UNDERSTANDING OF THE ORDERING OF NUMBERS
AND THE USE OF ABSOLUTE VALUE ON THE AXIS OF
REAL NUMBERS
Christos Pantsidis*, Fotini Zoulinaki*, Panayiotis Spyrou*,
Athanasios Gagatsis** & Iliada Elia**
* Department of Mathematics, University of Athens
** Department of Education, University of Cyprus
[email protected]
ABSTRACT
The aim of the present study is to investigate the role of the axis of real numbers in
the resolution of relative mathematical tasks by students of Senior High School.
The sample of the study consisted of 295 students of Grade 10 in Greece. The test
that was administered to the participants consisted of 8 mathematical tasks that
were related directly or indirectly with the axis of real numbers. In particular, the
tasks included the ordering of given numbers or numbers that emerged from the
solution of inequalities without absolute value and inequalities with absolute value,
on the axis of real numbers. Tasks which combined the ordering, the absolute value
and the projection of a point on the axis were also involved. Results showed that
there was a grouping of various tasks depending on the mathematical knowledge
that was required for the resolution of these tasks.
INTRODUCTION: THE ROLE OF REPRESENTATIONS IN THE
LEARNING OF MATHEMATICS
The ability to express an idea in various ways seems absolutely natural. The
combined use of different types of representations is a characteristic of human
intelligence (Duval, 1993). The need for a variety of semiotic representations in
teaching and learning Mathematics is usually explained through reference to the
cost of processing and the limited representation capabilities for each domain of
1
symbolism. The different ways of symbolism for a mathematical concept
complement one another.
Nowadays the centrality of representations in teaching, learning and doing
mathematics seems to become widely acknowledged. There is strong support in the
mathematics education community that students can grasp the meaning of
mathematical concepts by experiencing multiple mathematical representations
(e.g., Sierpinska, 1992; Lesh, Behr & Post, 1987). The NCTM’s Principles and
Standards for School Mathematics (2000) document includes a new process
standard that addresses representations and stress the importance of the use of
multiple representations in mathematics learning. Greeno and Hall (1997) maintain
that representations can be considered as useful tools for constructing
understanding and for communicating information and understanding. They
underline the importance of students’ engaging in choosing representations and
constructing representations in forms that help them see patterns and perform
calculations, taking advantage of the fact that different forms provide different
supports for inference and calculation. Similarly, Kalathil and Sheril (2000)
described ways in which representations can be useful in providing information on
how students think about a mathematical issue, and serve as classroom tool for the
students and the teacher. In mathematics instruction representations have a critical
role in that teachers can improve conceptual learning if they use or invent effective
representations (Cheng, 2000).
In elementary mathematics teaching and curriculum design, a representation that
plays an important role in the teaching of basic whole number operations, and
generally in arithmetic, is the number line (Klein, Beishuisen, & Treffers, 1998).
Despite the widespread use of number line diagrams as an aid to whole number
addition and subtraction, doubts about the appropriateness of using them have been
raised (Hart, 1981). Ernest (1985) supports that there can be a mismatch between
students’ understanding of whole number addition and their understanding of the
number line model of this operation. In fact, number line constitutes a geometrical
model, which involves a continuous interchange between a geometrical and an
arithmetic representation. Based on the geometric dimension, the numbers depicted
in the line correspond to vectors and to the set of the discrete points of the line.
According to the arithmetic dimension, points on the line can be numbered in a
way that measuring the distance between the points may represent the difference
between the corresponding numbers. The simultaneous presence of these two
conceptualisations may limit the effectiveness of the number line and thus hinder
the performance of students in arithmetical tasks (Gagatsis, Shiakalli, & Panaoura,
2003). However, Shiakalli (2004) argues that failure in arithmetic tasks involving
the number line may be due to lack of systematic instruction for using this model
rather than its inappropriateness. Gagatsis & Elia’ s (2004) research study
investigated the effect of the use of number line in a different context and
specifically in solving one-step additive problems, and compared it with the effects
of pictorial representations and the verbal description of the problems. It was found
2
that young pupils dealt with problems in verbal form and problems accompanied
with number line or decorative picture in a similar and consistent manner. In other
words, pupils overlooked the presence of the number line or the decorative picture
and gave attention only to the text of the problem. This kind of behaviour towards
number line was attributed to the difficulties caused by the divergence between the
conception of number within the context of the problem, as a quantity of items, and
the conception of number within the framework of the number line.
