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MINISTRY OF EDUCATION AND TRAINING HO CHI MINH CITY UNIVERSITY OF EDUCATION Nguyen Huu Loi CONCEPT OF EXPONENTIAL FUNCTION IN HIGH SCHOOL: SCIENTIFIC RESEARCH AND PEDAGOGICAL TRANSFORMATION Major: Principles and Methods in Mathematics Education Code: 62 14 01 11 SUMMARY OF DOCTORAL DISSERTATION OF EDUCATION SCIENCE Ho Chi Minh City – Year 2021 The dissertation was completed at Ho Chi Minh City University of Education Science supervisor: Dr Tran Luong Cong Khanh Assoc Dr Le Van Tien Reviewer 1: Assoc Dr Nguyen Chien Thang Reviewer 2: Assoc Dr Le Thi Hoai Chau Reviewer 3: Assoc Dr Duong Huu Tong The dissertation will be defended at the University Dissertation Judging Committee at: Ho Chi Minh City University of Education at … o’clock on February …, 2022 This dissertation can be found at: - National Library of Vietnam - Library of Ho Chi Minh City University of Education - Ho Chi Minh City General Science Library INTRODUCTION Reasons for choosing the topic 1.1 The issue of competencies development teaching, interdisciplinary integration, modelling and teaching through practical and experiential activities in Mathematics The Mathematics General Education Curriculum 2018 clearly states that the teaching orientation shifts from transmitting knowledge content to developing learners' competencies and qualities with attention to integration teaching content in Mathematics and science, other fields, practical and experiential activities This is deeply concerned by many domestic and foreign educators The next sections of the thesis show how the above problems are related to the exponential function topic 1.2 The concept of exponential functions In high school, the exponential function is used in Physics, Chemistry, Biology subjects as a tool At the university level, exponential function plays an important role in differential equation theory The exponential function has many applications in population, astronomy, economics, finance and banking, radioactive decay, etc Regarding the definition of the exponential function, there are many different processes Improving the concept of exponential function through interdisciplinary integration teaching, modelling or adding appropriate task types can help students realize the role of tools or means of the exponential function 1.3 Overview of the domestic and international research situation 1.3.1 research on competencies development teaching, interdisciplinary integration, modelling and teaching through practical and experiential activities in Mathematics Many domestic and foreign authors have studied the above issues to analyze the characteristics, find solutions for education reform, measure students' achievements more accurately, develop students' competence, and train pre-service teachers in the above-related knowledge; the modelling process is teaching through practical and experiential activities 1.3.2 Studies on the concept of exponential functions Many domestic and foreign authors have researched the characteristics and origin of exponential functions, problems and mathematical concepts related to exponential functions 1.3.3 Research orientation Studying exponential function simultaneously in the following aspects: Scientology, pedagogical transposition, teaching with the integration approach, modelling and experience 2 Scope of reference theory Some main theoretical bases: Scientific Research, anthropological theory (pedagogical transposition, institutional relations) and the case theory of mathematical Didactic; competencies development teaching, interdisciplinary integration teaching, modelling and teaching through practical and experiential activities in Mathematics Research discipline The formation and development of the concept of the exponential function, including the concept of the power of a number; exponential function's approaches; The pedagogical transposition of exponential function; The relevance of the exponential function concept to other mathematical concepts; Teaching exponential function in the direction of interdisciplinary, integrative teaching, modelling, teaching through teaching and learning activities and linking teaching with practice Purposes of Research Clarifying the scientific characteristics and the pedagogical transposition of the exponential function concept from scholarly knowledge into knowledge to be taught; Consider these characteristics from the perspective of integration teaching and modelling; Designing and organizing exponential function teaching is related to capacity development teaching, interdisciplinary, integrated teaching, teaching modelling and teaching through practical and experiential activities Research methods RESEARCH THEORETICAL BASIS SCIENTIFIC RESEARCH KNOWLEDGE RESEARCH IN OF EXPONENTIAL HIGH SCHOOL FUNCTIONS (Vietnam, France, USA) DESIGN RESEARCH SITUATIONS FOR TEACHING EXPONENTIAL FUNCTIONS Expexperiment Research procedure diagram Research hypothesis H1: Organization of teaching exponential functions through designed situations based on the results of scientific analysis and textbook analysis of exponential functions and proposed exponential teaching orientations to helping students develop modelling competency, enhancing the ability to apply mathematics to solve real-world problems in other fields, thereby providing students with meaning* of the exponential function * Represents the exponential growth of the actual phenomenon; represents the exponential attenuation relationship; represents an exponential growth relationship H2: The design and organization of teaching through practical and experiential activities of e and function 𝑦 = 𝑒 𝑥 based on applying the results of scientific analysis of expression and analysis of textbooks may foster knowledge, skills, and attitudes related to students' ability to solve practical problems and create opportunities for them to access the meanings* of exponential functions * Represent exponential growth relationship; represents an exponential attenuation relationship Research assignments - Analysis: works to form the theoretical basis; historical documents, university textbooks to clarify the scientific characteristics, task types, modelling problems, and integration related to exponential function; curriculums, high school textbooks of Vietnam, France, and the United States Survey teachers on teaching capacity development, interdisciplinary integration, modelling, teaching