Developing students’ mathematical literacy: PMRI schools revisited
Abstract
After a decade of PMRI implementation in primary schools in Indonesia, many concern about
sustainability of the innovation. This paper describes the study on the impact of PMRI
implementation in primary schools in Banjarmasin. The objective of the study is to find out
mathematics teaching practices carry out by teachers who involve in the PMRI project. These
teachers attend PMRI start-up workshop and participate in the followed activities such as followup workshops and quality-boost program. The question is whether PMRI teachers remain
perform well as standard of PMRI lesson. What are the different between PMRI and non-PMRI
classes? How do students in both classes behave? The study gives an evidence that the
implementation of PMRI contributes to the changing in the mathematics teaching and learning.
In PMRI class we find that the teaching is moving away from transmition of mathematical
procedure and factual knowledge to students ‘doing mathematics’. Students in PMRI class are
more active as compare to non-RME class. It reflects, more or less, human element of
mathematics teaching and learning. This type of mathematics learning is potential for
development of students’ mathematical literacy. Based on this finding, we suggest the
dissemination of PMRI to other schools. However, expanding PMRI to wider schools should be
accompanied by teacher preparation. The efforts made by the PMRI team has been pretty
effective. PMRI team conducted a series of workshops for teachers and lecturers. Further
implementation is needed to sustain schools and must be supported and facilitated by PMRI
center in each region.
Summary
Objectives
This paper elaborates the study on the impact of realistic mathematics education (RME) toward
students’ mathematical literacy. RME is a theory of mathematics teaching and learning which
originally developed in the Netherlands. It puts emphasize on students exploration to contextual
problems as vehicle to develop their mathematical understanding. The theory of RME has been
adapted to Indonesia context known as Pendidikan Matematika Realistik Indonesia (PMRI).
PMRI has attracted schools to join. These schools are labeled as PMRI partner schools. The
research is conducted in PMRI partner schools in Banjarmasin. The schools have been involved
in the PMRI project since the early phase of introduction. Some teachers from the schools have
participated in PMRI workshops, both locally and nationally. The main concern of this study is
whether the PMRI approach succeeded in changing the practice of mathematics teaching in its
target schools.
In the study we examined the differences in learning and of mathematical literacy of students
who learn mathematics lead by teachers who use PMRI and those who learn in conventional one.
The conventional way of mathematics teaching and learning is an instruction which is dominated
by teaching as telling method. In non-PMRI classes, teachers usually perform their lesson
following the sequence: opening – example – exercise – closing. Their lesson structure is
dominated by traditional ‘chalk and talk’ that put emphasize to teachers as the central of
instruction. While in the PMRI classes teachers tried to structure their lesson by emphasizing the
students’ learning. Although they perform it in a mechanistic way, the teachers always asked
their students to explain their thoughts, or comment on their responses, or facilitate discussion.
By doing so, students are expected to be able to build their mathematical literacy as it is needed
for their next learning as well as for their life.
Theoretical frameworks
By definition, mathematical literacy is the capacity of individuals to identify and understand role
of mathematics in life, make a reasonable consideration, use and engage with mathematics in
such a way that meets the needs of individuals as citizens in a constructive, concerned and
reflective (OECD, 2009). One of the goals of mathematics education in Indonesia is to develop
citizens who are capable to develop their own life and contribute to the development of the
nation. In the National Curriculum, it is stated that mathematics education may contribute to the
development of the nation character.
In fact, classroom teaching has a moral dimension. However, as long as mathematics is
understood, practiced and experienced “as a depersonalized, uncontextualized, non-controversial
and asocial form of knowledge” (Brown, 1996), then according to Falkenberg (2006) it is hard to
see the contribution of mathematics teaching to the character building, i.e. moral understanding
and moral imagination.
Falkenberg (2006) proposes to academic and professional mathematics education community to
give consideration to the ’human element’ in mathematics. He notes different forms of
‘humanizing of mathematics education’ which has taken different forms with different foci. For
instance, for some the focus is on the ‘human’ nature of mathematics (Hersh, 1997; Lakoff &
Nunez, 2000), some suggest a focus on the social responsibility of mathematics and mathematics
teaching (Skovsmore & Valero, 2001) and others suggest different ways of ‘humanizing’ the
curricular teaching of mathematics (Brown, 1996; Freudenthal, 1968; Katsap, 2002; Wheeler,
1975).
The suggestion to consider the role of mathematics teaching to students’ character building
might be bridged by its specific purposes like the development of mathematical literacy.
Learning math is not only aimed to help students understand mathematics as a stand-alone
subject and separately with the daily life of students, but also to help students to improve their
reasoning and ability to think mathematically for their own benefit as individuals and as part of
community.
