Using visual representations to support inductive reasoning and abductive reasoning of 15 year old students in discovering mathematical patterns

MINISTRY OF EDUCATION AND TRAINING HO CHI MINH CITY UNIVERSITY OF PEDAGOGY TRUONG THI KHANH PHUONG Specialization: Theory and Methods of Teaching and Learning Mathematics Scientific Code: 62.14.01.11 SUMMARY OF DOCTORAL THESIS ON EDUCATIONAL SCIENCE HO CHI MINH CITY– 2015 THE THESIS COMPLETED IN: UNIVERSITY OF PEDAGOGY HO CHI MINH CITY Supervisor: Assoc Prof Dr Le Thi Hoai Chau Assoc Prof Dr Tran Vui Reviewer 1: Assoc Prof Dr Vuong Duong Minh Ha Noi University of Pedagogy Reviewer 2: Assoc Prof Dr Nguyen Thi Kim Thoa Hue University of Pedagogy Reviewer 3: Dr Le Thai Bao Thien Trung Ho Chi Minh City University of Pedagogy The Thesis Evaluation University Committee: HO CHI MINH CITY UNIVERSITY OF PEDAGOGY Thesis can be found at: - General Science Library of Ho Chi Minh City - Library of Ho Chi Minh City Pedagogical University THE PUBLISHED WORKS OF AUTHOR RELATED TO CONTENT OF THESIS Truong Thi Khanh Phuong (2011), “Using dynamic visual representations to support inductive reasoning and abductive reasoning of students in the process of exploring mathematics”, Journal of Science – Hanoi national university of education, ISSN 0868-3719, No 05(56), pp 109-116 Truong Thi Khanh Phuong (2011), “The potential of open-ended problems in supporting students to develop abductive reasoning competency”, Journal of Education, Ministry of education and training, ISSN 0866-7476, No 276 (period 2-12/2011), pp 34-36 Truong Thi Khanh Phuong (2012), “The reflection of abductive and inductive reasoning through dragging manipulation in the dynamic geometry environment”, Journal of Science – Ho Chi Minh City University of education, ISSN 1859-3100, No 33 (67), pp 28-35 Truong Thi Khanh Phuong (2014), “Using Open-ended problems to enhance students’ abductive reasoning in mathematics classroom”, In Bulletin: Multilingual education and philology of foreign languages Almaty (Kazakhstan), ISSN 2307-7891, No 2(6), pp 84-91 Truong Thi Khanh Phuong (2014), “Creating open-ended problems to improve students’ abductive reasoning in mathematics classroom”, Journal of Sciences - Hue University, ISSN 1859-1388, Vol 99, No.11, pp 49-59 Truong Thi Khanh Phuong (2015), “Abductive reasoning and inductive reasoning in discovering sequence patterns – some theoretical and empirical analysis”, Journal of Science – Ho Chi Minh City University of education, ISSN 1859-3100, No (75), pp 16-28 Chapter INTRODUCTION 1.1 Introduction research issues The most common description about mathematics accepted by most mathematicians is: Mathematics is the science of patterns One of the ways to describe patterns is showing its rules through functions and relationships In particular, the process of looking for mathematical rules relates to two types of reasonable reasoning namely abduction and induction Reasoning and representation are also two of the eight capacity selected for evaluation in the program of international student assessment PISA 1.2 Demand for research and speech research issues Inductive and abductive reasoning, with its significance in helping students to explore math knowledge through discovering rules in patterns, should be paid more attention in math education Step into the early years of the 21st century, the trend of applying mathematics in most of the problems that students encounter in life is studied globalization People rarely using deductive reasoning because of their strict standards Again, abductive reasoning and inductive reasoning become effective tools for students when facing with real life problems The object that we are interested in this study is 15year-old students, who recently completed education program officially and need to choose between continuing high school program or become an independent citizen with a career for the future right now This transition period has an important significance when mathematical abilities were accumulated by students will have a big impact on the success of the students in some next school years and their career later Moreover, 15-year-old students were also subject of the program of international student assessment PISA, an educational assessment program is held periodically every years with the size of nearly 70 countries around the world including Vietnam In this trend, we choose: “Using visual representation to support inductive reasoning and abductive reasoning of 15-year-old students in discovering mathematical rules” as a topic of this thesis 1.3 Scope of research In this thesis, 15-year-old students mean the students who started attending grade 10 at Vietnam Specifically, the rules that we would like to focus in the field of algebra are relating to the term “number sequence” Until students were fifteen, they learned about the concept: “algebraic expressions”, “linear function”, “quadratic function”, ie they have enough knowledge to discover the linear function number sequence and the quadratic function number sequence Because students have not officially learn the concepts of arithmetic and multiplication so we have a chance to evaluate more