Based on the above, the role of representations, in general, and number line, in
particular, on mathematics learning and problem solving, have received extensive
research in the field of mathematics education concerning elementary school
students. Nevertheless, no attempt has been made to identify and explore the
empirical ground for justifying the use of the axis of real numbers in mathematical
tasks of algebra in secondary education.
The axis of real numbers is a particularly complex mathematic construction that
appeared along the course of mathematical development. It is expected to the
degree that this construction troubled great mathematicians so much, that it will, as
well, continue to cause difficulties in our students today. A question that arises is
how we can identify these difficulties. In secondary mathematics education, the
axis of real numbers is only used locally, in the representation of rational numbers,
in the resolution of inequalities etc. It is obvious, that the partial study of individual
mathematical topics cannot make known the essential substance of the axis nor the
difficulties of its understanding by the students. This is an exploratory study which
constitutes a first attempt to investigate the difficulties that arise when handling the
axis of real numbers. The use of the axis of real numbers is investigated in three
different contexts: First, the algebraic context involving inequalities, absolute
values etc.; second, the verbal context involving the process of problem solving;
and third, the geometrical context involving the connection of the axis with another
geometrical representation. We consider the examination of the problems that exist
compulsory, before a methodical presentation of the axis arises, in order to
understand how the students’ difficulties are related to this subject. This would
require a second study that would follow the present study.
THE RESEARCH
Purpose
The main aim of the study is to examine how students deal with the ordering of real
numbers and how they use the axis of real numbers, in order to represent specific
numbers (rational or irrational) as well as solutions of inequalities. The use of the
axis of real numbers can be comprehended in a variety of ways, as students’ mental
processes which are connected with the mathematical object that is taught are not
clear in all of their complexity. Identifying them can be useful to mathematics
instruction which aims at surpassing the different levels of difficulty which
students have to face in different stages of their thinking process.
3
Method
Participants
The sample of the study consisted of 295 first grade students of Senior High School
(Grade 10) from 9 different schools in the wider region of Athens in Greece. These
students were not familiar with the use of the axis of real numbers since this
representation is first introduced at the second grade of Junior High School.
However, they were familiar with the ordering of real numbers and the concept of
absolute value.
The research instrument and process
For the study’s needs a test was constructed and administered to the participants. It
consisted of 8 questions (shown in the Appendix). The first five tasks of the test
asked students to place real numbers on the axis, compare or sort given numbers
and verify that specific numbers satisfy certain inequalities. The sixth task required
representation of the length of a straight line’s segment that corresponded to the
distance between two specific numbers. The seventh task involved the
representation of an irrational number on the axis of real numbers. The last task
combined arrangement, absolute value and projection of a point on the axis of real
numbers.
An attempt was made to focus the attention of students both in the arrangement of
distinct numbers and to the inequality’s solutions, using the axis of real numbers.
The problems revolved around concepts that were familiar to the students, as
mentioned above.
During the administration process of the test relative explanations were not
provided. It was only pointed out students were to read the questions carefully and
answer afterwards. The time provided for the completion of the test, was an
instructive period (45 minutes).
Variables of the study and marking criteria
The variables of the research stand for students’ responses in all the questions that
have to do with arrangement, the representation of numbers on the axis of real
numbers, the comprehension and the proper handling of absolute value as well as
the way these relate. The variables of the study are presented below:
V1a: placing of numbers on the axis of real numbers.
V1b: reduction of numbers by a constant positive quantity.
V1c: placing of numbers on the axis of real numbers.
V2: finding of solutions of given inequality, from explicit finite set of real
numbers.
4
V3a, V3b, V3c, V3d, V3e, V3f: representation of solutions of inequality on the
axis of real numbers.
V4: use of ordering attributes.
V5: sorting of numbers in descending order, by using the corresponding symbol
(“>”).
V6: representation of the length of a straight line’s segment, that corresponds to the
distance of two explicit numbers.
V7a: representation of an irrational number on the axis of real numbers.
V7b: combination of arrangement, absolute value and projection of point on the
axis of real numbers.
V8a, V8b, V8c, V8d, V8e: combination of arrangement, absolute value and
projection of a point on the axis of real numbers.
