PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.PHÁT TRIỂN KIẾN THỨC VÀ THỰC HÀNH NGHIỆP VỤ CỦA GIÁO VIÊN TOÁN TƯƠNG LAI ĐỂ DẠY HỌC PHẦN THỐNG KÊ.
HUE UNIVERSITY UNIVERSITY OF EDUCATION NGUYEN THI HA PHUONG DEVELOPING THE KNOWLEDGE AND PROFESSIONAL PRACTICE OF PROSPECTIVE MATHEMATICS TEACHERS FOR TEACHING STATISTICS Major: Mathematics Teaching Methodology Code: 9140111 SUMMARY OF PH.D DISSERTATION Hue, 2022 The dissertation was completed at the University of Education, Hue University Supervisors: Assoc Prof Dr Tran Kiem Minh Reviewer : ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… Reviewer : ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… Reviewer : ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… The dissertation is defended at the Hue University Thesis Evaluation Committee Meeting at… o’clock .day month .year The dissertation can be accessed in the Library: The National Library of Vietnam Library of University of Education, Hue University PREFACE Developing knowledge for teachers to teach a certain mathematical content effectively is a significant research direction, especially in teacher training The work of Shulman (1986) was pioneering and greatly influenced the field of teacher knowledge research for teaching After that, many researchers have developed the work of Shulman (1986) to study the types of mathematical knowledge required to teach different topics Among those studies, Ball and her colleagues (Ball, et.al, 2008; Hill et al., 2008) made an essential contribution to this area of research by distinguishing six different types of knowledge that teachers need to teach effectively, called mathematical knowledge for teaching (MKT) Many studies have been based on the MKT model to evaluate and develop the types of knowledge teachers need to teach a particular topic effectively For example, Döhrmann et al (2012) based on the MKT model to develop a model to evaluate the professional competence of mathematics teachers in many countries within the framework of the international assessment program TIMSS Gonzalez (2014) discussed statistical knowledge to teaching by using the MKT model Wilkie and Clark (2015) designed mathematics tasks based on the MKT model to develop teachers' knowledge about teaching functional thinking In Vietnam, Le (2016) studied the knowledge of prospective mathematics teachers (PMTs) to teach functions Minh & Lien (2018) have started researching and applying the MKT model to evaluate and develop the mathematics knowledge of PMTs to teach the application of derivatives in economics However, knowledge for teaching statistics is still an unclear issue (Eichler & Zapata-Cardona, 2016) Teaching statistics has common characteristics in teaching and has its particularities (Cobb & Moore, 1997) Many researchers have focused on describing the knowledge needed to teach statistics to help teachers understand statistics deeply (Garfield & Ben-Zvi, 2009; Gfeller, Niess & Lederman, 1999; Leavy & O'Loughlin, 2006; Groth & Bergner, 2006; Jacobbe, 2008; Santos, 2013; Gonzalez, 2014; Sanchez-Sanchez & Gomez-Blancarte, 2015) Nowadays, in the studies on teaching statistics, there has been little research on teachers' knowledge for teaching statistics to their contextual teaching practice (Cobb and Moore, 1997) In particular, studies on the relationship of knowledge types placed in the context of cooperative teachings, such as lesson study (LS), have been a new approach that has still little exploited Recently, in Viet Nam, although there has been some research related to teaching statistics, almost no analysis has mentioned teachers' knowledge and professional development Besides, in the current general education curriculum, statistical content is only presented sporadically in some classes However, statistics is one of the three primary knowledge circuits included in the 2018 mathematics curriculum This shows that statistics have been focused on the 2018 mathematics curriculum Therefore, it is urgent and necessary to develop teachers' knowledge for teaching statistics The overarching objective of this study is to evaluate and develop knowledge types of PMTs for teaching statistics Specifically, we evaluated the types of content knowledge and pedagogical knowledge of PMTs for teaching statistical histograms, the measures of central tendency from statistical graphs, and the dispersion of the data variability on statistical graphs (detail bar charts and histograms) As a next step, a LS cycle was proposed by us from the improved LS cycle of Lewis (2009) This cycle helped PMTs to develop their mathematics knowledge step by step Finally, based on the obtained research results, we discussed and made recommendations with a scientific basis to innovate training content and development professional skills for PMTs at teacher training institutions in Vietnam In addition to the Introduction, the thesis is divided into chapters: In chapter 1, we have analyzed the studies related to mathematical knowledge for teaching and professional competence of mathematics teachers, especially studies based on the model of Ball et al (2008); reviewed of studies on teachers' knowledge for teaching statistics In addition, we have analyzed the epistemological characteristics and content of the topic of statistics in high school as well as in the mathematics teachers' training program, as a basis for developing and adjusting the MKT by Ball et al (2008) into the context of teaching