objectively the effects of visual representation to the students’ reasoning in discovering the rule of the number sequence Moreover, this is one of the interesting content because of simultaneous occurrence of inductive reasoning and abductive reasoning during the process of discovery and generalization the rule of number sequence Besides, we are also interested in the capabilities of students exploring rules in the field of geometry With 15-year-old students, we choose the geometry knowledge related to topics such as parallel and perpendicular relationship, polygons and circles which students were enrolled in the Geometry program in the class of grade 8, and the earlier of grade 10 On the other hand, we also want to consider visual representation forms created in learning environments that use computers and the dynamic geometry software To provide opportunities for students to discover the rules of mathematics in the field of geometry with the support of the dynamic visual representation, we choose the open-ended geometry problem as an object for exploitation and analysis in the experiments of this thesis 1.4 Mission research 1.5 Research questions Research question 1: What kind of reasonings are used in discovering the number sequence and what is the relationship between them? Research question 2: How does the visual representation describing the number sequence affect the students’ reasoning in discovering a general rule? Research question 3: How to use the visual representation to support abductive reasoning and inductive reasoning while discovering openended geometry problem? Research question 4: How to develop the ability of students to discover the rules of mathematical patterns by inductive reasoning and abductive reasoning? 1.6 Definition of key terms 1.7 Structure of the thesis Chapter LITERATURE REVIEW 2.1 Mathematic and plausible reasonings 2.1.1 2.1.1.1 Inductive reasoning Definition Reasoning that give a general hypothesis (not sure exactly) from verifying the correctness of the hypothesis for a number of specific cases (Polya, 1968, [68]; Cañadas & Castro, 2007, [23]; Christu & Papageorgiu, 2007, [27]) 2.1.1.2 The model of inductive reasoning Canadas & Castro (2009, [24]) offers seven-step model for the process of inductive reasoning: (1) Observation of particular cases; (2) Organization of particular cases; (3) Search and prediction the rule of patterns; (4) Conjecture formulation; (5) Conjecture validation; (6) Conjecture generalization; (7) General conjectures justification 2.1.2 Abductive reasoning 2.1.2.1 Abductive reasoning from the viewpoint of logic and philosophy of Peirce Peirce was the one who developed the concept of abduction and putting it into the system of reasoning The form of abduction: The surprising fact, C, is observed; But if A were true, C would be a matter of course; Hence, there is reason to suspect that A is true ([65]; 5.189) 2.1.2.2 Abductive reasoning from the viewpoint of J Josephson and S Josephson: J Josephson and S Josephson (1996, [39]) inherited the definition about abductive reasoning of Peirce and added to his model a stage: select hypotheses that yield the best explanation The new form is: D is a collection of data (facts, observations, givens) H explains D (would, if true, explain D) (1) (2) No other hypothesis can explain D as well as H does (3) Therefore, H is probably true (4) 2.1.2.3 Abductive reasoning from the viewpoint of problem solving of Cifarelli 2.1.2.4 The classification of abduction • By Eco • By Magnani • By Patokorpi Erkki (2006) divided abduction into four basic forms: • Selective abduction: Select a Rule from the available Rules that can explain the Conclusion • Creative abduction: When the available Rules not help to explain the observation, it should invent a new Rule that can explain the Conclusion • Visual abduction: Thinking during the process of observation to hypothesize a Case that can explain the Conclusion • Manipulative abduction: Doing the appropriate actions during the process of discovering in order to collect more data for hypothesizing a Case that can explain the Conclusion 2.1.2.5 Model of abductive reasoning 2.1.3 Distinguishing deduction, induction and abduction in mathematics 2.1.3.1 In terms of conditions for the occurrence and outcome of three types of reasoning 2.1.3.2 In terms of the purpose of using each kind of reasoning 2.1.3.3 In terms of the level of certainty of outcome The certainty of the conclusions by three kinds of reasoning descending from deduction to induction and finally abduction However, in terms of discovering new knowledge, knowledge inferred from deduction can be seen as the logical consequence from known axioms, so it can not extend the intellectual capital of the people With induction, new knowledge obtained as a generalization, which is expanding the scope of knowledge under the foreseeable trends With abduction, when the available knowledge does not explain the observations, new knowledge is formed Therefore, abduction help provides new ideas and expand our knowledge 2.2 Mathematical representation 2.2.1 Classification of mathematical representation 2.2.2 Visual representation 2.2.2.1 