Spring 1999, no.6

It is a pleasure for me as chairperson to welcome members and readers to this issue of the ALM Newsletter. Since ALM was formerly established in July1994 as an international research forum it has grown in stature and confidence through the work of its members, researchers and practitioners, in the field of adults learning mathematics. I am pleased to say that this work is continuing and is visible at our annual conferences, in our published proceedings, other academic publications, and textbooks and resources for learners and teachers.

Communication must have a high priority in any organization such as ours and I can assure members that we are continually looking for ways to improve that aspect of ALM as an international organization. The ALM newsletter is a key element in ALM’s communications strategy. It is an important vehicle for keeping members up to date; presenting news, reports, book reviews, members views, testing ideas and generally encouraging informed debate. It is important that we set and maintain high standards in every aspect of our work and I have every confidence in the professionalism of the new editorial team. There is evidence here in this issue of a new sense of purpose and competence that will benefit all members.

I congratulate the editorial team for their efforts on our behalf. We can all demonstrate our appreciation by supporting the editors in their work and submit items for inclusion in the newsletter. I look forward to interacting with members through the pages of the newsletter and meeting them in person at our annual conferences - the next one is scheduled for 8-10 July, at Sheffield Hallam University, U.K.

Prof. John O’Donoghue, Chair, ALM
Dept of Mathematics and Statistics, University of Limerick, Limerick, Ireland.
Fax: +353 61 334927. email: John.ODonoghue@ul.ie

An international survey of the numeracy abilities of adults is to be part of the International Life Skills Survey (ILSS) planned for the year 2001. This comparative survey is being jointly developed by Statistics Canada and by the United States’ National Center for Education Statistics (NCES), in cooperation with the Organisation for Economic Cooperation and Development (OECD). This article is based on a report prepared for the ILSS project by the Numeracy Working Group, comprised of individuals from Israel, Australia, Holland, United States, and Canada.

The ILSS project is a follow-up to the International Adult Literacy Survey (IALS), the world’s first large scale comparative assessment of adult literacy. In the IALS, adults from over 20 countries, have been tested based on survey methodology that combined household survey research and methods of educational testing. Using a similar approach involving home interviews, ILSS will test nationally representative samples of adults aged 16 and over in multiple countries. Tasks will assess performance in several skill domains, including Numeracy as well as Literacy, and Problem Solving; other variables will be assessed via a background questionnaire.

The inclusion of a Numeracy scale in the ILSS also offers a significant opportunity to develop a new conceptual framework for adult numeracy, which should be of interest to educators and researchers interested in the development and application of mathematical knowledge in purposeful contexts. To date the Numeracy Working Group has spent most of its time developing a background conceptual framework and a set of sample assessment items that fit the framework. Feasibility studies involving sample tasks from the Numeracy scale have started and results are expected in mid 1999.

Key motivations for conducting the overall ILSS are: to inform policymakers and educators regarding levels (distributions) of various skills, including of numeracy; to explore factors associated with observed skill levels (e.g., literacy); and to examine links between numeracy (or other skills) and important social variables, such as earnings, labor-force participation, unemployment, or health-related behaviors and educational background. Numeracy is becoming a growing concern for diverse education sectors, following its apparent low profile for many years. As countries increasingly attend to topics such as improving workplace efficiency and quality processes, to resulting lifelong learning needs, and to civic participation, it is seen as vital that nations have information about their citizens’ numeracy, among other skills, if they want to plan effective education and lifelong learning opportunities.

The concept of numeracy is specifically related to the dialogue about the goals and especially outcomes and impact of school mathematics education. More educators now encourage links between knowledge gained in the mathematics classroom and students’ ability to handle real-life situations that require mathematical or statistical knowledge and skills. However, while numeracy may be a key skill area, its conceptual boundaries, cognitive underpinnings, and assessment, have not received much scholarly attention so far.

It might be thought unnecessary to undertake a full numeracy assessment for ILSS since the IALS included the Quantitative Literacy scale. While there is a clear connection and relationship between numeracy and the IALS measure called Quantitative Literacy, there are significant differences, with numeracy covering a much wider breadth of mathematical skills and purposes.

