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

Truong Thi Khanh Phuong 2011, “Using dynamic visualrepresentations to support inductive reasoning and abductive reasoning of students in the process of exploring mathematics”, Journal of

-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

UNIVERSITY OF PEDAGOGY HO CHI MINH CITY

Supervisor: 1 Assoc Prof Dr Le Thi Hoai Chau

2 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

RELATED TO CONTENT OF THESIS

1 Truong Thi Khanh Phuong (2011), “Using dynamic visualrepresentations 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

2 Truong Thi Khanh Phuong (2011), “The potential of open-endedproblems 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.

3 Truong Thi Khanh Phuong (2012), “The reflection of abductive andinductive 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.

4 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.

5. Truong Thi Khanh Phuong (2014), “Creating open-ended problems toimprove students’ abductive reasoning in mathematics classroom”,

Journal of Sciences - Hue University, ISSN 1859-1388, Vol 99,

No.11, pp 49-59

6 Truong Thi Khanh Phuong (2015), “Abductive reasoning andinductive reasoning in discovering sequence patterns – some

theoretical and empirical analysis”, Journal of Science – Ho Chi Minh

City University of education, ISSN 1859-3100, No 9 (75), pp 16-28.

Chapter 1 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 internationalstudent assessment PISA

1.2 Demand for research and speech research issues

Inductive and abductive reasoning, with its significance in helpingstudents to explore math knowledge through discovering rules inpatterns, should be paid more attention in math education Step intothe early years of the 21st century, the trend of applying mathematics

in most of the problems that students encounter in life is studiedglobalization People rarely using deductive reasoning because oftheir 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

15-year-old students, who recently completed education programofficially and need to choose between continuing high schoolprogram or become an independent citizen with a career for thefuture right now This transition period has an important significancewhen mathematical abilities were accumulated by students will have

a big impact on the success of the students in some next school yearsand their career later Moreover, 15-year-old students were alsosubject of the program of international student assessment PISA, aneducational assessment program is held periodically every 3 yearswith 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 startedattending grade 10 at Vietnam Specifically, the rules that we wouldlike to focus in the field of algebra are relating to the term “numbersequence” Until students were fifteen, they learned about theconcept: “algebraic expressions”, “linear function”, “quadraticfunction”, ie they have enough knowledge to discover the linearfunction number sequence and the quadratic function numbersequence Because students have not officially learn the concepts ofarithmetic and multiplication so we have a chance to evaluate moreobjectively 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 simultaneousoccurrence of inductive reasoning and abductive reasoning during theprocess of discovery and generalization the rule of number sequence.Besides, we are also interested in the capabilities of studentsexploring rules in the field of geometry With 15-year-old students,

we choose the geometry knowledge related to topics such as paralleland perpendicular relationship, polygons and circles which studentswere enrolled in the Geometry program in the class of grade 8, 9 andthe earlier of grade 10 On the other hand, we also want to considervisual representation forms created in learning environments that usecomputers and the dynamic geometry software To provideopportunities for students to discover the rules of mathematics in thefield of geometry with the support of the dynamic visualrepresentation, we choose the open-ended geometry problem as anobject 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 discoveringthe number sequence and what is the relationship between them?Research question 2: How does the visual representation describingthe number sequence affect the students’ reasoning in discovering ageneral rule?

Research question 3: How to use the visual representation to supportabductive reasoning and inductive reasoning while discovering open-ended geometry problem?

Research question 4: How to develop the ability of students todiscover the rules of mathematical patterns by inductive reasoningand abductive reasoning?

1.6 Definition of key terms

1.7 Structure of the thesis

Chapter 2 LITERATURE REVIEW

2.1 Mathematic and plausible reasonings

2.1.1 Inductive reasoning

2.1.1.1 Definition

Reasoning that give a general hypothesis (not sure exactly) fromverifying the correctness of the hypothesis for a number of specificcases (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 theprocess of inductive reasoning: (1) Observation of particular cases;(2) Organization of particular cases; (3) Search and prediction therule of patterns; (4) Conjecture formulation; (5) Conjecturevalidation; (6) Conjecture generalization; (7) General conjecturesjustification

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 definitionabout 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) (1)

H explains D (would, if true, explain D) (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

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 do 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 actionsduring 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

2.3 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 sequence2.3.4 Reasoning used in discovering the rule of number sequenceWhen referring to the reasoning based on observing some similarparticular cases to a general result, people often think of inductivereasoning Abduction is not even mentioned in the analysis of theauthors Reid (2002, [72]), Canadas & Castro (2007, [23], 2009, [24])about the reasoning of students while discovering the rule of numbersequence However, we seemed to ignore the creative element in thisprocess - factors that Peirce pointed out as a characteristic ofabduction 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])

We also draw some following different points between abduction andinduction:

● Hoffmann's (1999, [38, p 272]) states: “Induction can not put arule from a set of data but only help decide in terms of quantitieswhat has been suggested by abduction” In other words, the

purpose of abduction is given a hypothesis to explain, but the purpose of induction to assess the scope of expansion of

hypotheses have been proposed

● Induction “infers the existence of phenomena such as we haveobserved in cases which are similar,” while abduction “supposessomething of a different kind from what we have directlyobserved, and frequently something which it would beimpossible for us to observe directly” (Peirce, [65, 2.640])

● Inductive indicates growth trend predicted for furtherobservations, abduction doesn’t directly interest in furtherobservation but only aims to explain the case which is going on

So, abduction occurs in the first stage when a hypothesis on theavailable data is proposed Inductive appears later when more casesare checked to determine if the hypothesis is true or not andconducting generalization

2.3.5 Conclusion of Research Question 1

Based on the pattern generalization scheme proposed by Becker &Rivera (2007, [19]) and seven-step model of inductive reasoningproposed by Canadas & Castro (2007, [24]), along with our study inrelation to the 15-year-old students (Phuong, 2009, [4]), we

developed a five-step theoretical process to discover the rules ofnumber 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 geometry problem

2.4.1 Open-ended problem

2.4.2 Open-ended geometry problem

2.4.3 Experimental mathematics

2.4.4 Dragging modes in dynamic geometry environment

Arzarello et al (1998, [14]) show the development of the seven

modes of dragging during the process of establishing predictions and

proving open-ended geometry problems in the dynamic geometryenvironment with the software Cabri Based on the similarity nature

of two dynamic geometry software Cabri and The Geometer’ sSketchpad (GSP), we focus on four basic 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

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 in discovering the rule of number sequence.Specifically, we would like to clarify the following issues: (1) Howdoes visual representation affect the strategies that students use toexplore the rule number sequence?; (2) Do students use visualrepresentation in the phase of verifying and generalizing hypothesis

by inductive reasoning?; (3) The level of abductive-inductivereasoning that students achieved

Study 2: Surveying the effects of dynamic visual representation on

the processes of inductive and abductive reasoning when studentsexplore open-ended geometry problems in GSP Specifically, wewould like to clarify the following issues: (1) In the course ofexploring open-end geometry problem, are induction and abduction

in paper and pencil environment different from in GSP geometryenvironment?; (2) How are abduction and induction reflected throughfour modes of dragging when students explore open-ended geometryproblems with dynamic visual representation?

3.1 Research design

The survey is used for Study 1 because it is suitable for collecting

information from a large number of cases

Case studies are used for Study 2 because it is suitable for the

research question “what?” and “how?”, in combination with the

method of treatment interview