statistics In chapter 2, we have detailed the model of mathematical knowledge types of Ball et al (2008) Specifically, we have clearly analyzed each knowledge component of the MKT model and mentioned a number of studies on the statistical knowledge model for teaching Next, we introduced LS and some professional development research for teachers based on LS In chapter 3, we presented a brief epistemology and history of statistics and some basic knowledge about bar charts and histograms, the measures of central tendency and variability of data in statistical graphs In addition, we have built sets of indicators to assess the knowledge of teachers to teaching statistical graphs, the measures of central tendency and variability of data in statistical graphs At the end of the chapter, the specific objectives of the study were set out and concretized into research questions The content of chapter details the research method of the thesis Specifically, we have given the context of the experiments; research objectives and methods; experimental procedures and data collection; research tools, a priori analysis of questionnaires and methods of analyzing empirical data Chapter presents the experimental results Here, we have focused on analyzing four types of knowledge: common content knowledge, specialized content knowledge, knowledge of content and student, knowledge of content, and teaching of PMTs in each practice and describe their knowledge progression while participating in LS Chapter 6, from the research results, we have presented in-depth analysis and explanations for the results obtained in chapter to answer the research questions posed In addition, we have discussed the contributions of the research and the applicability of its findings in teacher training Chapter RESEARCH PROBLEM 1.1 Overview of studies on teachers' knowledge for teaching 1.1.1 Shulman’s work 1.1.2 Fennema & Franke’s work 1.1.3 Model of subject knowledge in mathematics 1.1.4 Ball and her colleagues’s work 1.1.5 The role and importance of subject-specific pedagogical content knowledge 1.2 Research overview of teachers' knowledge for teaching statistics 1.3 Statistics in high school mathematics program in Vietnam 1.4 Teachers’ professional competencies studies in Vietnam 1.5 Current high school Mathematical teacher training program 1.6 Research problem Chapter THEORETICAL FRAMEWORK 2.1 Model of mathematical knowledge for teaching Based on Shulman's (1986) classification, Ball et al (2008) developed a theoretical framework for mathematical knowledge for teaching (MKT) to research and evaluation of different types of knowledge that mathematics teachers need to perform effective teaching This model includes two areas of knowledge: subject matter knowledge (SMK) and pedagogical content knowledge (PCK) In each of these areas, the authors divided into three different types of knowledge (Figure 2.1) Figure 2.1 Model of mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008) 2.1.1 Common content knowledge 2.1.2 Specialized content knowledge 2.1.3 Horizon content knowledge 2.1.4 Knowledge of content and student 2.1.5 Knowledge of content and teaching 2.1.6 Knowledge of Content and Curriculum 2.2 Statistical knowledge for teaching 2.3 Lesson study 2.4 Lesson study cycle 2.4.1 Baba's lesson study cycle (2007) 2.4.2 Lewis lesson study cycle (2009) 2.4.3 Murata lesson study cycle (2011) 2.4.4 Proposing the lesson study cycle Through the analysis of some LS cycles, we have proposed a suitable LS cycle for the experimental implementation of this study The purpose of our LS cycle was development mathematics PMTs' knowledge for teaching Each step of the curriculum has contributed to the development of specific types of knowledge For example, in the planning step, teachers have developed CCK, SCK when discussing content lessons, KCS when predicting student responses, or KCT when teachers formulate questions or design teaching activities Figure 2.6 Proposed lesson study cycle The steps of this LS cycle are described in detail as follows: + Identify topics and set common goals: the topics selected may not be entirely within the general curriculum, but maybe extra-curricular topics, application topics, or integrated topics in STEM education Teachers must determine the overall goal when choosing a teaching topic + Planning: The members of the LS team discuss the research lesson, prepare the same lesson plan, design teaching activities, build a system of closed and open questions, predict students' answers + Teaching practice and observation: A teacher in the LS group teaches the lesson The remaining members observe, take notes to comment The collected data focus on the logic, the appropriateness of the lesson plan, the teaching steps and the teacher's pedagogical ability, the teacher's analytical and feedback on the students' answers + Reflection and adjustment: After the lesson, the LS group discusses, gives suggestions, edits the lesson plan, and adjusts the teacher's methods if necessary The reflection step can be reflection after each teaching session or reflection at the end of the research process In the step of reflection and teaching practice, we can repeat it many times This improves and enhances knowledge and professional practice for PMTs During the LS cycle, teachers work together with their peers to learn about the topic, pedagogy, and student thinking They then apply what they have learned to develop a lesson plan The