through teaching and learning activities concerning exponential function teaching - Experiment to test the research hypothesis The dissertation contribution and significance Contribution: Clarifying the scientific characteristics of the exponential function concept, pedagogical transposition, the exponential function approaches, characteristics of each approach Improvements may be related to curriculum development associated with exponential function teaching; design exponential function teaching situations Theoretical significance: Clarifying some theoretical bases of teaching models in Mathematics; some didactic math theory; scientific characteristics of exponential function; exponential function pedagogical transposition Practical implications: Contributing a demonstration of the specific effects of pedagogical transposition on the exponential function case CHAPTER 1: THEORETICAL BASIS 1.1 Some theoretical bases on competencies development teaching in Mathematics Competence is applying knowledge, skills, and attitudes to perform well in professional work and solve practical problems There are five important components of mathematical competence: mathematical thinking and reasoning; Mathematical modelling; solving math problems; mathematical communication; using tools and means of learning mathematics Applying math knowledge and skills to solve practical problems posed in Mathematics is called practical problem-solving competence The researcher builds a rating scale according to component competence, criteria, and achievement levels to assess the capacity to solve practical problems Modelling competence is the ability to fully implement the phases of the modelling process to solve the problem posed in a real-life context Competencies development teaching in Mathematics has some objectives about characteristics; teaching content; organization method; teaching space; check; and educational products 1.2 Some theoretical bases for interdisciplinary integration teaching Integration is an activity in which it is necessary to combine, relate, and mobilize similar and similar elements and contents of many fields to solve and clarify problems and together Achieve many different goals at the same time Interdisciplinary is a combination of different disciplines in establishing or solving problems in certain situations or generating and developing new knowledge in the teaching process Interdisciplinary, integrative teaching is teaching-related knowledge of two or more subjects by an integrated method 1.3 Some theoretical bases for modelling teaching in Mathematics Mathematical modelling is the process of solving real-world problems using mathematical tools Teaching modelling focuses on applying mathematical knowledge to solve practical problems through modelling According to Nguyen Danh Nam (2015), there are seven steps of the modelling process: Find out, analyze the problem; Establish the relationship between hypotheses; modelling mathematical situation of the problem; solve math problems; Understand the actual meaning and solution of the problem; Retest the model; Announce, explain or improve the model to make it suitable for practice The researcher built a rating scale according to component competence, criteria, and achieved levels to assess modeling capacity 1.4 Some theoretical bases for teaching through practical and experiential activities in Mathematics Practical and experiential activities in Mathematics refer to activities organized to help students apply mathematical knowledge, skills, attitudes and experiences to solve practical problems Currently, some valid models are commonly used: Kolb's model (1984); Betts and Dalla's model, 1996) developed based on Kolb's model 1.5 Some theoretical elements of Mathematics didactic Didactic mathematics is concerned with constructing mathematical knowledge, the operation and the conditions of learning the knowledge in this subject Didactic interesting subjects include teachers, students, knowledge The scientific research or analysis in didactic math is what we aim to understand the arising and development of knowledge, in which the issues we need to clarify such as the conditions that allow the arising of knowledge, the meaning of knowledge, problems that give rise to knowledge, obstacles to knowledge formation About pedagogical transposition: According to Chevallard (1985), pedagogical transposition is defined: " Since being designated as an object to be taught, knowledge must undergo a set of changes to adapt, to be capable of being taught ability to occupy a place of teaching objects" The pedagogical transposition can be summarized according to the following diagram: Scholarly knowledge Knowledge to be taught Taught (Institution for (Institution for knowledge producing knowledge) transforming knowledge) (Teaching Institution) About task type: The concept of knowledge organization is formed from the view that mathematical activity is also a social activity Every human activity involves the performance of a task t of some type of task T Therefore, task, task type are concepts that make up a knowledge organization For example, The task "Solve equations 2x -1=0" belongs to the task type "Solve quadratic equations with one unknown" Extrinsic and intrinsic validation: Two forms of validation are commonly used in research: intrinsic validation and extrinsic validation Extrinsic validation evaluates experiential results between the experiential group and the control group through a test with a rather large number of "samples" Intrinsic validation differs from extrinsic validation in that we require that the experiment is based on a not too large "sample" and at the same time evaluate the experiential results through the comparison between the a priori and the posterior analysis A priori analysis is the researcher's construction of a model of a specific situation in reality (situation Sa is associated with the object of knowledge being studied) The posterior analysis reconstructs the real situation Sp that occurs when experientially deploying the case Sa This analysis will contrast what was expected in the a priori analysis with the data and relationships between the data collected in the evolution of the Sp situation, that is, the comparison between the Sa and the Sa situation (Bessot et al., 2009, p.219) CHAPTER 2: SCIENTIFIC RESEARCH OF THE EXPONENTIAL FUNCTION 2.1 Brief history of formation and development of exponential