Visualization 2.2.2.2 Visual representation describes the rule of number sequence 2.2.2.3 Dynamic visual representation 2.3 Discovering the rule of number sequence 2.3.1 Tasks of discovering the rule of number sequence 2.3.2 The cognitive levels in discovering the rule of number sequence 2.3.3 The strategies to discover the rule of number sequence 2.3.4 Reasoning used in discovering the rule of number sequence When referring to the reasoning based on observing some similar particular cases to a general result, people often think of inductive reasoning Abduction is not even mentioned in the analysis of the authors Reid (2002, [72]), Canadas & Castro (2007, [23], 2009, [24]) about the reasoning of students while discovering the rule of number sequence However, we seemed to ignore the creative element in this process - factors that Peirce pointed out as a characteristic of abduction Meanwhile, Canadas & Castro ([23]) has confirmed that hypothesis formation (step 4) is important and most often appear in the students’ paperwork This is clearly a task of abduction Some questions we posed: Does abduction participate in the process of discovering the rule of number sequence? If so, it appears at which step? Focus again on Peirce’ study of abduction, especially in the 2nd phase (from 1878 onwards), Peirce began to use the term “abduction” to refer to “the first starting of a hypothesis” (Peirce, [65, 6.525]) “Abduction merely the beginning It is the first step of scientific reasoning while induction is the concluding step” (Peirce, [65, 7.218]) 10 Chapter RESEARCH DESIGN To answer research questions and 3, we conducted two studies: Study 1: Surveying the effects of visual representation to abductive reasoning and inductive in discovering the rule of number sequence Specifically, we would like to clarify the following issues: (1) How does visual representation affect the strategies that students use to explore the rule number sequence?; (2) Do students use visual representation in the phase of verifying and generalizing hypothesis by inductive reasoning?; (3) The level of abductive-inductive reasoning that students achieved Study 2: Surveying the effects of dynamic visual representation on the processes of inductive and abductive reasoning when students explore open-ended geometry problems in GSP Specifically, we would like to clarify the following issues: (1) In the course of exploring open-end geometry problem, are induction and abduction in paper and pencil environment different from in GSP geometry environment?; (2) How are abduction and induction reflected through four modes of dragging when students explore open-ended geometry problems with dynamic visual representation? 3.1 Research design The survey is used for Study because it is suitable for collecting information from a large number of cases Case studies are used for Study because it is suitable for the research question “what?” and “how?”, in combination with the method of treatment interview 11 3.2 Research subjects Study 1: A pilot study was conducted on 78 students in two grade 10 classes of Le Loi high school (Gia Lai province) and Phong Dien high school (Hue City) The official study was conducted on 326 students of eight grade 10 classes in five high schools in Thua Thien Hue province Study 2: We selected eight students in grade 10T2 of Quoc Hoc high school and two teachers who are teaching math for this class as objects for this study The students will be divided into pairs, each pair work on a computer Two teachers will monitor the work of two groups of students and conduct interviews as necessary 3.3 Instrument Study 1: A research instrument was specifically composed for this study is Questions Test with some following criteria: (1) The number of tasks in each Question set: The Questions Test including six tasks of discovering the rule of number sequence is divided into two Questions Sets Each Questions Set has three tasks and will be completed by students within 30 minutes (2) This type of rule: The Question Test will have two number sequences related to the linear function rules and four remaining number sequences related to the quadratic function rules (3) Visual representation described number sequence: We used squares symbolizing the cover plate (or bricks) as a single form of visual representation for Question Test 12 (4) Question scenario: We try to provide a practical context so that each question becomes meaningful for students (5) Matching question: Each task relating to the linear function rule will correspond with a task relating to quadratic function rule (6) The structure of each task: With all tasks in the Question Test, students must perform two requirements: (1) proposing a rule in the general case and (2) describing how to find that rule This structure enables students to freely express different approaches to the task and we also have the opportunity to check whether students have done the process suggested in Figure 2.13 or not Study 2: In the pilot study, teachers introduced four dragging modes to students Then students have time to practice with the dragging modes to solve the following problem: Problem The perpendicular