Yüksel DEDE*
© 2005 E¤itim Dan›flmanl›¤› ve Araflt›rmalar› ‹letiflim Hizmetleri Tic. Ltd. fiti. (EDAM)
*Correspondence: Assistant Prof. Dr. Yüksel DEDE, Cumhuriyet University, Faculty of Education Mathematics Education Department of Primary Education 58140 Sivas, Turkey.
E-Mail: ydede@cumhuriyet.edu.tr & ydede2000@yahoo.com Educational Sciences: Theory & Practice
6 (1) • January 2006 • 118-132
Although affective or cognitive aims or cognitive factors contain af-fective factors, it is seen that cognitive aims usually take place a mo-re significant place in curriculum and textbooks. Yet, affective fac-tors have developed in the same way and shown their own effects in men’s lives. So, this kind of negative point of view towards educati-on’s affective aspect is inconvenient. Studies that focus on mathe-matics education and effective usually focus on topics such as attitu-des, beliefs, and motivation while ignoring values (Seah & Bishop, 2000). However, studies that focus on mathematics and values are li-mited. Nonetheless, values are the most important elements of mat-hematics learning and teaching (Seah, 2002). What are these values?
According to Brown (2001), identifying values is hard. For this, we need some concepts such as “good” and “bad” (Swadener & Soed-jadi, 1988). The word “value” has been used in different meanings.
“The value” of unknown in an equation, the “value” of listening a conversation and a moral “value” of an individual can be given as an example (Seah & Bishop, 2000). According to Swadener and Soed-jadi (1988), identify values as a concept or an idea about value of anything has been always difficult. Mattthews (2001) also sees them as leaders and means of behaviors. When looking these identificati-ons, it can be described as personnel choices considering value or importance of a behavior or idea, or general aims that are adopted or followed by an individual as a member of a society. Therefore, valu-es have reflected concepts or ideas about anything. Valuvalu-es can be ca-tegorized into two as aesthetic and ethical. Aesthetic values are bea-uty concepts but ethical values are about concepts which can be ex-pressed as good or bad. Ethical values are interested especially in go-od and bad sides of a behavior. This part of values forms a whole-ness with education. They cooperate with education and so they make society formation possible (Swadener & Soedjadi, 1988).
Mathematics and Values
Modern mathematics has a deductive-axiomatic structure and gene-rally shows a hierarchical construction. So, it is hard to understand a mathematical concept without being aware of its preliminary sub-jects. This structure of mathematics depends on undefined terms, definitions, and logical rules (Swadener & Soedjadi, 1988). Absolutist philosophers who see mathematics from this perspective appreciate it as an abstract science and think that it is interested in
generalizati-on, theory, and abstractions. So, mathematics is seen as a field which has no social choice and with which only a few people concerns. Ac-cording to this view, mathematics is value-free; that it is neutral (Bis-hop, 1988; Bis(Bis-hop, 2002; Ernest, 1991). In fact, mathematics is loaded with values. It is not neutral. Yet, values are generally taught impli-citly rather than expliimpli-citly in mathematics. However, values are rarely taken seriously at mathematics-related educational discussions and mathematics teachers are generally interested in operations that ha-ve only one answer. They don’t belieha-ve values teaching in mathema-tics lessons (Clarkson et al., 2000). Sam and Ernest (1997) classify the values about mathematics education into three as:
i) Epistemological Values: They are values which are about theore-tical side of mathematics learning and teaching such as accuracy, systematicness, and rationalism and also characteristics, appreci-ation, and acquiring mathematical knowledge (e.g., accuracy, be-ing analytical, rationalism and problem solvbe-ing).
ii) Social and Cultural Values: They are values that indicate hu-man’s responsibilities about mathematics education for society such as compassion, integrity, moderation, and gratitude.
iii) Personal Values: Values that affect a person as an individual or a learner such as curiosity, thriftiness, patience, trust, and creativity.
