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MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY HOANG LE MINH BUILDING A LEARNING ENVIRONMENT OF STATISTICS CONTENT IN HIGH SCHOOL IN THE DIRECTION OF TRAINING STATISTICS REASONING SKILLS FOR STUDE[.]
MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY - HOANG LE MINH BUILDING A LEARNING ENVIRONMENT OF STATISTICS CONTENT IN HIGH SCHOOL IN THE DIRECTION OF TRAINING STATISTICS REASONING SKILLS FOR STUDENTS Major: Mathematics Theory and Teaching Method Thesis Code: 9140111 PhD THESIS SUMMARY NGHE AN, 2023 The present study has been completed at Vinh University Science instructors: Assoc Prof Dr Nguyen Chien Thang Prof Dr Nguyen Van Quang Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended in the institutional review Board meeting at Vinh University, at ….h… on ………………., 2023 The thesis can be found at: The National Library of Vietnam Centre for Information - Library Nguyen Thuc Hao, Vinh University PREFACE I THE RATIONALE OF THE RESEARCH The importance of statistics and statistical reasoning skills for each individual has been confirmed by Vietnamese math educators through the content of the 2018 Math curriculum with the following requirements that students need to achieve when completing the high school program is “Improve the ability to collect, classify, represent, analyze and process statistical data; using statistical data analysis tools through central trend measurement and dispersion measures for ungrouped and clustered data samples; using statistical rules in practice; recognize random patterns, basic concepts of probability and the significance of probability in practice” (Ministry of Education and Training, 2018) In the process of implementing the new general education program, those who are working in the education sector will need many scientists to research and exchange together to be able to contribute to the successful implementation of the task of educational innovation without the need of many scientists social responsibility In this work, we focus on studying the knowledge of statistical reasoning skills on high school students, the manifestations and levels of statistical reasoning skills, statistical reasoning ability, and statistical reasoning skills statistics that students need to be formed and developed to meet their future job needs as well as be ready to deal with situations that appear in real life related to statistics We also research to design and propose a learning environment for statistical content formed from pedagogical measures to contribute to training students' statistical reasoning skills For that reason, we have chosen the topic “Building a learning environment of statistics content in high school in the direction of training statistics reasoning skills for students” II RESEARCH PURPOSES To research to build a learning environment of statistics content in order to train students in statistical reasoning skills through the teaching process at high schools To research on the viewpoints, principles, positions and roles of statistics content and the requirements to be achieved by students in building a high school Mathematics program From there, building a learning environment for statistical content in order to train students in statistical reasoning skills through the teaching process at high schools NEW CONTRIBUTIONS OF THE THESIS 3.1 From theoretical perspective - Systematizing theoretical issues related to statistical education in high schools, especially in statistical understanding, inference and thinking - Proposing a concept of statistical reasoning skills of high school students and the manifestations of this skill - Proposing a concept of a teaching environment that trains statistical reasoning skills and the core components of this environment - Proposing Rubric on the levels of the students' statistical reasoning skills - Proposing pedagogical measures to create a statistical learning environment in the direction of training statistical reasoning skills for students 3.2 From practical perspective - Support teachers in organizing teaching of statistical content in high schools with the expectation of improving teaching effectiveness - The application of pedagogical measures proposed in the thesis to teaching practice will contribute to innovating teaching methods and achieving the goal of teaching statistics in high schools THESIS STRUCTURE The thesis includes parts: Introduction, Conclusion, References, Appendix, and chapters: Chapter The theoretical and practical basis of training statistical reasoning skills for high school students Chapter Building a learning environment in the direction of training statistical reasoning skills for high school students Chapter Pedagogical experiments Chapter THE THEORETICAL AND PRACTICAL BASIS OF TRAINING STATISTICAL REASONING SKILLS FOR HIGH SCHOOL STUDENTS 1.1 Overview of research works on teaching statistics 1.2 Epistemological features of statistics 1.2.1 A brief history of statistics 1.2.2 The Relationship Between Statistics and Probability 1.2.2.1 Probability 1.2.2.2 Statistics 1.2.2.3 The relationship between statistics and probability It can be said that probability theory forms the theoretical basis for statistics If we separate Probability Theory from Statistics, Statistics will lose many important results brought by the Deductive Statistics section: then, people only stop at the experimental level, not generalizing to the total body That way, Statistics will no longer hold its great value, especially for practical matters People will not be able to use Statistics as an effective and sharp tool for analysis, prediction, in order to make correct and necessary judgments However, the relationship between probability theory and statistics is not only that probability theory is necessary for statistics There is actually a link in the opposite direction: Statistics, specifically Descriptive Statistics, are also necessary for the study of Probability Theory For example, one of the approaches to the concept of probability of an event is the concept of frequency - the concept of descriptive statistics 1.2.3 Sampling method 1.2.4 Characteristic numbers 1.2.4.1 Central trend-measurement characteristic numbers 1.2.4.2 Characteristic numbers measuring the degree of dispersion 1.3 Statistics in Vietnamese Mathematics Education program 1.3.1 Statistical content in the Education program 1.3.2 Expression of mathematical ability of high school students through statistical content 1.4 Statistical reasoning skills 1.4.1 Reasoning 1.4.2 Statistical reasoning There are divergent views among Mathematics educators about statistical reasoning According to the studies of Chervaney, Collier, Fienberg, Johnson, Neter, Benson and Iyer, they defined statistical reasoning as what students can with statistical content, that is: recall, recognize, analyze and differentiate between statistical concepts and experience using statistical