Correct responses were assigned a score of 1, partly correct responses (found some
but not all the solutions) were assigned a score of 0.5 and incorrect responses or no
responses were assigned a score of 0.
Data analysis
For the analysis and processing of the collected data based on students’
performance in the written tasks, Gras’s implicative statistical analysis has been
conducted by using a computer software called C.H.I.C. (Classification
Hiérarchique Implicative et Cohésitive) (Bodin, Coutourier & Gras, 2000). Gras’s
Implicative Statistical Model is a method of analysis that determines the
connections and the implicative relations of factors. It is worth noting that the
particular method of analysis has been widely used by several studies in the field of
mathematics education in the last few years. It is either employed as the main
statistical method (Elia & Gagatsis, 2003; Gagatsis & Shiakalli, 2004; Mousoulides
& Gagatsis, 2004) or it is integrated with other statistical techniques such as
Structural Equation Modeling (Gagatsis & Elia, 2004; Modestou & Gagatsis, 2004)
for the analysis of the data.
For this study’s needs, a similarity and an implicative diagram were produced from
the application of the particular analysis. The similarity diagram allows for the
arrangement of tasks into groups according to their homogeneity. The implicative
diagram involves implicative relations which indicate whether success in one task
entails success in another task related to the former one.
Results
Regarding the placing of numbers on the axis (V1a), high success rate was
identified (84.4%), as the students are familiarized with the position of the integers
on the axis of real numbers from the teaching of previous years.
5
The high scores that were observed in the tasks V3a (70.8%) and V3b (69%)
indicate that it is easy for students to solve inequalities. What’s more, students have
connected successfully the representation of these solutions with the use of axis of
real numbers. However, the solution of inequalities including absolute value caused
difficulties (V3c with rate of success 29.5% and V3d with rate of success 23.4%),
which appeared to be greater in the solution of quadratic inequalities (V3e: 9.8%
and V3f: 7.8%). The lowest score was observed in the task which asked students to
place the square root of number 2 in the correct position on the axis (V7b: 3.4%).
The overwhelming majority considered as precise placement of square root of 2,
either 1.41 or close to 2.
Figure 1 illustrates the Similarity Diagram that emerged from the implicative
analysis. It shows how students’ responses to the tasks are grouped together,
according to their similarity.
Group 1a
V
1
A
V
1
B
V
1
C
V
3
A
V
3
B
V
3
C
V
3
D
Group 1b
V
3
E
V
3
F
V
7
B
V
4
Cluster 1
V
2
V
6
V
7
A
V
5
V
8
C
V
8
A
V
8
E
V
8
D
V
8
B
Cluster 2
Figure 1: Similarity diagram of students’ responses to the eight tasks of the test
Based on the similarity diagram two clusters of variables are identified. The first
cluster includes a similarity group (Group 1a) which consists of the responses to
the tasks requiring the algorithmic resolution of inequalities (V3a, V3b, V3c, V3d,
V3e, V3f). A close similarity relation is formed between the solution of quadratic
inequalities (V3e, V3f) and the solution of inequalities with absolute value (V3c,
6
V3d). These variables are connected with V7b which involves the application of a
mathematical criterion (i.e. Pythagorean Theorem) for finding the exact value of
2 and V4 which is similar to V3, although its solution procedure is not
algorithmic. Group 1a also involves similarity connections with responses in task 1
(V1b, V1c) which are based on the placement of points on the axis of real numbers.
The variables of the tasks which asked students to represent a straight line segment
by applying the Pythagorean Theorem (V6), place approximately an irrational
number on the axis of real numbers (V7a) and verify that different numbers satsfy
inequalities form a second similarity group (Group 1b), which is also included in
the first cluster. In the first cluster one can also observe that placing numbers on the
axis of real numbers (V1a) connects with the responses to all the other tasks of the
same cluster. This indicates the fundamental nature of the ability to place numbers
on the axis of real numbers, which can be considered as a prerequisite to more
complicated activities making use of the axis.
The second cluster consists of the variables of the geometric tasks that combined
the arrangement, absolute value and projection of a point on the axis of real
numbers (V8). The formation of the second similarity group almost completely
separated from the first one indicates a compartmentalization of the tasks of the
test. Students approached in a completely distinct manner the tasks which involved
the use of another geometrical representation connected to the axis of real numbers
(Cluster 2), relative to the tasks involving the algorithmic resolution of inequalities,
placement of numbers on the axis of real numbers and application of the
mathematical criterion of the Pythagorean Theorem (Cluster 1).