Overall, numeracy is a multifaceted and sometimes slippery construct. Our basic premise is that numeracy is the bridge that links mathematical knowledge, whether acquired via formal or informal learning, with functional and information-processing demands encountered in the real world. An evaluation of a person’s numeracy is far from being a trivial matter, as it has to take into account task and situational demands, type of mathematical information available, the way in which that information is represented, prior practices, individual dispositions, cultural norms, and more.

Numerate behavior obviously includes the ability to calculate or manipulate symbols but is far from being limited to it. In a large-scale survey context, assessment of numerate behavior can be accomplished through tasks couched in realistic non-school settings, with limited usage of formal notations, and with significant presence of text-rich tasks, as well as of some tasks where opinions rather than computation are called for (e.g., when interpreting statistical messages). Yet, while the scale we envision may cover a broad mathematical terrain, it may still fall short of encompassing the full scope of numerate behavior, due to pragmatic considerations. Some aspects of people’s numeracy skills, such as those pertaining to problem-solving strategies, or to interpretive responses and their underlying reasoning processes, cannot be fully reliably and validly assessed with the methodology presently available in the ILSS.

Numerate behavior and its five facets

Numerate behaviour involves:

managing a situation or solving a problem in a real context

everyday life

work

societal

further learning

by responding

identifying

interpreting

acting upon

communicating about

to mathematical information

numbers

statistical data

measurements

money

time

shape

direction

pattern and relationships

that is represented in a range of ways

objects & pictures

numbers & symbols

diagrams & maps

graphs

tables

texts

formulae

and requires the activation of a range of enabling processes and behaviours

mathematical knowledge and understanding

mathematical problem solving skills

literacy skills

beliefs and attitudes.

Difficulty/complexity. To ensure a distribution of items at different difficulty levels, the complexity of items will be pre-estimated on the basis of five general factors gleaned from prior assessments of adult literacy or of mathematical skills: (1) Complexity of Mathematical information / data; (2) Type of operation / skill; (3) Expected number of operations; (4) Plausibility of distractors (including in text); (5) Type of match / problem transparency. These factors can determine, separately and in interaction, the difficulty level of most numeracy tasks. For some tasks, such as those that are more interpretive in nature, other factors that affect complexity will also be considered.

Background variables. To shed a broader light on factors related to the distribution of numeracy skills across the adult population, additional data will be gathered through a background questionnaire, regarding three topics: school mathematics experience; numeracy practices (e.g., use of calculators, getting help from others, activity structures); and dispositions (e.g., anxiety, confidence, interest). This is in addition to information about demographic variables and about literacy practices at home and at work that will be collected to support interpretation of results from all the ILSS scales.

Full assessment of adults’ numerate behavior requires further work on the conceptualization of some of the facets and components of adult numeracy, as well as grappling with a host of pragmatic challenges, such as translation to different languages that will retain task characteristics, training of interviewers regarding follow-up questions or scoring of partial responses, and more. It is hoped that this brief report will facilitate a dialogue on these and other issues raised among researchers and educators interested in mathematics education.

* The Numeracy Working Group is comprised of: Iddo Gal, University of Haifa, Israel; Dave Tout, Language Australia; Mieke van Groenestijn, Hogeschool van Utrecht, Netherlands; Mary Jane Schmitt, National Center for the Study of Adult Learning and Literacy, Harvard University, USA; Myrna Manly, El Camino College, California; Stan Jones, Statistics Canada.

There was a variety of interesting workshops, computer demos and poster sessions and we had a very interesting video-conference netmeeting with the Bridging Maths conference in Australia.

And of course everybody was happy to meet each other as old friends or for the first time.

We are glad that there are so many enthusiastic people in adult numeracy education. People encouraged each other enormously and many appointments have been made for starting joint-projects across the world.

