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Editor’s Note: There are many experiments that compare traditional instruction with interactive multimedia and web-based learning. This is a study teaching symbolic logic for college level students in Turkey. It employs discovery learning, real-life situations, and interactive visuals to positively influence the performance and attitudes of teachers and learners in 9th grade mathematics.
The Evaluation of Introduction Level Computer-Assisted Symbolic Logic Materials Based on Realistic Mathematics Education and Guided Discovery Learning Approach
Jale BINTAS and Mehmet Fikret GELIBOLU
Turkey
Abstract
Logic is the formal systematic study of the principles of valid inference and correct reasoning (Cristal, 2002). In this research to teach mathematical logic we used the “Realistic Mathematics Education” (RME) approach that considers mathematics as a human activity, and the “Guided Discovery Learning” approach which enables students
to construct knowledge by themselves. Also, computer assisted applications of the experiment draw students’ attention and provide individualized learning to students. The main purpose of this study is to determine the effect of developed computer assisted symbolic logic materials that are based on realistic mathematics education and guided discovery learning approaches. The research sample consists of 59 students chosen for the experimental and control groups from 9th grade high school students in Turkey. Selection was based on a quantitative sufficiency performance test that identified the general mathematical status of students. A pre-test post-test control group experimental design was used in the research. Quantitative methods were used to analyze the effect of experimental instruction on the pupils’ and teachers’ opinions. To measure and compare the achievement of students at the end of the experimental instruction process, we developed a 30 item post test to measure performance in logic in the 9th grade mathematics curriculum. Post-test scores were significantly different in favor of the experimental group. At the end of the research, data was gathered on students’ and teachers’ points of view. Results showed that using computer assisted instructional materials based on Realistic Mathematics Education and Guided Discovery Learning are more efficient than traditional education.
Keywords: logic education, realistic mathematics education, guided discovery learning, computer assisted instruction, worksheets
Introduction
In the technological world we live in, the value of knowledge is more and more significant. Strong societies are only possible if individuals keep up with the increasing educational needs. That is why it is a deep concern of all educational units to train creative, productive people, who can solve problems, analyze, synthesize, use technology effectively and know how to find knowledge. A major way to make people qualified is related to logic education.
Logic is the formal systematic study of the principles of valid inference and correct reasoning (Cristal, 2002). “The increasingly technical demands placed on people by the information revolution makes it all the more important that people understand basic logical principles of reasoning” (ASL Committee on Logic Education, 1995). On that point new approaches on education help us to meet the needs of a growing world. Logic has been taught learners in several disciplines. Logic was formerly a branch of philosophy; more recently it became an essential part of mathematics and computer sciences. We believe that people who understand mathematics and use logic, have more choices to give form to their future than those who don’t.
“Mathematics education provides needed knowledge and equipments which help individuals to understand physical world and social interaction. It helps to guess, analyze their experiences, and gets them to have a systematic approach and language to solve problems. It also facilitates creative thinking, and accelerates developing reasoning skills” (Bulut, 2005).
Researchers and educators are trying to explain how learning happens and they are developing many approaches, techniques, strategies, and methods to improve the education such as “Realistic Mathematics Education”, “Guided Discovery Learning” and “Computer Assisted Instruction”.
Realistic Mathematics Education (RME) is a teaching and learning theory in mathematics education that was first introduced and developed by the Freudenthal Institute in the Netherlands. This theory has been adopted by a large number of countries worldwide. Its important points are that mathematics must be connected to reality as human activity. Mathematics must be close to children and be relevant to every day life situations (Zulkardi, 1999).
Discovery Learning is an inquiry-based, constructivist learning theory that takes place in problem solving situations where the learner draws on his or her own past experience and existing knowledge to discover facts and relationships and new truths to be learned. Students interact with the world by exploring and manipulating objects, wrestling with questions and controversies, or performing experiments. As a result, students may be more likely to remember concepts and knowledge discovered on their own (Learning Theories Knowledgebase, 2008).
“Computer-assisted instruction” refers to instruction or remediation presented on a computer. This enhances teacher instruction in several ways. Computer programs are interactive and can illustrate a concept through attractive animation, sound, and demonstration. They allow students to progress at their own pace and work individually or problem solve in a group. Computers provide immediate feedback and differentiated lessons to capture the students’ attention, challenge students who are at different levels, and improve instruction for students with disabilities (The Access Center: Improving Outcomes for All Students K-8, 2008).
