Validity of inverted sentence. Wechsler + Amthauer. 59 %. 5. Importance of conjunction. Wechsler + Amthauer. 45 %. 6. Importance of general quantifier.
PERCEPTION OF THE IMPORTANCE OF BASIC LOGICAL CONNECTIONS AMONG PRIMARY SCHOOL PUPILS Vlastimil Chytrý, Roman Kroufek, Jaroslav Říčan J. E. Purkyně University in Ústí nad Labem (CZECH REPUBLIC)
Abstract At all types of schools, erroneous judgments among pupils can be observed. These judgments are not caused by poor knowledge of mathematics but by imperfections in pupils’ logical thinking. The ability of logical thinking based on the perception of basic logical connections is, in fact, very often neglected or misinterpreted. In this article, the questions regarding what logical connections are understandable for pupils of lower secondary school and whether it is possible to positively influence the level of awareness of these connections by targeted treatment are discussed. Also the question of the impact of logical connection awareness or of logical thinking levels of pupils on their school evaluation is discussed. As a result, some of the attributes of logical thinking can be significantly reflected in the pupils’ assessments and thus, affect their math grades. This research was based on a quantitative analysis of data obtained from the test of logical thinking. It was attended by 200 pupils from 13 different schools in the Czech Republic. Due to the interpretation of some of the conclusions of this investigation, the elements of qualitative research, such as interviews with pupils and direct observation, were supplemented. Keywords: logical connections, logical thinking, school assessment, math knowledge.
1
INTRODUCTION AND DEFINITION OF THE BASIC CONCEPTS
If the main focus is on the concept of logical thinking, it is, first of all, necessary to define thinking as well as mathematical thinking because logical thinking is often seen as an “element” of mathematical thinking or associated with critical thinking [11]. Thinking can be described as a cognitive psychological process that enables mediated recognition of reality and leads to the realization of general and essential features of reality. In comparison, mathematical thinking reflects more than just the ability to solve arithmetic problems or problems related to algebra. Mathematical thinking is a set of approaches and methods to look at individual objects and consider their numerical, structural or logical basis [2]. Logical thinking then is often seen as some kind of a sequential thinking [1] or, alternatively, as thinking that enables complex problem solving thinking [9]. In addition to the above mentioned, many others have dealt with issues of logical thinking, such as Hurley [5], Dowden [3].
1.1
Basic logical judgment and its testing
The judgments based on negation, conjunction, disjunction, implication, and general quantifiers or De Morgan’s laws can be included among basic logical judgments. These judgments can be written down as follows:
A B A∧ B A⇒ B A B
A∧ B
A
A
A∨ B
A⇒ B B⇒C
(∀x ∈ M )ϕ ( x) y∈M
A
ϕ ( y)
C
¬( A ∧ B )
¬( A ∨ B )
¬A ∨ ¬B
¬A ∧ ¬B
Proceedings of ICERI2015 Conference 16th-18th November 2015, Seville, Spain
A∨ B ¬A B
(De Morgan’s laws)
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ISBN: 978-84-608-2657-6
Basic logical connections are described in detail also by Plato (2013) in his book Elements of logical reasoning.
2
METHODOLOGY
2.1
Methods for testing basic logical connections
This research was carried out at fourteen elementary schools in the Czech Republic. A total of 290 respondents participated in the testing. Annex 1 shows a sample of the test used to obtain data for this article. The answers to the questions were evaluated as follows: •
0 – the pupil answered incorrectly,
•
1 – the pupil answered correctly.
