We live in an information society in which the need to collect, organise,
display and interpret data involves almost everyone as citizens and workers.
Statistical ideas are necessary in those areas of human interest that involve
either measurement or quantification. Most definitions of statistics indicate
that the subject is concerned with the collection, presentation, analysis
and interpretation of data. It is an important area of study in its own
right and is an essential tool for use in many kinds of research. The topic
of probability, which is always closely linked with statistics, readily
lends itself to being applied directly to the real world, while a great
deal of it can be understood by using only elementary mathematics and arithmetic.
At a more advanced level, most statistical theory is based on probabilistic
models, so that knowledge and understanding of probability theory is necessary
for any serious study of statistics. For example, knowledge of probability
is indispensable to the learning and understanding of statistical inference.
2. Literature Review
In the fields of statistics and probability, it is common knowledge that
many students lack understanding of basic concepts, a view that is confirmed
by a number of research studies (for example, Garfield & Ahlgren,
1988; Glencross, 1998). In Australia, research suggests that in terms
of the quality of students' learning of mathematics or statistics, there
is a structural relationship between students' conceptions of the subject
and their approaches to learning it (Crawford, Gordon, Nicholas &
Prosser, 1994; Gordon, 1995, 1997; Gordon, Nicholas & Crawford, 1996).
Support for this idea has been found in South Africa (Glencross, Kentane,
Njisane, Nxiweni & Mji, 1997; Mji, 1995, 1998a, 1998b, 1999).
Probabilistic ideas are often concretised with such common objects as
coins, dice and urns containing balls of assorted colours. Such methods
enable us to model real-life random phenomena. Nevertheless, it is well
known that probabilistic ideas are notoriously counter-intuitive and are
easily misunderstood. Many misunderstandings may occur because the language
of probability (more precise) is different from our day-to-day conversational
language (less precise) (Hawkins, Jolliffe & Glickman, 1992). For
example, Green (1982) has suggested that there are problems with regard
to words like 'possible', 'likely', 'certain', etc., while much has been
written about the ambiguities introduced into research by the wording
of the questions used (Evans, 1989; Pollatsek, Well, Konold, Hardiman
& Cobb, 1987; Shaughnessy, 1992). In addition to different cultural
backgrounds, students for whom English is a second language often have
language difficulties that may lead to misunderstanding of concepts. Of
interest to the writer is the possible influence of a second language
on the ability to learn and comprehend probability and statistics, because
almost all students at the University of Transkei are Xhosa-speaking and
communicate in English with varying degrees of competence. The 'specialist
language' of probability and statistics may itself be a barrier to students'
understanding and concept development. This barrier becomes even greater
when the specialist language is used in complicated sentence structures
(such as occurs in many mathematics and statistics textbooks), or in language
patterns with which many students may be unfamiliar (such as the use of
the passive voice). If the language used is not the mother tongue, then
the difficulties are certain to be compounded. Thus, if students are confused
by the use of statistics and probability words and phrases that are not
immediately meaningful to them, their confusion may be aggravated when
complex sentence structures and unfamiliar sentence patterns are also
used. In this way, the development of students' understanding of probabilistic
concepts is inhibited rather than aided by the specialist language.
3. Research Project: 1996-2000
The current research project is an attempt to gain insight into first
year students' conceptions of basic probability concepts and their approaches
to learning statistics. A pilot study was conducted in October 1996 with
25 volunteer first year students. Students were asked to complete a four-part
questionnaire designed (1) to assess their understanding of basic probability
concepts, (2) to find out what they think the subject Statistics is, (3)
how they study and learn Statistics and (4) what feelings they have toward
the subject. The results showed, inter alia, that the concepts of equally
likely events and probability as relative frequency were not sufficiently
well understood by many students and that the (incorrect) concept of probability
as frequency only was somewhat stronger. Students' conceptions of Statistics
showed that they had a fragmented view of the subject, while their approaches
to learning relied mostly on learning by surface level approaches such
as practicing lots of examples. Students' attitudes were predominantly
positive, although there was no correlation between attitudes and performance.
The questionnaire has been administered to three more groups of first
year students: 62 in 1997, 67 in 1998 and 58 in 2000. The results have
been consistently similar to those of the pilot study.
