Development
of Learner’s Mathematical Competences
Jozef Sekerák
Abstract. The content
of this paper is to try to answer following questions: To what extent does
mathematics teaching develop key competences of students? Are trends of
student’s key competences development implemented in current education so that
our school graduates stand the European competition? We searched for the
answers in a survey and in this work we want to present the results of the study.
In the introduction theoretical resources are
described, the term ‘key competences’ is defined and system of mathematical key
competences is outlined. In the next part of this paper we follow the survey
offering answers to asked questions. Goal of the survey is presented, hypotheses
are set and applied methods are characterised. We deal with survey realisation
and quantitative and qualitative evaluation and we introduce possible
improvements.
Key words: key competences, categories of mathematical
key competences, pedagogical survey.
Introduction
The more and more important goal of school systems is preparing scholars
to successfully stand the information oriented society requirements as well as
to take as many opportunities this society offers as possible. This goal leads
to the inspection of the content of education, educational methods and goals
which perforce instigates the interest in key competences. Education system
aims more and more at knowledge, capabilities and skills application, at
building positive attitude to a subject and learning. As a result
most of the countries re-defined their educational goals paying attention to
key competences and this appears to be the right and effective step.
1. Key Competence Term Definition
The term key competence cannot be
considered to be strictly pedagogical or psychological expression. This phrase
is used in technical as well as in common language from the beginning of 70s of 20th century. It got into the area
of education in the late 90s of 20th century. Despite this no satisfactory and
standard definition exists by now. Taking
all characteristics and facts resulting from all area definitions from various
specialists into consideration (see [1], [2], [4], [5], [9], [10], [12]) we get
the following: competences present the
unification of all knowledge, skills, capabilities and attitudes. Individual
competences enable their bearer act adequately in certain situation in specific
field of activity.
Key competences are those that are
usable not only in one but in more various fields of activity. They
present the part of knowledge, skills, capabilities and attitudes individual
takes on during the whole life. (See figure.1).
Figure.1
Key
competences could be held as multifunctional collection knowledge, skills,
capabilities and attitudes and could be characterized as follows:
Ø They have active, procedural character: they are formed on the base of personal practical
experience and activity and they are applied in praxis.
Ø They form a complex unit: they present unity of knowledge, skills,
capabilities, attitudes and other elements that were perceived more
independently by now.
Ø They are dynamic and developed on various levels: they change their quality during whole life. They do
not date as information or expertise but keep their ability to develop
(therefore they could be the base of lifelong education and personal
flexibility).
Ø They are the result of formal, non-formal and informal
learning: they are the result of
lifelong learning. They are not the matter of personal performance only but
they require favourable social and ecological surroundings.
2. Implementation of Key Competences into Mathematical Teaching
Identification and
Implementation of key competences into mathematical learning appear to be as
big problem as is the key competences term delimitation.
Many methodological and practical problems go along
key competences selection process. There is no worldwide accepted model in the
field of key competences identification. In general three main streams exist. One of
them takes the competences subjectively, i. e. it covers system of competences
that relate to specific discipline or school subject. The second one takes them
super-subjectively (cross-curriculum) and the third one is combined. When
identifying key competences super-subjective orientation took precedence. The
development of key competences is not connected with certain content or subject
but it relates to procedural aspect of curriculum.
We developed system of key competences (according to
[6], [8], [13]) in range of mathematics teaching into twelve categories.
Specific mathematical categories are included among them because we combined
super-subjective and subjective perception of key competences when creating
this system. There is a strict line between given categories as there are
mathematical key competences there that could be included in several
categories. This competence model is dynamic and open:
1.
Mathematics thinking and pondering.
2.
Mathematical terms, facts[1], claims and procedures.
3.
Use of symbolic, formal and technical terms,
relations and operations.
4.
Figuring and describing of mathematical objects and
situations, representation..
5. Asking
a question, problem determination and its solution.
6. Mathematical
modelling.
7. Mathematical
argumentation, proof.
8.
Use of tools.
9.
Communication.
10. Information handling competences.
11. Competence related to attitudes and
value system.
