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: 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


THE PUBLISHED WORKS OF AUTHOR
RELATED TO CONTENT OF THESIS
1.

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.

2.

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.

3.

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.

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 to
improve 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 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. 9 (75), pp. 16-28.


1
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 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


2
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 3 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,


3
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, 9 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


4
Chapter 2. 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:


5
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 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 actions
during the process of discovering in order to collect more
data for hypothesizing a Case that can explain the
Conclusion.


6
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


7
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]).


8
We also draw some following different points between abduction and
induction:
● Hoffmann's (1999, [38, p. 272]) states: “Induction can not put a
rule from a set of data but only help decide in terms of quantities
what 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 have
observed in cases which are similar,” while abduction “supposes
something of a different kind from what we have directly
observed, and frequently something which it would be
impossible for us to observe directly” (Peirce, [65, 2.640]).
● Inductive

indicates

growth

trend

predicted

for

further

observations, abduction doesn’t directly interest in further
observation but only aims to explain the case which is going on.
So, abduction occurs in the first stage when a hypothesis on the
available data is proposed. Inductive appears later when more cases
are checked to determine if the hypothesis is true or not and
conducting 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 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 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 geometry
environment with the software Cabri. Based on the similarity nature
of two dynamic geometry software Cabri and The Geometer’ s
Sketchpad (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


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 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 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.


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 1. 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 2. 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

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 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 1
● The first problem: students do 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
do 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 do not give


16
explanations why the rules described number sequence is a
linear function or a quadratic function.
● The fourth problem: Many students do 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 2
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 1. 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 2. 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 3. 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 4. 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

4.3.1. Conclusion of Research Question 2
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, 2 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 3


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 5. 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 do not provide a definitive
conclusion like a traditional proving problem. Secondly, these tasks
are not familiar with students and do 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 do 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
1. 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.