Although there is much computer software like Alfie, Aristotle, Plato, LogicCoach, Organon, and Tarski’s World for learning different types of logic, we strongly needed to design our own materials for a number of reasons, such as:
Existing materials are too specific and do not cover the symbolic logic curriculum of Turkey in 9th grade entirely. (Generally they are not about propositional logic or just focus on making proofs.)
Most materials are inappropriate for high school education level students.
Language or symbols in some materials are not used in Turkey.
Mostly, they are not user friendly.
Some of them need to install to a specific operating system or hardware architecture.
Some of them are not free of charge.
Thus, in this research we developed computer asisted symbolic logic instruction materials combining “Guided Discovery Learning”, and “Realistic Mathematics Education” approaches to compare with traditional education in an experiment.
Research Goal
The main purpose of this study is to determine the effect of developed computer assisted symbolic logic materials that are based on realistic mathematics education and guided discovery learning approaches.
In this research, we investigated whether or not developed symbolic logic instruction materials make a difference compared to traditional logic education in 9th grade mathematics.
Method
Quantitative analysis was used to determine the effect of experimental instruction on pupils’ and teachers’ opinions. A pre-test post-test control group experimental design was used in the research. The research was conducted in the fall semester of the 2007-2008 academic year.
Participants
The research population was randomly chosen from three high schools in Izmir city of Turkey. A pre-test reliability study was made on 151 students; the post-test reliability study was made on 259 students. Students were assigned to experimental and control groups according to pre-test scores. The experimental sample consisted of 59 students. Both pre-test and post-test were delivered to both the traditional group and the experimental group.
Content
Symbolic logic is being taught in Mathematics lessons in 9th grade in Turkey. And in the syllabus the following topics are being presented:
Terms and axioms
Premises, compound propositions
Number of states of compound propositions
Negation operator (
), equivalence, double negative eliminationConjunction, disjunction, implication, and bi-conditional (
) connectivesTautology and contradiction
De Morgan’s Theorems
The lesson content was presented to both control and experimental groups with different educational tools and approaches. While the control group received traditional education, the experimental group was trained with the developed materials under guidance of the teacher.
Application period
Both traditional and experimental groups received 12 hours of instruction over a period of three weeks.
Limitations
All the tests and tools were developed in the Turkish language. (It will be possible to add different language support in the future.)
Instructional materials and measurement tools
For the experiment we developed pre-test, post-test, educational software, and worksheets.
(For further details please visit: http://site.mynet.com/fikretgelibolu/logic/help.html)
Software
We designed the computer assisted materials using a Shockwave Flash Technology and web interface to provide user flexibility and compatibility. Passwords controlled access to each topic in the web interface (Figure 1) so the teacher could guide the natural flow of lessons. Some topics have prerequisites so it is essential for students to receive them in a predetermined sequence.
Figure 1. Web interface scheme
There are three kinds of instructor application when analyzed structurally:
Drag/drop applications (Figure 2)
Figure 2. Dragging
Turn on/off applications (Figure 3)
Figure 3. Electrical switches
Interactive tables (Figure 4)
Figure 4. Truth table
Worksheets
In addition to computer assisted software, we also developed RME and Guided Discovery Learning approach based worksheets as instruction tool.
Lesson example
First, students are told to follow the instructions in the computer assisted materials.
Figure 5. Application
In this material (Figure 5), students can drag and drop the premise sentences through the empty lines in order to make a conditional premise. Students check their premises: either the meaning of the becoming conditional sentence or the feedback of the software (Figure 6-7).
Figure 6. True premise
( If the bell rings then I open the door – True)
Figure 7. False premise
(If the bell rings, then I don’t open the door – False)
Then they fill the truth tables below. The last part of the truth table remains inactive until students fill the first two premises right as seen in Figures 8 and 9.
Figure 8. Preconditioned truth tables
Figure 9. Subsequent interaction
After the students fill the entire table correctly the feedback appears like in Figure 10.
Figure 10. Feedback – (Congratulations)
After this application students are expected to fill the worksheet 10 as shown with blue italics.