Avanti Books,
8 Parsons Green,
Boulton Road
Stevenage SG1, 4QG, UK
Tel +44 1438 350155
Fax +44 1438 741131

Introduction
The following looks at the difficulties basic numeracy students have in dealing with multiplication and leading from that, division. I am concentrating on multiplication as this best illustrates the problem. It is also the area where I have started to develop a way of addressing the problem. I feel that to start with I should make it clear the level of student with whom I am concerned. I work in Adult Basic Education departments of Further Education and Community Education in Bristol, England. The students I am looking at in this paper have very immediate numeracy needs, such as being able to handle money on a day to day basis more confidently. When they start they are generally able to count, add and by informal methods subtract numbers. They may have found means of tracking their money in broad terms, but very often cannot itemize and check all the details. The development of these skills, although it can involve time and hard work on the part of the student, are not, relatively speaking, too difficult to achieve.

Outline of problem
Many basic numeracy students find the concepts of multiply and divide more difficult. We tend to teach the four rules one after the other as if they follow a logical order directly building on the previously learned skill. To some extent this is the case but not totally so. Just because a student can add and take away does not mean that they have all the skills or understand all the concepts that are needed to go on to multiply and divide.

To add follows on from counting. The need to do this on a simple level is all around. How many people want tea and how many want coffee in the group? The idea of adding these together to get to the total number in the group is something that can be seen. Going on from that, to take away is also something with which they are familiar - if five want tea and there are ten in the group then five from ten is five who want coffee. Using money gives lots of examples of the need in very practical terms to add and take away.

The step on from this to multiply and divide is not as smooth as it may seem to those of us who have mastered these ideas and use them regularly almost without thinking. How much will five teas cost if one costs 50p: 5 x 5 = 25 so œ2.50 - no problem. The bill came to œ2.50 for five of us so that is 5 into 25 = 5 so 50p each.

The answer to this sort of sum can be arrived at by repeated addition or subtraction. And so many basic students carry on using these methods of arriving at the answer and not bothering with the multiply and divide. I will leave aside the question of whether they need to for the time being, (this would be a subject for another talk altogether), and proceed on the basis that the student wants or needs, for whatever reason, to be able to do these things.

In working with a number of students at this level over the last five -six years I have found over and over that students revert back to repeated addition to work out practical sums when, for instance doing money problems, and resist using multiplication.

At first I thought that they resisted it because of the problem of learning their tables; they could not call to mind the answer quickly enough, so went back to the sure way of adding. However I started to find that even where students had tried and who for a short time managed to learn the times tables they still did not use this knowledge.

The other answer was that the students needed to try out multiplication in a range of practical settings. This is the way most tutors I have spoken to would proceed and is the way that the standard text books are laid out. Encouragement of this by way of simulated exercises or actual situations where they could use multiplication outside college did not help them significantly. Even if they made a special effort to try to use multiplication very often this would be checked by adding, showing a lack of confidence in the method.

One major problem was not resolved, they were still not sure of when to use the knowledge. I have gathered together work from various students who are able to mechanically work out a multiplication sum when they are given it as an exercise clearly labelled multiplication. These people have also been able to learn some times tables, if not all of them up to ten times.

When faced with a realistic situation where some addition, some subtraction, some multiplication and some division are used and they have to select the most appropriate they are confused as to what is required when. Students would often try the sum two or three different ways hoping one of them would be right. They would usually get so bogged down in this that they lost sight of what they were trying to achieve.

This led me to think that it is not the mechanics of the calculation that is confusing but the actual concept of multiplication (and division) that is not understood. I have been trying to find a reason which may explain the difficulties people have with both of them which may also help with the use and mechanics of dividing.

Materials that address the problem
When I tried to see if I could illustrate the difference between multiplication and division by using diagrams or some other graphic/tangible way of understanding what is happening I started to see another problem the students have.

1. Organise them into groups of similar things.
2. Count the number of groups
3. Count how many are in each group
4. Work out how many are there altogether

When I was making this up I thought I had maybe made it too simple, but thought that if I introduced it as a preliminary stage then the students may accept it. However what I found was that this was a more difficult exercise than I had expected and that the students had real trouble with the grouping of things. It was not something they did. I talked to them about doing this sort of thing at home - i.e. sorting out their kitchen cupboards and the like to count up what is there. They would count each set of tins separately and not try to group them in any way. They also said that whenever they tried to do this sort of thing they would get half way through and start forgetting the numbers and would have to start again. So this seemed to be something to develop.