functions Before the 17th century, there were examples of exponential growth and decay The 1-to-1 correspondence between the exponential plus the exponent representing the exponential law here shows the implicit appearance of an exponential function to solve exponential growth and decay problems In another study, a Babylonian timetable of integer correspondences with its squares was found Although there is evidence that the origin for the appearance of the exponential function is very early, until the 17th century, the concepts of exponentials and algebraic notation did not develop 2.1.1 The period of antiquity to the 17th century 2.1.1.1 The problems that form the base of exponential function and related problems Documents show that in 2000 BC, the Babylonian interest rate problem appeared, which required the use of a compound interest table calculated at predetermined equal intervals At any time, the interest is calculated based on linear interpolation The problem creates the need to appear (implicitly) exponential function by a table of correspondences between the number of years and the proceeds for which the function expression to calculate is a powerful expression According to modern analysis, in this problem, exponential function takes the meaning of "representing exponential growth relationship" ; the problem has created an integrated connection between mathematics and the fields of finance and economics and presents a modelling perspective on a practical problem of compounding interest on a predetermined interest rate cycle Jacques Bernoulli studied the compound interest problem in 1685 with a more "smooth" interest cycle than the Babylonian problem; there has been a development of the compound interest problem, and although not yet reached the final result, they were aiming to construct exponential function by limits This study demonstrates the practical significance of the exponential function concept, with the defined set being the set of positive integers At the same time demonstrates the view of modelling a practical problem of calculating the smallest possible periodic compound interest In another field, since ancient Egypt, people have also recognized phenomena in biology with exponential growth Accordingly, the exponential law is understood as a one-to-one correspondence between an arithmetic process (an arithmetic progression starting from and having positive integer difference) and a geometric process (an exponential with positive integer multiples) However, the problem does not have many opportunities to introduce exponential function 2.1.1.2 Early studies on exponential functions The development of its symbols shows the development of the concept of the exponent When a part of the concept of exponentiation is discovered or extended, the symbolic notation is also born, which may improve the existing one to be more suitable As shown above, initially, from using symbols to represent numbers with powers of to using letters and words to represent numbers with higher powers (3, 4, 5, …), and then the digits In one case, the exponential function (or at least some properties of the exponential function) is detected through locus constructed in geometry Thus, from the ancient period to the 17th century, the concept of exponentiation, exponential function and studies on them appeared Concepts appear implicitly in real-life problems and situations, astronomy or explicitly in related topics such as curve representation of physical applications, fund modelling However, the mathematical theories at that time could not construct the complete definitions as today 2.1.2 The period from the 17th century now 2.1.2.1 The development of the concept of exponent At this stage, the development of exponentiation was first noted by expanding the concept of exponentiation from positive integer exponents to negative integer exponents and fractions In 1740, the imaginary exponent introduced by Euler was a special case of today's complex exponent Concept development process exponential phase by the schema: The power exponent: positive integer negative fraction imaginary 2.1.2.2 The development of the concept of a function In the 18th and 19th centuries, function definitions were formed However, the literature does not show that the exponential function is defined from another function That shows that the exponential function is only completed when there is a complete definition of exponentiation 2.1.2.3 The concept of limits for the definition of exponent When studying the concept of limit, including kinetic view, the definition is approached in the direction of the kinematics of limit concept to define an analytic problem as exponential expansion When the concept of defining exponentiation is complete, at this point, there is a definition of a function, which x shows that exponential function is also present through the expression y a 2.1.2.4 A study to solve practical problems by modelling exponential function The pedagogical problem of Nicolas Magnin, Marc Rogalski to introduce exponential function through the evolution of a phenomenon whose rate of increase is proportional to it, or by the differential equation y ' = ky The study aims to link the use of exponential function in physics, chemistry, and biology curricula 2.2 Exponential functions at the university level 2.2.1 Exponential Functions in Advanced Mathematics Textbook, Volume 2: Differential Calculus – Common Functions, Guy Lefort, University of Saigon, 1975 * Procedure included in exponential: neper logarithm The power base e Exponential base a The power base a At the same time, based on the continuity of the Nepe logarithmic function, the continuity of the exponential function on the set of real numbers is also confirmed indirectly * The scope of impact of the concept of the exponential function: The practical application examples show that the exponential function is a modelling of phenomena in nature, physics, chemistry, etc * Related objects: Nepe logarithmic function, inverse function, exponentiation, primitive * Problems related to exponential functions: Problems related to exponential function include increasing and decreasing phenomena In this type of task, the exponential function takes the meaning of "representing an exponential growth relationship" 2.2.2 Exponential functions in the textbook Les Logarithmes et leurs applications, André Delachet, Presses Universitaire de France, 1960 * Process of inputting