´Now I Get It!´ : Developing a Real-World Design Solution for Understanding Equation-Solving Concepts

Abstract

Strong conceptual understanding contributes to mathematics learning. Manipulatives (i.e., hands-on learning tools that allow for mathematical concept exploration through different senses) can facilitate students' understanding of mathematical concepts when used meaningfully. However, a body of research has demonstrated that although teachers have considered manipulatives pedagogically beneficial, when it comes to everyday classroom practice, they often prefer traditional teacher-centred and paper-and-pencil instruction.

This doctoral research aims to develop a manipulative and its appropriate use to promote not only students' understanding of mathematical concepts, but also classroom adoption. Solving one-variable linear equations in primary school classrooms was used as a case study. An educational design research (EDR) approach was used throughout three phases of a 6-year enquiry: initial research, concept development, and design development. Phase 1 (initial research) was undertaken to gain a theoretical and contextual understanding and investigate existing manipulatives. In Phase 2 (concept development), four manipulative concepts were generated based on the Phase 1 findings. Each concept was then evaluated in terms of its pedagogical benefits and compatibility with school and classroom practice. During Phase 3 (design development), informed by the Phase 2 findings, a design solution (i.e., a tangible manipulative allowing physical input and providing digital output, student worksheets, teacher guides, and class activities) was developed. The developed design solution was then implemented and evaluated in classrooms.

Empirical research was conducted in Finnish comprehensive schools. Altogether, 18 teachers, 98 primary school students, and 65 lower secondary school students took part in different phases of the research. The data were collected using mixed methods, including class interventions, paper-based tests, thinking aloud, questionnaires, and interviews. Qualitative and quantitative data collected from various methods and data sources were simultaneously analysed and then compared and combined to holistically understand the research results.

Together, multiple iterations (of investigation, design, and assessment) resulted in practical and theoretical outcomes. The research-based design solution, which promotes students' understanding of equation-solving concepts and classroom practice, is the practical outcome of this research to directly improve educational practice. Additionally, the research contributes to three types of theories: domain theories, design frameworks, and design methodologies.

The first theoretical outcome is a <i>domain theory</i> yielding two types of knowledge, that is, context and outcomes knowledge. The <i>context</i> knowledge describes the challenges and opportunities of using manipulatives in mathematics classrooms, as well as strengths and limitations of existing manipulatives. The <i>outcomes</i> knowledge describes outcomes of implementing the design solution: the developed tangible manipulative accompanied by the instructional materials enhanced students' understanding of equation-solving concepts through discovery learning, social interaction, and multimodal expression of mathematical thinking; the manipulative is likely to be adopted in the classroom because of its pedagogical benefits and compatibility with school and classroom practice. The second theoretical outcome is a <i>design Jramework</i> for real-world educational technologies. Content, pedagogy, practice, and technology should be taken into consideration when designing real- world educational technologies to ensure their educational benefits, utilisation, adoption, and feasibility. The third theoretical outcome is a <i>design methodo/ogy</i> built on firsthand experience from undertaking this EDR. The guidelines for conducting EDR guides how to embrace opportunities and overcome challenges that may emerge.

This research contributes to a link between research and practice in mathematics education. It provides researchers with knowledge of how multimodal interaction with manipulatives enhances mathematics learning and guidelines for conducting EDR. It guides educational designers to take various aspects into consideration when designing educational technologies to improve real-world practice. Moreover, this research also has practical implications. First, it encourages teacher educators to prepare pre- and in-service teachers for successful incorporation of manipulatives in their mathematics classrooms. Second, it guides practitioners on how to support their students to benefit from manipulatives. Third, it urges schools to support the acquisition and utilisation of manipulatives. Finally, it calls on school curricula to encourage the use of manipulatives in the mathematics classroom to promote students' conceptual understanding.