bisectors of the sides of a quadrilateral ABCD form a quadrilateral HKLM a) Drag ABCD, consider all different configurations of the quadrilateral HKLM? b) Can the quadrilateral HKML be a point? Which property of the quadrilateral ABCD is necessary so that this situation occurs? In the main study, we used two open-ended geometry problems as follows: Problem Let ABCD be a quadrilateral Outside of this quadrilateral, construct four squares whose sides are respectively AB, BC, CD, DA Let M, N, P, Q be correspondingly the centers of these 13 squares In general, does the quadrilateral MNPQ has any special property? Problem Arbitrarily given three points A, M, K B is symmetric with A through M, C is symmetric with A through K, D is symmetric with B through K Drag M and makes predictions about the possible shapes of the quadrilateral ABCD In which conditions for ABCD be a rectangle? 3.4 Collecting data 3.5 Data analysis Study 1: To count data, we make a coding scheme Table 3.3 The coding scheme for abductive strategies Category Arithmetic Geometric Undefined Code Description 11 Comparison 12 Repeated substitution 13 Solving the equation 14 Guess and check 15 Compare unit with overall 21 Constructive 22 Deconstructive 23 Reconstructive 24 Figure-round reversal 2123 Constructive followed by reconstructive 31 Correct rule using an indeterminate strategy 90 Incorrect strategy 99 Blank 14 We also offer a Classification of abductive-inductive reasoning level based on “reasonable” and “best” of hypothesis: ● Level 1: The hypothesis suggests that students absolutely did not recognize similar characteristics between visual representations or in the collected data ● Level 2: The hypothesis has not shown a link between the number of cards (number of seats) of each visual representation with the size (width) respectively They only showed some elements of the available terms developing in accordance with a rule, but these elements are not enough to describe the entirety rule of the number sequence ● Level 3: The hypothesis only explains the rule found between available specific cases but not shown that this hypothesis will be generalized for the entire number sequence ● Level 4: The hypothesis suggests that students recognized the rule with reasonable justification, though still not optimally meet requirements of the problem (eg recursive strategy is not really effective when calculating the terms at any position of number sequence) ● Level 5: The hypothesis has two elements: “reasonable” (fully explained) and the “best” (may develop into function rule) Study 2: Based on the transcript was recorded by Screen Recorder software, we analyze reasoning corresponding to the dragging modes that students manipulate on dynamic visual representation 3.6 Restrictions 15 Chapter VISUAL REPRESENTATION SUPPORTS INDUCTIVE REASONING AND ABDUCTIVE REASONING 4.1 Effect of visual representation on discovering number sequence by inductive reasoning and abductive reasoning 4.1.1 The abductive strategies used in discovering number sequences 4.1.2 The classification of the level of abduction-inductive reasoning in discovering number sequences 4.1.3 Summary from empirical result of Study ● The first problem: students not really understand the meaning of the variables n In particular, most of the misunderstanding of the meaning of variables n appear in the arithmetic abductive strategies Meanwhile, the confusion about the meaning of the variables n in the geometry abductive strategies is that students not use the selected variable or use more than one variable, but the mean of the variables in the formula is still guaranteed ● The second problem: When moving from exploring number sequence under the rule of a linear function to the rule of a quadratic function, students trend to continue using recursive strategy if they had success with the linear function before However, almost students were not able to success with this strategy It appears that this approach has hindered them to think of other strategies ● The third problem: When using arithmetic abductive strategies like Solving the equation, Guess and check, students not give 16 explanations why the rules described number sequence is a linear function or a quadratic function ● The fourth problem: Many students not get the benefit from the visual representation Most students encounter the following obstacles: - When only interested in numerical data, students tend to think of recursive strategy rather than other strategies This is detrimental to students when finding the rule in quadratic functions number sequence - The most popular arithmetic abductive strategy that students use is Guess and Check According to Radford (2008, [71]), this strategy does not promote algebraic thinking - When converting a number sequence described by visual representation into a number sequence described only by number data, students will not have the facility to test a hypothesis for the unknown cases 4.2 Dynamic visual representation supports inductive reasoning and abductive reasoning in exploring open-end geometry problem 4.2.1 Inductive reasoning and abductive reasoning in dynamic geometry environment 4.2.2 The