RahayuCondroMurti Yogyakarta State University
[email protected] Abstract
Indonesian children face cultural crises nowadays. Learning primary mathematics is hoped to remedy, and even find a way out of that crisis. Constructivist theory, based on Vygotsky views, provide a framework to empower cultural elements and values in order to enhance kids’ capabilities of mathematics and social interaction too. This article aims at introducing Vygotsky ideas on learning and culture, applying them briefly in primary mathematics learning, and propose some examples of learning primary mathematics based on Indonesian cultural elements like banana leaves and temples’ complexes.
Keywords: Indonesian primary mathematics, culture-based learning
1. Introduction
Indonesian children nowadays are surrounding with multi-facet cultural crisis, as globalization unfold its negative consequences and impacts, especially in information and lifestyle assimilation.
Indonesian children today prefer to play different types of virtual realities through their Play-stations and tablets, rather than get involved into real games that involve real social contexts. Such trend is not only attributed to our kids in Indonesia, but also extends itself to kids all over the globe. Games in monitors appear more attractive as if have a magical power! That magical power deprived our kids from knowing and practicing their cultural heritage in shape of traditional games like dakon, gobaksodor, which start to die out! Traditional games nowadays are isolated in marginal areas like far-reach villages. Moreover, internet, far-distance communication over it and weak awareness among kids, resulted in negative impacts such as fraud, sexual abuse, and an overall decline in moral standards and practice among kids. Kids’ education trends usually find their roots in the structural changes in the social-economic and social-political environment. Technological breakthroughs, that occur in context of moving from agriculture-based to industrial and services-based economy and the accelerated ill-controlled democracy, resulted in that big picture of kids’ weak cultural connections.
From another hand, TIMSS (Trends in International Mathematics and Science Study) and PISA (Program for International Student Assessment) trends show weak performance of Indonesian kids in mathematics. TIMSS 2012 indicates the weak cleverness capabilities of Indonesian kids. One of that ‘weak capabilities’ is
shown through ill-application of mathematics’
lessons in real life contexts such as story-based problem solving. Primary mathematics, in context of sustainable development, aim to support connecting education and real life contexts.
Mathematics, within such frame, should be described clearly, articulated systematically and logically. If not, primary mathematics will appear as a heavy burden that disconnecting kids from their reality, and pushing them, more and more, into virtual realities in their monitors! If not, primary mathematics will lose its high potential role, although mathematics is surrounding us all the day through different kinds of modeling, measuring, calculations and transactions in different fields of real life.
Be aware of such trends and their roots, primary education demands, more than any time before, creative teachers with wide cultural awareness and integrated capabilities. One basic duty of creative teachers is to rebuild kids’
capabilities in connecting mathematics to real life contexts through story-based problems. Rebuilding those capabilities in kids, who are in concrete operational phase of mental development, should be oriented into articulating new concepts meaningfully. Meaningfulness, in such regard, means reconnecting kids with their selves, with their social and cultural environments. Primary mathematics, then, will not only mean ‘learn to calculate’, but also mean ‘learn to be one’s self’,
‘learn to work’, and ‘learn to live together’.
2. Primary Mathematics Education
2.1. Philosophies of Mathematics Education Hans Freudenthal [3] proposes that there are four types related to philosophies of mathematics education, which are mechanistic, structuralist, empiristic, and Realistic. From a mechanistic point
of view, man is a computer-like instrument, that can be programmed by drill to perform, on the lowest level, arithmetic and algebraic, even geometric operations, and to solve applied problem.
From a structuralist perspective, well-structured system of mathematics shall be taught. Whereas empirist shows the world as a reality, where man can acquire useful experiences, which are broad- mindedly interpreted. In Realistic instruction the learner is given tasks that proceed directly from reality.
Cultural-based primary mathematics education has similar foundation with the realistic point of view, where it always tries to connect mathematics with reality. Freudenthal [3] suggests that realistic primary mathematics education has two main bases: (1) mathematics must be connected to reality, (2) mathematics should be seen as human activity. Be aware of those principles, learning mathematics in Indonesian primary schools should be in service of the Indonesian kids, connecting them to the real contexts of Indonesian life, and helping them solving problems related to those contexts, primary education should not far from Indonesian kids’ daily activities and lives.
