Ch. Lemonidis, D., Polytidis (2010). The implementation of an electronic model in teaching negative numbers and their operations. 13 th International Conference ICT in the education of the Balkan countries Varna, June 17 - 19, 2010. Balkan Society for
Pedagogy and Education.
The implementation of an electronic model in teaching negative numbers and their operations
Charalambos Lemonidis
Dimitrios Polytidis
Abstract
The knowledge of negative numbers and performing operations among them is one of the topics within the mathematics that school children find very difficult to deal. The purpose of this study is to use technology in order to develop software with negative and positive electric charges which represent signed numbers and their operations. 54 first-grade high-school children participated in this study. The results revealed that the group that was taught the negative numbers and their operations with the help of the software program scored higher than the group which was taught with the traditional way.
I.
Introduction
Understanding negative numbers and performing operations with negatives is a task that students find difficult. The origin of this difficulty is epistemological and the number of natural phenomena that can represent operations with negative numbers is limited. Usually in education the textbooks use visual representations or models, such as the number line, scales, the time line, vectors and everyday life representations or models such as temperatures, money or objects to explain the operations which incorporate negative numbers (Kilhamn, 2008). Linchviski & Williams (1993) conclude in their work that at least subtraction with negative numbers can be understood through models, although not a single model but a multiplicity of models.
Gallardo (1995) suggests teaching negative numbers using discrete models, where whole numbers represent objects of an opposing nature rather than using the number line. Kilborn (1979) points out that some teachers use several different models simultaneously during a lesson and that these models seem to confuse the students.
Ball (1993), on the other hand, states that no representation captures all aspects of an idea and “teachers need alternative models to compensate for imperfections and distortions of any given model” (p. 384). She articulates a dilemma when she asks whether she confuses the children by letting them explore multiple dimensions of negative numbers, introducing several different representations. Some work has been done on the use of metaphorical reasoning when dealing with negative numbers
(Chiu, 2001; Stacey et al., 2001).
Extracted from the world of physics and chemistry, the atom with the positive charged protons and negative charged electrons is a good model for the representation of negative numbers and operations. We have used this model, with the negative and positive charges, in an electronic environment in a way that could be suitable for the introduction of negative numbers and operations to High School students.
The model that was used in this study is considered appropriate and valuable for the students as it is cohesive and presents all the operations involving negative numbers. The main hypothesis of this study is that the use of technology will be a

more effective way to present the positive and the negative charges to students which consequently will have better performance in calculating signed numbers .
II. Description of the electronic model
The implementation of the proposed electronic model was uploaded into a website, using the XML stylesheet and web designing technology with potential of incorporating Flash interfaces. Both the interface and the program are designed to deploy into two axes: the horizontal and the vertical.
In the horizontal axis and at the top of the webpage there are different cards of the chapters (introduction, addition, subtraction and multiplication)
The vertical axis represents the chapters dedicated to the different calculations.
Through this axis, the main page arises, where the students will practice in order to understand the calculations of the negative numbers.
In the introduction card the negative numbers are presented theoretically, based on the planetary model of atom – already known to students through physics- in order for students to better understand the way these operate as well as to activate the student‟s preexisting knowledge.
Moreover, the two essential to the program elements are introduced both verbally and graphically (+ is a proton, - is an electron). In addition, the basic theoretical concepts of negative numbers are presented, even if they have already been taught to students.
In the second chapter (addition), there is an introduction of the sub - activities that students have the opportunity to do if they choose the vertical right axis. Initially, students will read the math problem, and then they will first add the protons and then the electrons by moving both of them from an outside stack into a bucket, which represents the area where calculations take place.
This progressive procedure aims at an empirical understanding of the operation involving the addition of signed numbers.
In the same time, a conceptual transformation is attempted: transformation of protons and electrons proceedings into mathematical symbols and relations. Students are urged to implement the conceptual mathematic operations, that is transfigure the empirical actions into abstract mathematical symbols involving signed numbers.
In a similar way, there is presented a presentation of the remaining chapters of the
“subtraction” and the “multiplication”, where the students are urged in a progressive way to deal with a problem, initially empirically – through the proton-electron model
– and then to try evolving it into a mathematical formula. There are four exercises for each operation.
III. Methodology
The present research was conducted to 13-year-old schoolchildren who attend the first grade classes in a high school located in the city of Skydra, Pella. In order to test the effectiveness of the electronic program students were divided in two groups, one being the experimental (54 students) and the other serving as the control group (52 students).
Through the school curriculum, approved by the Greek Ministry of Education, negative numbers are taught in a period of 10 academic hours. The control group followed this program. The experimental group though, within this 10 week period, was taught in a different way, following the next steps: the first step included three academic hours in the subjects of addition, subtraction and multiplication of signed