implementation of the lesson plan in the real classroom context makes the case study an effective way to improve the teacher's pedagogical content knowledge However, many researchers have shown that although the process of performing LS seems simple, conducting research effectively is not an easy task (Chokshi & Fernandez, 2004; Yoshida, 2008) For LS effective, teachers need to develop additional skills such as how to research instructional materials, plan lessons, observe and collect data on student learning, reflect on lessons, and participate to discuss the lesson with colleagues The researchers believe that in order for the research to be effective and successful, teachers need to have enough time to study the content, teaching materials and anticipate students' reactions during the research and must perform these exercises more purposeful and explicit (Yoshida & Jackson, 2011) 2.5 Research on mathematical knowledge for teaching through lesson study Chapter MATHEMATICAL KNOWLEDGE FOR TEACHING STATISTICAL 3.1 Brief history and epistemology of the concept of statistics 3.2 Bar charts and histograms 3.2.1 Bar charts 3.2.2 Histograms 3.2.3 Common confusions between bar charts and histograms 3.3 Comparison of the measures of central tendency from statistical graphs 3.4 Teachers' mathematical knowledge for teaching histograms 3.5 Teachers' mathematical knowledge for teaching the measures of central tendency from statistical graphs 3.6 Teachers' mathematical knowledge for teaching variability of data in bar charts and histograms 3.7 Objectives of the study The overall objective of this study is to evaluate and develop the knowledge and professional practice of teachers for teaching statistics The specific objectives of this study are: • Objective 1: Giving the characteristics of PMTs’ MKT for teaching histograms; examines the progression of PMTs' professional knowledge and practice on teaching histograms through conducting LS • Objective 2: Giving the characteristics of PMTs’ MKT for teaching the measures of central tendency (mean, median, mode) from bar charts and histograms; examines the progression of PMTs' professional knowledge and practice on teaching the measures of central tendency through conducting LS • Objective 3: Giving the characteristics of PMTs’ MKT for teaching variability of data in bar charts and histograms; examines the progression of PMTs' professional knowledge and practice on teaching variability of data through conducting LS • Objective 4: Provide scientific and meaningful proposals to innovate training content and develop professional skills for PMTs in Mathematical teacher training institutions in Vietnam today 3.8 Research questions For each research objective, the following research questions were asked: • Question 1: How are the characteristics of the PMTs' mathematical knowledge for teaching histograms? How has PMTs' knowledge and professional practice in teaching histograms progressed through the implementation of the lesson study? • Question 2: How are the characteristics of the PMTs' mathematical knowledge for teaching the measures of central tendency (mean, median, mode) from bar charts and histograms? How has PMTs' knowledge and professional practice in teaching the measures of central tendency (mean, median, mode) from bar charts and histograms progressed through the implementation of the lesson study? • Question 3: How are the characteristics of the PMTs' mathematical knowledge for teaching variability of data in bar charts and histograms? How has PMTs' knowledge and professional practice in teaching variability of data in bar charts and histograms progressed through the implementation of the lesson study? • Question 4: From the research results of the topic, how can we make suggestions to reform the training content and develop the professional skills for PMTs in current teacher training institutions? Chapter RESEARCH METHODS In this chapter, we present the context of the experiments, research tools, participants, data collection and analysis methods of the study 4.1 Context In Vietnam, our research can be considered as a pioneer in applying the MKT model to teaching statistics, and we combine LS to develop knowledge to teach statistics to PMTs The experiments of the study were conducted on PMTs who are studying in the 3rd and 4th years at two universities of education of the University of Da Nang and Hue University We designed experiments on the content of statistics teaching as follows: Table 4.1 Summary table of the number of people participating in the experiments Experimental content The first experiment Characteristics of the PMTs' mathematical knowledge for teaching histograms Time (school year) 2017-2018 University of Education 84 Hue The number University of University of Participants Education Da 24 Nang University Total 128 The second experiment Characteristics of the PMTs' mathematical knowledge for teaching the measures of central tendency (mean, median, mode) in statistics graphs 2018-2019 The third experiment Characteristic s of the PMTs' mathematical knowledge for teaching variability of data in statistics graphs The fourth experiment Applying LS to develop knowledge to teach histograms, the measures of central tendency in statistical graphs, variability in statistical charts 2019-2020 2019-2020 75 46 69 72 144 118 documents such as lesson plans, versions of lesson plans prepared by LS groups, and final evaluation sheets of PMTs participating