Contributed by:

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

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

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

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.

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.

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).

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

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).

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

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.

Anderson, Ј. (2009). Mathematics Curriculum Development and the Role of Problem Solving.

ACSA Conference http://www.acsa.edu.au/pages/images/Judy

Břehovský, J., Eisenmann, P., Novotná, J., & Přibyl, J. (2015). Solving problems usingexper-

imental strategies. In J. Novotna, H. Moraova (eds.). Developing mathematical language

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.

Anderson, Ј. (2009). Mathematics Curriculum Development and the Role of Problem Solving.

ACSA Conference http://www.acsa.edu.au/pages/images/Judy

Břehovský, J., Eisenmann, P., Novotná, J., & Přibyl, J. (2015). Solving problems usingexper-

imental strategies. In J. Novotna, H. Moraova (eds.). Developing mathematical language

10.
Process Approach to Learning and Teaching Mathematics 215

and reasoning (Proceeding of International Symposium Elementary Math Teaching)

(72 – 81) Prague, the Czech Republic: Charles University, Faculty of Education.

De Jong, T., Ainsworth, S., Dobson, M., Van der Hulst, A., Levonen, J., & Reinmann,

P. (1998). Acquiring Knowledge in Science and Math: The Use of Multiple Representa-

tions in Technoogy Based Learning Environments. In van Someren, M.W., Reimann,

P., Boshuizen H.P.A., de Jong, T. (ed.), Learning with Multiple Representations 9 – 40.

Amsterdam: Pergamon.

Demetriou, A., Platsidou, M., Efklides, A., Metallisou, Y., & Shayer, M. (1991). The Devel-

opment of Quantitative-relational Abilities From Childhood to Adolescence: Structure,

Scaling and Individual Diffence. Learning and Instruction, Vol. 1., 19 – 43. Great Britain:

King’s College, London University.

Dindyal, J., Eng Guan, T., Tin Lam, T., Yew Hoong, L., & Khiok Seng, Q. (2012). Mathe-

matical Problem Solving for Everyone: A New Beginning, The Mathematics Educator,

Vol. 13, No. 2, 1 – 20.

Duval, R. (2002). The Cognitive Analysis of Problems of Comprehension in the Learning of

Mathematics. Mediterranean Journal for research in Mathematics Education, 1(2), 1 – 16.

Gilly, M., Blaye, A., & Roux, J.P. (1988). Elaboration de constructions cognitives individualles

en situations socio-cognitives de resolutions de problemes [Elaboration of individual

cognitive constructs in the socio-cognitive situations of problem solving]. In: Mugn,

G., Perrez, J.A.( eds.): Psicologia social del desarollo cognitivo. Barcelona: Anthjropos.

Griffin, S. in Case, R. (1997). Re-thinking the Primary School Math Curriculum: An

Approach Based on Cognitive Science. Issues in Education, 3(1), 1 – 49.

Labinowicz, E. (1989). Izvirni Piaget [The Original Piaget]. Ljubljana: DZS.

Novak, J.D. & Musonda, D. (1991). A twelve-year longitudinal study of science cocept

learning. American Educational Research Journal.

Mallet, D.G. (2007). Multiple representations for system of linear equations via the com-

puter algebra system Maple. International Electronic Journal of Mathematics Education

2(1), 16 – 32

Maričić, S., Špijunović, K., & Malinović Jovanović, N. (2013). The Role of Tasks in the

Development of Students’ Critical Thinking in Initial Teaching of Mathematics. In:

Novotna, J. & Moraova, H. (eds.). Task and tools in elementary mathematics (Proceedings

of International Symposium Elementary Math Teaching). Prague, the Czech Republic:

Charles University, Faculty of Education. 204 – 212.

Shulman, L. (1987). Knowledge and Teaching: Foundations of a New Reform.

Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum

documents. Journal of Mathematical Behavior, 24, 341 – 350.

UNESCO (2012). Challenges in basic mathematics education. United Nations Educational,

Scientific and Cultural Organization, Paris.

Van de Walle, J.A., Karp, K.S., & Bay-Williams J.M. (2013). Elementary and middle school

mathematics: teaching developmentally. Boston [etc.]: Pearson.

Vygotsky, L. (1978). Mind in society: The develepment of higher psychological processes.

Cambridge. MA: Harward University press.

and reasoning (Proceeding of International Symposium Elementary Math Teaching)

(72 – 81) Prague, the Czech Republic: Charles University, Faculty of Education.

De Jong, T., Ainsworth, S., Dobson, M., Van der Hulst, A., Levonen, J., & Reinmann,

P. (1998). Acquiring Knowledge in Science and Math: The Use of Multiple Representa-

tions in Technoogy Based Learning Environments. In van Someren, M.W., Reimann,

P., Boshuizen H.P.A., de Jong, T. (ed.), Learning with Multiple Representations 9 – 40.

Amsterdam: Pergamon.

Demetriou, A., Platsidou, M., Efklides, A., Metallisou, Y., & Shayer, M. (1991). The Devel-

opment of Quantitative-relational Abilities From Childhood to Adolescence: Structure,

Scaling and Individual Diffence. Learning and Instruction, Vol. 1., 19 – 43. Great Britain:

King’s College, London University.

Dindyal, J., Eng Guan, T., Tin Lam, T., Yew Hoong, L., & Khiok Seng, Q. (2012). Mathe-

matical Problem Solving for Everyone: A New Beginning, The Mathematics Educator,

Vol. 13, No. 2, 1 – 20.

Duval, R. (2002). The Cognitive Analysis of Problems of Comprehension in the Learning of

Mathematics. Mediterranean Journal for research in Mathematics Education, 1(2), 1 – 16.

Gilly, M., Blaye, A., & Roux, J.P. (1988). Elaboration de constructions cognitives individualles

en situations socio-cognitives de resolutions de problemes [Elaboration of individual

cognitive constructs in the socio-cognitive situations of problem solving]. In: Mugn,

G., Perrez, J.A.( eds.): Psicologia social del desarollo cognitivo. Barcelona: Anthjropos.

Griffin, S. in Case, R. (1997). Re-thinking the Primary School Math Curriculum: An

Approach Based on Cognitive Science. Issues in Education, 3(1), 1 – 49.

Labinowicz, E. (1989). Izvirni Piaget [The Original Piaget]. Ljubljana: DZS.

Novak, J.D. & Musonda, D. (1991). A twelve-year longitudinal study of science cocept

learning. American Educational Research Journal.

Mallet, D.G. (2007). Multiple representations for system of linear equations via the com-

puter algebra system Maple. International Electronic Journal of Mathematics Education

2(1), 16 – 32

Maričić, S., Špijunović, K., & Malinović Jovanović, N. (2013). The Role of Tasks in the

Development of Students’ Critical Thinking in Initial Teaching of Mathematics. In:

Novotna, J. & Moraova, H. (eds.). Task and tools in elementary mathematics (Proceedings

of International Symposium Elementary Math Teaching). Prague, the Czech Republic:

Charles University, Faculty of Education. 204 – 212.

Shulman, L. (1987). Knowledge and Teaching: Foundations of a New Reform.

Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum

documents. Journal of Mathematical Behavior, 24, 341 – 350.

UNESCO (2012). Challenges in basic mathematics education. United Nations Educational,

Scientific and Cultural Organization, Paris.

Van de Walle, J.A., Karp, K.S., & Bay-Williams J.M. (2013). Elementary and middle school

mathematics: teaching developmentally. Boston [etc.]: Pearson.

Vygotsky, L. (1978). Mind in society: The develepment of higher psychological processes.

Cambridge. MA: Harward University press.