Problem-Driven Teaching: Estimating the Population from a Sample ^†

Abstract

Today, mathematics teachers are realizing that there are many benefits of problem-driven teaching, but also face a number of challenges, such as a lack of confidence, not knowing how to design high-quality problems, and not knowing how to deal with the challenges that emerge in problem-driven lessons and statistics classes. To address these challenges, we propose a model for problem-driven teaching which includes three stages: the preparation of the problems; the implementation of teaching; and the evaluation and reflection. Accordingly, we select the topic of ’Estimating the population from a sample’ in the General Senior Secondary Textbook, Mathematics, Compulsory Book 2, to design problem-driven instruction and as an application of problem posing. In the instruction, we pose several interesting problems related to the topic. These problems can successfully stimulate students’ interest in learning statistics. Through these problems, the mathematics teacher can guide the students to think actively and to 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 can also improve teachers’ problem-posing skills.

1. Introduction

Problems are at the heart of mathematics and statistics. The 23 most influential mathematical problems posed by David Hilbert inspired a great deal of progress in mathematics science [1]. Einstein was convinced that posing new questions and exploring new possibilities for old problems from new angles need creative imagination and lead to real advances in science [2]. In 1980, Paul Halmos wrote “I do believe that problems are the heart of mathematics”. Both Cantor and Klamkin strongly believed that posing a problem is as important or more valuable than solving it [3]. Therefore, posing mathematical and statistical problems is more important than solving problems.

This paper is organized as follows: In Section 2, we give the definition of problem posing, and briefly introduce the related research. In Section 3, we provide the model of problem-driven teaching, then give an application of the model in Section 4. Finally, we provide the discussion in Section 5, and the conclusions in Section 6.

2. Literature Review

2.1. Definition

Before the 1980s, mathematical problem posing was not considered an independent activity but was seen as a part of problem solving. Since the 1990s, problem posing has begun to be viewed as an independent mathematical activity. Silver [4] gave a classical definition of problem posing as the generation of new problems based on a given situation or the reformulation of given problems, which is generally accepted by the mathematics education community. Another definition of problem posing refers to several related types of activities that engage teachers and students in formulating (or reformulating) and expressing a problem according to a given situation [5]. The former definition focuses on the source of posing new problems; the latter definition tends to present problem posing as a teaching or learning activity.

2.2. Related Research

Mathematical problem posing is positioned differently in different studies, such as focusing on problem posing as a cognitive activity, as a learning goal, or as an instructional approach [6].

Many national curriculum standards for mathematics include problem posing as a teaching goal. For instance, in China, the mathematics curriculum standards for senior high school advocate improving students’ ability to discover problems, pose problems, analyze problems, and solve problems [7]. We already know that the curriculum standards place great emphasis on the importance of problem posing; that there is much research showing the positive effects of mathematical problem posing on students’ interest in mathematics and creativity in mathematics, as well as on self-efficacy and problem solving; and that there are many specific strategies or techniques for helping students to pose (better) mathematical problems, such as the strategies of “what-if-not” [8], chaining, systematic variation, and symmetry [9]. There are also studies on the process, phases, or stages of problem posing, such as Cruz’s model of process (educative needs, goal, problem formulation, problem solving, and problem improving) [10]; the five phases (setup, transformation, formulation, evaluation, and final assessment) [11]; the five types of activities (situation analysis, variation, generation, problem solving, and evaluation) [12]; and, most recently, Cai’s problem-posing task-based instructional model [13]. However, there is still a need to unfold the following directions: the design of problem-posing tasks in mathematics textbooks; teachers’ design of problem-posing-based lessons; teachers’ implementation of problem posing in mathematics classrooms; and the assessment of problem posing in teachers’ teaching and students’ learning, teachers’ problem-posing beliefs and performance, students’ problem-posing performance and influencing factors, and how problem posing promotes mathematics.

3. The Model of Problem-Driven Teaching (PDT)

Today, mathematics teachers are progressively realizing that there are many benefits of PDT, but they also face many challenges, such as a lack of confidence, not knowing how to design high-quality problems, and not knowing how to deal with the challenges that emerge in problem-driven mathematics and statistics classes. Therefore, to answer these open questions, after dissecting the models (stages, phases, and processes) in the existing literature, combined with a delicate diagnosis of the first author’s teaching experience, we aim to propose a ready-to-use model of PDT which includes three stages:

• The preparation of the problems (generating new problems based on textbooks; generating new problems based on mathematical, scientific, and life situations; and imagining the solutions of the prepared problems);
• The implementation of teaching (teachers teaching based on prepared problems and posing new problems in real time, and students solving them or posing new problems);
• The evaluation and reflection stage (evaluating the quality of teaching and the quality of problems, improving the instruction).

We divided the problems into two categories: mathematical and non-mathematical problems, such as heuristic problems and prompts.

4. Application

Accordingly, we selected the topic of “Estimating the population from a sample” in the General Senior Secondary Textbook, Mathematics, Compulsory Book 2 (People’s Education Press, 2019) [14] to design a lesson/instruction that fulfills problems based on the model of PDT. This topic includes studying how to design a frequency distribution table and frequency distribution histogram using sample survey data and how to compute the average and percentile of the sample, therefore further going on to estimate the numerical characteristics of the population.

4.1. The Routine Situation

The textbook [14] proposed the following problem situation:

To save water, a city plans to implement a tiered water pricing system for residential domestic water costs, i.e., to determine a standard α for the average monthly water consumption of residents, specifically, the portion of water consumption up to α will be charged at a flat rate and the portion above α will be charged at a negotiated rate. What do you need to do to try to determine a more reasonable standard so that the majority of residential customers are not affected by their water fees? Because of the large number of residential customers in the city, a sample survey is generally used to estimate the distribution of average monthly water consumption of residential customers in the city by analyzing sample observations. Therefore, the overall population is all residential customers in the city, the individual is each residential customer and the variable surveyed is the average monthly water consumption of residential customers. By a random sample survey, a data table of the average monthly water consumption of 100 residential customers was obtained, see

Table A1. A random sample data table of the average monthly water consumption of 100 residential customers (tons) (cf. [14], pp. 192–193).