support of dynamic visual representation to inductive reasoning and abductive reasoning in dynamic geometry environment 17 4.2.2.1 Reflection of inductive and abductive reasoning through the dragging modes 4.2.3 Summary from empirical result of Study Based on experimental results, we summarize a process of using four dragging modes to propose, verify and generalize hypothesis by inductive reasoning and abductive reasoning while exploring openended geometry problems in GSP dynamic geometry environment ❖ Phase Discovering randomly At this stage, the students use combination random dragging and dragging on the particular cases to explore some interesting properties when observing numerous different instances of dynamic visual representation ❖ Phase Detecting invariant Detecting a property T In experiments of this thesis, two cases occurred: a) T always happens to all different instances of visual representation T is detected through random dragging, but sometimes students discovered T first by dragging on the particular cases b) T appears only in some specific cases not to be determined yet In this case, T might be suggested by the question in the problem or can be a surprising result that students want to discover further Students use random dragging, dragging on the particular cases to realize that in some specific circumstances, the property T will be maintained 18 ❖ Phase Proposing hypothesis by abductive reasoning - In the case a): The hypothesis is stated as: “In terms of the problem, the property T always happens” - In the case b): Students use maintain dragging to affirm a set of points D so that when you drag a point on visual representation to one of the points of this set, properties T is maintained Using maintain dragging and dragging with trace activated to mark the collection points D This collection can be realized as a geometric locus Q The hypothesis is stated: “If the point D is located on locus Q then the property T is satisfying” ❖ Stage Verify/refute hypotheses by experiment through inductive reasoning - In the case a): Students use random dragging associated with the measurement tools and calculations tools of GSP to verify the hypothesis by experiment - In the case b): Students uses linking dragging to link point D to locus Q in order to confirm the hypothesis by experiment 4.3 Summary of chapter 4.3.1 Conclusion of Research Question The experimental results showed that there are low number students who come up with a right rule (11,95%-22,16% for the quadratic function and 54,5%-61,64% for the linear function) Four issues affecting the results they achieved were: (1) students misunderstand the meaning of variables n; (2) students tend to use the recursive strategy without regard to other alternatives; (3) students only used 19 visual representation in data collecting phase for the given case without regarding to the structure contained in this visual representation; (4) students does not test the hypothesis for unknown terms in number sequence The analysis of experimental results indicates that paying attention to the regularity of the visual representation can help students avoid four issues mentioned above, thereby reducing errors in the results Also, the difference in the amount of equivalent function rules generated by the arithmetic abductive strategies and by the geometry abductive strategies of the six tasks shows that interesting in visual representation helps students have many different views about the rule, especially with quadratic function number sequence Another important aspect is that students using visual representation during discovery process often show that they have a reasonable hypothesis through clear explanation on the structure of the visual representation Students also have the facility to test a hypothesis by inductive reasoning based on the description of the next terms or even the general terms of the number sequence through the construction of the corresponding visual representation Therefore, we think that interested in visual representation is one way to reduce the answer in low level 1, and put them on a higher level in the classification of inductive-abductive reasoning level that we established 4.3.2 Conclusion to Research Question 20 Experimental results in this study show that there is a relationship between dragging modes that students manipulate on visual representation and the corresponding forms of reasoning Chapter DEVELOPMENT THE ABILITY TO EXPLORE MATHEMATICAL PATTERNS FOR STUDENTS BY ABDUCTIVE AND INDUCTIVE REASONING 5.1 Abduction and induction in mathematics activities at school 5.2 Tasks help develop abductive and inductive reasoning Through two empirical studies of this thesis, we get that finding rule in number sequence and discovering open-ended geometry problem are two kinds of tasks that promote students to use abductive reasoning and inductive reasoning because of two common characteristics First, these tasks not provide a definitive conclusion like a traditional proving problem Secondly, these tasks are not familiar with students and not have a clear process to ensure the right solution Those are basic characteristics of an openended