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In the research, a quasi-experimental model was applied and the experimental group received the process approach to learning and teaching mathematics, which builds on the cognitive-constructivist findings of the educational profession about learning and teaching mathematics. In the control group, the transmission approach prevailed. In the research, the question was answered of what impact the implementation of the process approach to learning and teaching mathematics has on the learner’s knowledge, which can be tested and assessed.
Keywords: a process approach to learning and teaching, mathematics, basic and conceptual knowledge, solving simple mathematical problems, complex knowledge
1.
Amalija Žakelj
Process Approach to Learning
and Teaching Mathematics
DOI: 10.15804/tner.2018.54.4.17
Abstract
In the research, a quasi-experimental model was applied and the experimental
group received the process approach to learning and teaching mathematics,
which builds on the cognitive-constructivist findings of educational profession
about learning and teaching mathematics. In the control group, the transmis-
sion approach prevailed.
In the research, the question was answered of what impact the implementa-
tion of the process approach to learning and teaching mathematics has on the
learner’s knowledge, which can be tested and assessed.
Students in the experimental group (EG) performed significantly better in
basic and conceptual knowledge, in solving simple mathematical problems, and
in complex knowledge than those in the control group. Results of the research
have also shown that there are statistically significant correlations between
individual areas of mathematical knowledge. The correlations between the
areas of knowledge are from medium high to high, indicating that conceptual
knowledge correlates significantly with solving simple mathematical problems
and with complex knowledge.
Keywords: process approach to learning and teaching, mathematics, basic
and conceptual knowledge, solving simple mathematical problems, complex
knowledge
2.
Process Approach to Learning and Teaching Mathematics 207
The purpose of teaching mathematics is not just to transmit mathematical
knowledge – the opposite is true: the basic purpose is to make students discover
mathematics, think, and build it. To learn mathematics means doing mathematics
by solving and exploring it. But the findings of international evaluations point to
deficient knowledge of mathematics and poorly developed competences, because
of which the question of the quality of learning and teaching mathematics is
persistently raised. The findings also warn that in the practice of mathematical
education formal teaching prevails, oriented to techniques of memorising rules,
which students often do not understand. Students do not manage to see the links
between new knowledge and previously acquired concepts, they are not able to
connect mathematics with everyday life, in their work they are not autonomous,
and they often just repeat certain activities or procedures (UNESCO, 2012).
Although it has been emphasised since the eighties of the past century that the
teaching of mathematics should include solving problems and point to the use of
mathematics in everyday life, in reality it seems that this kind of teaching has not
actually come to life (Dindyal et al., 2012) and that this continues to be one of the
unattainable goals of teaching mathematics (Stacey, 2005).
Basic mathematics education is still too often boring because: it is designed as
formal teaching, centred on learning techniques and memorizing rules, whose
rationale is not evident to pupils; pupils do not know which needs are met in the
mathematics topics introduced or how they are linked to the concepts familiar to
them; links to the real world are weak, generally too artificial to be convincing and
applications are stereotypical; there are few experimental and modelling activities;
technology is quite rarely used in a relevant manner; pupils have little autonomy in
their mathematical work and often merely reproduce activities (UNESCO, 2012).
To overcome the above-mentioned challenges, changes in teaching practices
must be made consistently with the stated goals. As early as 1987, Shulman (1987)
found that the teacher needs not only a good methodological and substantive
knowledge of the topics he teaches, but also a substantive pedagogical knowledge,
i.e., awareness of how students construct knowledge of individual contents. The
teacher who knows how the student constructs knowledge, the teacher who
possesses substantive pedagogical knowledge prepares activities that build on
students’ pre-knowledge, on linking knowledge, he introduces concepts and content
gradually. The notions of both learning and teaching, in turn, significantly influ-
ence the individual’s understanding, perspective or interpretation of the context of
3.
208 Amalija Žakelj
learning or teaching. The basic assumption of the teacher’s operation is promoting
the quality of learning, which leads to students’ quality knowledge.