Bishop (1996) classifies values taught in mathematics into three dif-ferent types by making them more specialized than that of Sam and Ernest. They are general educational values, mathematical values, and mathematics educational values (cited in Bishop et al., 1999).
a) General Educational Values
General educational values are values that help teachers, schools, culture, society, and students to improve. Generally, they contain ethical values such as good behavior, integrity, obedience, kind-ness, and modesty (Bishop et al., 1999; FitzSimons et al., 2000).
Warning a student who cheated during an exam can be an examp-le (Seah & Bishop, 2000).
b) Mathematical Values
Mathematical values are those that reflect the nature of mathema-tical knowledge. They are produced by mathematicians who have
grown up in different cultures (Bishop et al., 1999). Proving Pytha-gorean Theorem in three different ways and their appreciation are examples to mathematical values (Seah &Bishop, 2000). Culture stands as a powerful determiner of mathematical values. Research shows that basis values of all cultures have not been shared. So, mathematics teachers work in different cultures do not teach the sa-me values, even if they have taught the sasa-me curriculum (Bishop et al., 2000). Bishop classifies mathematical values taught in Western culture into three categories as complementary of each others (1988; cited in Seah & Bishop, 2000).
These values are listed below:
i) Rationalism-Objectivism: Rationality indicates the values that people have about mathematics. According to this value, mathe-matics has the ideas which depend on theory, logic, and hypot-hesis (Bishop et al., 2000). Shortly, rationalism value shows a de-ductive logic which concerns about only correctness of results and explanations. Objectivism value shows objects and symbols which are instruments to concretize mathematics that has an abs-tract language (Bishop et al., 1999; Seah & Bishop, 2000).
ii) Control-Progress: Control value shows that mathematics be app-lied, not only on phenomena about its nature but also on prob-lems and solutions in social areas (Seah & Bishop, 2000). Mathe-matics’ results have correct answers that can always be controlled (Bishop et al., 1999). However, mathematics with its other aspect is open to progress every time and it can be used in other fields especially in school lessons.
iii) Openness- Mystery: Openness value shows discussing and analy-zing mathematical theorems, ideas, results, and argumentations.
And such a situation leads us to reach corrects and to find new theorems (Seah & Bishop, 2000). Mystery value indicates mathe-matics own relation, pattern and surprises in its own nature. Such as; dividing every circle’s perimeter into its diameter gives the same number (π number) or Pythagorean triangles that have 3, 4, 5 or 5, 12, 13 cm edge length gives always a multiple of 60 when they are multiplied with each other. Mathematics has always such kinds of mystery and surprise in itself (Bishop et al., 1999).
c) Mathematics Educational Values
Teaching mathematics educational values may show differences ac-cording to countries, cities, school types and grades. For example;
choice of problem solving strategies may show differences according to the environment. So, the number of mathematics educational va-lues can increase to that rate. In this paper, five complementary mat-hematics educational values will be emphasized. These are;
i) Formalistic view- Activist view: Formalistic view value shows the deductive and receptive learning values of mathematics, while activist view value shows its intuition and discovery learning;
that is to say, its inductive sides.
ii) Instrumental understanding/learning-Relational understan-ding/learning: Instrumental learning indicates learning rules, operations and formulations in mathematics education and their applications to special questions. Relational learning shows disp-laying the relationships among concepts and forming appropria-te graphics.
iii) Theoretical knowledge-Relevance: Mathematical education’s theoretical value suggests teachings mathematics at theoretical basis and far from daily events. Relevance value shows the im-portance of mathematical knowledge in solving daily problems.
Daily problems and demands show different at societies and cul-tures. Thus, mathematics can provide special solutions to cultu-ral needs and demands.
iv) Accessibility -Special: These values indicate doing and preparing mathematical activities by either everyone or just by people who has talent in it.
v) Evaluating - Reasoning: Students are asked to realize the steps of knowing, applying routine operations, searching solving prob-lem, reasoning and communicating in order to solve a problem.
The first three of this five steps demonstrate using mathemati-cal knowledge about evaluating an unknown answer; while the last two demonstrate the capability of using mathematical know-ledge, reasoning more and the ability of spreading the knowled-ge (Seah & Bishop, 2000).