12. Personal and interpersonal competences.
We will spend more time on some of the categories in the following
sections of the paper when evaluating the survey.
Now several other methodological and
diagnostic questions come to mind. How to develop key competences? How to judge
and evaluate the level of the development?
Problems
solving is considered to be the most suitable mean of developing and
diagnostics of competences in mathematics. So we concentrated to creating of
suitable problems.
Not
all competences could be represented by problems, mainly in affective area. On
the other handsome problems represent several competences. This is one of the
reasons why it is suitable to count given competence categories system into
wider classes [8], [13]:
1. reproduction level competences – reproduction of learned material, making
routine calculations and procedures and resolving routine problems,
2. connection level competences – integration, connection and simple spreading
of known material, modelling, connection of more methods that are known to
student.,
3. reflection level competences – developed thinking, argumentation,
abstraction, generalization and modelling used for new unknown contexts,
original mathematical attitude, connection of several and more complicated
methods, the look inside a problem infiltration into the core of the
mathematics.
As you can see simple
levels are based on the type of cognitive needs that are necessary to solve
various mathematical problems. Mathematical competences mentioned above need
not to be included in one class of competences only. Classes form conceptual
continuity, beginning with simple reproduction off acts and calculation
capabilities, followed by skills to take over various sources when solving
problem and finishing with „mathematization“ of real-life problems. In this
case we take the hierarchy as follows: needs(e.g. in form of mathematical
problems) require class 3 competences are mostly more difficult than those that
require class 2 competences what in fact doesn’t mean that class 2 competences
are necessary for all class 3 competences.
3. Key Competences Level Diagnostics Tools
It
is fully logical and completely right to try to recognize the level of key
competences along their development and not only during the education process. To
be able to judge the level of key competences of a pupil he/she has to
prove by a performance that he/she has this certain competence, and this
performance must be measurable. According to M. Romainvillea [11] level
evaluation of the competences in fact means forming of believes based on
deduction. Person who performs the evaluation has to judge according to
student’s performance, what is the possibility that the pupil assumes certain competence.
D. Curtis
and R. Denton [3] identified four general attitudes when evaluating the
level of control of key competences:
Ø
Holistic
review of level of these competences by teacher.
Ø
Review
according to batch of various papers and projects of a certain pupil.
Ø
Evaluation
in range of real-world reminding situations.
Ø
Evaluation
using standard tools appointed for measurement of key competence level.
Pedagogical diagnostics offers number of
methods. According to attitudes mentioned above we took the following out of
this set:
1.
working sheet method (various kinds of didactic tests),
2.
observation method,
3.
dialogue method,
4.
questionnaire method,
5.
analyze
of scholars’‘ papers,
6.
other
methods, according to the goal of probe.
According to the probe difficulty of attitudes that will often be the
subject of our search it is not suitable, in most cases, to focus just to one
method. More mentioned diagnostics methods need to be used. Man needs to gain knowledge
about existing possibilities to be able to decide for a certain method or
methods that are adequate for given problem. This selection is positively influenced by
precise formulation of diagnostics goals and adherence of basic methodological
rules.
4. Survey
Various monitor and
survey results show that despite the amount of mathematics skills Slovak
Republic scholars hold according to curriculum they do not have the capability
to use these skills in real-world situation (see the results of monitors: PISA 2003, TIMSS 2003). According
these surveys we asked questions said in the introduction.
The goal of the survey
is to determine whether and to what extent current mathematics teaching develops
or suppresses scholar’s key competences and state the importance to concentrate
to key competences, specifically their gaining and improvement. According to
set goal we worded two hypotheses of this survey.
H1: Current mathematics teaching doesn’t
suppress nor develops key competences of pupil at adequate level. Scholars
develop mostly their reproductive level competences, they develop their connection
level competences less and do not develop their reflection level competences at
all.
H2: Education subjects consider gaining and developing of key competences to
be useful in one’s present as well
as in one’s future.
To verify our hypotheses we used
working sheets that diagnose the level of some scholars‘ competences and anonym
form questionnaires. We used working sheets to verify hypothesis H1 and questionnaires
served to verify hypothesis H2 and as complementary method to working sheets
one.