the exponential function: Nepe Logarithmic Exponential e Real power to base e Real power to base a Exponential to base a Similar to the textbook Advanced Mathematics, this textbook defines exponential functions based on Nepe logarithmic Therefore, the properties of the exponential function are inferred from the properties of the logarithmic neper, which includes the continuity of the exponential over the set of real numbers * No task types found (due to no related assignments) 15 4.1.2 Situations 4.1.2.1 Scenario The probiotic B Subtilis reproduces by cell division with a generation time (the time from the birth of a cell until the number of cells in the population doubles) of 30 minutes Assume that the probiotic population is not killed during the fission process With the number of beneficial bacteria in the population, at first, is one beneficial bacteria How many hours does it take for an initial B Subtilis beneficial bacteria cell to divide so that the number of beneficial bacteria obtained is respectively: a) 220 beneficial bacteria b) 22020 beneficial bacteria (Bui Phuong Uyen et al., 2020), modified) 4.1.2.2 Scenario When a plant or animal dies, it stops absorbing carbon-14 (radiocarbon) and begins the decay of carbon-14 After t years of death, the total amount of y (grams) of carbon-14 remaining in an organism is expressed as a function t of y a.(0,5) 5730 , where a (gram) is the original amount of carbon-14 What percentage of carbon-14 is released each year since the organism died? (Bui Phuong Uyen et al., 2020), with adjustments) 4.1.2.3 Scenario 3: (biological situation) A function can approximate the number of eggs at each age of a Leghorn x hen y f x 179, 0,89 52 , where x is the hen's age (in weeks) and x 22 How many eggs can a chicken lay at the age of 2.5 years? Know that one year has 52 weeks 4.1.2.4 Scenario 4: (Business situation) A Honda Civic RS car at the time of purchase costs 900 million VND Its residual value p (million VND) over time using x (year) can be approximated by the function: p( x) 900.(0,9) x After how many years, the residual value of the car is 531,441 (million VND)? 4.1.3 Experimental organization process Experimental internal validation on 12th-grade students at Nguyen Thi Minh Khai High School, City Ho Chi Minh, studied the exponential function topic Phase 1: Students are introduced to the modelling process of author Nguyen Danh Nam to solve practical problems; Students work individually for 15 minutes on the worksheet The goal is to create an environment for students to solve situations and approach teaching and learning Phase 2: Repeat the situation solving in phase on each group of students in order for students to exchange, discuss, present their individual ideas, absorb the good ideas of the group, and agree on group work 16 Phase (20 minutes): The groups present their work in turn, receive comments, and the teacher comments and corrects errors Phase 3: Students discuss, teachers intervene to limit the group of students to complete the work, this is the phase of legalizing knowledge Phase 4: two individual math problems to test the ability of mathematical physics, students' attitudes about solving practical problems by math (experiential sheet: Appendix of the thesis) 4.1.4 Analyze pre-experiment situations 4.1.4.1 Scenario a) Pedagogical intent of the situation: Knowledge of cell division in biology The terms: "generation time of bacteria", "beneficial bacteria" are clearly explained in the problem The problem yields "representing the exponential growth of real phenomena" by the exponential function b) About mathematical modelling capacity: developing mathematical modelling capacity for students through applying the mathematical modelling process to solve practical problems 4.1.4.2 Scenario Pedagogical intent of the situation: The problem belongs to the field of archeology Related knowledge: radioactive decay Developing problemsolving ability by the modelling process means "representing the exponential attenuation relationship" of the exponential function 4.1.4.3 Scenario 3: Cases and survey students' progress in solving practical problems using mathematical models a) Analyzing the problem of case 3: The situation integrates with biology; students need to change 2.5 years into weeks Determine the formula for calculating the number of eggs laid b) Predicting difficulties that students will encounter:- Failure to determine the correct age of chickens (in weeks); - Incorrect understanding of the condition leads to incorrect determination of the value of ; - Students cannot evaluate the results of the problem 4.1.4.4 Scenario x Analysis of situational problem 4: p ( x) 900 (0,9) Students need to determine the correct value of x to p x 531, 441 million VND; students need to solve the exponential equation correctly, the results need to be rounded accordingly (the number of years is an integer) 4.2 Research 4.2.1 Objectives: test students' understanding of the exponential function and apply it 17 to practice in various fields Validate hypothesis H1 4.2.2 Situations 4.2.2.1 Scenario 1: Determine the exponential function's value at a point related to the banking and finance sector Situational content: Mr A has deposited money in bank B, and now Mr A requests to withdraw the entire deposit with interest Due to the software's numerical display format error, the bank reported the total amount withdrawn in a formula but could not write the result in a regular format Mr A received a notice that the amount of money in his account is billions of dong Question: Please rely on the data table below to help Mr A know the amount to be received (accurate to VND) The amount Mr A received is: VND Please explain your choice of the answer above (check only one of the four boxes that you think is the most appropriate): A Use the bottom x of the calculator to check again, you get the result Same as above B Choose the term in the last row, 3rd column of the table below, and use up to decimal places C With rQ and r increasing to , the range of 3r starts to stabilize at the answer you choose D Another opinion (specify the reason for the choice) If possible, write a few lines to guide an 11th grader (who has not learned exponentiation with irrational exponents) on how to determine 3√2 from what you've learned 4.2.2.2 Scenario 2: help students expand their understanding of knowledge related to subjects, help students be more interested in