If a pupil did not answer a question at all, a blank sign was used to encode it. This method of encoding allows the following interpretation of the results: an arithmetic average of the measured values is an appropriate point estimation of parameter p of alternative distribution, i.e. the probability that a randomly selected student answers the question correctly. If, for example, a student is assigned 0.73 then this value refers to the fact that the student is able to correctly answer each item with a probability of 73%. To guarantee maximum validity, the test was based mainly on standardized tests, such as the Wechsler or Amthauer tests. The final test was passed to experts from different fields, such as psychology or didactics of mathematics. Therefore, mainly the content validation was used. The following Table no. 1, shows on what types of logical connections the individual items of newly created test are focused, what tests were used when the new test was created and how successful the students were in solving it. Table no. 1 - The success of the respondents + the source of the inspiration. Test items
Used tests
Success of students
1. Importance of conjunction
Wechsler
33 %
2. Importance of disjunction
Wechsler
24 %
3. Importance of implication
Amthauer
72 %
4. Validity of inverted sentence
Wechsler + Amthauer
59 %
5. Importance of conjunction
Wechsler + Amthauer
45 %
6. Importance of general quantifier
Amthauer
51 %
7. Negation of quantifiers
Wechsler
6%
8. “Zebra Puzzle” tasks
Wechsler + Amthauer
73 %
9. Negation of complex proposition
Wechsler + Amthauer
8%
The respondents achieved the worst results in items seven and nine which are focused on the negation of quantifiers and negation of complex propositions. Both of these items need to be analyzed separately. In case of quantifier negation, the quantitative research did not enlighten why the success rate is so low and, therefore, the qualitative investigation was acceded. It turned out that for the pupils aged 12 – 13 years it is very difficult to work with the word “exist” and incorporating the existential qualifier into the answer. In this age they already had knowledge of what it means when something exists but they were not able to work with the existential quantifier. This corresponds to the view of Frege [4]. He argues that the incorrect comprehension of logic is caused by the way our everyday language correlates with the logic. Therefore, it is not enough to simply use “mathematical” terms. It is also necessary to use them with the correct meaning [10]. Katrin Přikrylova [8] dedicated her diploma thesis to this issue in great detail.
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The negation of complex propositions proved to be too difficult for students. The main reason is that for its solution it is necessary to use more simple judgments. This type of judgment can be equally challenging. That conclusion is based on the findings obtained during the pilot testing on the 250 respondents.
A∧ B ⇒ C ¬C ¬A ∨ ¬B
¬(∀x) ϕ ( x)
ϕ ⇒γ ¬γ
(∃x) ¬ϕ ( x)
¬ϕ
Modified sentence
For example, for the left judgment it is necessary to chain the following judgments:
A⇒ B ¬( A ∧ B )
A
¬A ∨ ¬B
B
A⇒ B ¬B ⇒ ¬A
For interest, the Table no. 2, presented below shows the success of the students that are older than 13 years (the grammar school students are not included since they achieved much better results and thus the overall results would be distorted). The students considered in this sample were eight and ninth graders from primary schools and students from secondary medical schools. The reason for choosing these schools is that the students with minimum lessons of mathematics and technical subjects were deliberately sought out. Table no. 2 - Success of the respondents according to gender. Test items
Total Boys
Girls
1. Importance of conjunction
70 %
82 %
2. Importance of disjunction
70 %
82 %
3. Importance of implication
84 %
84 %
4. Validity of inverted sentence
81 %
84 %
5. Importance of conjunction
87 %
78 %
6. Importance of general quantifier
78 %
84 %
7. Negation of quantifiers
30 %
33 %
8. “Zebra Puzzle” tasks
92 %
82 %
9. Negation of complex proposition
48 %
40 %
From the Table no. 2, it is obvious that it is not possible for boys to reach significantly different results than girls. The items that had significantly different values are highlighted. It is interesting, that in item 1 the better results were achieved by girls while in item 5 by boys even though both are focused on conjunction.
3 3.1
RESULTS Dependence of school evaluation on pupil’s ability of basic logical connections
It is interesting to observe to what extent the pupil’s ability of basic logical connections correlates with his or her school evaluation. If the test of logical thinking is considered as a whole it shows that this indication can be confirmed at the one percent level of significance. More about this was written in the article [6].