Mathematics textbooks are also main teaching tools. In some cases, mathematics textbooks are perceived as the mathematics
curricu-lum (Seah, 2000). They have also conveyed the values. Textbooks studied at both primary and secondary Turkish public schools can be published at special publishers and also at National Educational Publishing. The appropriateness of these textbooks for curriculum has been checked by the Instruction Council which assigned to the Ministry of National Education. The textbooks that are approved from this council can be used in primary and secondary schools.
The choice of these textbooks shows differences according to teac-hers who are employed there. So, at public schools, textbooks pub-lished at different publishers can be taught. Therefore the questi-on answered in the present research is how much mathematics edu-cational values take place in mathematics textbooks taught in 2004-2005 educational term at Turkish public high schools. For this pur-pose, the answer for the question below was searched
* How much importance is given to mathematics and its educatio-nal values in mathematics textbooks taught in the 9th, 10th, and 11th grades at Turkish public high schools?
Method
This study is limited with searching only high school mathematics textbooks. So, three textbooks were randomly chosen for each mat-hematics textbooks taught in the 9th, 10th and 11th grades. Howe-ver, total twelve textbooks were taken into study by three textbo-oks published by the Ministry of National Education. Each of the-se books was named as A9, B9, C9, D9, A10, B10, C10, D10, A11, B11, C11, D11 special codes and during research, these books we-re named with these special codes. Hewe-re, number 9 stands for 9th grade, number 10 stands for 10th grade, and number 11 stands for 11th grade. In searching textbooks, semantic content analysis was used. Semantic content analysis is a method that finds out the ma-in subject areas ma-in material content, its dimensions, and special sub-dimensions about these areas and sub-dimensions (Tavflanc›l &Aslan, 2001). In this study, mathematics and its educational values are ta-ken into consideration as general areas and sub-areas which enter to these areas (see table 1).
The pairs of rationalism-objectivism, control-progress and open-ness-mystery studied under general category of mathematics valu-es were invvalu-estigated by the help of the method below:
In rationalism value, logical connectors such as action-reaction, cau-se and effect have been cau-searched within exercicau-se, problem and example solutions and their numbers cited have been determined.
However, examples, exercises and problems which were explained by mathematics deductive logic an abstract language have been eva-luated within this category. Graphics, figures etc…that concretize its abstract language have been investigated. Example, exercise and problems; which don’t give students any freedom and have impera-tives within the scope of a direction, have been considered for con-trol value, while progress value examples, exercises and problems;
which give freedom to students, suggest usage of mathematics at ot-her fields and have analogy, model and etc…have been searched.
Hard and complicated examples, exercises and problems that disp-lay mysteries and surprises of mathematics are investigated within the scope of mystery value, while easy ones that can be done by stu-dents are evaluated within the scope of openness value.
For formalistic view among mathematics educational values, the si-tuations; in which teaching has been done by deductive approach-that is to say by teachers and textbooks- have been detected, while for activist view, the situations; in which there are inductive and discovery teaching, have been detected. In relevance value, examp-les, exercises and problems which emphasize that mathematics is about daily-events are investigated and in theoretical knowledge, the ones; which emphasize that mathematics is only mathematics,
Table 1
General categories and sub-categories in semantic content analysis
General Categories Sub-Categories
Rationalism-Objectivism
Mathematical Values Control -Progress
Openness - Mystery
Formalistic view- Activist view Instrumental-Relational Understanding Mathematics Educational Values Relevance-Theoretical
Accessibility-Specialism Evaluating - Reasoning
are investigated. Subject, explanations, examples, exercises and problems; in which only rules, operations and formulations were used, are studied in instrumental learning and in relational learning, the ones; which demonstrate the relations between concepts, are studied. Also in accessibility value, examples, exercises and prob-lems which can be understood by every student easily are focused on. In evaluation, exercises and problems given at the end of topic are looked at and the questions that require routine operations si-milar with the topic examples are evaluated. For reasoning value, questions that lead students to think about and different from pre-vious ones are searched. Here, for example; a question related as
“f (x) = 2x-3 is given defined in R. What is f (2) = ?” Contains both instrumental understanding/learning value and accessibility value.