Figure 2 presents the Implicative Diagram that was produced by the application of
the implicative analysis of the data. Implicative relations are identified among all
the variables of task 3 (algorithmic solution of inequalities). This can be attributed
to the algorithmic character of this group of tasks which provides an
understandable affinity. Moreover, success to task 3e entails success to task 3c
which in turn entails success to 3b. This implicative chain is indicative of the
different levels of difficulty of the particular tasks, since task 3e involved the
resolution of a quadratic inequality, task 3c required the resolution of an inequality
with an absolute value and task 3b involved the resolution of a simple inequality.
This observation provides support to the different success rates of students in the
particular tasks, presented above. Implications are also formed among students’
responses to tasks 1a, 1b and 1c which are of the same nature, as well as, between
8a and 8e, which have a common feature: they involve equalities and refer to the
same geometric representation. The two chains of variables, each representing a
different ability as regards the use of the axis of real numbers (V3: algorithmic
resolution of inequalities and V1: placement of points), are connected by an
implicative relation between V3e and V1b. No implicative connections appear
among the other tasks of the test.
7
V7B
V3F
V3E
V3D
V3C
V4
V8B
V7A
V1C
V8A
V8E
V6
V8D
V2
V1B
V5
V8C
V3B
V3A
V1A
Figure 2: Implicative Diagram of students’ responses to the eight tasks of the test
To sum up, the findings based on the implicative diagram are in line with the
findings derived from the similarity diagram. Both diagrams illustrate in an explicit
way the special character of algorithmic questions in task 3. The same is observed
for the questions in task 1 (1a, 1b, and 1c). The similarity diagram provided
additional information on students’ behaviour towards the tasks: the existence of
compartmentalization between tasks in a geometric context and tasks in an
algebraic or verbal context, even though all tasks involved the use of the axis of
real numbers.
DISCUSSION
According to the preceding Gras diagrams and our observations, a potential
classification of mathematic activities concerning the axis is presented in Figure 3.
This classification does not have a hierarchical character. Tasks of placing numbers
on the axis of real numbers are strongly associated with tasks of the algorithmic
solution of inequalities. Both types of tasks are ultimately related to tasks involving
the application of a mathematical criterion, i.e. Pythagorean Theorem and the
verification that a number satisfies an inequality. The connection of the axis with
another geometrical representation leads students to a completely different
approach. Thus, tasks involving the use of the axis of real numbers in a geometrical
context are distinguished from the aforementioned tasks.
8
A possible cause for students’ difficulties is that the geometrical character of the
axis of real numbers is not distinguished during mathematics teaching. Indeed, as
shown in the classification which is based on the statistical analysis, there were no
connections between all the proposed groups of tasks, as the substantial
understanding of the nature of the axis of real numbers would require.
Placement
of points
Algorithmic
resolution
Geometrical
representation
Other tasks
related to the
axis
Pythagorean
TheoremMathematical
criterion
Verification of a
number as the
solution of an
inequality
Figure 3: Classification of tasks involving the axis of real numbers
Possible instructive activities would focus on the unification of the two different
groups of tasks. Moreover, a future study referring to the object of the axis of real
numbers could have the following objectives:
•
The first objective concerns a historical study that would explicitly
display the scientific character of the real axis as well as the scientific
9
obstacles that are related with this as they result from various mathematical
texts.
•
The second objective is the proposition of a model that takes into account
not only the semiotic character of the axis and the scientific knowledge
related to it but also the cultural character of the object “axis of real
numbers”.
•
The third objective is related to the interpretation of students’ difficulties
which concern the real axis’s mathematical tasks according to the
historical study and the scientific obstacles that are related with this.
•
The fourth objective concerns the construction of a test involving a
greater range and variety of tasks for examining students’ abilities in using
the axis of real numbers in order to obtain a more valid classification of the
included tasks.
•
The fifth objective concerns the planning of appropriate instructive
intervention which would result in the appearance of mutual relations
between all the proposed instructive activities in order to ensure an
essential understanding of the nature of the axis of real numbers by the
students.