studying, understand the lesson, and apply it to practice better Case content: In 2018, city A had a population of 13,000,000 people The annual natural population growth rate is 1.2% Every year, 84,000 people come from other places to live in city A The number of people leaving city A to live elsewhere is negligible In the socio-economic development plan of city A, by 2030, all people living in the city will receive free medical care It is estimated that by 2030, the cost of medical care per person per year will be 30 million VND In order to well implement the plan, the city needs forecast data in the coming years, including data on population Question 1: To estimate the amount of money that city A needs to spend in 2030 for medical care for its people as planned, one needs to calculate the city's population in 2030 Know the formula for calculating the population of city A is: P ( t ) = 13,000,000 e (0.012+0.006) t , where 13,000,000 is the population of city A in 2018, e is the natural logarithmic base, 0.012 is the natural population 18 growth rate, 0.006 is the mechanical population growth rate, t is the period (years) from 2018 Based on the data table, please choose the line that you think is the correct and meaningful calculation From there, it shows how much money the city will have to spend in 2030 Explain this choice Solution The line you choose is (write the row number in the data table) Million VND Amount of money: Explain: Question 2: Set r = 0.012 + 0.006 = 0.018 Another formula to calculate the population in years: P1 (t ) 13.000.000.(1 r )t , is the number of years ( t =0 corresponding to 2018); the value 0.006 is the mechanical population growth rate The formula to calculate the population by quarter (3 months) is: r P4 (t ) 13.000.000.(1 ) 4t t is the number of quarters The formula to calculate r the population by month is: P12 (t ) 13.000.000.(1 )12t , t is the number of 12 rt n r nt r r months With lim Pn (t ) P(t ) , Pn (t ) (1 ) (1 ) Please tell n n n me: P(t) can calculate population growth in what periods Why? (Based on the expression lim Pn (t ) P(t ) ) n Answer: P(t) can be used to calculate population growth over time: Because the: Question 3: a) Draw a graph of exponential function s (without exact calculation) representing population growth P1 (t ) 13.000.000(1 r )t and P(t ) 13.000.000ert b) Please comment on the geometry of the graphs just sketched c) According to the above comment, please state the meaning of each graph for the phenomenon of population growth Answer: a) Sketch the shape of the graph 19 P1 (t ) 13.000.000(1 r )t P (t ) 13.000.000e rt b) Comment on the geometry of the graphs: The graph of the function P1(t): The graph of the function P(t): c) The meaning of the graph for the phenomenon of population growth: Graph of the function P (t): Graph of the function P (t): 4.2.3 Experimental organization process 12th-grade students at Nguyen Du High School, City Ho Chi Minh studied the exponential function 4.2.3.1 Scenario 1: Phase 1: making individual vote number Phase 2: make a group of questionnaires (Quotes: Appendix of the thesis) 4.2.3.2 Scenario Question 1: Question as an individual, answer sheet as a group Learn the ability to handle real-life situations related to different fields (population, amount) Students need to recognize the unit of calculation: person, million dongs Phase 3: form Phase 4: form (Appendix of the thesis) Question 2: Personalize vote (Appendix 6) Help students better understand the calculation of the population for all periods Question 3: Personalize vote (Appendix 6), find out the concept of students (through the graph of a function) about the continuity of exponential function in the case of population growth 4.2.4 Analyze pre-experiment situations 4.2.4.1 Scenario Answer 1: There are two answer strategies (True/False) Answer 2: Strategy S1A : answer option A Strategy S1B : option B Strategy S 1C : answer option C Strategy S1D : answer option D Predict the probability that strategy S1A is the highest, followed by S1B, strategy S1C S1D is a strategy with little chance of happening 20 4.2.4.2 Scenario – question Strategy S2.1A : line strategy with the oddest numbers Strategy S2.1B : line strategy number (optimal strategy) Strategy S2.1C : strategy other than line (randomly selected strategy of students) 4.2.4.3 Scenario – question Strategy S2.2A : monthly-quarterly-yearly strategy Strategy S2.2B : monthly strategy Strategy S2.2C : quarterly strategy Strategy S2.2D : strategy year Strategy S2.2e : strategy every interval (strategy expected) Predict the strategy S2.2A has the highest chance of happening, then the strategies S2.2B , S2.2C , S2.2D have the same chance of occurring, and finally the strategy S2.2E 4.2.4.4 Scenario – question Strategy S2.3A : curve graph or solid curve Comments: Graph P1 (t) is a broken line, graph P(t) is the solid curve, P1 (t) covariates (population increase); P(t) covariate (population increases with continuous-time) Strategy S2.3B : the graph is a solid curve Comment: P1 (t) is the solid curve; P(t) is the solid curve, P1 (t) covariates, P(t) covariate Strategy S2.3C : draw the correct graph Predict that strategy S2.3A has the highest chance of happening, followed by strategy S2.3B and finally strategy S2.3C 4.3 Research 4.3.1 Objectives: Expand students' understanding of e numbers, functions y e x and applications; develop practical problem-solving capacity; gives students two meanings of the exponential function: "representing exponential growth relationship" and "representing exponential decreasing relationship" ; Test hypothesis H2 4.3.2 The experiential process includes: Teaching through practical activities and experiences was designed in grade 12 of Nguyen Thi Minh Khai High School when studying exponential function 4.3.3 Practical activity and experience with the case "Discovering e-numbers and their applications." Form of working in groups (6 groups) The process is designed according to the model of D Kolb with four activities 4.3.3.1 Activity What is the e number? 