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The text below focuses on what the correlation is between the ability of each pupil’s logical judgment and his or her school evaluations. This testing was attended by all respondents from elementary school for a total of 207 respondents. Given the fact that the individual items focused on basic logical connections are evaluated in dichotomy (see above) and school assessment is on an ordinal scale (1– 2 5) the Pearson χ –test for contingency tables was used for this investigation. The following factual hypothesis was tested: Hv: The pupil’s school evaluation is dependent on his or her ability of basic logical judgments. This hypothesis corresponds with the null hypothesis saying that the two variables are independent. In the following Table no. 3, the values of correlation coefficients and also values of p-level for individual logical connections are presented. This testing was performed on a total of 168 respondents aged 12 – 15 years. Table no. 3 - Dependence of individual items on school evaluation. Test items
Correlation coefficient
Values of p-level
1. Importance of conjunction
-.099
p=.2055
2. Importance of disjunction
-.154
p=.0494
3. Importance of implication
-.164
p=.0376
4. Validity of inverted sentence
-.052
p=.5075
5. Importance of conjunction
-.299
p=.0001
6. Importance of general quantifier
-.219
p=.0049
7. Negation of quantifiers
.179
p=.0229
8. “Zebra Puzzle” tasks
-.071
p=.3722
9. Negation of complex proposition
-.143
p=.0685
It is interesting that all the correlations are negative. These obtained values evoke the idea that a pupil’s school evaluation does not reflect his or her ability in correct logical reasoning. However, it should not be forgotten that a correlation coefficient significantly different from zero is not a proof of functional relationships between variables. Considered the removal of remote values and elimination of possible subsets of objects the only conceivable option for the possibility of influencing the overestimation of the correlation coefficient is the influence of a third variable. The items with statistically significant dependence are marked in bold. It is in these items where it can be concluded that with the consideration of correlative coefficients the school evaluation does not reflect a pupil’s ability of basic logical connections, rather to the contrary.
3.2
Development of pupil’s ability of basic logical judgments
The issue of to what extent it is possible to positively influence a pupil’s ability to work with the basic logical connections by playing mathematical or logical games was already addressed previously. It was discussed in detail in the book [7]. Now, it should be only mentioned the percentage differences in the success in the test (annex 1) during testing before and after the thirteen weeks of playing such games. The games were played for one lesson each week.
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Table no. 4 - Differences in respondent’s understanding of basic logical connections. Values before the period of playing games
Values after the period of playing games
Difference
1. Importance of conjunction
37 %
54 %
17 %
2. Importance of disjunction
41 %
51 %
10 %
3. Importance of implication
20 %
20 %
0%
4. Validity of inverted sentence
73 %
78 %
5%
5. Importance of conjunction
61 %
46 %
-15 %
6. Importance of general quantifier
17 %
15 %
-2 %
7. Negation of quantifiers
5%
5%
0%
8. “Zebra Puzzle” tasks
90 %
89 %
-1 %
9. Negation of complex proposition
39 %
59 %
20 %
Bold fonts are the ones where the difference was statistically significant. In almost all cases the pupils’ understanding of basic logical connections was improved. What is considered positive is the fact that the biggest change occurred in the area focused on negation of complex propositions. It is also interesting that item number five was influenced negatively in a statistically significant manner.
4
CONCLUSION
It has been shown that the reason for misunderstanding of basic logical connections is caused by incorrect expression of pupils and by the way the language and logic is interrelated. The word “exists” is used very rarely by pupils and, if so, in an incorrect meaning. General and existential quantifiers are not even understood as two antagonistic concepts. Also interesting is the extent to which “the level of logical thinking” reflects in the pupil’s school evaluation. Albrecht’s [1] idea that on the basis of specific training in logical thinking, “smarter” people refusing to reject quick answers such as “I do not know.” or “It’s too hard.” is considered plausible. Such people can, therefore, immerse deeper into their own thought processes, achieve better understanding of the methods used and in that way find the correct solution. It has been proved that it is possible to positively influence the pupils’ understanding of basic logical connections. This can be achieved even on the basis of only thirteen hours devoted to playing mathematical and logical games. However, it cannot be forgotten that this game playing must have set rules and it is important for the teacher to express him or herself during this course correctly.