Research Methodology
The purpose of the research
In the research, we sought to answer the question of how the implementation
of the process approach to learning and teaching mathematics, which had been pro-
duced on the basis of the theoretical knowledge of children’s mental development,
also of recent findings about the child’s thinking, and the knowledge of social cog-
nition, of learning and teaching mathematics, influences the student’s knowledge
that can be tested and assessed. In this we based on the theory of developmental
psychology, which studies the development of concepts from the point of view
of the developmental stage of the child’s thinking (Vygotsky, 1978; Labinowicz,
1989; Gilly et al., 1988) and took into account more recent cognitive-constructivist
findings of learning, which emphasise learners’ activity in the learning process
(Maričić et al., 2013; Van de Walle et al., 2013; Břehovský et al., 2015). The process
approach to learning and teaching mathematics is characterised by experiential
learning, discovering and exploring mathematics through mathematical and life
challenges, and by developing reading learning strategies as the integrating activity
of learning and teaching.
We wanted to determine whether the students in the experimental group (EG),
who had received the process approach to learning and teaching mathematics,
performed better in basic and in conceptual knowledge (PR), in solving simple
mathematical problems (EP) in complex knowledge (ZP) than the control group.
Three research hypotheses were formulated.
Research hypotheses:
H1: In the selected contents block in basic and conceptual knowledge (PR),
the experimental group will perform significantly better than the control
group.
H2: In the selected content blocks in solving simple mathematical problems
(EP), the experimental group will perform significantly better than the
control group.
H3: In the selected content blocks in complex knowledge (ZP), the experimental
group will perform significantly better than the control group.
4.
Process Approach to Learning and Teaching Mathematics 209
Research method
The model of quasi-experiment was applied and the experimental model process
approach to learning and teaching mathematics was introduced in the experimental
group, whereas in the control group the transmission approach prevailed. Because
the model without randomisation was applied—opportunities for the use of mod-
els with randomisation are rather limited in schools—the students’ most relevant
factors were controlled at the beginning (overall learning performance, marks in
Slovenian and in mathematics, education level of parents).
Research sample
In the experimental group (EG), there were 190 eighth grade pupils and in the
control group (CG), 220 eighth grade pupils of Slovenian basic schools. All the
students participating in the research were at the age between 13 and 14 years.
Data gathering and processing
The students’ performance in dependent variables was assessed with knowledge
tests, the content structure of which was: dependent and independent quantities,
percentage, direct proportion, inverse proportion, and equation. The situation
before and after the introduction of the experimental factor was recorded empir-
ically, namely with initial and final tests of knowledge. The knowledge tests that
had been adapted as a measurement instrument were used to determine basic
and conceptual knowledge, solving simple mathematical problems, and com-
plex knowledge. The initial and final tests of knowledge were preliminarily first
attributed measurement characteristics: objectivity, difficulty, reliability, discrim-
inativeness, and validity. Results of the initial and of final tests were processed
with the use of multivariate factor analysis. The Guttman split-half coefficient of
reliability for the initial test was 0.82 and for the final test 0.87. The discriminative
coefficients for individual items at the initial test ranged from 0.38 to 0.700; while
the discriminative coefficients for individual items at the final test ranged from
0.29 to 0.72.
To determine the significance of the differences between the students of the
experimental and control groups and to determine the significance of differences
within the experimental group at the end of the experiment, the following sta-
tistical techniques were applied in data processing: descriptive statistics, testing
the homogeneity of the sample, factor analysis, one-way analysis of variance, and
multivariate analysis of variance.
5.
210 Amalija Žakelj
Results and interpretation
The results of the research show that the experimental group performed
statistically significantly better in basic and conceptual knowledge, in solving
simple mathematical problems and in complex knowledge than the control group
(Table 1, Table 2).
Table 1. Average performance of students according to areas
of knowledge in the initial and final tests
INITIAL TEST FINAL TEST
N performance in % x SD N performance in % x SD
EG PR 101 61 % 16.4 5.40 101 68 % 6.8 2.50
EP 101 39 % 12.6 6.90 101 83 % 9.9 2.21
ZP 101 25 % 3.1 1.92 101 54 % 15.9 7.59
CG PR 130 52 % 13.5 6.50 130 55 % 5.5 1.60
EP 130 31 % 10.0 6.50 130 73 % 8.8 3.16
ZP 130 23 % 2. 8 2.56 130 38 % 11.7 6.90
Legend: x – average number of points, SD – standard deviation, N – number of students, PR – basic
and conceptual knowledge, EP – solving simple mathematical problems, ZP – complex knowledge.
Table 2. The significance of performance differences between
the control group and experimental groups by areas of knowledge
Sum of squares (dif. III) df Average of squares F Sig.