in the lesson The collected data focused on examining the impact of PMTs' participation on PMTs' mathematical knowledge for teaching In this study, we used the MKT model of Ball et al., (2008) as the basis for analyzing PMTs' mathematical knowledge for teaching Based on the sets of indicators, the characteristics of knowledge types, and teaching content, we proposed indexes to evaluate PMTs' knowledge of teaching Data analysis was carried out continuously during the experiment We summarized the evaluations of the manifestations PMTs' knowledge types participating in the study in each step of the lesson study In addition, the data were analyzed based on the PMTs' own reflections Questions asked teachers to describe what they had learned, and their own progression of knowledge in discussions or throughout the entire LS cycle The data sources collected have shown a high degree of uniformity Chapter RESEARCH RESULTS In this chapter, we analyze and evaluate the results in experiments 1, 2, and to see the current status of PMTs' knowledge for teaching histograms, the measures of central tendency from statistical graphs, and the variability of data in statistical graphs Then, we analyze the LS steps of the PMTs in experiment to see the progress of their knowledge A number of comments and suggestions are given to us to improve the training of PMTs 5.1 Orientation to analyze research results The results of this study were obtained in all types of knowledge according to the MKT model of Ball et al., (2008) However, we only focus on analyzing types of knowledge including Common Content Knowledge (CCK), Specialized Content Knowledge (SCK) in the field of Subject Matter Knowledge (SMK); and Knowledge of Content and Student (KCS), Knowledge of Content and Teaching (KCT) in the field of Pedagogical Content Knowledge (PCK) The scope of knowledge in our research focuses on the teaching of statistical graphs, the measures of central tendency, and variability of data in statistics graphs 11 5.2 Prospective mathematics teachers’ knowledge for teaching histograms and their professional practice progression 5.2.1 Prospective mathematics teachers’ common content knowledge progression for teaching histogram through lesson study a) Prospective mathematics teachers’ common content knowledge for teaching histograms Common content knowledge (CCK) is evaluated through exercises and 2.1 in drawing histograms in two cases where the data tables have equal and unevenly divided intervals The chart below shows the answer data of exercise and exercise 2.1 about CCK Figure 5.1 Diagram showing data about CCK in experiment From the chart above, it can be seen that PMTs are only familiar with plotting histograms with data tables with equally divided intervals In the exercise, 82.03% of PMTs gave the correct answer Still 1,56% of PMTs misrepresent the data on this chart (see table 5.1) With exercise 2.1, which required drawing histogarms for a data table with unevenly divided data, only 3.13% of PMTs could draw the correct chart to fully represent the data In particular, up to 75% (codes and 0) PMTs did not respond to this task or gave the wrong answer (see table 5.2) This shows that PMTs did not have full knowledge of histograms in particular than in the case of unevenly divided data intervals b) Prospective mathematics teachers’ common content knowledge progression for teaching histogram through lesson study We evaluated the CCK progression of PMTs based on the CCK indexes built in Table 3.1 In the first step of LS, the topic "Bar charts 12 and histograms" was suggested by the researchers Although this study focused on knowledge of histograms but bar charts were also mentioned to distinguish the difference from histograms During the topic discussion and goal setting, PMTs noticed their mistakes in drawing the chart in exercise 2.1 Only PMT H34 drew the correct shape of the histogram because she had read the material outside the curriculum She only knew how to plot histograms but did not fully understand the nature of histograms which was the frequency of the data represented by the area of the columns PMTs in both groups showed gradual development and refinement of content knowledge about bar charts and histograms at the lesson planning step The PMTs in group discussed how to determine the height of the data columns of histograms and what the vertical axis of histograms represents GVTTL D47: Why does the vertical axis of the graph in exercise represent frequency while in exercise 2.1 it does not represent frequency What does the vertical axis of the histogram in lesson 2.1 represent? GVTTL D61: The essence of histograms is that frequencies are represented by the acreage In exercise 1, the data intervals are equally divided, so the height of the data columns is also proportional to the area So, the height of the columns representing the frequency is still correct GVTTL D65: Then, the vertical axis of the chart in exercise 2.1 does not represent anything, it is just a scale of the ratio between data columns Through the exchange observation of group 1, the researcher reconfirmed histogram knowledge for PMTs The researcher noted that: "histograms in the case of unevenly divided data intervals may not need to use a vertical scale, but if a vertical scale is used, it is the density scale of the data." The PMTs in group discussed a lot about the distinction between the two types of bar charts and histograms, the difference between the two types of figures, and what types of data are represented in bar charts and histograms GVTTL H34: Do bar graphs represent composite layer data? How can we distinguish these two types of graphs? 