concepts to solve real-world problems Lovett has provided a rather detailed overview of Statistical Inference, considering Statistical Reasoning as belonging to one of the following three approaches: theoretical research, experimental research and classroom study (from Hoang Nam Hai, 2013) Dani Ben – Zavi, Joan Garfield and Gal all agreed: Statistical inference is how people reason with statistical ideas and make statistical information meaningful Statistical reasoning also means understanding and interpreting statistical procedures, being able to fully interpret statistical results (Nicolxki X M, 2002) According to Hoang Nam Hai (2013): Statistical reasoning is a type of inference based on statistical data to identify, interpret, analyze and draw statistically significant conclusions, discover statistical laws of a crowd of the same type 1.4.3 The relationship between statistical reasoning and statistical understanding and statistical thinking 1.4.3.1 Statistical understanding Statistical understanding is the ability to understand statistical information, grasp and use the language, tools, and basic concepts of statistics: - Statistical understanding includes the basic skills which are used to understand statistical information or research results, such as organizing data, constructing and representing tables, and working with other representations of data - Statistical understanding includes understanding concepts, terms, and symbols, and understanding how to use probability as a measure of uncertainty 1.4.3.2 Statistical reasoning Statistical reasoning is a way of reasoning with statistical ideas and making statistical information become meaningful This involves making interpretations based on data sets, representations of data, or statistical summaries of data 1.4.3.3 Statistical thinking Statistical thinking is the process of reflecting in general the characteristics of statistical data to draw conclusions and rules on a crowd of statistical phenomena, serving the activities of people's lives through using the knowledge of statistical probability theory 1.4.3.4 The relationship between statistical understanding, statistical reasoning, and statistical thinking UNDERSTANDING REASONING THINKING Figure 1.3 The point of view has the independence and interference between the three regions (delMas R C., 2002) UNDERSTANDING REASONING THINKING Figure 1.4 Knowledge is the foundation for developing statistical reasoning and thinking (delMas R C., 2002) We believe that statistical understanding is the premise and foundation for students to develop their statistical reasoning and statistical thinking abilities Statistical thinking and statistical reasoning are interrelated and they can be used interchangeably to represent cognitive activities of the same kind Reasoning is generally considered a type of thinking Statistical thinking and statistical inference are interrelated and they can be used interchangeably to represent cognitive activities of the same kind Inference is generally considered a type of thinking While statistical literacy can be viewed as understanding and interpreting the statistical information presented, statistical inference can be viewed as the ability to solve specific problems through tools and concepts concepts learned, and statistical thinking is the ability to go beyond what students are taught in the course, which is the ability to ask questions, investigate problems, and related data in a particular context In order to achieve the goal of the teaching process, it is necessary to have many influencing factors and flexible pedagogical measures to suit each content, context and target audience But first, teachers need to understand the levels of the students' cognitive processes, the specific expressions that can be achieved by the learners, so that they can apply flexible pedagogical methods , suitable to influence that cognitive process to achieve teaching goals If we want to evaluate students on cognitive ability and mathematical thinking, but only ask students to identify, describe, or rephrase, then that is only the level of cognitive ability 1.4.4 Comparing statistical reasoning and mathematical reasoning First, we consider the similarity of these two types of inference Statistical reasoning and mathematical reasoning are both separate instances of human reasoning Therefore, they are procedural and follow the rules of inference Any inference generally has a logical structure A=>B, where A is the premise, B is the conclusion (Nguyen Duc Dong, Nguyen Van Vinh, 2001) Rossman, Chance, and Medina (2006) describe statistics as a science using mathematics Therefore, when students make statistical reasoning and mathematical reasoning, both use mathematical knowledge Second, we look at the difference between statistical inference and mathematical reasoning According to Polya (1995), there are two types of inferences: demonstrative reasoning and rational inference Proving inference is a particularity of mathematical reasoning based on general logic rules, which determine that if the premises are true, then the conclusion is also true Meanwhile, statistical inference based on the collected data sample, makes a conjectural conclusion about all the individuals in a population; Generalizing from the characteristics of a relatively small sample to infer the characteristics of a sometimes very large population makes it impossible to avoid the risk of making mistakes (Duong Thieu Tong, 2003) Therefore, the conclusions drawn from statistical inference only guarantee a certain degree of confidence, which means that statistical inference is a form of logical inference There is another difference between these two types of inference, which is contextuality in statistical inference Although data may seem like numbers, Moore (1992) argues that data are “contextual numbers” And unlike mathematics, where context obscures the underlying structure, in statistics context provides meaning to numbers and data cannot be meaningfully analyzed without careful consideration their context: how they are collected and what they represent (Cobb & Moore, 1997) 1.4.5 Skills Although there are many concepts related to skills, the concepts of skills are basically unifying that skill is the ability to apply human knowledge and understanding to perform a job that is to produce the desired result 1.4.6 Statistical reasoning skills From analyzing and learning about the above statistical inference concept, we can understand: Statistical reasoning skill is the ability to draw conclusions, make predictions about all survey elements based on information and statistical data collected from samples and presented in the form of articles, tables or graphs That process is reflected in the human mind, filtered, linked, analyzed, transformed to perceive the real world and draw meaningful statistical conclusions 1.5 The reality of teaching and learning statistics in high schools In fact, the