4.1. Working Sheets
201 elementary school scholars and
74 high school pupils were involved to working sheet testing in 2006. We tried
to learn scholars‘ capability level to activate their mathematical competences
to solve problems not only to find out their mathematical knowledge level by
this testing. This attitude to scholar’s mathematics capability level
diagnostics is in contrast with traditional school testing that is usually
narrower.
Working sheets problems
are in full correspondence with high school mathematics curriculum. They are
concentrated to geometry mostly because this area involves logical and abstract
and functional thinking and also imagination and intuition. According to demandingness
we divided the problems into three level classes mentioned above – problems
diagnosing reproductive level competences (1st level), problems diagnosing connection
level competences (2nd level) and reflexion level competences (3rd level).
Content subject of the testing were communication competences (the ability
to express oneself to the question of mathematics content and argumentation and
proof and ability to take notes in a symbolic way and to understand those
notes), information handling competences
(working with texts and images and capability to criticise information (veracity)
and to evaluate information (their worth) and qualitative character information
involved in diagrams and ability to modify information) and problem solving competences (capability
to think and conclude logically and functionally
and knowledge of terms and capability to apply them and knowledge of algorithms
and procedures and ability to use them and mathematization and
demathematization and representation and description and ability to define the
problem and find fitting strategy and apply it and capability to use tools and
have the information about the range of their usage).
We used composite scoring to
evaluate the successfulness of working sheets. Following percentual success
scale to verify hypothesis H1 was set: 
- does not develop, 
- slightly develops, 
- develops, 
- markedly develops. These intervals follow the rate of
problems requiring competences at certain level.
We present
a problem from working sheet Triangles Congruity as an example.
Problem 1 (Problem Z.7 from working sheet Triangles Congruity)
Determine the way to
measure the distance of two trees if their direct join is not available (see
Figure 1). Use the figure and describe briefly the way to do this.
Figure.1
Solution and phenomena
(competence) analyze:
The most important key competence is the
ability to use mathematics when establishing, formulating, solving and
interpreting problems in various situations. The problem is appointed to the
diagnostics of level of some problem solution diagnostics and some
communication competences.
|
Solution |
Tracked phenomena (competences) |
Points |
|
Our task is to find the way how to
measure the distance of two trees if direct join is not available. We make
a use of triangles congruity. Sketch:
Suggested solution description: We‘ll use e.g. two ropes of known
and same length. We stretch the first rope from the tree A and the
second one from the tree B the way they will cut themselves in halves. The
ends of the ropes indicates the two points we called A‘ and B‘ in the
figure. The distance between trees A and B is the same as the one
between points A‘ and B‘. Proof:
|
– Determine basis (define the
problem), find fitting strategy for problem solution – problem solution
competences, reflection level. – Figure usage, sketching – problem
solution competences, reproduction level. – brief procedure description and
solution accuracy provement – ability to express oneself in writing to the
question of mathematics content, argumentation and proof – communication
competences, reproduction and connection level. |
2pts 1pt 3pts |
Statistical evaluation results are presented
table 1.
Table 1: Final
evaluation of working sheets successfulness.
|
Final evaluation of working sheets |
|||
|
Competences |
Percentual expression |
||
|
1st. level |
2nd level |
3rd level |
|
|
Comunication |
35,46% |
17,93% |
7,94% |
|
Information handling |
50,30% |
44,72% |
22,44% |
|
Problem solving |
58,98% |
33,61% |
10,45% |
|
Total |
52,26% |
35,60% |
11,94% |
Received results
analyze:
-
Pupils
don’t have problems on the level of routine operations or problems where
information is simply defined. They know how to use basic algorithms,
procedures and how to find simple strategy for problems solutions. This
indicates that scholars shouldn’t have difficulties to solve problems according
to instruction.
-
Mathematics
teaching does develop the problems solutions on the reproductive level. The
other levels competences are not developed though.
-
Scholars
have little difficulties with problems where information is defined not that
simply or where they need to use more sources of information. Our pupils lack
some practical skills e.g. graph reading or figure reading and interpreting
read information even if these are common information and explanation tools.