13 GVTTL H09: Bar charts not represent composite layer data, but rather discrete qualitative and quantitative data The composite layer data is represented on histograms GVTTL H17: Bar charts have separate columns and histograms have adjacent columns GVTTL H09: So, when to use bar charts and when to use histograms to represent data? GVTTL H34: Depending on the given data type choose the appropriate chart format Typically, bar charts represent discrete qualitative or quantitative data, and histograms represent data as composite layers data According to the observational data, we found that the questions of the PMTs were clarified After that, PMTs assigned to design lesson plans In versions of the lesson plan, PMTs clearly demonstrated how to plot the bar graph and histogram in the case of equal and unequal composite layers Besides, they designed exercises that demonstrated the nature of the distribution graph The examples and exercises included in the lesson plan in the later versions are highly practical, helping learners to understand the meaning and apply bar graphs and histograms to real problems According to the observational data, we found that the questions of the PMTs were clarified After that, PMTs assigned to design lesson plans In versions of the lesson plan, PMTs have clearly demonstrated how to plot the bar graph and histogram in the case of equal and unequal composite layers Besides, they have designed exercises that demonstrate the nature of the distribution graph The examples and exercises included in the lesson plan in the later versions are highly practical, helping learners to understand the meaning and apply bar graphs and histograms to real problems Figure 5.2 Illustrate PMT D40's response to CCK in the interview form 14 The above analysis showed that CCK knowledge to teach bar charts and histograms of traditional teachers has been enhanced when participating in the discussion and lesson planning process This is also confirmed by the teaching staff through the interview form at the last step of LS 5.2.2 Prospective mathematics teachers’ specialized content knowledge progression for teaching histograms through lesson study a) Prospective mathematics teachers’ specialized content knowledge for teaching histograms b) Prospective mathematics teachers’ specialized content knowledge progression for teaching histogram through lesson study 5.2.3 Prospective mathematics teachers’ knowledge of content and student progression for teaching histograms through lesson study a) Prospective mathematics teachers’ knowledge of content and student for teaching histograms b) Prospective mathematics teachers’ knowledge of content and student progression for teaching histogram through lesson study 5.2.4 Prospective mathematics teachers’ knowledge of content and teaching progression for teaching histograms through lesson study a) Prospective mathematics teachers’ knowledge of content and teaching for teaching histograms b) Prospective mathematics teachers’ knowledge of content and teaching progression for teaching histogram through lesson study 5.3 Prospective mathematics teachers’ knowledge for teaching the measures of central tendency in statistics graphs and their professional practice progression 5.3.1 Prospective mathematics teachers’ common content knowledge progression for teaching the measures of central tendency in statistics graphs through lesson study a) Prospective mathematics teachers’ common content knowledge for teaching the measures of central tendency in statistics graphs b) Prospective mathematics teachers’ common content knowledge progression for teaching the measures of central tendency in statistics graphs through lesson study 15 5.3.2 Prospective mathematics teachers’ specialized content knowledge progression for teaching the measures of central tendency in statistics graphs through lesson study a) Prospective mathematics teachers’ specialized content knowledge for teaching the measures of central tendency in statistics graphs b) Prospective mathematics teachers’ specialized content knowledge progression for teaching the measures of central tendency in statistics graphs through lesson study 5.3.3 Prospective mathematics teachers’ knowledge of content and student progression for teaching the measures of central tendency in statistics graphs through lesson study a) Prospective mathematics teachers’ knowledge of content and student for teaching the measures of central tendency in statistics graphs b) Prospective mathematics teachers’ knowledge of content and student progression for teaching the measures of central tendency in statistics graphs through lesson study 5.3.4 Prospective mathematics teachers’ knowledge of content and teaching progression for teaching the measures of central tendency in statistics graphs through lesson study a) Prospective mathematics teachers’ knowledge of content and teaching for teaching the measures of central tendency in statistics graphs b) Prospective mathematics teachers’ knowledge of content and teaching progression for teaching the measures of central tendency in statistics graphs through lesson study 5.4 Prospective mathematics teachers’ knowledge for teaching variability of data in statistics graphs and their professional practice