I feel that patterning and maybe other related special skills are something that these student are not able to do, at least do well, and instinctively. I therefore developed the grouping idea. Exercises either written or practical where students are encouraged to group objects have started to show some benefits to the students. As yet though I have not developed an adequate range of materials to fully develop these skills.

One of the difficulties is trying to find things that an adult would be happy to use. Some find the use of counters and so on a bit childish, although in the right environment this can be incorporated into the classroom session. It seems that these people may well have missed out on some of the play that many of us did as children. It is with things such as building blocks and other maybe more boy-like toys that we develop these skills. This is an area I would like to explore more in the future.

In looking into this I have found little help from the standard text books; even ones which have a generally good practical approach to numeracy. The accreditation appropriate for this level of student does not encourage the development of these skills either. Counting in batches is one example from the Number-power scheme which touches on this but it is one small part of a unit. Although we may try to avoid teaching strictly in line with the requirements of the accreditation schemes it is a fact of life in most centres that tutors have to pay regard to their requirements and often are expected to tutor students to pass these courses. I am concerned that the lack of attention to the spacial/organising skills referred to here are being neglected, only to prevent the student going beyond the lowest levels of achievement.

In general these materials are all geared towards a numerical or mechanical understanding of multiplying and dividing. The spacial skills come later when students go on to measuring and geometry. But maybe some of these skills are needed at an earlier stage.

I have indicated the problems students have with their tables and using multiply and divide. Making the work relevant and showing practical examples has not necessarily helped to make the decisions about what to use any easier.

Conclusion
Although I have concentrated on these few students who may be a minority of those we see as tutors I feel that these points have a wider importance. Firstly because I am not convinced that all those who appear to understand multiply and divide and who have passed exams or gained accreditation which purports to show that they are able to do these things are actually able to go out and use these skills to make life easier for themselves. Often tutors take the mechanical knowledge to be an illustration of the student learning how to do them. They may have a mechanical skill that in given situations where the need to do them are clearly spelt out they can appear to apply them successfully.

Secondly even those who can do these things and can use multiply and divide at a simple level may not have all the resources they need to move on further with maths. At some point the understanding may break down or they will find it much more difficult to do other things - fractions and so on. These tasks may be made much easier if the patterning and spacial skills I have touched on can be more fully developed.

Altogether, encouragement gathered understood difference accreditation achievement resources. However they would usually count out money by grouping like coins together, probably because this is commonly seen being done by others. Very often the money would then be counted consecutively and the reason for grouping not actually used.

This paper was prepared for the ALM-5 Conference and is part of the 1998 ALM-5 Conference Proceedings.

News and Events

Adult Numeracy Network Meeting - Teaching and Learning Mathematics using Real-world Contexts
The Adult Numeracy Network (ANN), an affiliate organization of the National Council of Teachers of Mathematics, will hold its fifth annual meeting on Wednesday, April 21, 1999.

To register: Send cheque and registration form (available from http://www.std.com/Newbury/anpn/) to Jan Phillips, William Rainey Harper College, 1200 W. Algonquin Road, Platine, IL 60067-7398, USA.

23rd Conference of the Inter-national Group for the Psychology of Mathematics Education (PME)
When: Sunday 25 July - Friday 30 July 1999

Cost: Approx. $US350, which includes meeting registration, an excursion, lunches, conference dinner and one-year membership.

For more information and to register contact PME23 Conference Secretariat, Palex Tours, PO Box 33626, Haifa, 31336, Israel. Ph: + 972 4 8524254 Fax: + 972 4 8522491. Email: pl@netvision.net.il or visit the web site at http://members.tripod.com/~IGPME/pme23/index.html

Special Numeracy Issue
Numeracy issue of Literacy & Numeracy Studies: An International Journal in the Education and Training of Adults. Volume 8 Number 1, 1998.

The articles in this issue of Literacy and Numeracy Studies provide resources for numeracy teachers and researchers to fashion responses to the socio-political pressures which influence what and how they teach, research and theorise.