-
Information
handling competences seems to be slightly developed but not enough to reach
adequate level.
-
Scholars
have bigger difficulties with problems where written description of the
solution or result interpretation is required which relates to communication
competences. Overall communication competences in competences rating came out
as the worst (specifically argumentation).
These competences show to be not developed.
-
The
outcome of these results is that the mathematics teaching does not develop
pupil’s key competences at adequate level. Even the reproductive level
competences are average although in working sheets there were no problems they
wouldn’t meet during mathematics classes. Using this diagnostic method
Hypothesis H1 was proved.
4.2. Questionnaires
Questionnaires were appointed to
pupils and mathematics teachers at elementary and high schools. Respondents
were given a chance to evaluate mathematics teaching from the point of
view of developing of some key competences as well as a need of their gaining
and developing by means of five value scale. A respondent assigned
a value to each competence in a certain way using a set of
values 
, where -2 means that mathematics teaching strongly suppress
given competence, -1 means, it suppress it, 0 it doesn’t suppress it, built
doesn’t develop it either, 1 it develops it and 2 means that mathematics
teaching strongly develops given competence. In case of evaluation of a need
of assuming and developing of given competences respondent was given the same
value scale where -2 means it is absolutely useless to assume and develop given
competence, -1 means it is rather useless, 0 means I’m not sure to decide, 1 it
is useful and 2 means it is very useful to assume and develop given competence.
Questionnaires were filled in April
2006 and survey counted 445 respondents, 89 teachers and 356 scholars. 46 teachers
and 195 scholars were from elementary schools and 43 teachers and 161 students
were from high schools (training institutions, technical schools and grammar
schools).
43 competences in total were listed
in questionnaires. Because of meaningly closeness of partial expressions we divided
these competences into following six bigger groups: communication competences, information handling competences, perceptual
competences, problem solving competences, personal competences and
interpersonal competences.
Communication
competences and information handling competences have
several forms of demonstration. It is not only development of good phrasing capability but willingness to listen to other people and take their opinion into consideration. Quality
and efficiency of subject‘s work with information sources is based on these
demonstrations. Information handling competences exhibit mainly in ability to gain information and to ponder various
data sources and criticise gained information.
All mentioned
competences are closely bound to perceptual
competences, problem solving competences and personal competences in range
of which we prefer to organize self perceptual process and to
be responsible for self learning, involvement, tenaciousness, ability to manage
the insecurity and situation comprehensiveness , ability to solve problems and
capability to find new solutions.
Interpersonal competences are the most important
competennces in the society. We
meet their demonstrations daily and occurance of most of them is essential.
That means behaviour expressions like solidarity,
empathy, tolerance even altruism, cooperation ability and team work. There
are only few occupations where you don’t need to cooperate with someone.
We set up following value scale for certain
options to verify hypothesis H2: 
- strongly suppresses, it is absolutely useless; 
- suppresses, useless; 
- does not suppress, nor develops; cannot decide; 
- develops, useful; 
- strongly develops, very useful.
Statistical evaluation
results are listed in the following histogram.
Gained results analyze:
-
As
expected, mathematics teachers are bit more optimistic in development of key
competences perspective than are the scholars. Likewise teachers address the
importance to the need to assume and develop key competences more than pupils.
-
Mathematics teaching slightly develops Communication competences while it is necessary
to assume and develop these competences. Yet teachers do consider these
competences to be very important. On the other hand according to the pupils’
mathematics teaching does not influence these competences. However the school
develops their ability to express
themselves in writing it does not develop partial competences as to take
part in a discussion and express their own opinion, to defend their own
opinion, to listen to other people and consider their opinion or to present the results of their own work.
-
According
to respondents mathematics teaching does not influence information handling competences even though they consider them to
be important. Pupils positively evaluated the development of information assumption competence that
they consider to be very important as do the teachers. However, development of
following competences was judged very negatively: to ponder various information sources (the most fitting information
source selection), criticize assumed information and evaluate
gained information. According to most of the scholars’ mathematics teaching
suppress mentioned competences even though they do consider them to be
important.