To test the reliability of the study, textbooks searched have been given to different people and all the results which were held by them are compared. And also, all the sections of textbooks are not examined; only certain topics of each mathematics textbook were analyzed. Logic, sets and relation, functions and treatment topics in 9th grade textbook, trigonometry, logarithm and induction topics in 10th grade textbook, and limits, derivative and integral topics in 11th grade textbook were analyzed. The selection of these topics of each textbook done by random method. Thus, total of the 2570 pa-ges in the 12 textbooks were examined.
Results
In this part; for example, explanations about the situations in which 9th grade mathematics textbooks within the scope of research car-ried mathematical and mathematics educational values will take place. The distribution of the numbers of each mathematical valu-es and their complementary pairs in 9th grade textbooks searched according to each subject is given at table 2.
When looking at table 2, it can be seen that total of 150 pages in A9 textbook were analyzed. For example in A9 textbook, in logic to-pic it can be seen that rationalism value (148) is emphasized more than objectivism value (23), control value (36) is emphasized more than progress value (7) and openness value (64) is emphasized mo-re than mystery value (29). For the distribution of mathematical va-lues in other two subjects, table 7 can be looked at. When generally
Table 2Distribution of mathematical values for some topics in 9th class mathematics textbooksMathematical ValuesTextbookTopicsPage numbersComplementary Value PairsComplementary Value PairsComplementary Value PairsRationalismObjectivismControlProgressOpennessMysteryLogic1-37148233676429Sets38-711421772310727Relation,A9function and72-1502484616862358treatmentTotal150538862761640664Logic1-261979459679Sets27-6319022137-14330Relation,B9functionand treatment64-1422315418021968Total140682853621140647Logic8-4111525132812911Sets42-8216313176-15113Relation,C9function83-14822122242-242-and treatmentTotal14049960550852224Logic1-2410511116109213Sets25-5214124165-1614Relation,D9function53-11634047384-3651and treatmentTotal11658682665106181General Total54623053131853451952136
looking at A9 textbook, rationalism value (538) is emphasized mo-re than objectivism value (86), control value (276) is emphasized more than progress value (16) and openness value (406) is empha-sized more than mystery value (64).
Two examples about rationalism-objectivism values in A9 textbook are given below:
*A = {a,b}, B = {3,4,6} sets are given. Let’s show AxB ≠ BxA (A9, p.75, relations, function and treatment, rationalism).
* Write of negative the propositions are given below.
i) Elif gathered flowers or Gökhan didn’t play football.
ii) I threw a stone and didn’t break the window (A9, p.16, sets, objectivism).
For the distribution of mathematical values in other the textbooks, table 2 can be looked at. Furthermore, from table 2, it can be seen that total of 546 pages in 9th grades textbooks were analyzed. When looking at 9th grades textbooks in general, it is obvious that rationa-lism (2305) is conveyed more than objectivism (313), control (1853) than progress (45) and openness (1952) is conveyed more than mys-tery (136). These data indicate that in each of these four textbooks investigated, rationalism, control and openness values have been emphasized more. This situation proves that 9th grades mathema-tics textbooks have been prepared in the way that doesn’t keep stu-dents out of textbooks and take care about mathematics’ good si-des, mysteries and being abstract.
Mathematics educational values that are conveyed by 9th grade mathematics textbooks are given at table 3:
When looking at table 3, in teaching logic topic in A9 textbook, we can observe that formalistic view (100) is emphasized more than ac-tivist view (0), theoretical value (54) than relevance value (26), ins-trumental understanding (74) than relational understanding (10), accessibility value (64) than specialism value (20) and finally evalu-ation (71) than reasoning value (0). And in the topics of sets, relati-on, function and treatment, similar situation attracts our attention.