1 1 1.2 1.2.3 1.2.3.4 a) Approximate S b) Complete the following table x 101 102 1 1 x x 103 104 105 106 21 c) Graph the function y = ex: Complete the following table of values: x -2 -1 x y=e - Graph the function y e x with the points according to the table above 4.3.3.2 Activity Observe, give feedback Help students realize that the calculated values are approximate The teacher calls the groups to present and poses some questions to help students understand clearly the approximate value of the special number e, and the number e can be represented in many ways 4.3.3.3 Activity Conceptualize the abstract problem The teacher shows the students to discuss, and the teacher concludes the following content: S 2,7182 The history of mathematics has recorded special numbers: , e, i The number e or the base e is the number that x 1 approximates the expression 1 as x increases; Definition of the number x x e; Graph y e 4.3.3.4 Activity Application, active testing Students two problems in the fields of economics and medicine Problem 1: An's friend plans to deposit 50 million into the bank to spend on college The bank proposes two options for An: Option 1: the bank pays a compound interest rate of 6%/quarter Option 2: compound interest continuously 4%/quarter In my opinion, which option should An choose to have more money with a deposit period of 10 years? Knowing that compound interest is calculated nt r A A0 , the formula calculates continuous n rt compounding interest A A0 e Problem 2: For medical research on the harmful effects of drugs, a function y f ( x) 5e 0,4 x is used to approximate the amount of milligram y of a drug in the patient's blood after x hours of use Ask how many milligrams of the drug are in the patient's blood after hours of use Students must perform activities out of phases: Phase 1: Students work individually for problems and in 15 minutes Phase 2: Students discuss in groups (6 groups) problems and for 20 minutes Phase 3: Teacher summarizes, evaluates and institutionalizes by the formula 22 4.3.4 Post-experiment survey 4.3.4.1 Survey problem: The air pressure p (mmHg) on the plane decreases as it increases in altitude and is calculated by the function p(h) 760e 0,145 h , h is the plane's altitude above sea level Calculate the air pressure when the plane is km above sea level 4.3.4.2 Survey question: Please rate your participation in the activity by placing an X in the corresponding box (Specifically: Very good; 4: Better than group members; 3: Average; 2: Not as good as team members; 1: Not helping the group) Point evaluation Reviews section Enthusiastic responsibility for learning activities The spirit of cooperation, respect and listening in the group Give valuable input to the team Contributing to the finished product Performance of the work involved 4.5 Conclusion Chapter Studies and have validated hypothesis H1; study valid hypothesis H2 has been proposed CHAPTER 5: RESEARCH RESULTS 5.1 The results of research 5.1.1 Results of students' work through problem 5.1.1.1 Phase Regarding results, level accounted for a fairly high evaluation criteria (37.93% - 65.52%), and only two tasks achieved the maximum level in all evaluation criteria Regarding perceived competencies: a) Capacity 1: 71 % could not find a way to solve the problem (37.93% of students could not establish a relationship, 34.48% of students showed very little problem solving) Some modelling strategies perform quite well at levels 3, 4: diagram of time relationship - number of cells, number of dissociations - number of cells, the distance between dissociations - number of cells; then, one part converts to the correct formula, another part converts to proportional, exponential, the rest has not yet detected the problem direction b) Capacity 2: Level 1, 2: accounting for a high percentage (70%) c) Capability 3: 65.52% of students did not complete the assignment (wrong, unable to it); 6.9% of students make mistakes in presentation and calculation; 27.59% of students check the results and give answers More than 23 70% of students did not answer the question for the situation despite having the correct answer 5.1.1.2 Phase There were three groups (50%) reaching the maximum criteria, group (16.67%) still could not find a reasonable solution, two groups had small errors The data show that group work gives better results because each individual has reached phase Group can write the formula for calculating the number of cells and solve it Groups 2, 3, and share the same idea of deducing by proportionality to find the number of hours for a cell to divide to reach a given number Group could not determine the relationship, could not find the modelling, could not solve it Through the results of phase 2, we achieve the goal for students to exchange and discuss based on having learned the problem in phase 5.1.1.3 Phase When listening to your group's presentation, the groups that have not yet found the answer understood the problem and solved the problem Particularly for group 6, they wondered why they could not use the formula for calculating population growth recorded in the textbook 5.1.2 Results of students' work through problem 5.1.2.1 Phase Compared with problem 1, the percentage of students at level is still high in the criteria (31.03% to 92.76%), but increases at level 4, students complete the test, the standards other lice at level increased Meaning: Students learn from previous problems Capacity 1: Level 1: 31.03% (lower than problem 1), level significantly increased (41.38%) Students have experience from problem Ability 2: Solve mathematical problems posed in the built model More than 50% of students are at level 1; they still have not found an appropriate math operation to solve Ability 3: Explain the correctness of the problem results Although some exercises show the correct formula, more than 50% of students not have a situational answer 5.1.2.2 Phase In phase 2, the groups achieved quite good results in capacity Group could not present the modelling The proportion of groups rated at levels and for both criteria is 75% or more 5.1.2.3 Phase 3: Through group presentation, students find the focus: % of carbon-14 released after each year is equal to the initial mass minus the 24 remaining mass after one year (then t=1) dividing the initial mass, understand formula meaning, state the meaning of each solution and implementation steps 5.1.3 Post-experiment survey results 5.1.3.1 The results of the assessment of skills based on the scale Ability 1: Students have established the relationship between the assumptions that gradually improve through survey problems Students are assessed mainly at level 4, achieving over 65% for both problems Students have better mathematical language skills; the percentage of students who build complete math problems in problems and is 68.97% and 86.67% Some students