IT PR 3.092E-05 1 3.092E-05 .000 .993
IT EP 1.046 1 1.046 2.420 .121
IT ZP .792 1 .792 1.733 .190
FT PR 6.134 1 6.134 36.345 .000
FT EP 7.785 1 7.785 16.769 .000
FT ZP 3.162 1 3.162 6.369 .010
Legend: initial test, basic and conceptual knowledge (IT PR), final test, basic and conceptual
knowledge (FT PR), initial test, solving simple mathematical problems (IT PR), final test solving
simple mathematical problems (FT PR), initial test, complex knowledge (IT ZP), final test, complex
knowledge (FT ZP).
Compared to its initial state, after the introduction of the experimental factor
into the learning process, the experimental group progressed significantly in
solving simple mathematical problems and in complex knowledge (Table 3).
6.
Process Approach to Learning and Teaching Mathematics 211
Table 3. The significance of differences on the initial and final tests
by areas of knowledge in the experimental group
Sum of squares (dif. III) df Average of squares F Sig.
PR 0.464 1 0.464 0.582 0.447
EP 46.470 1 46.481 60.960 0.00
ZP 5.682 1 5.660 6.480 0.01
Legend: basic and conceptual knowledge (PR), knowledge that allows for solving simple mathemat-
ical problems (EP), complex knowledge (ZP)
The first hypothesis, H1, was confirmed with the results of the research, which
show that in basic and conceptual knowledge after the introduction of the experi-
mental factor, the experimental group obtained statistically higher results than the
control group (EG: 68 %, CG: 55 %, p=0.00).
Basic and conceptual knowledge, which covers the knowledge and under-
standing of mathematical concepts, was tested with the recognition of concepts,
determining the relations between data, analysing, proposing examples and
counterexamples, etc. In verifying understanding, attention was paid to compos-
ing the task in such a way that allowed the student to really demonstrate his/her
knowledge. This is the reason why in this kind of tasks mathematical procedures
were, as a rule, not included.
At the time of the experiment, the students of the experimental group built their
knowledge and deepened understanding through various activities of representing
concepts, which includes pictures, diagrams, symbols, concrete material, language,
realistic situations, shaping conceptual networks, etc. As early as in 1991, also
Novak & Musonda (1991) attracted attention to the significance of conceptual
networks in shaping concepts with understanding, emphasising that based on
students’ correct and wrong presentations, the teacher can analyse their knowledge
and determine wrong and correct conceptual images (ibid.) and based on the
findings, guide students in upgrading and transforming knowledge.
Similarly, Griffin & Case (1997) and after them Duval (2002) stated that the
teaching of mathematics that is based on exploring diverse representations of
a definite mathematical concept and that encourages students to fluently and
flexibly transit between a variety of representations is more efficient and allows for
a better understanding of mathematical concepts than the teaching that does not
enable this. De Jong et al. (1998) emphasise that in teaching mathematics handling
diverse representations fluently and also transiting between them (e.g., knowing
how with concrete material to compute a given calculation and to “translate” the
7.
212 Amalija Žakelj
calculation into symbolic record) and from offered representations selecting the
one appropriate for the representation of a definite concept (e.g., representation of
adding three-digit numbers with tens units is a more appropriate representation
than representing computing in the 1 000 range with non-structured material)
is important for the student’s successful and productive interaction with diverse
representations. In addition to what has been mentioned, the use of diverse rep-
resentations of mathematical concepts satisfies the needs of learners with different
styles of learning (Mallet, 2007).
The second hypothesis, H2, was confirmed with the results of the research,
which show that after the introduction of the experimental factor in solving simple
mathematical problems the experimental group obtained statistically significantly
higher results than the control group (EG: 83 %, CG: 73 %, p = 0.00). In solving
simple mathematical problems, the experimental group also significantly pro-
gressed in relation to its initial state (Table 3).
Taking into account the results concerning progress in solving simple mathe-
matical problems, where mathematical procedures had to be meaningfully applied
such as computing procedures, drawing diagrams, production of tables, solving
simple one-stage textual tasks, the fact must be emphasised that the students in
both the experimental group (83 %) and the control group (73 %) demonstrated
satisfactory knowledge.
Douglas (2000) states that for learning algorithms and computing procedures as
well as for solving problems, the understanding of concepts is crucial. We learn to
solve problems faster and better if we understand the basic concepts. A conclusion
can be drawn that the advantage of the students of the experimental group in
solving simple mathematical problems also lies in the acquisition of diverse expe-
riences in the learning of concepts. Introducing procedures when the student has
not yet thoroughly acquired the basic concepts inherent in the procedure implies
learning by memorising. In this case, how well the procedure is going to be learnt
depends on the number of repetitions of the procedure. Such knowledge is, how-
ever, short-lived and quickly forgotten; it is also not transferable and applicable,
e.g., to solving problems.