teaching of statistical content in Vietnamese high schools needs a lot of changes In our daily lives, we often encounter uncertain election results, collapsed bridges, stock market downturns, unreliable weather forecasts, false predictions about population growth, inefficient economic models and other manifestations of uncertainty in our real world (OECD, 2003, 2009) Faced with such uncertain sources of information, the question is “How we make the right judgments?”, or “What competencies does each citizen need to handle? sources of this information?” In today's modern life, a citizen needs skills different from those of previous decades, in which statistical skills are considered as important as writing and reading skills for each person An objective reason that statistical content has not received the proper attention of teachers and students in the past is due to the Covid-19 epidemic situation, teaching and learning have been affected by the Ministry of Education and Training (2020) issued the Official Dispatch on load reduction (CV3280/BGDĐT – GDTrH dated August 27, 2020) in which 50% of the time for teaching statistical content is reduced By observing the current learning situation and exam form, we can see that the statistical knowledge achieved by high school students after graduation is not high, the interest of students is not much, some Studies closely related to statistics such as data science have only had a few open universities recently (Even universities in the Central region opened a major after data science after 02 years, but too few candidates registered, so in 2023 in the enrollment notice, they had to stop recruiting) Conclusions of Chapter In fact, in the world with today's amount of information, people need knowledge and skills about statistics, students need to be equipped with this knowledge fully and early, this is even more important confirmed through the content of Mathematics in the general education curriculum in 2018 with the increase of the duration of the statistical content Researching the works on the levels of statistical competence, we have synthesized and clarified the concepts and relationships between the levels: statistical understanding, reasoning and thinking We have also pointed out the specific manifestations of these levels, and at the same time focused on further analyzing the students' statistical reasoning skills This is the level that we think is suitable for the purpose of teaching the General Education Curriculum 2018 We also investigated and surveyed the reality of teaching statistics in high schools, there were many results that were not as expected Because of objective and subjective reasons, both teachers and students now pay little attention to statistical content, leading to the formation of this skill for students is still limited From the school year 2022-2023, the new General Education Curriculum will be implemented, which is very necessary and a favorable condition for both teachers and students to change their views and traditional teaching methods There will be many difficulties and challenges for those who implement teaching statistical content, but we have already predicted that and have taken the necessary preparation steps In chapter 2, we will propose pedagogical measures to build a positive learning environment to train statistical reasoning skills for students Chapter BUILDING A LEARNING ENVIRONMENT OF STATISTICS CONTENT IN HIGH SCHOOL IN THE DIRECTION OF TRAINING STATISTICS REASONING SKILLS FOR HIGH SCHOOL STUDENTS 2.1 The need to build a learning environment to train students' statistical reasoning skills The goals and content of teaching Mathematics in the 2018 General Education Program are built in the direction of developing quality and capacity, so the teaching method must also change accordingly Therefore, the process of teaching Mathematics in general, including statistical content, should comply with the following basic requirements (Ministry of Education and Training, 2018): First: It must be suitable with the cognitive process of students (from concrete to abstract, from easy to difficult); attach importance to the logic of mathematical science and at the same time pay attention to the approach based on the experience capital and experience of students; Second: Mastering the spirit of "taking learners as the center", promoting the positivity, self-discipline, paying attention to the needs, cognitive abilities, and different learning styles of each individual student; organize the teaching process in a constructivist direction, in which students can participate in exploring, discovering, reasoning and solving problems; Third: Flexibility in applying active teaching methods and techniques; combine skillfully and creatively with the application of traditional teaching methods and techniques; combine teaching activities in the classroom with hands-on experiences, applying mathematical knowledge into practice The lesson structure ensures a balanced and harmonious ratio between core knowledge, applied knowledge and other components Fourth: Using sufficient and effective means and equipment for teaching at least as prescribed for Mathematics; can use self-made teaching aids suitable to the learning content and the students' subjects; increase the use of information technology and modern teaching facilities and equipment in an appropriate and effective manner; The characteristics of the 2018 Math Program in general, and the Statistics content in particular, as indicated above, require basic orientations in teaching this content in high schools In our opinion, the major orientations in teaching and learning statistical content in the 2018 math program are: Orientation 1: Choosing examples in teaching statistics to create opportunities for students to form and develop their capacity Orientation 2: Applying active teaching methods and techniques to teaching statistical content Orientation 3: Applying methods and forms of testing and assessment towards developing students' learning capacity in teaching statistics content Orientation 4: Using information technology in teaching statistics The reality of teaching statistical content as described in Chapter needs a big change if it wants to meet the requirements of the General Education Program in Mathematics 2018, positive and result-oriented new perspectives Teaching that meets the requirements of educational goals is essential The above orientations both show the need to build a learning environment to overcome the reality of teaching statistical content and also contribute to identifying the main components of that environment We believe that educational methods need to be seriously studied and experimented before being applied, the learning environment to practice literacy skills for students is one of the pedagogical methods suitable to the trend education direction of the world and will play a solve that problem, attach that problem to the