Worksheet 10
p premise: The temperature of water is
q premise: Water boils.
If the truth value of the premises given upside is 1 (true) then write down the premises and their truth values below.
 | : The temperature of water is not |
 | : Water doesn’t boil. |
| If the water is 100°C, then water boils. | |
 | If the water boils, then the temperature of water is |
 | If the water is 100°C, then water boils and if the water boils, then the temperature of water is |
p | q | p Ž q | q Ž p |
|
1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
“If and only if the temperature of water is 
”. So fill the table below considering the sense of the sentences.
p | q | p Ū q |
1 | 1 | 1 |
1 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | 1 |
What kind of relations are there between the premises 
and 
?
As a result we expect that students discover the last columns of last two tables upside have the same values. Thus they are expected to write “the premises “If the water is 
.
As it can be seen in the instance materials, learners use real life examples in the materials. While teaching premises, logical connectives etc, students always use relations of real objects, events or what they perceive realistic as been referred in RME. Also through guided discovery learning approach, expected results are not ever given directly in the materials; students always discover the answers by following steps on their own. These are especially the distinguishing ways of the materials we developed.
Pre-test (quantitative sufficiency performance test)
In order to compare the introductory information of groups, we developed a 32 itemed pre performance test about general mathematical knowledge. Questions are chosen from previously applied governmental examinations to enter high schools or take scholarships in
Table 1
Pre-test reliability analysis results
N | S | Maximum | Median | Mean | Alpha |
151 | 4.88 | 30 | 19 | 19.3 | .77 |
Post-test (logic issue performance test)
At the end of the experimental instruction process, to measure and compare the achievement of students, we developed a 30 itemed performance posttest about logic issue in the 9th grade math curriculum. Draft items are analyzed and rearranged considering validity and reliability as on the pre-test. And tested on 259 students from three high schools. Statistics are shown in table 2.
Table 2
Post-test reliability analysis results
N | S | Maximum | Median | Mean | Alpha |
259 | 8.06 | 29 | 18 | 16.93 | .92 |
Findings
This research was conducted to test the instruction with developed instructional materials, and compare it with traditional instruction. Two random class were chosen as experiment and control group for that purpose. Pre-test was applied to both groups and analyzed using independent samples t-test in SPSS. No significant difference on introductory mathematical knowledge of students between groups could be found as shown in table 3. Thus both groups were considered equal about quantitative sufficiency.
Table 3
Equivalence of groups
Pre-test | N |
| S | df | t | p |
Experiment Group | 29 | 21.51 | 3.46 | 57 | 1.28 | .20 |
Control Group | 30 | 20.26 | 4 |
*p>0.05
After the approximate 3 week instruction, post-test was applied to groups, and gathered data was statistically analyzed using independent samples t-test in SPSS.
Table 4
Final status of groups
Post-test | N |
| S | df | t | p |
Experiment Group | 29 | 25.55 | 2.22 | 51.56 | 2.38 | .02 |
Control Group | 30 | 23.83 | 3.23 |
*p<0.05
As can be seen in table 4, post-test scores were significantly differentiated in favour of experiment group.
Pupils’ opinion on Realistic Mathematics Education approach and Guided Discovery Learning based computer assisted mathematical logic instruction
A questionnaire with nine open-ended questions was used for this analysis. Twenty four (24) students from the experiment group participated in the questionnaire voluntarily. The majority of the pupils assessed their performance as positive. They enjoyed the topic on both content and application aspects. Almost all of the pupils found assistive materials in the lessons to be useful and believed they benefited from the assistive learning materials.
“Computer assisted materials are useful. They draw our attention and make us learn better.” (Student 16)
“I believe that they’re (worksheets) a good source (of learning) for better understanding.” (Student 7)
“Using the related materials make us visually perceive better.” (Student 22)
When asked whether they would like to change something, students didn’t recommend any modification in materials, but they did suggest the possibility of adding a smart board and mobile applications.
Teachers’ opinion on Realistic Mathematics Education approach and Guided Discovery Learning based computer assisted mathematical logic instruction
To learn teachers’ opinions about the learning materials and logic education we used a questionnaire which includes 15 open-ended questions. 9 mathematics teachers from 4 different schools participated in the questionnaire after seeing the materials. Majority of the teachers find logic topic difficult, abstract, and essential, which teaches systematic and correct thinking.