numbers in the computer lab. These three lessons were taught using the necessary software, seated by 3 students to each of the 9 personal computers.
A month later, all children had to fill in a questionnaire where was tested the student‟s capability in solving problems with whole numbers. The purpose of this action was to compare the performance of both groups as well as the influence that the program had in teaching children the negative numbers.
IV. Results
Table 1 shows the performances of the experimental group and the control group in eight exercises using the signed numbers.
Exercises
1. (–6) – (–8)
Experimental Group
33(61%)
45(83%)
Control Group
12(23%)
33(63%)
2. (–2).(–3)
3. ( + 5 ) + ( - 9 )
4. ( -10 ) – (- 6 )
5. (–2) – (–5)
46(85%)
43(80%)
28(52%)
46(85%)
33(63%)
30(58%)
17(33%)
31(60%)
6. (–4) . (–2)
7. ( - 3 ) . ( + 5 )
8. (-5) + ( - 4)
Mean of Overall Success
45(83%)
44(81%)
(76%)
29(56%)
27(52%)
(51%)
Table 1. Performances of students in experimental group and control group
As it is shown in the above table students belonging to the experimental group, who were taught the whole numbers using the software program scored higher in all exercises. The mean score of the experimental group in all eight exercises (Μ=6.11,
SD=2,13) is significantly higher (t=4,76, DF=104, 2-tailed p=0,000) than the mean score of the control group (Μ=4.07, SD=2,26).
In both groups of children, two problems were proposed from which the first had four sub-questions.
Problem 1.
Four groups of students participate in a game using questions. Each group will have an extra score for each correct answer and will be scored negatively for each wrong answer. After four rounds of questions the groups received the following points: 1a. Group A: +4, 1b. Group B: -2, 1c. Group C: -2, 1d. Group D: 0.
How did each group answer in every round?
Problem 2.
A thermometer outside of a house shows -5 degrees Celsius; inside the house the thermometer shows +20 degrees Celsius. What is the difference of the temperature?
Table 2 shows the percentages of success of students of both groups in the problems, involving the signed numbers.
Problem
Problem 1α (+4)
Problem 1β (-2)
Problem 1γ (-2)
Problem 1δ (0)
Problem 2
Mean of Overall Success
Experimental Group
45(83%)
28(52%)
26(48%)
24(44%)
39(72%)
(60%)
Control Group
31(60%)
16(31%)
14(27%)
13(25%)
15(29%)
(34%)

Table 2. Performances of the students of the experimental and control group in problems solving.
From the table 2, it is revealed that the students belonging to the experimental group scored higher in all exercises and the total score of this group compared to the control group was almost double (60% -34%). The mean score of the experimental group in five problems (Μ=2.79, SD=1,87) was significantly higher (t=2,82, DF=104,
2-tailed p=0,006) from the mean score of the control group (Μ=1.76, SD=1,86).
Significant role in this difference played the fact that the students from the experimental group had better knowledge of the exercises of the signed numbers
Comparison of students’ performance with international evaluations
Among the eight exercises we asked from our students to perform, two of them
(5&6) (see table 1.) were also asked in a similar study in G. Britain (CSMS, 1981) and two (1&2) in another study which was conducted in 18 European countries
(SIMS, 1989). In the study entitled “Secondary Mathematics and Science” (CSMS,
1981), that took place in G. Britain, where 14-year-old students were taught positive and negative numbers, the question: Find the result of the exercise (-2)-(-5) (exercise
5), reveals success rate 44% (Hart, 1981, p. 83). In the same exercise as it is shown in table 1, the students of the control group had a 33% success rate compared to the experimental group who had a success rate of 52%.
In the exercise (-4).(-2) (exercise 6), the experimental group had a success rate of
85%, the control group 60% compared to the CSMA study were the success rate was
76%.
Regarding the results of the study which was published in the Second
International Mathematics Study (SIMS) where there was examined the performance of 18 European countries, in the exercise 1, (-6)–(-8) the mean success rate was 49%
(Robitaille, 1989, p. 63). In the same exercise as it is seen in the table 1, the students from the control group achieve a success rate of 23%, compared to the performance of the students belonging to the experimental group which achieved a success rate of
61%.
Regarding the exercise 2, (-2).(-3) in the study that was implemented in the 18
European countries of the SIMS the mean success score was 67%, compared to the experimental group 83% and the control group 63% (table 1).
In conclusion, the success of the experimental group was higher regarding the mean success score in both studies (G. Britain and European countries), while the performance of the control group was lower again in both studies.
These differences between the experimental and control group, reveal the usefulness of the proposed model in teaching negative numbers at schools. The use of computers assists in the process of understanding abstract thinking using the negative numbers and it makes it easier for the students to use this model in solving various exercises and tasks.
V. Conclusion
In this study, using technologies, a new model was constructed, bearing concepts taken from physics, having some conventions. Protons and electrons are used as representations of numbers, the action of addition is construed as “+”, whereas the sustraction one as “-”, and the neutralization between them is thought as 0, through the neutralization of protons and electrons. This model becomes easily familiar to the

students‟ constructions without subjecting them to further conceptions and techniques in order to understand the concept of negative numbers.
The strategies that students used for the problem and exercise solving were heavily affected by their experiences through the model. This comportment showed that, they were influenced by the real state of the model and they interpreted the operations based into a realistic view that was presented. So, the students have in their disposal a realistic model, through which they can be driven to a pure symbolical interpretation of the operation. On the other hand, the students that were taken through the current classical teaching did not have an intermediate realistic model to help them conceptualize the negative numbers. They were restricted in a purely algorithmic management of the negative numbers resulting in difficulties and mistakes in problem solving.
This software implementation to students as well as the comments that the students made during the use of this software, in an adjacent questionnaire, gave us many clues about the software‟s amelioration. There is the potential of making the software more open and functional, by replacing the numbers with variables, so that various exercises can be implemented according to the will of the user. In addition, the software can be extended to the remaining lessons of teaching signed numbers, especially in division. Having those improvements done, a completed software can be implemented to teach the whole subject of negative numbers.
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