9.0	13.6	14.9	5.9	4.0	7.1	6.4	5.4	19.4	2.0
2.2	8. 6	13.8	5.4	10.2	4.9	6.8	14.0	2.0	10.5
2.1	5.7	5.1	16.8	6.0	11.1	1.3	11.2	7.7	4.9
2.3	10.0	16.7	12.0	12.4	7.8	5.2	13.6	2.6	22.4
3.6	7.1	8.8	25.6	3.2	18.3	5.1	2.0	3.0	12.0
22.2	10.8	5.5	2.0	24.3	9.9	3.6	5.6	4.4	7.9
5.1	24.5	6.4	7.5	4.7	20.5	5.5	15.7	2.6	5.7
5.5	6.0	16.0	2.4	9.5	3.7	17.0	3.8	4.1	2.3
5.3	7.8	8.1	4.3	13.3	6.8	1.3	7.0	4.9	1.8
7.1	28.0	10.2	13.8	17.9	10.1	5.5	4.6	3.2	21.6

.

4.2. The First Stage

In the stage of preparing problems, the teacher needs to understand the given situation, identify information and discover or create problems about quantity, space, or logic that may exist, and generate new problems based on the given situation. The teacher gives different possible mathematical, scientific and life situations, generates new problems based on these situations, and imagines solutions to the prepared problems.

4.3. The Second Stage

In the stage of the implementation of teaching, this should take place in the classroom. According to the presetting in the first stage, teachers should teach based on prepared problems and pose new problems in real time, and students should solve them or pose new problems based on the problem situation envisaged by the students themselves. The teacher may give appropriate prompts for the problem-posing task or give good examples of problems so that students can engage in problem posing and formulate their problems correctly and positively. Here, the problems posed by teachers and students are not linear and can be disordered and crossed. To enable students to pose effective problems, teachers should introduce some strategies such as “what-if-not”, chaining, systematic variation, symmetry, analogies, associations, considering the counter-problems, extensions, reductions, specializations (considering special cases), generalization, varying the givens, and imitation. To record the lessons, information technology such as video and audio recordings can be used. Students are also asked to fill in the notebook with their posed problems, which the teacher can collect for analysis and evaluation, and assess after the bell rings.

4.4. The Third Stage

There are three different aspects to the evaluation and reflection stage: (1) the evaluation of the teacher’s problem posing; (2) the evaluation of the student’s problem posing; and (3) the evaluation of the quality of the teacher’s teaching. The evaluation of the quality of the posed problem considered the fluency, flexibility, and profundity [15].

4.5. Possible Problems

We demonstrate the possible problems that teachers and students can suggest.

• If the $\alpha$ is too large or too small, what are the effects? Why?
• There are many households; how do you survey them?
• What questions can be asked about the data (Table A1)? (Teachers introduce the frequency distribution table and the frequency distribution histogram.)
• The teacher introduces the concept of range and then poses the problem:
  If you modify (add or delete) some of the data in the table, but do not change the maximum and minimum values, find the range.
• What should be a reasonable number of groups? Can the group size be very large or very small?
• Why is the interval closed on the left and open on the right in

  Table A2. The table of Frequency distribution (cf. [14], p. 194).

  Groups	Count	Frequency	Frequency/Group Size
  $[1.2,4.2)$	23	0.23	0.077
  $[4.2,7.2)$	32	0.32	0.107
  $[7.2,10.2)$	13	0.13	0.043
  $[10.2,13.2)$	9	0.09	0.030
  $[13.2,16.2)$	9	0.09	0.030
  $[16.2,19.2)$	5	0.05	0.017
  $[19.2,22.2)$	3	0.03	0.010
  $[22.2,25.2)$	4	0.04	0.013
  $[25.2,28.2]$	2	0.02	0.007
  Total	100	1

  , except for the last one?
• Why is it necessary to group them equidistantly? What happens if not?
• Does changing just some of the data change the number of groups?
• What kinds of problems can be posed when looking at Table A2? For instance, why does the sum of frequencies equal 1? Even if we modify (add or delete) some data, how about the sum?
• What kinds of problems can be posed when looking at Figure A1? For instance, calculating the sum of the areas of all the rectangles.
• If some of the data have been modified (added or deleted), calculate the sum of the areas of all the rectangles.
• Why is it a dromedary? Why are the rectangles on the right side very short?
• Calculate the plurality of the sample and compare it with Table A1.
• Calculate the median of the sample.
• Calculate the quartiles of the sample.
• Calculate the three-quarters of the sample.
• Calculate the mean of the sample.
• If the data are divided into 10 equally spaced groups, design the frequency distribution table and the histogram of the requency distribution, and find the plural, median, and mean.
• Design new problem situations and present the corresponding data and problems.
• ...

5. Discussion

More investigation is needed to examine whether students’ mathematics performance, ability to pose problems, interest in learning mathematics, and creativity improve after they experience PDT. Later studies may establish a more scientific and rational system of evaluation indexes. The assessment of teaching quality may take into account students’ attitudes to learning, learning outcomes, and academic performance, which is still being conceptualized.

6. Conclusions

In the lesson/instruction, several interesting problems were posed by teachers or students. These problems can successfully stimulate students’ interest in learning statistics. Through these problems, the mathematics 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 can also improve teachers’ problem-posing skills.