interest of students, when creating a need to find a way to solve the problem, not merely the repetition of knowledge mode and mode of operation, that is, requires students to have adaptation to certain situations Problem solving is done through question-answering activities If there is no question, no problem, there is no scientific knowledge An effective and active statistics classroom can be viewed as a learning environment to develop in students a deep and meaningful understanding of statistics and to help students develop their ability to statistical thinking and reasoning This is a learning environment to develop statistical reasoning skills By viewing it as a learning environment, we emphasize that it represents more than a textbook, activities, or exercises that we provide to students That environment is a combination of real data, activities and cultures in the classroom, discussion, technology, teaching methods and a variety of forms of assessment The principles of instructional design described by Cobb & McClain, 2004 (according to Joan B., Garfield J., Ben-Zvi D 2008) recommendations for application to the teaching of statistics are the use of real data, classroom activities, technology, and assessment From that, we believe that the learning environment for students to practice literacy skills has the following model: Diversifying assessment methods in teaching to develop statistical reasoning skills Applying suitable teaching methods for training statistical reasoning skills Using real-life data in teaching statistics Increase the use of information technology to support teaching and learning to train statistical reasoning skills Figure 2.3 Learning environment for statistical content to practice statistical reasoning skills A change in the method of teaching statistical content in high schools is necessary, because it is a task that teachers need to implement if they want to achieve the goals of the 2018 Math education program With the above proposed model, students will receive the overall impact in the process of receiving statistical knowledge, the teacher plays the role of creator, observer, support, and control the student's learning process in order to guide the students' learning process to develop statistical reasoning skills for students The learning environment for training statistical reasoning skills made up of the above factors can be considered as a positive learning environment to 10 develop students' deep and meaningful statistical understanding, to help them develop their thinking and statistical reasoning skills For example, the collection of data from practical problems is a very important premise that determines the process of analyzing and drawing statistical conclusions and predictions Due to the limited amount of study time in the program, the textbook only provides a brief introduction to the problem of collecting and describing statistical data The data in the textbooks given are usually available, hypothetical, so the students' statistical reasoning skills are less interested in training, mainly remembering and applying formulas to calculate certain characteristics To create a positive learning environment to practice statistical reasoning skills, in teaching, teachers can divide the class into small groups and assign the task of collecting data before coming to class In order to create and practice statistical reasoning skills for students, teachers need to create a positive learning environment as analyzed and built with the above components An effective and active statistics classroom will be a learning environment that can deepen students' understanding of statistics, and at the same time develop their ability to think and make statistical inferences That type of classroom is a kind of learning environment that trains statistical reasoning skills 2.3 Pedagogical measures to build a learning environment to train students' statistical reasoning skills 2.3.1 Measure 1: Organizing for students to search and exploit practical data in daily life to teach statistical content in high schools 2.3.1.1 Proposal basis 2.3.1.2 Purposing and meaning of the measure a Using real data in teaching statistics contributes to the completion of some necessary mathematical knowledge and skills for students to be able to apply their knowledge to practice b Finding and using real data in statistics teaching situations helps students build knowledge in a better way c Using real data in teaching statistics contributes to the development of intellectual abilities for students d Exploiting and using real data in teaching statistics helps students develop statistical reasoning skills 2.3.1.3 Method to perform There are many ways to collect data sets to serve the teaching of statistics, the convenience of accessing data today also greatly supports teachers in using real data Teachers also need to be aware that questions used with datasets will interest learners and not all datasets will interest all students, so data should be used from a range of different contexts 11 In teaching statistics at high schools, it is necessary for students to be able to solve statistical problems by themselves by collecting and describing data millet in life The process of applying statistics to solve practical problems can be described in four steps: Collecting data Analysis and interpretation Data organization Data representation Proposing some practical data mining options that teachers can guide students to exploit for the lesson: - Proposing some real-life data utilizing options that teachers can guide students to exploit for the lesson: - Exploiting data from daily real life (one month's electricity consumption, book purchase amount, shopping amount, ) Exploiting data from the General Statistics Office (https://www.gso.gov.vn/), the provincial statistical office from district and commune statistical officers, - Exploiting data and figures from the internet, official websites of reputable organizations - Self-survey, collect and build data - Developing from existing textbook exercises, making connections to local realities where you live (e.g athletic performance metrics, or height and weight measurements that can be collected in class, in school the student is attending) Teachers can refer to some suggestions on practical issues that students are interested in in Appendix This measure is usually organized as follows: Teachers organize groups or individual students to collect practical data The teacher is the one who gives the topics, then the teacher guides the group on how to collect data, how to record data, how to investigate, find data on books, the internet, according to the following process: Step 1: Suggesting motivation to help students learn the needs and statistical significance of the topic, orienting them how to approach actual data Step 2: Guiding and organizing for students or groups of students to search and exploit sources to get actual data Step 3: Guiding students or groups of students to synthesize and classify the collected data into tables of data, analyze and process data with the support of information technology 12 Step 4: Organizing for students or groups of students to present the group's obtained products, which may require clarification of the level of achievement compared to the requirements of the set topic 2.3.2 Measure 2: Selecting and applying active teaching methods and techniques suitable for training students' statistical reasoning skills in teaching statistical content in high schools 2.3.2.1 Proposal basis 2.2.3.2 Purpose and meaning of the measure Traditional statistics classes often not have much discussion, information is "given" through lectures and questions This is different from teaching in which students answer each other's questions and learn to ask questions and defend their arguments In a learning environment to practice literacy skills, the use of activities and technology allows a new form of classroom language, students participate more in exchange, focus debate on important statistical ideas important - Concretize the educational orientation of "Students-centered", enhance the initiative and interest in learning for students during class hours, and improve teaching effectiveness - The purpose is to make students better understand the statistical meanings and results after the calculation steps, students understand what that number means in reality The meaning of the measure is the connection between mathematics and practice, after the formulas, after the results are meaningful conclusions Help students see clearly the relationship between mathematics and practice, the application of mathematics in life 2.2.3.3 Method to perform To implement classroom models and create an environment for learning statistical content, teachers should note: - Using questions that encourage students to speculate and infer, including wrong answers (students can make mistakes) in addition to questions with correct answers; - Asking students to explain their arguments and back up their answers, using numbers featured in inferential reasoning and the direction of the data Then ask other students to confirm whether or not they agree and why; - Creating a positive classroom environment, students feel comfortable, secure expressing their views This can be done if teachers encourage students to boldly present their own conjectures, and other students comment and test hypotheses using tools and software Students are encouraged to reason to find answers, should not reprimand or impose or frame answers, teachers should focus on thinking processes, or the path that students take to reach conclusions (1) Teaching discovery and problem solving in Math helps math knowledge (concepts, theorems, consequences, properties, ) to be formed as a 13 result of the process of students actively thinking to solve a problem, it is not declared by the teacher Teachers can organize the following activities: Identifying the problem Planning to solve the problem - Implement the plan - Checking, evaluating and concluding Applying the teaching of discovery and problem solving to teaching statistical content is appropriate because statistical problems often have practical elements characteristic numbers and graphs to draw conclusions or judgments Solving statistical problems/problems helps students firmly grasp statistical knowledge, fluently use formulas to calculate, and understand the meaning of characteristic numbers (2) Teaching of mathematical modeling is teaching how to build mathematical models of reality, aiming to answer questions and problems arising from reality Teaching of mathematical modeling is mathematics teaching through modeling teaching Basically, the modeling process is done through four main steps: - Step 1: Build an intermediate model (qualitative model) of the problem, i.e identify the most important factors (characterizes the system under consideration) and establish rules, reflect the relationship between them or the rules to follow - Step 2: Build a mathematical model for the problem under consideration, that is, describe it in mathematical language for the intermediate model For the initial problem to be solved, there can be many mathematical models, depending on which factors are important and which relationships are emphasized when building the intermediate model - Step 3: Use mathematical tools to survey and solve problems formed in step two Here, in the established mathematical model, it is necessary to choose or build a suitable solution method In this step, we can use computer software to support performing math operations and solving mathematical problems - Step 4: Analyze and verify the results obtained in step three In this section, it is necessary to determine the appropriateness of the model and calculation results with the original problem to be solved To determine the degree of relevance, we can apply some specialized analytical method associated with the original problem The modeling process that we mentioned above is based on the point of view: each practical problem can correspond to many theoretical models (the number of steps can be changed, the model can be different to suit the actual problem), the performer's task is to build a mathematical model to find an acceptable answer - say acceptable because in reality there is not always only one answer, but normally there are many answers that are suitable for different circumstances of the problem Therefore, teachers need to be flexible and have creative application in specific teaching situations 14 Statistics are closely related to practice From the main steps of the abovementioned mathematical modeling process, we propose a process of teaching mathematical modeling in order to train students' mathematical skills, including the following steps: - Step 1: Build an intermediate model Teachers raise practical problems in accordance with statistical knowledge specified in the program; HS identifies the problem - Step 2: Build a mathematical statistical model for the practical problem under consideration - Step 3: Use mathematical and statistical tools to solve the model found in step - Step 4: Analyze and check the results in step (3) Teaching mathematics through experiential activities is model-based teaching associated with experiential learning theory introduced in 1971 by Kolb D According to this theory, knowledge is created through transformation experience Teachers organize for students learning activities in the form of real experiences, specifically aimed at the formation of new mathematical knowledge That knowledge can be a new concept, a new formula or a theorem, a way of proof, etc Either students experience learning materials, learning materials for Math, or can also participate in outdoor experiential activities to apply knowledge This process consists of steps: Concrete Experience - Observation, Reflection - Generalization, Abstraction - Active experimentation The project-based teaching method can be conducted in four stages as follows: Stage 1: Prepare the project Stage 2: Implement the project implementation plan Stage 3: Writing and reporting the results of the project Stage 4: Evaluation of project implementation results 2.2.3 Measure 3: Exploiting information technology to support teaching statistical content in high schools to train students' statistical reasoning skills 2.2.3.1 Proposal basis 2.2.3.2 Purpose and meaning of the measure - Contributing to helping students access IT, towards using IT to support their study and work later, this is also one of the teaching orientations - Reduce complicated and time-consuming calculations, to focus more on the nature and meaning of statistical content - Contributing to changing the form and means of teaching and learning: the form of teaching and learning through the medium is the computer Teaching with the support of IT is a new trend in education in our country today - For teaching Mathematics in high schools, information technology contributes to improving the potential of teachers by providing them with modern 15 facilities (such as the Internet, electronic dictionaries, electronic books, etc.) email, e-mail, math software, study groups, group connections with parents and colleagues, ); Technology contributes to innovating teaching and learning methods, innovating teaching methods in a more positive direction 2.2.3.3 Method to perform Video tutorial using excel to practice basic statistics calculation: https://www.youtube.com/watch?v=3lQ-eFhYhrk&t=17s During the teaching process, teachers easily stop back and forth, supported with additional backgrounds, effects, typography, and many actions to link the lecture content or emphasize information to orientation, suggesting students to explore and solve problems Teachers save time lecturing, introducing or demonstrating new knowledge content This makes it easier for students to absorb the lesson In preparing lessons with the applications of information technology, teachers themselves must regularly update their professional knowledge and the ability to apply information technology in teaching from which it is easy to generate new ideas in the lecture, thereby self-improvement skills Teaching with the support of IT is currently required by the Ministry of Education to be included in educational programs at all levels The use of MS Excel software to teach statistical content at high school level, especially statistical content in grade 10, is completely consistent with the level of students and teachers In addition to MS Excel, there are many other software that support better, but students need additional knowledge about that software compared to MS Excel, which is more complicated and difficult Along with increasing the use of problems with practical content, it helps students see the applications of Statistics and Mathematics in practice more, helping students to have a more positive spirit and attitude to study Strengthening practical problems with the support of MS Excel to help improve the practicality in teaching statistical content is necessary and appropriate to the situation 2.2.4 Measure 4: Using methods and tools to assess students' statistical reasoning skills in the process of teaching statistical content in high schools 2.2.4.1 Proposal basis 2.2.4.2 Purpose and meaning of the measure - Using a diversity of assessment forms in the teaching - learning process to create cohesion, the relationship between the evaluation elements and the whole educational process, making the assessment a force that affects and regulates the learning process adjust the process of forming knowledge of students - Through different forms of assessment, teachers can obtain information about students' progress and efforts in learning and training, helping students to maximize their abilities, ensuring timely fairness and objectivity Through the 16 assessment results, students also have their own learning plans to be able to adjust their individual learning, towards better learning outcomes 2.2.4.3 Method to perform In the process of preparing to implement the 2018 educational program, the training modules to equip teachers to meet the requirements of the new program have been elaborately compiled by the authors There are also effective and useful assessment methods in the process of teaching statistics at high schools that teachers can rationally apply such as: Evaluating through the project This is a method to evaluate the ability to link, systemize knowledge, skills and transform, apply to solving tasks This method will take a relatively long time, the project's products can be presented in many different forms such as reports, presentations, lectures, Evaluating through profile Profile is a collection of evidence about the person being evaluated to determine whether they have a certain ability or not This is a deliberate and meaningful assessment method, not a random collection Evaluating through reports Ask the evaluator to write a task performance report in a uniformly prescribed format More valuable than writing essays in class, because it is based on actual task performance activities Evaluating by the question and answer allows learners to demonstrate their level of understanding through direct response to the evaluator's questions and process of answering questions (intonation, expression, confidence level ) Observational method is used to evaluate the behavior of individuals by recording a lot of information as a basis for analyzing, explaining and concluding the attitude of the person being evaluated Observation is conducted in a structured manner (with time to prepare, with a clear purpose) The reason for the observation, the aspect of the observation, and the information observed should be clearly defined Experimental process of round and round 2, in addition to the end-ofchapter assessment by scores, we combine methods of student assessment with the product of group work, assessment through the process of attending classes of students Students, refer to the channel of mutual assessment of individuals in the group, evaluate the student's participation in learning, the initial results are remarkable, when the members are informed in advance of the evaluation forms such as: At the same time, they experienced in the statistical reasoning environment created by the teacher, the students were very interested and the lesson was lively and effective (specific results are available in Chapter Pedagogical Experiment and the Appendix) Conclusion of Chapter In chapter 2, we study and clarify the role and impact of the environment on teaching The environment plays a huge role in the teaching process and affirms that teachers can use pedagogical measures to create a better learning 17 environment for students We have also built a model to describe that expected environment We have also studied