Are qualitative and quantitative methodologies irreconcilable? This paper firstly explores the difference between the two methodologies, followed by arguments that view ”research strategies” as their common basis, and that they are reconcilable by in-depth delving cyclically. The second section of this paper posits a model of mixed methodologies, which illustrates the complementary possibilities of quantitative and qualitative methodologies, with examples from a study on children's emotional responses to mathematical problem- solving. The model describes methods to establish research focuses and questions, design research tools, collect data, analyze data, and write research results and discussions in mixed methodologies. The data analysis methods include asking questions, interrogating data, quantifying qualitative data, qualifying quantitative data, and making links. The final section discusses implications and suggestions of using mixed methodologies in the research of education and educational psychology.