Problem-Driven Teaching: Estimating the population from a sample

Problems are at the heart of mathematics and statistics. The posing of mathematical and statistical problems is more important than the solving of problems. Today, mathematics teachers are realizing that there are many benefits of problem-driven teaching, but they also face a number of challenges, such as a lack of confidence, not knowing how to design high-quality problems, and how to deal with the challenges that emerge in problem-driven mathematics and statistics classes. To answer these questions, we propose a model for problem-based instructional design which concludes with three stages: The stage of preparation of the problems (generating new problems based on textbooks; generating new problems based on mathematical, scientific, and life situations, imagining solutions of the prepared problems); The stage of implementation of teaching (teachers teach based on prepared problems and pose new problems in real-time, and students solve them or pose new problems), evaluation and look-back stage (evaluating the quality of teaching and the quality of problems, improving the instruction). Accordingly, we select the topic of 'Estimating the whole from a sample' in the 'General Senior Secondary Textbook, Mathematics, Compulsory Book 2 (People's Education Press, 2019)” to design a problem-driven instruction and as an application of problem posing. In the instruction, we will pose several interesting problems related to the topic. These problems can successfully stimulate students' interest in learning statistics. Through these problems, the maths teacher can guide the students to think actively, and pose and solve the problem themselves, thus understanding the related concepts and methods in statistics. In addition, the design model applied here is a ready-to-use tool for teachers in designing specifically problem-driven instruction and also can improve teachers' problem-posing skills.