problem Through developing a closed problem into an openended problem, we can provide more opportunities for students to use plausible reasoning by the following reasons: ● Tackling open-ended problems requires students to propose an appropriate strategy, to choose an adequate hypothesis and eliminate redundant assumptions, to select knowledge or rules that may be used along with the reasons for that choice ● Supporting the viewpoint of Foong (2002, [36]) that the problem “lack of clear information, lack of a permanent process to ensure 21 the right solution, and lack of a standard for evaluating solutions achieved”, we can create opportunities for students to explore different options in a situation, generalizing the results by inductive reasoning, or choosing a best answer based on available knowledge and the individual explanation ● Some open-ended problems not provide enough data This makes it difficult for students who want to use deduction to apply known formulas or algorithms However, students are required to expand their knowledge by suggesting hypothesis based on incomplete data, or adding data to create a new problem ● Open-ended problems with the supporting of dynamic geometry software (such as GSP, Cabri ) will make easier for students to explore, hypothesize, verify or disprove the hypothesis 5.3 Creating open-ended problems to develop capabilities by exploring math reasoning and inductive abduction 5.3.1 Posing problem 5.3.2 Investigating problem 5.3.3 The problem led to the formation of new concept, new rule 5.3.4 Predicting mathematical properties from drawings 5.3.5 Fiding rules in patterns 5.3.6 Changing the familiar requirements in textbooks 5.3.7 Solving real-life problem 22 CONCLUSIONS OF THE THESIS The results of the thesis In terms of theoretical ● Distinguish three types of reasoning: deduction, induction and abduction in mathematical contexts, showing that two types of reasoning used in combination to discover the rule of number sequence are induction and abduction ● Propose the process of discovering the rule of number sequence by abductive reasoning and inductive reasoning ● Recommend the classification of levels abductive-inductive reasoning fit the performance of students in mathematics discovery process via empirical part of this study ● Proposing activities practicing inductive reasoning and th abductive reasoning in grade 10 math class ● Show the evidence that visual representation is an effective tool for students in the process of exploring and generalizing number sequence by induction and abduction ● Summarizing the process using the dragging modes to discover the open-end geometry problems by inductive reasoning and abductive reasoning from experimental studies In terms of practical ● The system of examples in the thesis can be a source of reference for teachers and students to practice inductive reasoning and abductive reasoning while exploring mathematics 23 ● Propose the ways to design of open-ended problem from the problems of the textbook Note for teachers From the experimental results of Study 1, we give some notes on pedagogy for teacher when practicing these tasks in the classroom: ● Same rules for a number sequence can have many different abductive ways to explain Teachers should introduce students to the other abductive ways than the one that students was used ● Teachers should point out to students that even if they come up with a hypothesis to explain the given circumstances, they still should verify this assumption for other cases before generalized hypothesis by inductive reasoning ● Research offer strong evidence for the advantages of geometry abductive strategy under the arithmetic abductive strategy in establishing the general rules for the quadratic function number sequence Therefore, it is necessary to introduce to students geometry abductive strategy Besides, teachers should provide problems to encourage students using visual representation as a tool to discover the rule of number sequence and recognize the meaning of variable in that rule From the experimental results of Study 2, we also had some notes for teachers: ● When using or designing the open-ended geometry problems to giving students the chance of exploring mathematic, teachers 24 should choose the situations to require students to coordinate both visual abduction and manipulative abduction ● When students propose hypotheses from the observations on dynamic visual representation, teachers should ask students to test a hypothesis by experiment before choosing the last one ● The hypothesis comes from the observation and verification on visual representation need to be proven seriously ● In four dragging modes, maintain dragging and linking dragging are quite new Therefore, teachers need to create an environment for students to practice more with this two dragging modes Recommendations for program and textbooks One of the important results on mathematics education in England for 20 years is that: students constantly “invent” the mathematical rules to explain the “patterns” they found around them (Askew & Wiliam, 1995, [15]) Moreover, in the report on the future of national math education (NRC, [55], 1989), the International Research Council said that in this era, doing mathematic goes further than only use deduction and calculating but it involves observing the rules, checking the hypotheses and estimating the results This shift is proposed to implement not only on the curriculum but also on the teaching method Therefore, we believe that students need practice with patterns more depending on their age and their level of awareness [...]