-
In
case of perceptual competences (to organize self perceptual process and to
be responsible for self learning) teachers‘ and scholars‘ evaluations do
correspond. Yet the pupils evaluated the mathematics teaching more positively. According
to respondents mathematics teaching develops these competences and their assumption
and development is necessary.
-
We
count the following among the problem
solving competences: ability to
define the problem, to ask the question, capability to outline and describe; ability
to experiment, to examine; capability to solve problems; to find new solutions;
to make projects; to manage mathematical and model tools; to understand graphs,
diagrams and tables; to use various tools and instruments. Pupils
criticized mathematics teaching in the point of view of these competences.
There were only two cases the value was higher than 1 and it was in case of
following competences: to manage
mathematical and model tools and to understand graphs, diagrams and tables.
Most of the pupils evaluated mathematics teaching not influencing development
of problem solving competences and that they do consider these competences to
be important and teachers even consider them to be very important. The result
in general was that mathematics teaching slightly influences development of
problem solving competences and their assumption and development are important.
-
We
included competences: to make decisions and
ability to make use of experience that are
also used in problem solving but are not organic part of the solution itself to
the group of personal competences. Other
competences like: to organize one’s self
work, to be able to work individually, to be tenacious in case of difficulties,
ability to manage insecurity and situation
comprehensiveness, to be critical, flexible by changes or to judge the risks
influence the general ability to
solve problems and that is why it is important and sometimes necessary to have
these abilities. According the respondents evaluation mathematics teaching does
not influence to the development of personal competences even though they are
important. This conclusion corresponds with pupils‘ one. Teachers differ only
in opinion of a need of assuming and developing of these competences. According
to them these are very useful.
-
Interpersonal competences got the worst result. Mathematics
teaching absolutely does not influence these competences even though they are
important. Yet it slightly supress leader
abilities and competence to show solidarity.
It is closely related to the competence of ability
to assert oneself. According to respondents mathematics teaching does not
influence this competence.
The results of the survey in fact
confirmed stated hypotheses where we suggested that current mathematics
teaching does not suppress but does not
develop key competences of a pupil at adequate level either(56,43% out
of all respondents‘ answers was in interval of 
and 25,12% of solutions of
working sheets was in interval of ), and that assuming and developing of key competences is
useful at present as it is in the future (78,37% out of all responses war in
interval of 
). Teachers
even consider development of key competences to be very important, so do most
of the pupils.
We even verified hypotheses by one
selection Wilcoxon test at the relevance level of 
. Hypothesis H1 was not rejected at the relevance
level. That’s why we consider it to be true. Hypothesis H2 was
refused at this relevance level in favour of alternative hypothesis that could
be interpreted the following way: „Assuming and developing of key competences
is more than useful in the one’s life“. That in fact does not challenge given
hypothesis H2.
Teachers‘ questionary shows that key
competences problems are not really in pedagogical society attention centre,
not even pedagogical science deals with this problems at adequate level. That
is why forming of key competences at present is only an occasional phenomenon
and it is just kind of minor product of our education system. Only
one teacher out of 89 gave positive answer to the question no.4: „Have you had
a chance to learn about key competences problems?“.
After
survey realization during dialogues with respondents most of the commented was
the naming of the competences that the names are too long and complicated and
hard to understand for larger group of pupils. This was considered to be the
imperfection of the questionnaires.
Conclusion
Development
of key competences is a new phenomenon that could help to find the answers
for the questions related to status and to the future of education. A need
to equip young people with necessary key competences and to better the level of
achieved education presents a necessary step when society prosperity escalation
even according to Europe Parliament and Counsel of
In
the end we present two more reasons why it is important to deal with key
competences problems in mathematic teaching:
1. According to our opinion it is possible to change the status of
mathematics respecting today’s and future needs of the society, so it means to
re-define the curriculum of mathematics teaching only using the paradigm of key
competences.
2. Our existing experience show that mathematics teaching respects key
competences development and builds good base for whole pupil structure
development.
Used
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