were evaluated at levels 2, because: they had not recognized all the mathematical elements, incorrectly identified the objects in the model, and were unable to transfer all the assumptions about the mathematical form and reexpressed them Ability 2:The ability to use math knowledge is sensitive; most of them are good at solving math problems The rate of students solving problems at levels and for both problems is over 90%, a few students solve math at level (because they not say the problem well, so they not know how to solve it) Capability 3: Students show a fairly stable ability to analyze and understand the meaning of the solution The rate of students correctly answering questions for problems and is 58.62%, 82.76%, a few students have difficulty giving answers to problems and 4, in which, the error of Students are in lesson 3, because they focus too much on the condition x 22 to deduce the age of chickens, many students make this problem wrong From another perspective, the research results lead to the affirmation that the students' computational ability is better developed through experiential teaching of situations and as follows: From experiential problems 1, 2, students already know the steps to solve a problem using modelling However, the level of achievement of the student's mathematical ability criteria is still low; over 70% of the assignments are at levels and in the criteria With the experiential phase, after doing individual work, students can work in groups, give presentations, discuss in class, create conditions for students to learn from experience, practice and develop mathematical skills After the experiment, the achievement of the criteria of the students has markedly improved, shown in problems and Accordingly, the percentage of students is evaluated mainly at level for the criteria Even the situational scores were above 65% In addition, students' ability to recognize mathematical knowledge to be used is more sensitive 90% of students can solve math problems in both problems In addition, the results of the two problems after the experiment also showed that the students showed quite a stable ability 25 to analyze and understand the meaning of the solution to answer the problem 58% of students correctly answered questions for problems and Thus, it can be concluded that teaching students through designed situations contributes to helping students develop their mathematical skills and enhance their ability to apply mathematics to solve practical problems in other fields, providing the meanings of the exponential function 5.1.3.2 Results of attitude assessment (through interview questions) 65.52% of students answered "liked" the problem-solving lesson by modelling (learn more new information outside the subject); 34.5% of students are "normal" or "dislike" such a class (who is not good at Math, has long disliked this subject) The form of real math students often encounter, such as interest rate and cell division, so when there are more lessons in biology - agriculture, archeology, car value calculation, students feel interested Because of the newness of knowledge and the breadth of mathematics in many fields, many students want to take similar lessons, although there are still some difficulties, such as using the language of mathematics to describe and present the answer In summary, through experiential problems, in general, students' mathematical ability has improved Experiments have proved the correctness of the research hypothesis H1 5.2 The results of research Experiments show that students can calculate the values of exponential function well, but many students have not shown their understanding of the value of the exponential function in specific real-life situations In the following situation, students have more problem-solving skills considering the relationship with other fields, but students have not connected well the problem results with other fields (their's field) The situation is sociological) Therefore, students need to practice many math problems related to many different fields and interdisciplinary subjects For example, the results of the exercise of question in experiential case are as follows: Strategy S2.1A Strategy S2.1B Strategy S2.1C Student code 17 55 Percentage (%) 22,3% 5,3% 72,4% S2 1A : line selection strategy that many of the oddest S 2.1B : line strategy number for the optimal answer S 2.1C : strategy of selecting lines other than (random) Predict that S2.1A has the highest chance of occurring, followed by S2.1C , and then S2.1B However, the students who chose the most S2.1C , in which the number of students who chose line number had 28 students with the explanation that lines from onwards were stable at the resulting value (showing math skills with takes into account relationships with other sectors but is not 26 appropriate for this situation) S2.1B very few students choose (students have not been able to connect the results of the problem with the field of sociology) Research tested hypothesis H1 5.3 The results of research When groups participate in experiential learning activities, the majority understand the requirements of the problem and have the expected results For example, all groups can compute the sum S required by the problem On the other hand, in question b) the problem requires calculating the expression value x 1 at x 10 , most of the other groups' results are close to 2.71828 but x one group gives the same result This is a valid prediction, so even though the result is incorrect, you have made initial guesses that the number e or the natural x 1 base e approximates the expression 1 as x increases The results also x show that the teaching process through experiential activities in groups of students has the following advantages: Students understand the work as well as the assigned tasks; teaching activities ensure the requirements set forth; Group discussions are very lively, students actively contribute ideas, generate arguments to help students develop their ability to explain arguments Activities such as observation, feedback; Conceptualizing the abstract problem; Actively applying and experimenting with helping students better understand the problem to be solved, assessing students' practical problemsolving ability, creating a sense of active learning, excitement, cooperation in learning, self-expression, develop problem-solving capabilities The study has achieved the set objectives and verified