progression 5.4.1 Prospective mathematics teachers’ common content knowledge progression for teaching variability of data in statistics graphs through lesson study a) Prospective mathematics teachers’ common content knowledge for teaching variability of data in statistics graphs b) Prospective mathematics teachers’ common content knowledge progression for teaching variability of data in statistics graphs through lesson study 16 5.4.2 Prospective mathematics teachers’ specialized content knowledge progression for teaching variability of data in statistics graphs through lesson study a) Prospective mathematics teachers’ specialized content knowledge for teaching variability of data in statistics graphs b) Prospective mathematics teachers’ specialized content knowledge progression for variability of data in statistics graphs through lesson study 5.4.3 Prospective mathematics teachers’ knowledge of content and student progression for teaching variability of data in statistics graphs through lesson study a) Prospective mathematics teachers’ knowledge of content and student for teaching variability of data in statistics graphs b) Prospective mathematics teachers’ knowledge of content and student progression for teaching variability of data in statistics graphs through lesson study 5.4.4 Prospective mathematics teachers’ knowledge of content and teaching progression for teaching variability of data in statistics graphs through lesson study a) Prospective mathematics teachers’ knowledge of content and teaching for teaching variability of data in statistics graphs b) Prospective mathematics teachers’ knowledge of content and teaching progression for teaching variability of data in statistics graphs through lesson study Chương EXPLANATIONS, DISCUSSION AND CONCLUSIONS In this chapter, we focus on discussing research results to answer the research questions posed Then, we state some contributions of the topic and directions for further research of the topic 6.1 The evolution of prospective mathematics teachers’ knowledge for teaching histograms through lesson study We provided an initial survey of PMTs’ knowledge for teaching histograms The results of the study showed that PMTs did not clearly 17 understand the nature of histograms, how to choose the right chart type to represent the data table, were confused between bar charts and histograms Therefore, PMTs made many mistakes in data representation or data interpretation on bar charts and histograms Studies also showed that PMTs' pedagogical content knowledge is lower than expected for the teaching aspects of statistical graphs The results of LS showed that PMTs' SMK and PCK are significantly improved when working, exchanging with each other and researchers The PMTs themselves involved in the study also recognized the nature of their learning and gave positive feedback on the knowledge they received when participating in LS 6.2 The evolution of prospective mathematics teachers’ knowledge for teaching the measures of central tendency in statistics graphs through lesson study The results of the study showed that PMTs did not clearly and accurately understand the mean, median or its significance represented on histograms The findings have also shown that the SCK of PMTs is also inadequate to represent this topic content well in a specific teaching context For the KCS and KCT knowledge types of PMTs, the results were not as expected Most PMTs have not been able to design lessons about the mean, and median on histograms in detail and specificity as well as the ability to predict possible answer situations of students to have can be effectively applied in teaching practice The progression in both SMK and PCK of PMTs has been improved through each step of the implementation process Specifically, PMTs participating in the LS not only have to discuss the content knowledge of mean and median expressed in the form of bar graphs and histograms, but they also have to focus on KCS and KCT these contents This has helped them gain a deeper understanding of teaching content related to the study lesson In the last lesson of the period, the researchers found that the knowledge types of PMTs were quite fully demonstrated through the steps of the teaching process PMTs have also noticed and given positive feedback on the evolution of their own MKT They acknowledged the difficulties in 18 implementing LS But they also confirmed that joining LS was a very good opportunity to equip themselves with more knowledge and skills to develop their professional capacity while increasing their love and interest in teaching and learning mathematics 6.3 The development of prospective mathematics teachers’ knowledge for teaching variability of data in statistical graphs through lesson study The results show that PMTs' CCK for teaching variability of data in bar charts has been evaluated as good, but the knowledge on the histograms has been the opposite Most PMTs made the common mistake of considering the variability of the data on the histograms based on the difference in height of the data columns This also means that they made the mistake of not clearly distinguishing bar charts and histograms Besides, many PMTs had SCK not reaching the expected level They were very embarrassed in giving evidence for the mistakes of students This shows that PMTs' SMK was not really complete in terms of variability of data and their representation, especially on histograms The analysis and