the guiding principles of teaching methods for the 2018 Mathematics program and then applied them to the teaching of statistical content The four proposed pedagogical measures to be able to create a learning environment to practice literacy skills for students are: Measure 1: Increase the use of real data in daily life to put into teaching - learn statistics; Measure 2: Apply active teaching methods and techniques suitable for training statistical reasoning skills; Measure 3: Strengthen the application of information technology to support teaching statistics in order to practice statistical reasoning skills; Measure 4: Apply some methods and tools to evaluate students' statistical reasoning skills We have also developed the bases and procedures for implementation and examples to be able to implement the above pedagogical measures We will detail this content in Chapter The content of chapter is presented in articles 3,4,5 in the section "published works related to the thesis" Chapter PRACTICAL PEDAOLOGY 3.1 Purpose and hypothesis of pedagogical experiment 3.1.1 Purpose of pedagogical experiment 3.1.2 Pedagogical experimental hypothesis 3.2 Evaluation methods 3.2.1 Survey results of teachers for proposed pedagogical measures 3.2.2 Qualitative assessment 3.2.3 Quantitative assessment 3.3 Organization of pedagogical experiments 3.3.1 Experimental content 3.3.2 Experimental deployment Pedagogical experiment was conducted at Thanh Hoa Ethnic Minority Boarding High School (round - school year 2020 - 2021, time April 2021) and Dao Duy Tu high school - TP Thanh Hoa (2nd round - academic year 2021 2022, April 2022) 3.3.3 Experimental method 3.3.4 Methods of evaluating pedagogical experimental results 3.3.5 Pedagogical experiment lesson plan 3.4 Processing and evaluating the results of pedagogical experiments 3.4.1 Qualitative assessment results After studying and using the measures developed in chapter of the thesis, the experimentation teachers all have the opinion that it is not difficult to apply these measures, especially the solutions Exploiting and using real data in the 18 recommendation is close to reality, suitable to students' knowledge and interests The use of Excel in teaching statistics has reduced the calculation time of teachers and students to increase the time for discussion, group work, review and consolidation of basic concepts to help them master and be able to operate Use acquired knowledge to solve real-world problems With the control class, when teaching according to the usual lesson plan, they encountered problems in statistics like in textbooks, they still did not have interest in learning the content of statistics, did not see the application of statistics Statistics on life leads to a lack of knowledge and skills in statistics, when encountering the tasks assigned by the teacher after the lesson, the children face more difficulties and embarrassment than the control class 3.4.2 Quantitative assessment a) Results of the first experiment at Thanh Hoa Province Ethnic Minority Boarding High School, April 2021 Table 3.9 Analysis of test scores of students in grades 10E and 10G at Thanh Hoa Province Ethnic Minority Boarding High School, school year 2020-2021 Control class Experimental class Mark Number of Number of Rate (%) Rate (%) Students Students 0 0% 0% 0% 0% 0% 0% 7% 3% 7% 0% 13% 7% 11 37% 17% 17% 12 40% 17% 23% 3% 10% 10 0% 0% Amount 30 100% 30 100% Sample mean Sample Variance Sample Standard Deviation Coefficient of variation X ÐC 6.13 X TN 7.0 S ÐC 2.19 S TN 1.66 S ÐC 1.48 STN 1.29 V 0, 22 V 0,18 19 * Comparing the mean score of the experimental class and the control class The Null Hypothesis H0: The average score of the experimental class is equal to the average score of the control class The Alternative Hypothesis H1: The average score of the experimental class is higher than the average score of the control class The significance level is α = 0.05 * Comparing the mean score of the experimental and control classes of 10th grade Thanh Hoa Province Ethnic Minority High School Looking up the normal distribution table, we have the critical level Z = 1, 64 We have: Z0 = xTN − xÐC STN S2 + ÐC N TN N ÐC = 7.0 − 6.13 = 2, 42 2.19 1.66 + 30 30 As we see Z Z thus Ho should be rejected, and Null Hypothesis H1 is accepted Thus, the average score of the experimental class is higher than that of the control class, which means that the teaching method in the direction of creating an environment to practice statistical reasoning skills for students is more effective b) Results of the second experiment at Dao Duy Tu High School - Thanh Hoa City, April 2022 Table 3.13 Analysis of test scores of students in grades 10A1, 10A3 at Dao Duy Tu High School, Thanh Hoa city, academic year 2021-2022 Control class Experimental class Number Mark Number of of Rate (%) Số HS Students Students 0 0% 0% 0% 0% 0% 0% 0% 0% 4% 2% 13 27% 7% 11 23% 13% 17% 11 24% 19% 15 33% 8% 20% 10 2% 0% 20 48 Amount 100% 45 X ÐC 6.52 Sample mean 100% X TN 7.40 S 2.21 S 1.41 Sample Variance Sample Standard SÐC 1.49 STN 1.19 Deviation Coefficient of VÐC 0.23 VTN 0.16 variation * Comparing the mean score of the experimental class and the control class The Null Hypothesis H0: The average score of the experimental class is equal to the average score of the control class The Alternative Hypothesis H1: The average score of the experimental class is higher than the average score of the control class The significance level is α = 0.05 * Comparing the average scores of the experimental and control class of grade 10, Dao Duy Tu School, Thanh Hoa Looking up the normal distribution table, we have the critical level Z = 1, 64 We have: Z0 = xTN − xÐC STN S2 + ÐC N TN N ÐC 2 ÐC TN = 7.40 − 6.52 1.41 2.21 + 45 48 = 3,11 As we see Z Z thus Ho should be rejected, and Null Hypothesis H1 is accepted Thus, the average score of the experimental class is higher than that of the control class, we can accept that the teaching method in the experimental group proved to be more effective than the teaching method in the control group in training statistical reasoning skills for students students in high school From the obtained results, we have the following remarks: - The mean score of the experimental group is higher than that of the control group, the standard deviation S has a small corresponding value, so the obtained data are less scattered, so the mean has high reliability S2TN < S2ĐC and STN < SĐC showed that the dispersion in the experimental group decreased compared to the control group - The percentage of weak and poor students in the experimental group decreased a lot compared to the control group In contrast, the percentage of students achieving good and good grades in the experimental group was higher than in the control group Conclusions from the experiment: - The teaching plan in the direction of paying attention to training statistical reasoning skills for students is feasible 21 - When applying pedagogical