2. Learning Mathematics in Primary School Learning development usually classified into 4 phases. According to Piaget, in Muijs [4], learning occur through 4 stages: (1) sensorimotor that occur from age 0-2 years, (2) pre-operational that occur from age 2-7, (3) concrete operational that occur from age 7-12, and (4) formal operational that occur from 12 and above. Pupils in primary school, based on Piaget conception, are in the ‘concrete operational’ phase, where student can understand a concept through material application or real settings. Learning mathematics in primary school, based on Piaget’s, should not be of complex abstract nature, but of simple, yet systematic, and concrete nature.
Learning mathematics is a process of interaction among students and teacher, which make use of its all aspects of learning environment, in order to achieve curriculum objectives, including the improvement of learning process in an optimal way. To a learning process, in order to be improved, students should be taught in a way that always connect mathematics new concepts with previous ones, especially when we understand mathematics as a process of continuous building of pattern and relationships. In this regard, Deboys [1]
proposes that one of mathematics priorities is to
‘equip children to thing for themselves’. Deboys words may be find their application in a curriculum practice that helping students to build their own
knowledge independently, in corporation with teacher and/or his/her friends, making use of concrete media and real contexts, which result in high motivation and consistent behavior of valuing mathematics lessons due to their positive role in daily life around the kids.
3. Culture in Primary Mathematics Learning Saifer [6] defines ‘culture’ as a ‘way of life’, which can be connected to ‘socially transmitted habits, customs, traditions, and beliefs that characterize a particular group of people at a particular time’. According to Saifer also, the way we do learning, solving problems and teaching others are affected deeply by culture. Primary mathematics, building on Sifer views, cannot be separated from culture spheres around pupils, and primary mathematics’ aim could be stated as helping pupils in dealing effectively with daily life problems they face. Following are reviews of main views on culture-based learning of primary mathematics.
3.1. Vygotsky Theory of Culture & Learning Guided with his psychology background, Vygotsky developed constructivist social learning. According to Vygotsky, cognitive capabilities are built through constructive process that is articulated independently by learners. Lev Vygotsky could develop his constructivist views building on Piaget ideas on knowledge internalization of learners.
Vygotsky widened Piaget ideas by including factors of social interaction and collaboration that occur during learning process. Whereas Piaget believes that mental development process includes 4 main phases (sensorimotor, pre-operational, concrete operational and formal operational), Vygotsky realizes that process as a complex and continuous one that occur along the life line and that is difficult to defined in shape of particular stages [ ]. In this regard, Vygotsky criticized Piaget’s views on the relationship between learner’s motivation and cognitive development. Piaget suggested that motivation comes from within the learner and pushes her to learn and interact with her environment, then develop her cognitive capabilities. In the other hand, Vygotsky believed that social interaction is the most defining factor in the process of cognitive development in context of learning. According to Vygotsky, learning process runs efficiently and effectively if learners learn cooperatively with their peers, in supportive environment, and guided by more capable person, like teacher. Following are the central concepts within Vygotsky framework of relationship between culture and learning.
3.1.1. Culture
According to Vygotsky, social environment has an important role in articulating children knowledge.
Children usually learn through language, songs, arts and games. To analyze one’s cognitive structure, culture and history must be taken into consideration. Vygotsky stressed, also, the defining role individual plays in constructing, actively, his own knowledge.
3.1.2. Language
Language plays fundamental role in the development of cognitive capabilities of human being. In this regard, Vygotsky stresses the importance of the psychological mechanism of knowledge internalization through language.
Students construct knowledge, i.e. ways of understands and solving problems, in social environments through language. Through language, learners, including kids in primary school, negotiate the meaning of different experiences they go through.