4. Research Project: 2001
For 2001, a modified version of the questionnaire will be administered
to first year students and semi-structured focused interviews will be
conducted with a number of students. The interviews, which will be video-recorded,
will be used for several purposes. First, to explore more deeply students
understanding of probability concepts using questions selected from the
questionnaire. These will be supplemented with suitable apparatus (e.g.,
spinners). Students will also be interviewed about how they study and
learn Statistics. Almost all our students are Xhosa-speaking for whom
English is a second language. The interviews will also be used to explore
some cultural beliefs (related to random events and probability) that,
on the basis of my interactions in workshop situations with students and
local junior secondary teachers, may play a role in the learning of probability.
The video-recorded interviews will be conducted in both English and Xhosa.
This will permit a careful exploration of students' understanding of probability
concepts and related cultural beliefs. Written transcripts will be made
of the interviews, together with translations from Xhosa into English.
It is anticipated that this approach will allow for in-depth probing of
students' ideas in a non-threatening way.
Crawford, K., Gordon, S., Nicholas, J. & Prosser, M. (1994) Conceptions
of mathematics and how it is learned: the perspectives of students entering
university. Learning and Instruction, 4, 331-345.
Evans, J. St B. T. (1989) Bias in Human Reasoning, Causes and Consequences.
Essays in Cognitive Psychology, Lawrence Erlbaum Associates Inc.
Garfield, J. & Ahlgren, A. (1988) Difficulties in learning basic concepts
in probability and statistics: implications for research. Journal for
Research in Mathematics Education, 19, 44-63.
Glencross, M. J. (1998) Understanding of chance and probability concepts
among first year university students. In L. Pereira-Mendoza, L. S. Kea,
T. W. Kee & W-K. Wong (Eds.) (1998) Proceedings of the Fifth International
Conference on Teaching Statistics. Voorburg: ISI Permanent Office, Vol.
Glencross, M. J., Kentane, L. H., Njisane, R. M., Nxiweni, J. G. &
Mji, A. (1997) Conceptions of science subjects and how they are learned:
views of first year students. In M. Sanders (Ed.) (1997) Proceedings of
the Fifth Annual Meeting of the Southern African Association for Research
in Mathematics and Science Education. Johannesburg: University of the
Gordon, S. (1995) A theoretical approach to understanding learners of
statistics. Journal of Statistics Education (Online), 3(3).
Gordon, S. (1997) Students' orientations to learning statistics. In F.
Biddulph & K. Carr (Eds.) Proceedings of the Twentieth Annual Conference
of the Mathematics Education Research Group of Australasia Inc., University
of Waikato, New Zealand: MERGA, 192-199.
Gordon, S. Nicholas, J. & Crawford, K. (1996) Psychology students'
conceptions of a statistics course. In Puig & Gutierrez (Eds.) Proceedings
of the Twentieth Conference of the International Group for the Psychology
of Mathematics Education. Valencia: University of Valencia, Vol. 3, 11-18.
Green, D. R. (1982) Probability Concepts in 11-16 year old Pupils. (2nd
ed.) Loughborough: Centre for Advancement of Mathematical Education in
Technology, University of Technology.
Hawkins, A. Jolliffe, F. & Glickman, L. (1992) Teaching Statistical
Concepts. London: Longman.
Mji, A. (1995) First year university students' conceptions of mathematics
and approaches to learning the subject: a phenomenographic study. Unpublished
MEd dissertation, University of Transkei, Umtata.
Mji, A. (1998a) Conceptions of learning: the view of undergraduate mathematics
students. Psychological Reports, 83. 982.
Mji, A. (1998b) Prospective teachers' conceptions about, and approaches
to learning mathematics. Journal of the Southern African Association for
Research in Mathematics and Science Education, 2, 72-76.
Mji, A (1999) Understanding learning: a survey of undergraduate mathematics
students' perceptions. South African Journal of Higher Education, 13,
Pollatsek, A., Well, A. D., Konold, C., Hardiman, P. & Cobb, G. (1987)
Understanding conditional probabilities. Organisational Behavior and Human
Decision Processes, 40, 255-269.
Shaughnessy, J. M. (1992) Research in probability and statistics: reflections
and directions. In D. A. Grouws (Ed.) (1992) Handbook of Research on Mathematics
Teaching and Learning, New York: Macmillan, 465-494.