“I consider that it (logic) is essential because it teaches systematic and correct thinking.” (Teacher 4)
“I find it (logic) crucial because it orientates students to think abstract and improves their thinking.” (Teacher 6)
Teachers stated that using computer assisted education and real life instances might be the most appropriate techniques for mathematical logic education.
“I think materials such as visual contents which include real life examples should be used with computer assisted education.” (Teacher 9)
Teachers found the developed computer assisted materials useful because they provided a permanent incentive and concrete learning experiences. They also mentioned that worksheets were useful for control and permanence of instruction.
“These (learning) materials structured the way which draw students’ attention and motivate. So they (learning materials) will be useful.” (Teacher 3)
“Worksheets provide permanency (on learning) when students discover and find out by themselves” (Teacher 3)
Teacher’s ideas about their lessons were usually parallel to the techniques used in the developed materials, thus they believe lessons should be assisted by such materials.
Almost all of the teachers had a positive attitude about using assistive learning materials and they stated that they were not willing to make any changes in their lessons.
Conclusion
“Realistic Mathematics Education” and “Guided Discovery Learning” approaches based, computer assisted developed logic instruction materials’ efficacy was tested in this research. Following a three week experiment, student achievement in the posttest favored the experimental group which was instructed by developed materials. It was understood that teaching mathematics as a real life activity by using real life examples on abstract matters, and instructing the lessons using discovery techniques including interactive applications, influence students more positively than traditional education. Also students and teachers mentioned that they were quite interested in using developed logic instruction materials. Nonetheless, teachers resist making any change in their lessons because they cannot anticipate the effect on their own students, or they already use some good instructional materials with proven success.
Instruction of logic is essential because it facilitates critical thinking. Logic instruction contributes factors which develop scientific and consistent reasoning skills and the ability to evaluate events that happens around us. This research did not test the persistence of learning, but use of a discovery model may have a positive effect. Further studies are required to verify and extend the findings of this study.
References
ASL Committee on Logic Education (1995). The Bulletin of Symbolic Logic, Vol. 1. Retrieved September 9th, 2008 from http://www.math.ucla.edu/~asl/bsl/01-toc.htm
Bulut, S. (2005). MEB İlköğretim Matematik Dersi Öğretim Programı ve Kılavuzu, 6–8. Sınıflar, MEB Devlet Kitapları Md. Ankara, Turkey (Turkish National Education Ministry Primary School Mathematics Curriculum Guide for Grades 6-8)
Crystal, D. (2002). The New Penguin Encyclopedia,
Learning Theories Knowledgebase (2008). Discovery Learning (Bruner) at Learning-Theories.com. Retrieved August 10th, 2008 from http://www.learning-theories.com/discovery-learning-bruner.html
The Access Center: Improving Outcomes for All Students K-8 (2008). Computer-Assisted Instruction and Writing. Retrieved August 10th, 2008 from http://www.k8accesscenter.org/training_resources/computeraided_writing.asp
Zulkardi, 1999, How to design lessons based on the realistic approach?. Retrieved August 30th, 2007 from http://www.geocities.com/ratuilma/rme.html
Appendix
Developed materials and the instruction manual are available on http://site.mynet.com/fikretgelibolu/logic/help.html
About the Authors
| Dr. Jale BINTAS Position: Associate Professor at Computer Education and Instructional Technologies Department of Ege University Faculty of Education Bachelors’ Degree: Mathematics - Masters Degree: Applied Mathematics - Ege University Institute of Natural Sciences (1988) PhD: Applied Mathematics - Ege University Institute of Natural Sciences (1994) E-mail: jale.bintas@ege.edu.tr |
| Mehmet Fikret GELIBOLU Position: Research Assistant at Computer Education and Instructional Technologies Department of Bachelors’ Degree: Computer Education and Instruction Technologies - Masters Degree: Computer Education and Instruction Technologies - Ege University Institute of Natural Sciences (2008) PhD: Computer Education and Instruction Technologies - Gazi University Institute of Educational Sciences (in progress) E-mail: fikretgelibolu@selcuk.edu.tr |