the correctness of the research hypothesis H2 5.4 Chapter conclusion Students' advanced mathematical skills can generally meet the research objectives set out to experiment with mathematical modeling There is an improved understanding of real-world problems and their application to exponential function-related problem-solving The level of understanding of the exponential function and the ability to apply it to handle real-life situations related to the fields is increased, realizing the ability to solve problems with interdisciplinary integration between Math and other subjects Experiential activities show that students are active, interested and proactive in class In addition, the experiment has validated the hypotheses H1, H2 of the thesis 27 CONCLUSIONS AND RECOMMENDATIONS Achievements - Important scientific features: The problem of compound interest appeared in Babylonian times, solved by a pre-calculated compound interest table with linear interpolation Meaning exponential function: "represents exponential growth" In addition, there is a problem in growth biology that follows an exponential rule with related concepts of "arithmetic process" and " geometrical process" The exponential solution technique corresponds to an arithmetic progression with a geometric progression and the addition and double operations Meaning the exponential function "represents the exponential growth of the actual phenomenon" - Early studies on exponential function by Nicole Oresme (15th century), Chuquet, Ramus (1569) Powers to real exponents were then defined by Johann Bernoulli, powers to imaginary exponents by L Euler Finally, powers with irrational exponents are defined by the limit - There is a very diverse pedagogical transposition for the exponential function - The teaching models in the current Vietnamese textbooks have not given adequate attention and have not created favorable conditions for students There are difficulties for students to solve real-world problems by using mathematical models The teacher survey results show that most teachers are interested in the above teaching models, but the level of approach and application is not high, and teachers still face certain difficulties This speaks to the necessity and importance of teacher training activities The thesis has validated two research hypotheses H1, H2 Limitations The thesis has not designed and implemented many teaching and learning situations to help students discover new knowledge to meet the new educational curriculum The experiments were carried out in Ho Chi Minh City; therefore, some results are still local, the representativeness of the subjects participating in the experiment is not rich Recommendations Regarding teaching students: strengthening practical problems applying mathematical modelling to help students develop problem-solving and mathematical modelling skills, helping students to mobilize good mathematical knowledge to solve related problems related to the fields, especially economics, 28 financial management This is also an opportunity to help students approach the meanings of the exponential function Foster and update full teaching models for teachers at high schools as well as students at teacher training institutions Further research directions i) The exponential function has a close relationship with the logarithmic function Therefore, this research direction also needs attention ii) Research teaching exponential functions, including using information technology to help students discover new knowledge iii) Research on the problem of examining and evaluating exponential function content to develop students' capacity LIST OF PUBLICATIONS Nguyen Huu Loi, Tran Luong Cong Khanh (2021), Mathematics teachers' perspectives on teaching by modelling and experiential learning , Eur opean Academic R research , Vol VIII, Issue 10/January 2021, ISSN 2286-4822, p.6241-6261 Nguyen Huu Loi, Tran Luong Cong Khanh, Le Van Tien (2020), Connecting mathematics and practice: a case study of teaching exponential functions , European Journal of Education Studies – Vol 7, ISSN: 2501 1111, ISSN-L: 2501 1111, DOI:10.46827/ejes.v7i12.3424, p.612-624 Tran Luong Cong Khanh, Le Van Tien, Nguyen Huu Loi (2020), An Analysis Of The Concept Of Exponential Functions In History And Textbooks In Vietnam , The International Journal of Engineering and Science (IJES), Volume 9, Issue 11, Series II , Pages PP 23-28, ISSN (e): 2319-1813 ISSN (p) : 20-24-1805, DOI:10.9790/1813-0911022328, p.23-28 Nguyen Huu Loi, Tran Luong Cong Khanh (2019), Two sides of the dialectical impact of the development of mathematical concepts: the case of exponential functions , April 2019, ISSN 1859-3100 - Scientific journal of Ho Chi Minh City University of Education HCM, Vol 17, No (2020):211-221 Tran Luong Cong Khanh, Nguyen Huu Loi (2017), A scientific survey on the expansion of the concept of exponentiation , Proceedings of the 6th International Conference on Didactic Mathematics, 4/2017, ISBN 978-604947 -988-5, pp.147-153 Nguyen Huu Loi (2017), Forms of expression of concepts and processes of teaching concepts in high schools – the case of exponential concepts , Proceedings No Scientific conference of graduate students and researchers Student (Scientific Journal of Ho Chi Minh City University of Education), Publishing House of Ho Chi Minh City University of Education HCM Tran Luong Cong Khanh, Nguyen Huu Loi (2016), Exponential functions: An epistemological and comparative study between Vietnam and France , Scientific Journal of Ho Chi Minh City University of Education HCM, 4/2016 (82), ISSN 1859-3100, pp.61-70 Nguyen Huu Loi (2014), Some scientific factors on exponential functions , Scientific conference of Faculty of Mathematics and Information Technology, University of Pedagogy of Ho Chi Minh City Ho Chi Minh ... Calculus – Grade 12, Division C, E – Hachette Publishing House 1997 (Mathématiques – Analyse – Terminales C et E Hachette 1997) - The process of inputting into an exponential function: Logarithmic... mathematical knowledge, skills, attitudes and experiences to solve practical problems Currently, some valid models are commonly used: Kolb's model (1984); Betts and Dalla's model, 1996) developed... states that the teaching orientation shifts from transmitting knowledge content to developing learners' competencies and qualities with attention to integration teaching content in Mathematics
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