PME Stochastics Teaching and Learning Working Group

Letter No 6 - May 1997

Dear Friends,

First, we must apologise for some delay in getting this Newsletter out. John's scholarship has finished and the pressures involved in working on a thesis, earning some money, and living life have been rather hard to balance. Kath and Carmen were also rather busy.

People Coming to Lahti

We look forward to seeing many of you in Lahti. By now you will know whether your paper has been accepted for PME, but as yet there is not a full list on the WWW, so we cannot include it here. .Please, let us know your plans. But those who can not attend are still very welcome to contribute their thoughts.

As far as we know, the following people with an interest in stochastics will be attending:

Carmen Batanero (Spain)

George Bright (USA)

Claude Gaulin (Canada)

Juan Godino (Spain)

Alan Graham (UK)

Anne Hawkins (UK)

Michel Henry (France)

Bernard Parzysz (France)

Anna Sfard (Israel)

John Truran (Australia)

Kath Truran (Australia)

Bernd Wollring (Germany)

Please tell us of any others whom you know are coming.

Overheads

Some of you will have seen John’s letter in PME News, November 96, where he commented that overheads used for talks needed to be large (minimum 18 point, preferably 24 point), and that they should summarise the main ideas, rather than merely reproduce the text. Our Working Group (and PME in general) has people from many different language bases, and it is important that we help those who are not strong in English. So we encourage you to take special care with your overheads to make communication across language groups as easy as possible.

If people want to present or ask questions in their own language, we shall do our best to arrange translations. We do not want people to be left out because they are not strong in English.

Programme for Three Working Group Meetings at Lahti

There will be three meetings of the Working Group, not four as previously announced. So the programme will be revised in the following way.

Session 1 Tue pm (120 minutes)

General Introduction, Circulation of Material, &c

Report on Recent Summary Publications

Borovcnik & Peard

Shaughnessy Garfield & Greer

Gal & Garfield

Do we have any volunteers to provide a 5 minute summary (maximum) of one of these publications?

Brief Presentations

Any New Members of the Group who are not presenting full papers would be welcome to provide a 5 minute summary of their current work at this stage. Please let us know if you would like to do so, and perhaps bring a couple of overhead transparencies.

Main Item - Building an Improved Data Base for Stochastics Research

Deciding on Small Groups to work on Data Base

Session 2 Fri 1400 - 1600 (120 minutes)

Full Group discussion on best form of Publication

Continue Small Group Work on Data Base

Carmen will be talking with the Advanced Mathematical Thinking Working Group about our Work

Session 3 Sat 0830 - 1000 (90 minutes)

Small group work (30 minutes)

General Full Group forum and making of plans

Questions for the Working Group

We are agreed that some form of Annotated Bibliography could be a constructive project. It needs to be of value to new and current researchers. It needs to be deeper than a straight-forward bibliography, valuable though that might be. It also needs to be very accessible. Another point to discuss is the advantages and disadvantages of building a specific research bibliography or making it also accessible to practising teachers. This decision will influence the purpose style and content, of our work.

Publication

It is probably best to prepare this as an electronic publication, but with the opportunity for a hard-copy version and/or a disk version to be available where these are seen as preferable. We shall need to discuss this further at Lahti.

Another model we might consider is something like the “Featured Reviews” of the American Mathematical Society. These are especially commissioned reviews of what are seen as very important papers or books.

Universities have often not thought about whether an electronic form of publication is equivalent to a hard-copy form in terms of giving staff credit for the research they have done. If any of you can find out clear information about the policies of your universities that would be of great help to us. It will certainly influence what we decide to do

Format

Carmen has prepared a draft model layout for references, &c. We will bring copies of this to Lahti for discussion and new suggestions and any modifications which we might decide to make can be sent to you electronically after the conference.

Purpose

This is the critical matter. The two main potential readership groups are researchers and teachers. Few books attempt to talk to both at the same time, so that our decision about this is of great importance. The work must add something new to the scene, and must make it easy for new workers to enter the field. Hopefully it will help to make more sense of the rapidly increasing work in the area.

We also think it would be a good idea to include a chapter which comments on the many bibliographies currently available.

Bias

Jane Watson has written to emphasise the pitfalls of summarising in as fair a way as possible because all of us have our biases. Her comments lead to the desirability of having all reviews looked at carefully by people of different persuasions.