ACKNOWLEDGMENT This paper was partly supported by the Grant Agency of the University of J. E. Purkyně in Ústí nad Labem, grant SGS “The Analysis of level of logical thinking”.
REFERENCES [1]
Albrecht, K. (1984). Brain building: easy games to develop your problem-solving skills. 1. Owl Books Ed. Englewood Cliffs, N.J.: Prentice-Hall.
[2]
Devlin, K. (2012). Introduction to Mathematical Thinking. Leland Stanford Junior University.
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[3]
Dowden, B. (1993).. Logical reasoning. 10th ed. Belmont, Calif.: Wadsworth Pub. Co., 404, 89 p. ISBN 05-341-7688-7.
[4]
Frege, G. (1992). O smyslu a významu. Scientia et Philosophia., 4 (1), s. 33 – 75.
[5]
Hurlez, P. J. (2012). A concise introduction to logic. 11th ed., international ed. Australia: Wadsworth Cengage Learning, 706. ISBN 9781111185893. (Dostupné z: http://emilkirkegaard.dk/en/wp-content/uploads/A-Concise-Introduction-To-Logic-Hurley-7thed.pdf)
[6]
Chytrý, V., Kroufek, R. & Janover, J. (2015) Individual’s logical thinking and its development through games at elementary school In: Fleischmann, O., Seebauer, R., Zoglowek, H. & Aleksandrovich, M. [eds.] The Teaching profession: New Challneges - New Identities. Lit Verlag GmbH & Co. KG, Wien.
[7] Chytrý, V. (2015). Logika, hry a myšlení. Ústí nad Labem: Univerzita J. E. Purkyně v Ústí nad Labem. Plato, J. (2013). Elements of logical Reasoning, Cambridge: Cambridge university press, 271 pp, ISBN 9781-107-61077-4. Str. 108. [8]
Přikrylová, K. (2015) Logical conectives in sentiment analysis. Nepublikovaná diplomová práce, Praha, Karlova univerzita.
[9] Labouvie-vief, G. (1992) A neo-Piagetian perspective on adult cognitive development. In R.J. STERNBERG & C.A BERG (eds) Intellectual Development, New York, Cambridge University, Press. [10]
Pimm, D. (1987) Speaking Mathematically: Communication in Mathematics Classrooms. London: Routledge.
[11]
Vaidya, A. & Erickson, A. (2011). Logic & critical reasoning: conceptual foundations and techniques of evaluation. Dubuque, IA: Kendall Hunt, , 221 p. ISBN 9780757588068. (Dostupné z: http://www.sjsu.edu/people/anand.vaidya/courses/c4/s2/Logic-and-Critical-ReasoningBook.pdf)
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ANNEX 1) Logical links a) The natural numbers are 1, 2, 3, 4 etc. Write down the first five natural numbers divisible by three and four at a time. …………………………………………………………………………………………... b) Write down the first five natural numbers divisible by three or four. ……………………………............................................................................................... c) Is the following statement true? If a number is divisible by six, it is even.
YES - NO
d) Is the statement reversed to the previous one true?
YES - NO
2) Write down the numbers that are even, and fall into the interval between three and sixteen, included. …………………………………………………………………………………………………………… ………………………………………………………………………. 3) Answer to the following statements. Base your replies on the expressions used in these satements. a) If you know that each mammal drinks milk and a dolphin is a mammal, what conclusion do you draw from that? ...................................................................... ………………………………………………………………………………....... b) Let´s take this statement: Each child has at least one friend. When is this statement not right? ............................................................................................................................................... ............................................................................................................... 4) We keep three dogs in the flat, and each dog has its own bed. Alik lies in Bertik´s bed and Rex is not in his own. In which bed is Bertik? ...................................................................................................................................................... ........................................................................................... 5)
If I do the homework and the training is cancelled, I will go to see my friend. What does the fact mean that I did not go to see my friend? Answer in full sentences. In your reply make use namely of the expressions used in the assignment.
……………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………… ………………………………………………
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