The interweavement among different areas of knowledge is also indicated by
the statistically significant correlations between them. The correlations between
the areas of knowledge range from medium to high and show that conceptual
knowledge is significantly related to solving simple mathematical problems and
complex knowledge (Table 4).
8.
Process Approach to Learning and Teaching Mathematics 213
Table 4. The correlations between basic and conceptual knowledge
and between complex knowledge and solving simple mathematical problems
FT EP FT ZP
FT PR Pearson coefficient 0.44** 0.69**
Legend: FT PR – basic and conceptual knowledge on final test, FT EP – solving simple mathematical
problems on final test, ** the coefficient is statistically significant at the level of 1 % risk, * the coeffi-
cient is statistically significant at the level of 5 % risk
It can be concluded that the advantage of the students of the experimental group
in solving simple mathematical problems as well as in complex knowledge—as will
be shown below—also lies in the acquisition of a variety of experiences in learning
concepts, which has a positive impact on efficient learning of procedures and solv-
ing problems. Solving problems, in turn, is an important skill that is indispensable
in life, as it involves analysis, interpretation, reasoning, anticipation, assessment,
and reflection, so it should be the main goal and fundamental component of the
mathematical curriculum (Anderson, 2009).
The third hypothesis, H3, was confirmed with the results of the research, which
show that after the introduction of the experimental factor in complex knowledge,
the experimental group obtained statistically significantly higher results than the
control group (EG: 54 %, CG: 38 %, p = 0.01). In complex knowledge, the experi-
mental group also significantly progressed in relation to the initial state (Table 3).
Complex knowledge, which covers solving problems, was tested with solving
complex tasks (multistage textual problems), analysing the problem situation,
generalising, substantiating, etc. Detailed analysis of the results by items shows
that neither the students of the control group nor those of the experimental group
successfully solved textual tasks, they especially experienced difficulties in solving
algebraic problems, generalisation, and using formal mathematical knowledge.
With the task “Compute what percentage of the figure is shaded,” the ability of
solving problems at the symbol level was tested. The text of the textual task was
accompanied with a picture of a rectangle, a part of which was shaded. The data
of the lengths of the sides were given at the symbol level, with variables. Very
few students solved the task at the symbol level. Most students solved the task
by choosing concrete data—some by measuring, others by drawing a grid and
defining the surface unit, some also came to an approximate result by estimation.
The students’ lower results in complex knowledge can partly be explained with
the findings of Demetriou et al. (1991), who developed four tests for the deter-
mination of the level of development of the cognitive system and understanding
9.
214 Amalija Žakelj
of mathematical concepts, among other things also a test for the definition of the
stage of formal-logical thinking and algebraic abilities. The essential development
of integrating the four calculus operations happens at the age of 13–14, and the
development of algebraic abilities at the age of 14–15. The introduction of abstract
algebraic concepts (e.g., the concept of a variable) is possible when the develop-
ment of algebraic abilities has been completed. The introduction of these concepts,
though, must still be linked to concrete objects (ibid.).
It can be concluded that the path do deeper knowledge, which is applicable and
complex, is neither easy nor fast, it is conditioned both on the student’s cognitive
development and on the quality learning and teaching.
Concluding findings
The issue of examining the impact of approaches to learning and teaching
on learning performance is an extremely demanding and complex one. In our
research, we focused on three levels of mathematical knowledge: basic and concep-
tual knowledge, solving simple mathematical problems and complex knowledge.
As evident from the paper, there are substantiated reasons for the assertion that
the implementation of the process approach to learning and teaching mathemat-
ics, which we have produced ourselves on the basis of the theoretical knowledge of
the mental development of children and recent findings about children’s thinking,
significantly contributes to the quality of learning and teaching mathematics and
to students’ academic achievement.
A positive impact of the process approach to learning and teaching mathe-
matics is recorded both in the understanding of concepts and solving problems
and in learning algorithms and calculation procedures. The research results show
that mathematical conceptual knowledge is significantly related to solving simple
mathematical problems and complex knowledge; learning with understanding,
however, is a long lasting process associated with the cognitive development of the
student and with quality teaching.
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