In any case, it is essential to have a good refereeing process for professional reasons.

Comprehensiveness

Anne Hawkins has emphasised the enormous nature of the task which we are undertaking. And Jane Watson has observed that it may be quite difficult to obtain a comprehensive coverage of all relevant topics. Both of these issues will need to be addressed. But we will almost certainly have to think small to start with.

Some members of the group have recently finished a research or dissertation on a specific topic or are currently carrying it out. A list of specific topics to review and of people willing to cooperate in the review of these topics would be another starting point to our work.

Content

We have presented some ideas in this newsletter, but need to do more thinking about these issues. At the end of this newsletter, I attach copies of four possible forms of reviews - one each by Joan Garfield, Carmen, Kath, & John. We would welcome comment on these. We have a few more examples from Joan, Carmen & Kath, and these will be brought to Lahti as a basis of further discussion.

What to Bring to the Working Group

We will bring copies of the reviews which we already have, together with a summary of our thinking so far. (A first draft of this summary will come out in the next newsletter.)

We would be very grateful if other members could bring copies of papers which they see as very important papers in the development of the subject. Bringing 5 copies would allow a small group to work on the paper in detail. If you could let us know what you are able to bring that would be good.

Next Newsletter

The Trurans will be leaving Australia on 1 July. We shall send out one more Newsletter before we go. If you have any special issues to raise, please contact one of us by 15 June 97.

Carmen Batanero <batanero@goliat.ugr.es>

Kath Truran <Kath.Truran@unisa.edu.au>

John Truran <jtruran@arts.adelaide.edu.au>

SOME EXAMPLES OF POSSIBLE BIBLIOGRAPHIC REVIEWS

These examples have been chosen to illustrate a range of approaches and practices. No attempt has been made to standardise them. We would appreciate feed-back on the styles, the content, and what else might be included. What sort of articles have not been included. We are looking for a fairly standard style which will achieve the aims set out above. Hopefully, by the end of the Lahti Conference we will have a model which is sufficiently clear that it will enable different groups and individuals to work within it on different topics. All examples are drafts only and are not for reproduction or quoting.

Carmen Batanero - This article reminds us, among other things, the fact that a lot of relevant work is to be found with psychology, rather within mathematics or statistical education. (Editorial comments are by John.)

Crocker, J. (1981). Judgment of covariation by social perceivers. _Psychological Bulletin_, 262-292.

KEYWORDS: Association; basic steps in the judgment normative model; research survey

In this article the author identifies a normative model for making covariation judgment, which consist of six steps starting from deciding what data are relevant and going to using the judgment as a basis for prediction and decision. Potential sources of error in social perceivers' covariation judgment are identified at each step, and research on social perceivers' ability to follow each step is reviewed.

1. Deciding what data are relevant to the covariation judgment: One must have measurements of the corresponding values of each variable across a set of cases which should represent the range of possible variables for each variable. However, research suggests that individuals regard positive confirming cases as most relevant when making judgments.

2. Sampling cases: According to the model the instances sampled should be representative of the population as a whole. But when a naive observer samples cases, the samples of cases serving as a data base is likely to be a non-random sample. The relevance of sample size is not always recognized as some people believe in the "law of small numbers".

3. Classifying instances: Once cases have been observed, they must be assigned values on the variables of interest and classified as confirming or disconfirming cases. One factor that introduces systematic bias in this process is the perceiver's initial expectations, which may influence how an ambiguous event is interpreted as well as the credibility of instances that disconfirm those expectations and even the credibility of the entire sample of cases. In addition, the categories a perceiver has used in the recent past can influence the perceiver's interpretation of subsequent information. Finally, negative confirming cases might be encoded as cases that are unrelated to the covariation in question.

4. Recalling evidence and estimating the frequencies of confirming and discomfirming cases. Here, the major issue is that instances which fit the perceiver's expectations have an advantage over other instances in frequency estimates and recognition memory. This has been shown by research on "illusory correlation".

5. Integrating the evidence. Once social perceivers have accumulated information about instances, they must combine the evidence to make an assessment of the degree of covariation in the observed instances. This step is the one most clearly specified by the statistical model for assessing covariation. The author distinguished relationships between binary and non-binary variables. For non-binary variables, he only considered the case of numerical variables, where the correlation coefficient can be applied. He noticed that very few studies have investigated naive estimates of covariation between numerical variables and that available studies suggest that naive estimates seem to be sensitive to the actual degree of covariation. The author presented a comprehensive summary of people's strategies and performance in judging association between binary variables (2 x 2 contingency tables) that have provided mixing results. Some studies show that people perform poorly, while other suggest that under the right conditions people are remarkably good at such a task. Two factors may influence the strategies used to estimate covariation: the nature of the covariation question itself and the perceiver's expectations.