... Recorder software, we analyze reasoning corresponding to the dragging modes that students manipulate on dynamic visual representation 3.6 Restrictions 15 Chapter 4 VISUAL REPRESENTATION SUPPORTS INDUCTIVE REASONING AND ABDUCTIVE REASONING 4.1 Effect of visual representation on discovering number sequence by inductive reasoning and abductive reasoning 4.1.1 The abductive strategies used in discovering number... ABDUCTIVE AND INDUCTIVE REASONING 5.1 Abduction and induction in mathematics activities at school 5.2 Tasks help develop abductive and inductive reasoning Through two empirical studies of this thesis, we get that finding rule in number sequence and discovering open-ended geometry problem are two kinds of tasks that promote students to use abductive reasoning and inductive reasoning because of two common... induction and abduction in mathematical contexts, showing that two types of reasoning used in combination to discover the rule of number sequence are induction and abduction ● Propose the process of discovering the rule of number sequence by abductive reasoning and inductive reasoning ● Recommend the classification of levels abductive -inductive reasoning fit the performance of students in mathematics discovery... support of dynamic visual representation to inductive reasoning and abductive reasoning in dynamic geometry environment 17 4.2.2.1 Reflection of inductive and abductive reasoning through the dragging modes 4.2.3 Summary from empirical result of Study 2 Based on experimental results, we summarize a process of using four dragging modes to propose, verify and generalize hypothesis by inductive reasoning. .. dragging modes in GSP (built from seven modes of dragging in Cabri): random dragging, maintain dragging, dragging on special cases, linking dragging 2.5 The research relating to this topic in Viet Nam 2.6 Sum of Chapter 2 10 Chapter 3 RESEARCH DESIGN To answer research questions 2 and 3, we conducted two studies: Study 1: Surveying the effects of visual representation to abductive reasoning and inductive. .. of this study ● Proposing activities practicing inductive reasoning and th abductive reasoning in grade 10 math class ● Show the evidence that visual representation is an effective tool for students in the process of exploring and generalizing number sequence by induction and abduction ● Summarizing the process using the dragging modes to discover the open-end geometry problems by inductive reasoning. .. algebraic thinking - When converting a number sequence described by visual representation into a number sequence described only by number data, students will not have the facility to test a hypothesis for the unknown cases 4.2 Dynamic visual representation supports inductive reasoning and abductive reasoning in exploring open-end geometry problem 4.2.1 Inductive reasoning and abductive reasoning in dynamic... seven-step model of inductive reasoning proposed by Canadas & Castro (2007, [24]), along with our study in relation to the 15- year- old students (Phuong, 2009, [4]), we 9 developed a five-step theoretical process to discover the rules of number sequence in Figure 2.13 Figure 2.13 Process of discovering the rule of number sequence by abductive reasoning and inductive reasoning 2.4 Discovering open-ended... inductive in discovering the rule of number sequence Specifically, we would like to clarify the following issues: (1) How does visual representation affect the strategies that students use to explore the rule number sequence?; (2) Do students use visual representation in the phase of verifying and generalizing hypothesis by inductive reasoning? ; (3) The level of abductive -inductive reasoning that students. .. hypotheses by experiment through inductive reasoning - In the case a): Students use random dragging associated with the measurement tools and calculations tools of GSP to verify the hypothesis by experiment - In the case b): Students uses linking dragging to link point D to locus Q in order to confirm the hypothesis by experiment 4.3 Summary of chapter 4 4.3.1 Conclusion of Research Question 2 The experimental ... Vietnam In this trend, we choose: Using visual representation to support inductive reasoning and abductive reasoning of 15- year- old students in discovering mathematical rules” as a topic of this... occurrence and outcome of three types of reasoning 2.1.3.2 In terms of the purpose of using each kind of reasoning 2.1.3.3 In terms of the level of certainty of outcome The certainty of the conclusions... INDUCTIVE REASONING AND ABDUCTIVE REASONING 4.1 Effect of visual representation on discovering number sequence by inductive reasoning and abductive reasoning 4.1.1 The abductive strategies used in discovering