When looking at A9 textbook in general, it can be said that forma-listic view (681) is conveyed is more than activist view (0), theore-tical value (360) than relevance value (41), instrumental understan-ding (395) than relational understanunderstan-ding (10), accessibility value (369) than specialism value (52) and lastly evaluation (234) than re-asoning value (2). Two examples about relevance- theoretical knowledge values in A9 textbook are given below:
Table 3Distribution of mathematical values for some topics in 9th class mathematics textbooksMathematical ValuesTextbookTopicsComplementaryComplementaryComplementaryComplementaryComplementaryValue PairsValue PairsValue PairsValue PairsValue PairsFormalisticActivistRelevanceTheoreticalInstrumentalRelationalAccessibilitySpecialismEvaluationReasoningViewviewKnowledgeunderstandingunderstandingLogic100-26547410642071-Sets98-107484-1262855-Relation,A9function255-5232237-17941082and treatmentTotal681-4136039510369522342Logic163-249487557542-Sets84-21186124-12131918Relation,B9function128-720123932242-1583and treatmentTotal375-52481450374203629111Logic53-161121048108460-Sets78--170148221482285-Relation,C9function106-12142132215-113-and treatmentTotal237-174964653247124258-Logic45-81171108125860-Sets37--1801522815327104-Relation,D9function112--34630323431228-and treatmentTotal194-86435653862136392--General Total1487-110198018751171881148117513
* The value of correctness of q, r and s is 1, the value of correct-ness of p and t is 0. Draw the electric circuit which expresses [(p∧q)∨r]∧(s∧t). Determine if the current passes in the electric circuit? (A9, p. 19, logic, relevance)
* If p’ : (π > 3), write p proposition and find the correctness va-lue of p (A9, p. 5, logic, theoretical knowledge).
Similarly, in examining B9, C9 and D9 textbooks in general, it has been fixed that formalistic view (375, 237, 194) is cited is more than activist view (0, 0, 0), theoretical value (481, 496, 643) than relevan-ce value (52, 17, 8), instrumental understanding (450, 465, 565) than relational understanding (37, 32, 38), accessibility value (420, 471, 621) than specialism value (36, 24, 36) and lastly evaluation (291, 258, 392) than reasoning value (11,0,0). When considering whole books in general, that similar conclusions got is seen. Total mathematics educational values emphasized in all of the textbooks are; formalistic view: 1487 - activist view: 0; theoretical knowledge:
1980 - relevance: 110; instrumental understanding: 1875 - relational understanding: 117; accessibility: 1881 - specialism: 148; evaluation:
1175 - reasoning: 0. In fact, these data aren’t very surprising. Becau-se in 9th grade textbooks above, it has been fixed that mathemati-cal values; which are far from daily events, has an abstract language are dominant. Therefore, it is not surprising that similar mathema-tical values have been emphasized more in the same textbook whi-le teaching logic, sets, and relation, function and treatment topics.
Similarly, in examining 10th and 11th grades mathematics textbo-oks in general, that similar conclusions got is seen.
Discussion
We reached these findings as a result of the research that investigates whether the 9th, 10th and 11th grade mathematics textbooks convey mathematics and its educational values in Turkish public high schools.
It is fixed that 9th, 10th and 11th grade mathematics textbooks emphasize rationalism more than objectivism, control than progress and openness more than mystery. And also it is fixed that 9th, 10th and 11th grade mathematics textbooks emphasize formalistic view more than activist view, theoretical knowledge than relevance, ins-trumental understanding than relational understanding, and
acces-sibility than specialism and evaluation than reasoning among mat-hematics educational values.
The findings of this research show some similarities and differences with the findings of a research that examines whether Seah and Bis-hop’s (2000) Singapore and Victoria mathematics textbooks convey mathematics and its educational values or not. At the result of Seah and Bishop’s (2000) research, objectivism, control and mystery mat-hematical values are emphasized more than rationalism, progress and openness in both Singapore and Victoria mathematics
The findings of this research show some similarities and differences with the findings of a research that examines whether Seah and Bis-hop’s (2000) Singapore and Victoria mathematics textbooks convey mathematics and its educational values or not. At the result of Seah and Bishop’s (2000) research, objectivism, control and mystery mat-hematical values are emphasized more than rationalism, progress and openness in both Singapore and Victoria mathematics