Topics covered include Numeracy as Social Practice, Numeracy practices of young unemployed people, Generic Numeracies on the Shop Floor.

Available from:
Centre for Language and Literacy,
University of Technology,
Sydney, PO Box 123,
Broadway NSW 2007, Australia.

Review - Measuring Up

Measuring Up is an interactive [software] program for teaching and applying metrics and measurement to real-life contexts through the learning areas of decimals, fractions, scales and the metric system.

The program is aimed at learners who have basic skills in adding, subtracting and multiplying whole numbers. Learners at higher levels would find the learning activities relevant and valuable for practice and revision purposes. The program is suitable for independent and group learning, with a minimum of assistance for those needing computer training.

In the four Learning Areas, a skill or concept is introduced and modelled using real life or visual models, then an opportunity is provided for practice before the learner proceeds to the next skill knowledge area. A learner can repeat a modelling or practice activity at any time. In the section on adding decimals, the program models three ways to add, including estimation. The skills / knowledge areas have several explanations within real-life contexts; the explanations are very clear.

The five Scenes for Applying Skills and Knowledge allow learners to apply or assess what they have learned by interacting in the following real-life situations: the Supermarket, the Barbeque, the Material Shop, the Mail Room and Sports Day. The scenes involve such activities as weighing packages, measuring ribbon, calculating prices of meat, vegetables and cloth, deciding the best buy, ranking athletes according to speed or distance, and interpreting recipes.

The activities provide the opportunity to deepen understanding of decimals and fractions, and plenty of practice in basic operations with decimals. Estimating skills are reinforced throughout the activities, along with practice in measuring and managing money. Above all, the activities are fun and truly related to everyday life.

When a learner has completed all the activities in one of the Scenes, the program generates a Summary. The Summary identifies the skill areas in which the learner has demonstrated competence as well as those requiring further practice. The learner may print the Summary to provide a permanent record of the learner’s performance.

The program has clear menus and distinct buttons for navigating through the activities with instructions in both verbal and written form. In the Learning Areas, the learner is given specific feedback, such as "No, type two numbers after the decimal point", or simply, "No, that’s not right. Try again." Unfortunately, as in any similar interactive program, it cannot interpret answers; one answer is accepted as correct.

Learners will find Measuring Up very flexible. They can decide which activities they will do, and in which order. The program is completely self-paced, and provides numerous repetitions. The examples and practice exercises are different on each occasion that the learner accesses them. Accompanying the CD-ROM is an easy-to-read and thorough 24 page manual. It also includes a "Learner’s Record Sheet" and suggested order for activities.

Measuring Up is an excellent program in all respects. As a computer program, it is accessible and appealing, and as a numeracy program, it successfully integrates the learning of skills and concepts with concrete and real-life situations. As this is an Australian program, the learner must adapt to the speaker’s accent and some unfamiliar vocabulary; however, these are of minor concern. A major plus is that the money and measurement systems are the same as we use in Canada.

The content of Measuring Up was written by Dave Tout and Beth Marr, authors of several outstanding works in adult numeracy.

Review contributed courtesy of Alpha Ontario. The review was published in CONNECT (a newsletter for adult literacy and technology), published from the Continuing Education Centre in Ottawa, Canada.
Available on-line at: http://www.nald.ca/connect/may98/page9.htm

Measuring Up is available in DOS and Windows formats from:
Protea Textware Pty Ltd
PO Box 49, Hurstbridge
Victoria 3099 AUSTRALIA
Tel: +61 3 9714 8660 Fax: +61 3 9714 8644
email: protea@mpx.com.au
web: <http://www.proteatextware.com.au
or
Avanti Books,
8 Parsons Green, Boulton Road
Stevenage SG1, 4QG, UK
Tel +44 1438 350155 Fax +44 1438 741131

Adults Learning Maths – A Research Forum (ALM) is an international research forum bringing together researchers and practitioners in adult mathematics/numeracy teaching and learning in order to promote the learning of mathematics by adults.

ALM was formally established at the Inaugural Conference, ALM-1 in July 1994 as an international research forum with the aim to promote the learning of mathematics by adults through an international forum which brings together those engaged and interested in research and developments in the field of adult mathematics* teaching and learning.