3.1.3. Proximal Development Zone
Vygotsky [7] suggests that things kids do today with guidance of others, will be done by their selves tomorrow. According to Vygotsky, the development of learner kids could be classified into two levels: (1) actual development level, and (2) potential development level. Actual development level, what Vygotsky calls also as ‘intra-mental capability’, is the capability of a kid to carry out different tasks independently. Potential development level, what Vygotsky calls also as
‘inter-mental’ capability, is the capability of a kid to carry out different tasks with guidance of adult people or in collaboration with his more capable peers. The distance between inter-mental capability and intra-mental capability is called, according to Vygotsky, ‘proximal development zone’, which could be defined as a set of functions and capabilities that did not develop completely and still developing.
Vygotsky explains that learning process occurs where kids carry out tasks that they have not done before but still could be reached. Zone of ‘not done yet’ and ‘could be done’ is what Vygotsky calls
‘proximal development zone’; a zone that is above the recent zone of development. According to Vygotsky, higher mental functions, in general, appear within discussions and cooperation among individuals before that functions be absorbed into cognition of these individuals.
3.2. Some Examples of Primary Mathematics Culture-based Learning
Honigmann in Koentjaraningrat[ ] realizes that there are 3 main cultural elements: (1) ideas, (2) activities, and (3) artifacts. Ideas covers concepts,
wishes, values, and norms. Activities mainly indicates models of behaviors that dominate in one society. Artifact indicates man-made product. In addition, one cannot deny the deep effect nature has on man life, and consequently, his culture.
Following are some experiences of the writer in applying cultural elements to primary mathematics learning.
3.2.1. Fractions Culture-based Learning
Culture and social environment are the most fundamental factors that affect the articulation of learner’s knowledge, including primary school students. Cultural components that are most appropriate for children learning are songs, language, arts, local games and local objects.
Banana leaves could be used as learning media in explaining to kids how to add two fraction numbers, either pure fractions or mixed fractions.
Besides making primary mathematics learning clearer and more concrete, using banana leaves
making mathematics closer to kids’ daily lives. It is better to start with explaining adding pure fractions, where the numbers’ values are less than 1. Below are steps followed to add two pure fraction numbers, for example
.
(Rahayu, 2013)
(1) Take two square banana leaves, sized 20 x 20 cm (size may be adapted to class considerations).
(2) Divide one of the leave into 3 equal parts, then draw multiple lines on of its parts, in the surface of the leaf, by a marker, in order to show part that we have.
(3) Take the other leave, divide it into 4 equal parts, then draw multiple lines on one of its parts, in the surface of the leaf, with a marker, to indicate of the part that is to be added. Do not forget to draw in different color from color used in the first leaf.
(4) Weave the first leaf with the second one, till shape a woven mat as explained in figure below.
Fig. 1, Banana leaf mat shape in learning adding fractions
(5) The previous process will result in 12 small squares, which constitute a mat. Look carefully at lines at the leaves, based on the type of lines, which are now constitute connected boxes, count how many parts that have drawn lines. Yes! There are 7 boxes.
(6) Parts with drawn lines are 7 out of total 12 boxes.
(7) That means the result of adding fractions:
Adding two fraction numbers using banana leaves could explain the concept behind ‘adding’, which is to unifying two things. Banana leaves, as a learning media, could be substitute of ‘cake’ in the classic story based problem where ‘Dewi has of a round cake, then her mother gave her of the same cake, so, how many parts Dewi has in total?’. Seeing directly banana leaves’ parts being divided then added, students become aware why . Students are not only asked to unifying the denominators, but also to compare the result they found out to the real movements outside there.
From a realistic point of view: mathematics notate real world.
3.2.2. Culture-based Integers Learning
To a primary school student, exploring integers (whole numbers) with negative values is a new abstract conceptual experience, so, it is better to a teacher to use a concrete media to make his students understand properly that concept. Media that is available easily are white and black buttons.
White buttons stand for positive whole numbers, whereas black ones stand for negative whole numbers. Teacher may attribute cultural concepts to each color; for example, white is attributed as
‘good’ and black is attributed as ‘bad’, where such attribution is a cultural symbol that still alive in Indonesian society, even in early aged children.
Example:
For -7+5 could be solved by aligning black and white buttons as follows,
To carry out adding task above, teacher shows to pupils how many white buttons that do not have couples, i.e. (-2).