Methods
In the study, data are collected using several instruments. The main instrument is the
Mathematical Literacy Test (TeLiM) (Hadi, et al. 2010). The instrument is administrated to
students from PMRI and non-PMRI classes. Students’ works are scored using a certain rubric.
Hereinafter, t-test is used to know the different of both groups. To obtain a clearer picture about
the learning process of students, their paper-and-pencil works are analyzed, which includes their
competences to think mathematically, model, solve problem, and utilize mathematics
representation such as symbols and formal notations.
In addition, to reveal how both groups learn, they are given worksheet which contains a
contextual problem to be discussed in groups. The activities are video recorded. The data in the
form of video footage is complimentary in analyzing their paper works. Video analysis is also
used to determine how the interaction takes place among students in group discussion. We
examine their performance in arguing, questioning, and communicating.
Results
The analysis of learning activities on PMRI and non-PMRI classes resulted in the following
differences (Table 1).
Table 1 Difference of mathematics learning between PMRI and Non-PMRI classes
Aspects of
observation
Center of
learning
PMRI class
Non PMRI class
Students as the center of learning.
The teacher act more as facilitator
and motivator.
Teacher dominates the learning activity
from beginning till the end.
The use of time
Time for learning is more effective
as it is used by students for thinking
and discussing in small groups.
Class
management
Variety of
students’
problem
solving strategy
Teacher’s
acceptance to
students’
mistakes
Students’
expression of
own ideas and
concepts
Students are learning in small
groups.
Strategies used by students in solving
problem are diverse.
Time for thinking and discussion is
limited since it is used by the teacher to
explain the lesson, while students do a
note taking of story problem which is
actually written in the book.
Learning is conducted in classical
manner.
Students’ solutions are more or less the
same.
To have a mistake is not a taboo, and Students are afraid to make mistake.
it should not be ridiculed. Students
Every wrong comment would get bad
are more willing to express ideas
consequence of being laughed by
without fear of being laughed.
classmates.
Students are given opportunity to
Students are not given opportunity to
explain ideas and argument to others. explain their ideas. When teacher asked
Each argument is appreciated and
a student to come to the board to write
commented by others.
his answer, teacher does not ask him/her
to explain to his/her classmates.
The PMRI class performs better in term of the general course of learning, teacher and students
activity. In addition, students’ behavior is also difference between both groups. Students in
PMRI class dare to express ideas, more diverse in using problem-solving strategies as compared
to students in a class of non PMRI. Moreover, in a PMRI class students are able to communicate
their ideas, and not afraid to make mistakes.
Based on qualitative analysis of students’ works, we find a difference in the ability to solve
problems. PMRI students solve problems better, and more varied. In general, students in PMRI
class scored better than students of non-PMRI. The average score obtained by students of PMRI
class is 33.15 while the Non-PMRI class average score is 21.89 from the maximum score of 80.
By using the t-test, the difference in scores of the two groups is significant.
References
Brown, S.I. (1996). Towards humanistic mathematics. In Allan J. Bishop et al. (Eds),
International handbook of mathematics education (2 vols., pp 1289-131). Dordrecht:
Kluwer.
Falkenberg, T. (2006). Moral education in the teaching of mathematics. In J. Novotna, H.
Moraova, M. Kratka, and N. Stehlikova (Eds.), Proceeding of the 30th Conference of the
International group for the Psychology of Mathematics Education, Vol 3. Prague: Charles
University, Faculty of Education.
Freudenthal, H. (1968). Why to teach mathematics as to be useful? Educational Studies in
Mathematics, 1(1), 3-8.
Hadi, S., Danaryanti, A., Ansori, H., Zani, M.Y., and Arifin, B. (2010). Perbandingan Literasi
Matematika Siswa Sekolah Dasar yang Diajar dengan PMRI dan Diajar dengan Cara
Konvensional. Laporan Penelitian, Universitas Lambung Mangkurat, Banjarmasin.
Hersh, R. (1977). What is mathematics, really? New York: Oxford University Press.
Katsap, A. (2002). Humanizing mathematics: The humanistic impression in the course for
mathematics teaching. Humanistic Mathematics Network Journal, 26, 12-19.
Lakoff, G., & Nunez, R.E. (2000). Where mathematics comes from: How the embodied mind
brings mathematics into being. New York: Basic Books.
OECD (2009). PISA 2009 Assessment Framework: Key Competencies in Reading, Mathematics
and Sciences. OECD.
Skovsmose, O., & Valero, P. (2001). Breaking political neutrality: The critical engagement of of
mathematics education with democracy. In Bill Atweh, H. Forgasz & B. Nebres (Eds.),
Sociocultural research on mathematics education (pp. 37-55). Mahwah, NJ: Lawrence
Erlbaum.
Wheeler, D. (1975). Humanising mathematics education. Mathematics Teaching, 71, 4-9.