* Within ALM we understand the term ‘mathematics’ to include ‘numeracy’.

ALM is a forum for experienced and first-time researchers to come together and share their ideas and their reflections on the process as well as the outcomes of research into hitherto neglected area of adults learning mathematics. ALM puts people in touch with each other, providing a framework for collaboration and helping to stimulate and develop research plans. We are especially keen to encourage practitioners to undertake research.

Since 1994, ALM has gone from strength to strength and now has 140 members in 19 countries.

What does ALM offer?
ALM membership brings with it opportunities to:
• contribute to an international forum of researchers and practitioners in the field
• share concerns, insights and research at ALM annual conferences, and to attend at a reduced rate
• receive ALM newsletter (free)
• receive ALM conference proceedings (free of charge to conference delegates). These proceedings constitute the most significant and authoritative collection of papers on adults learning mathematics available today
• network, electronically and otherwise, with practitioners and researchers in the emerging field of adults learning mathematics.

ALM Officers
Chair: Prof. John O’Donoghue, University of Limerick
Secretary: Dr Diana Coben, University of London
Treasurer: Sylvia Johnson
Membership Secretary: Sue Elliott, Sheffield Hallam University

Join ALM today!
ALM is actively seeking to expand its membership worldwide. Membership is open to all individuals and institutions who subscribe to its aims. For details contact Sue Elliott, Membership Secretary at the Centre for Mathematics Education, Sheffield Hallam University, 25 Broomsgrove Road, Sheffield S10 2NA, UK email: S.Elliott@shu.ac.uk or your regional ALM membership agent:

AUSTRALIA Dr Janet Taylor, OPACS, Uni. of Southern Queensland, Toowoomba, Australia. Email: taylor@usq.edu.au

BRAZIL Eliana Maria Guedes, Dept. of Architecture, Mathematics and Computing, UNITAU, University of Taubaté, Sao Paulo, Brazil. Email: emg@aquarius.com.br

DENMARK Tine Wedege, IMFUFA, Roskilde Uni., PO Box 260, 4000 Roskilde, Denmark. Email: tiw@mmf.ruc.dk

NEW ZEALAND Barbara Miller-Reilly, Student Learning Centre, The University of Auckland, Private Bag 92019, Auckland, N.Z. Email: Barbara@math.Auckland.ac.nz

REPUBLIC OF IRELAND Prof. John O’Donoghue, Dept of Maths and Statistics, University of Limerick, Limerick, Ireland. Email: John.ODonoghue@ul.ie

THE NETHERLANDS Mieke van Groenestijn, Hogeschool van Utrecht, PO Box 14007, 3508 SB, Utrecht, The Netherlands. Email: Mieke.v.Groenestijn@feo.hvu.nl

UNITED KINGDOM Sue Elliott, Centre for Mathematics Education, Sheffield Hallam University, 25 Broomsgrove Road, Sheffield S10 2NA, UK email: S.Elliott@shu.ac.uk

USA Dr Katherine Safford, Saint Peter’s College, Kennedy Boulevard, Jersey City, NJ 07306, USA.
Email: SAFFORD_K@spvxa.spc.edu

Membership fees
Individual     stg £15
Institution     stg £30
Student/unwaged   stg £3
Low waged (minimum)

Editorial Committee
Brian Cann
Mieke van Groenestijn Hogeschool van Utrecht
Tom McDonald
Dave Tout Language Australia

The views expressed in individual articles are those of the authors and do not necessarily represent the views of ALM or of the editorial committee.

We would like to encourage members to submit items to the newsletter. These should be sent to:
Mieke van Groenestijn, Faculty of Ed., Hogeschool van Utrecht, PO Box 14007, 3508 SB, Utrecht, The Netherlands.
Email: Mieke.v.Groenestijn@feo.hvu.nl

Baked Beans	Peas	Tomato soup
Baked Beans	Mushroom soup	Tomato soup
Peas	Mushroom soup	Baked beans
Peas	Mushroom soup	Tomato soup

	ADULTS LEARNING MATHS NEWSLETTER
	No. 6 Spring 1999