measures in the experimental class, it is possible to improve the cognitive level, statistical reasoning skills for some weak students 3.4.3 Some comments The experiment process, through the survey and reality, we found that the current situation of teaching statistical content in high schools has not been paid enough attention, the objective reason is the situation of the Covid-19 epidemic in recent years, so The Ministry of Education and Training has issued Official Letter 3280 to reduce the load, the entire Chapter V - Statistics in the program has only periods, moreover, the content of statistics is not included in important exams (National High School Exam), so teachers often only assign tasks to students to read on their own (and as a result, many students may have overlooked this interesting and practical content) The implementation of experiments is relatively convenient, because the majority of teachers and high school leaders are aware of the position and role of statistical content in the General Education Program 2018 (will start from the school year 2022) -2023 with early grades), many teachers are interested in the increase in content and some new content of statistics Conclusion of Chapter In order to verify the correctness of the proposed scientific hypothesis, to test the feasibility and effectiveness of pedagogical measures, we organized a pedagogical experiment through rounds in two academic years 2020-2021, 2021-2022 at Thanh Hoa Ethnic Minority High School and Dao Duy Tu High School - Thanh Hoa City We have processed the data and analyzed the experimental results both quantitatively and qualitatively, the results are reliable and initially shown to be effective Specifically: - Regarding the feasibility of pedagogical measures through experiment and survey, teachers all affirmed that it is not difficult to implement and have positive effects, just need teachers to invest, research and prepare serious lesson - The hours in experimental classes are lively, students actively discuss and work in groups, many children are interested in statistical content and tend to learn about data science Students see the application of mathematics in practice, using mathematics to solve real-life problems - The learning results of the experimental class showed that the students of the experimental class understood the lesson, initially formed statistical reasoning ability and got better results than the control class, clearly demonstrated through the scores, through the level of completion of the learning task The proposed pedagogical methods have a positive impact on teachers and students 22 CONCLUSIONS AND RECOMMENDATIONS Conclusion Thesis has the following major contributions: 1.1 Synthesizing and analyzing theoretical and practical problems of environmental factors affecting the teaching process, perspectives on teaching statistics in high schools in the direction of training statistical reasoning skills for students From there, there are proposed pedagogical measures to build a positive statistical learning environment towards the expected educational outcomes 1.2 Studying the perspective of building the 2018 General Education Curriculum to see that the program approaches modern educational trends, statistical content is enhanced, general orientations on teaching methods to create the qualities and competencies of students are both specific but still ensure openness and flexibility for teachers and can be flexibly applied to the real situation to be able to train students' statistical reasoning skills in the best way 1.3 Through the investigation of the current situation, the thesis has also pointed out some difficulties and limitations in the current teaching of statistics at high schools The thesis has also proposed pedagogical measures, implementation steps, and examples when implementing to contribute to overcoming those difficulties and limitations The pedagogical measures of the thesis are anticipating the educational trend of the 2018 general education program when this program is widely deployed from the 2022-2023 academic year 1.4 The results of the thesis were initially tested and confirmed the science and feasibility in practical implementation through two rounds of experimentation in two school years with two different high schools in Thanh Hoa province with the following results: reliable results From the above results, it can be concluded: The scientific hypothesis of the thesis is acceptable, the research task of the topic has been completed, the contributions of the thesis can be deployed and applied in practice teaching statistics in high schools in the near future Recommendations 2.1 The implementation of the 2018 general education curriculum has clarified the important role and position of statistical knowledge for each student, which places a higher responsibility on each teacher in teaching We believe that teachers and Mathematics educators can continue to conduct research in the direction of the topic with teaching content at specific school levels The thesis can be viewed as a reference 2.2 Although statistics and mathematics are very close, there are still certain differences, it is necessary to have thematic instructions and training for the current teaching staff and prepare the content of mathematics pedagogical students, and statistical teaching methods to meet the requirements of the 2018 general education curriculum 23 LIST OF PUBLICATIONS RELATED TO THE THESIS Hoang Le Minh (2017), The role of the Statistical reasoning learning environment in teaching mathematics in high schools, Education & Society Magazine, Special Issue, 04/2017, p.48-51 Nguyen Chien Thang, Hoang Le Minh, Dao Quoc Dung (2017), Building a statistical reasoning learning environment for students, Journal of Education, Volume 408, 2-6/2017, p 41-45 Hoang Le Minh (2021), Training statistical understanding, statistical reasoning and statistical thinking for high school students, Vietnam Journal of Educational Sciences, Volume 47, 11/2021, p.24-28 Hoang Le Minh, Nguyen Chien Thang (2022), Teaching mathematical modeling to train statistical reasoning skills for high school students, Journal of Education, Volume 22, Special Issue 1, March 2022, p.29-33 Nguyen Chien Thang, Hoang Le Minh (2021), The basic orientations of teaching statistics content at high school in the Mathematics Program 2018, Vinh University Science Journal, (registration certificate, code ED10-2021) Scientific conference: Hoang Le Minh (2019), Statistical content in the build of the 2018 Mathematics curriculum and some measures to build a learning environment to promote statistical thinking for students, Scientific Workshop on Researching and Teaching Mathematics to meet current educational innovation requirements, Vinh University Hoang Le Minh (2021), Statistical content in the high school math program in Vietnam, Proceedings of the inter-university scientific conference at Hong Duc University