We believe that only by a multidimensional approach we can have an exact idea of
the nature of the object “axis of real numbers” and of the student’s difficulties
related to it.
REFERENCES
Bodin, A., Coutourier, R., & Gras, R., CHIC : Classification Hierarchique
Implicative et Cohesitive-Version sous Windows – CHIC 1.2. Rennes: Association
pour le Recherche en Didactique des Mathematiques, (2000).
Cheng, P.C.H., Unlocking conceptual learning in mathematics and science with
effective representational systems. Computers and Education, 33, 109-130, (2000).
Duval, R., Registres de Representation Semiotique et Fonctionnement Cognitif de
la Pensee. Annales de Didactique et de Sciences Cognitives, 37-65, (1993).
Elia, I., & Gagatsis, A., Young children’s understanding of geometric shapes: Τhe
role of geometric models, European Early Childhood Education Research Journal,
11 (2), 43-61, (2003).
Ernest, P. The number line as a teaching aid. Educational Studies in Mathematics,
16(4), 411-424, (1985).
Gagatsis, A., & Elia, I., The effects of different modes of representations on
mathematical problem solving. In M. Johnsen Hoines & A. Berit Fuglestad (Eds.),
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Proceedings of the 28th Conference of the International Group for the Psychology
of Mathematics Education (Vol. 2, pp. 447-454). Bergen, Norway: Bergen
University College, (2004).
Gagatsis, A., & Shiakalli, M., Translation ability from one representation of the
concept of function to another and mathematical problem solving. Educational
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(5), 645-657, (2004).
Gagatsis, A., Shiakalli, M., & Panaoura, A. La droite arithmétique comme modèle
géométrique de l’ addition et de la soustraction des nombres entiers. Annales de
didactique et de sciences cognitives, 8, 95-112, (2003).
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Hart, K. M. (Ed.), Children’s understanding of mathematics. London: John
Murray, (1981).
Kalathil, R. R., & Sheril, M.G., Role of students’ representations in the
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Erlbaum, (2000).
Klein, A. S., Beishuisen, M., & Treffers, A., The empty number line in Dutch
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Mathematical Education-Topic Study Group 2: New Developments and Trends in
Secondary Mathematics Education (pp. 87-94). Copenhagen, Denmark: ICME-10,
(2004).
Mousoulides, N., & Gagatsis, A., Algebraic and geometric approach in function
problem solving. In M. Johnsen Hoines & A. Berit Fuglestad (Eds.), Proceedings
of the 28th Conference of the International Group for the Psychology of
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College, (2004).
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APPENDIX: THE TASKS OF THE TEST
1. Μark numbers -4 and 3 on the axis of real numbers.
Then subtract 3 units. Name their new positions A and B respectively
2. Which of the following numbers {-5, 0,-9, -
5 3
,- ,-6, 2} satisfy the relations
2 2
Χ<-6 or Χ>-4
3. Mark on the following axes the solutions which correspond to each inequality.
Χ-1<3
. . . . . . . . .
-4 -3 -2 -1 0 1 2 3 4
.
Χ-1>3
. . . . . . . .
-4 -3 -2 -1 0 1 2 3 4
|Χ|-1<3
. . . . . . . . . .
-4 -3 -2 -1 0 1 2 3 4 5
|Χ|-1>1
. . . . . . . . . .
-4 -3 -2 -1 0 1 2 3 4 5
Χ2-1<3
. . . . . . . . . .
-4 -3 -2 -1 0 1 2 3 4 5
Χ2-1>3
. . . . . . . . . .
-4 -3 -2 -1 0 1 2 3 4 5
4. Given that -4<Χ<2 find the possible values for Χ-4.
5. Sort the following numbers in a descending order, using the symbol ¨ >¨
-
1
1 4 4
,+ ,
,3
3 3 3
6. Numbers -2 and 3 are given. Draw on the axis the straight line’s segment AΒ,
the length of which represents the distance d (-2, 3).
12
7. Place the number
2 on the axis of real numbers:
a) Approximately.
b) Precisely.
8. If point A can be anywhere on the circle then mark the following sentences: (T)
for True and (F) for False.
a) Χ=2
b) Χ≤2
Α (Χ,Ψ)
Ψ
c) -2≤Χ≤2
-2
d) |Χ|≤2
e) |Χ|=2
13
Ο
Χ
2