collection of data have shown that PMTs' MKT for variability of data on bar charts and histograms has improved through the process of participating in the LS PMTs' knowledge for variability of data has been reinforced through the steps of planning and working discussions with each other and the researchers LS has promoted the progression of SCK Besides, their ability to practice teaching has also been cultivated through the revision of lesson plans and practice of teaching research lessons They have also recognized that good teaching requires considering many aspects of teaching and learning, and requires extensive content integration and understanding of pedagogical knowledge 6.4 Recommendations on innovating professional training for prospective mathematics teachers We propose to supplement and strengthen the content related to mathematical knowledge aspects for teaching in the pedagogical training modules In particular, even specialized content knowledge modules should involve a deep understanding of the epistemology related to mathematical topics to enable PMTs to grasp the essence of 19 concepts and understand how to apply the knowledge to the current high school mathematics program In addition, the teaching method of these modules should also be improved in the direction of capacity development so that PMTs can practise more specific skills in addition to equipping content knowledge for teaching For the teaching of statistics, we also suggest that it is necessary to rethink the way of teaching statistics Teaching methods need to go deeper into the nature of statistical content In addition, it is necessary to form and train PMTs with sufficient statistical analysis and processing skills to meet the requirements of statistics teaching capacity in the 2018 general education curriculum Through the study, we suggest that this LS process can be applied in the training process of PMTs at university level Specifically, LS should be included in teaching practice modules to help improve professional understanding and contribute to professional capacity development Moreover, joining LS will create a good collaborative academic environment for PMTs to work together on thematic lessons and practice cooperation and teamwork skills for the participants From the analysis in the experimental process, we propose a more compact knowledge framework for MKT in order to be more convenient for application in the training process of PMTs Figure 6.1 Proposing mathematical knowledge model for teaching in teacher training 20 6.5 Contribution of research From the research results obtained, the thesis has made theoretical contributions to mathematics education in general and in particular in training mathematics teachers In addition, the thesis has also provided empirical evidence for the theoretical basis that has been built with the content of teaching statistics Specifically, the contributions are follows: Firstly, the thesis has researched and clarified the types of knowledge necessary to teachers for teaching based on the MKT model of Ball et al (2008) Since then, we have analyzed epistemological aspects to build sets of indicators to assess the knowledge types of teachers for teaching histograms, the measures of central tendency in statistics graphs and variability of data in statistics graphs Secondly, the thesis has applied the sets of indicators that have been built based on the MKT model of Ball et al (2008) to assess PMTs' knowledge types to teach histograms, the measures of central tendency in statistics graphs, and variability of data in statistics graphs at the Universities of Pedagogy From the analysis of the obtained data, the thesis initially provided empirical evidence about the level of knowledge that PMTs have gained about teaching statistics in high schools Third, the study has explored some of the LS processes of the researchers (Baba, 2007; Lewis, 2009; Murata, 2011) and from that, the proposed LS process is suitable for the objectives of the study In this process, the practice step of teaching and reflection can be repeated many times, which creates a good opportunity for PMTs to develop math knowledge for teaching Fourth, the thesis has contributed empirical evidence showing the positive influence of the proposed LS process on the progression of PMTs' mathematical knowledge types for teaching statistics, especially CCK, SCK, KCS, and KCT Participating in LS also shows that this is a good opportunity for PMTs to work, exchange with colleagues, and accumulate the results of the whole group to gain good experiences for development their personal 21 Fifth, the thesis shows the feasibility and initial effectiveness of applying LS in training and knowledge development for teaching and professional practice of PMTs Finally, the thesis has made suggestions of practical significance in teaching Theoretically, the thesis has proposed a mathematical knowledge model for teaching consisting of knowledge components TCK, KCC, KCS, and KCT for easy application in the training program of PMTs at the university In terms of training, the thesis has proposed changing the training program, supplementing and enhancing the content related to knowledge aspects in the teaching methods modules, and professional development modules In addition, the thesis also gives suggestions to improve the teaching method of specialized subjects, contributing to improving the quality of teacher training to meet the current