In carrying out adding, subtracting, multiplying, and dividing, teacher could use also the traditional
‘dakon’ game, where using it as a learning media, rather than explaining the concept, it has a cultural added value, which is sustaining traditional team games.
3.2.3. Learning Mathematics in Monumental Settings (Temples)
Temples spread all over the Indonesian archipelago, and could be a rich learning source in primary mathematics. For example, Prambanan temple is one of very famous Indonesian temples’ complexes located in Yogyakarta (Middle Java). Prambanan was established by kings of Sanjaya’s kingdom in the 9th century. Prambanan is labeled as the most beautiful temple in the world and located about 17 km to the east of the city of Yogyakarta.
Many mathematics concepts and lessons could be learnt from Prambanan as a ‘cultural media’.
Following a table that explain some of thoses concepts and lessons.
Table 1
Learning Mathematics in Prambanan Temple No. Artifacts of
Prambanan Temple
Related Mathematics Lessons
1 Counting 8 hands of
LoroJonggrang, 4 heads of Brahma 2 Identifying shapes Stoned beams,
rectangle floor tiles, …etc
3 Adding There are 240
units of the Temple (big 16
& small 224) 4 Subtracting Almost all units Fig. 2, black and white buttons
aligning as a learning media to discover the concept of negative numbers
Fig. 3,Prambanan temple
have heavy, moderate or small damage.
5 Tessellation/Installing floor tiles
Installing floor tiles alongside the garden or in the temple’s halls.
6 Map The map of the
Temple’s complex
7 Distance 17 km to the east
from Yogyakarta and 53 km to the west from Solo
8 Time The restoration
of the Temple started on the 20th of December 1953, stopped in 1977, started again in 1982, and continued till 1991.
9 Directions There are 4
doors of the Complex, matching the principle 4 directions.
10 Symmetry and equal structures
There are 224 units of the Temple that have the same size and structure.
11 Angles Almost all types
of angles are spread all over the temple
12 Area Total area of the
Complex, area of each unit, … etc.
Table above just represents a sample of lessons that could be connected to Temples as a cultural media of learning mathematics in primary school.
Moreover, those lessons, and may be others, could be more developed by teachers as they invite their students to visit temples and get involved directly in observation, measurement, comparing and other study activities, besides get familiar with traditional culture, its values and achievements.
4. Conclusions
Learning primary mathematics based on culture has an important role to play in remedying different symptoms of cultural crisis surrounding Indonesian kids nowadays. Cultural-based mathematics learning should relate mathematics to Indonesian culture richness, and could be done in many context and towards many objects. Learning primary mathematics using local objects (like banana leaves) and traditional monuments (like temples) has a potential role to enhance students’
mathematical capabilities and awareness of their original culture; its values and its achievements.
References
[1] Deboys. Mary, Pitt. Eunice, Line of Development in Primary Mathematics, London:
The Blaackstaff Press, 1996
[2] Driscoll. Marcy P, Psychology of Learning for Instruction. Needham, Ma: Allyn & Bacon,1994 [3] Freudenthal. Hans, Revisiting Mathematics Education, London: Kluwer Academic Publishers, 2002, pp.134-135
[4] Koentjaraningrat, Pengantar Ilmu Antropologi, Jakarta: PT Rineka Cipta, 1990, pp.186-187 [5] Muijs. Daniel & Reynolds. David, Effective Teaching, London: Sage Publication Ltd, 2008, pp.24-25
[6] RahayuCondroMurti, DaunPisangsebagai MediaPembelajaranKonsepPenjumlahanPecahanpa daPendidikanMatematikaRealistikBerbasisBudaya, FakultasIlmuPendidikan UNY, ISBN: 978-979-26- 1970-6, 2013, pp. 194-201
[7]Saifer. Steffen, Culturally Responsive Standards-Based Teaching, United Kingdom:
Corwin, 2011
[8]
Vygotsky. L.S, Mind and society: The development of higher mental processes.Cambridge, MA: Harvard University Press. 1978