology, the context, mathematics, and learning mathematics. Lastly, studentsí experience was under the constant influence of the social structure of the classroom community. While meaningful mathematics learning remained a recurring theme throughout the data set, other themes coexisted with the emergence of taken-as-shared mathematical conceptions.
Mathematically speaking, the teaching experiment led to the initial establishment of four major mathematical practices among the participants. First, factoring was recognized as a complex mathematical procedure beyond routine number manipulations. Second, relatively prime numbers were gradually established as numbers that do not share common factors. Third, modular arithmetic was approached from the perspective of remainders. Fourth, the unit group, which integrates prime numbers, relatively prime numbers, and modular arithmetic, was investigated as a mathematical structure full of puzzling and interesting patterns.
Taking this study as a paradigmatic case, future research efforts should explore the value of the PiC-T sequence with other student populations, including preservice and inservice mathematics teachers, and further consider its theoretical implication for instructional design involving other fundamental ideas of mathematics within the theory of Realistic Mathematics Education, especially when technology is an integral part of instruction. Throughout the text, all participantsí names are pseudonyms that are intentionally chosen to reflect their ethnic backgrounds.