educational innovation requirements 6.6 Limitations of the research and development direction of the topic Within the framework of a topic, due to limited time and capacity conditions, our research still has certain limitations Specifically, the study has only been surveyed and carried out on a scale at two pedagogical universities in the central part of Vietnam, without conditions to expand to schools nationwide The research has only been conducted on PMTs, not on teachers who are teaching in high schools to compare the knowledge types of these two research subjects On the other hand, the research has only bean carried out on some content on the topic of statistics, which is one of the three main knowledge circuits in the 2018 general education curriculum However, we have also achieved certain results in teacher evaluation and training Specifically, we have surveyed and made initial judgments about PMTs' knowledge of teaching statistics and have successfully applied the LS process to develop knowledge types for teaching them From there, the topic can be expanded in the direction of understanding mathematics knowledge to teach PMTs about some other topics in statistics, as well as some other fields in mathematics In the next studies, the topic can be further expanded in 22 terms of the number and scope of research Besides, the topic can also be extended to the audience of mathematics teachers (teaching in high school) to consider their knowledge related to teaching some mathematics topics in high school and can compare knowledge between PMTs and incumbent teachers The topic may also have longer-term studies with PMTs who participated in the research when they became full-time teachers in high schools Moreover, when analyzing the final survey of PMTs participating in the LS process, we found that belief factor was mentioned by the PMTs This also suggests a new direction for us to research on factors affecting the teaching process such as beliefs, motivation or educational context of teachers Another exploration direction also suggested in the research process is to examine the influence of teachers' mathematics knowledge on students' learning outcomes on a specific mathematics topic in high school The above are the future research directions of the topic that can be developed 23 LIST OF RESEARCH WORKS Phuong, N T H., Lien, L T B., & Thuy, N T M (2017) Developing professional skill for high school mathematics teachers by using visual illustrations in teaching toward the constructivism theory, Journal of Educational Sciences, University of Education, Da Nang University, No 58, 74-84 Le Thi Bach Lien, Nguyen Thi Ha Phuong, Tran Kiem Minh (2018) A cognitive approach to the evaluation of prospective mathematics teachers’ professional competencies, Proceedings of 5th International Conference on Language, Society and Culture in Asian Contexts, 926-937 Nguyen Thi Ha Phuong, Tran Kiem Minh (2019) Prospective mathematics teachers’ mathematical knowledge for teaching histograms in statistics, Vietnam Journal of Education, No: Volume (December 2018) (English version), tr 195-201 Nguyen Thi Ha Phuong (2019) Apply the model of mathematical knowledge for teaching in the training of prospective mathematics teachers in pedagogical universities, Vietnam Journal of Education, No: 7/2019, tr 240-245 Nguyen Thi Ha Phuong, Tran Kiem Minh (2019) Developing prospective mathematics teachers’ mathematical knowledge for teaching histograms, Proceedings of the 1st International Conference on Innovation in Learning Instruction and Teacher Education - ILITE Editor: Nguyen Thuy Linh, Hoang Thi Quyen, 344-352 University of Education publishing house, Ha Noi Nguyen Thi Ha Phuong, Tran Kiem Minh (2020) Teachers’ knowledge for teaching: Exploring prospective mathematics teachers’ knowledge of measure of central tendency from statistical graphs, Proceedings of the 3rd International Conference on teacher education renovation (ICTER 2020): Teacher competencies for education 4.0 Editor: Hoang Duc Nguyen, pp 415-428 Thai Nguyen University publishing house, Thai Nguyen Nguyen Thi Ha Phuong, Tran Kiem Minh (2021) Prospective Mathematics Teachers’ Professional Development through Lesson Study: The Case of Teaching Measures of Central Tendency from Statistical Graphs, Journal of Educational Sciences, University of Education Ha Noi, No: 66 Issue 3, 187-200 Nguyen Thi Ha Phuong (2021) Prospective mathematics teachers’ mathematical knowledge for teaching mean and median from data histograms, Journal of Educational Sciences, University Hue, No: 130 (6B), 17-33 Nguyen Thi Ha Phuong, Tran Kiem Minh (2021) Professional development through lesson study: prospective mathematics teachers’ knowledge for teaching variability in statistical graphs, Proceedings of the 2nd International Conference on Innovation in Learning Instruction and Teacher Education - ILITE 2021, p 228-245 University of Education publishing house, Ha Noi 24 TT SCIENTIFIC PROJECTS AND THEMES HAVE BEEN PARTICIPATED Level topic Year (Central, Responsibility Research topic title start/year Ministerial, to participate complete sectoral, in the topic university) Developing the capacity of teaching mathematics and Ministry 2018 - 2020 Chairman statistics to students level topic of mathematics pedagogy Developing knowledge and teaching capacity for Prospective State-level mathematics teachers 2021-2023 Member topics to meet the requirements of the new general education curriculum