6. Using the covariation estimate to take other judgments and decisions. The following factors operate against normatively appropriate use of covariation estimates: the tendency for predictions to be non regressive and the tendency to infer a causal relationship from information about a covariation.

Joan Garfield - This paper emphasises affective issues which are particularly important in stochastics learning as well as aspects of assessment. It also reminds us that good electronic journals are already with us.

2. Gal, I., and Ginsburg, L. (1994)

"The Role of Beliefs and Attitudes in Learning Statistics: Towards an Assessment Framework"

Journal of Statistics Education [Online], 2(2) Available by e-mail: archive@jse.stat.ncsu.edu Message: send jse/v2n2/gal

While many teachers of statistics are likely to focus on transmitting knowledge, many students are likely to have trouble with statistics due to non-cognitive factors, such as negative attitudes or beliefs towards statistics. Such factors can impede learning of statistics, or hinder the extent to which students will develop useful statistical intuitions and apply what they have learned outside the classroom. This paper reviews the role of affect and attitudes in the learning of statistics, critiques current instruments for assessing attitudes and beliefs of students, and explores assessment methods teachers can use to gauge students' dispositions regarding statistics.

Kath Truran - this article reports research into teachers and teaching, as well as illustrating the sort of material which can come out of study for a further degree.

Edwards, Roger (1996) ‘Teaching Statistics: Teacher Knowledge and Confidence’ in Clarkson, P.C. (ed.) Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA) (pp. 178 - 185) Victoria: Deakin University Press.

This paper reports on research in progress that explores primary school teachers’ ideas about statistics and the teaching of statistics. Data on teachers’ attitudes and beliefs was collected through interviews using a semi-structured interview format and focused on getting teachers to talk about their views of statistics and its teaching. Belief and attitude scales were introduced about 35 minutes into the interview and teachers were encouraged to talk aloud as they completed two scales: a beliefs and attitude scale (BAS) assessing levels of agreement with statements about statistics and teaching: and a confidence rating scale (CRS) asking teachers to rate on a numerical scale their confidence in teaching particular topics or concepts in statistics.

Issues related to teachers’ lack of statistical training, confidence in teaching and their views about essential knowledge for teaching statistics were explored. The writer claims that this study indicates a number of implications for teacher training in the teaching of statistics. He claims that it would be more useful to raise research issues based on what teachers do understand and how this impacts on teaching and learning.

John Truran - this review takes a “classic” article from a prestigious journal. The article looks at integrating mathematical and teaching issues. It also points out that even fine articles may have important weaknesses and also that their valuable recommendations may be neglected by subsequent generations.

Freudenthal, Hans (1974) The Crux of Course Design in Probability Educational Studies in Mathematics 5: 261 - 277

This article by a man who was distinguished and respected both as a mathematician and as a mathematics educator examines the formal mathematics of probability in an endeavour to decide what is appropriate for probability teaching in schools, and how it might best be taught.

The approach is formal and symbolic and so does not invite careful attention from non-specialists. But Freudenthal raises critical issues about the divergence between the traditional view that probability is a function of events or statements about events and the axiomatic view which sees it as a measure defined on a system of sets. He argues that the traditional view is of far more value, but that it is often constrained by failing to emphasise the generality of its claims. He argues that this may be overcoming by emphasising that probability theory makes statements about “stochastics”, rather than about sets. In this way the statements are generalised to wide fields of application rather than to narrow fields such as, for example, a tossing of a single coin.

Unfortun, his mathematical definition of a stochastic does not match his informal one. Sometimes he seems to mean that it is the outcome of a random trial, but he also refers to is as “a chance experiment” and later as an ordered triple comprising a domain (sample space), a set of subsets of the domain, and a function which ascribes probability values to these subsets.

This paper is of great importance in spite of its inconsistencies because it argues that the traditional approaches found in most textbooks are imprecise, and that the approach advocated is one which can be easily used in the classroom. It has been seriously neglected, and deserves much greater attention.