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ED 362 767 CE 064 882

TITLE Numeracy. Viewpoints: A Series of Occasional Paperson Basic Education. 16.

INSTITUTION Adult Literacy and Basic Skills Unit, London

(England).REPORT NO ISSN-0-:466-20989

PUB DATE May 93NOTE 39p.

AVAILABLE FROM Adult Literacy and Basic Skills Unit, KingsbourneHouse, 229/231 High Holborn, London WC1V 7DA,England, United Kingdom (2.50 British pounds pluspostage).

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EDRS PRICE MF01/PCO2 Plus Postage.DESCRIPTORS Adult Basic Education; Adult Literacy; *Arithmetic;

*Basic Skills; Foreign Countries; Literacy Education;Mathematics Instruction; *Mathematics Skills;*Numeracy; Performance Contracts; PostsecondaryEducation; Unemployment

IDENTIFIERS Australia; United Kingdom

ABSTRACTThe six articles in this issue tackle topics that

relate to the role and function of numeracy in adult life. "ReadingComprehension in Written MathemAtics Problems" (Sue Wareham) examinesthe levels of reading ability required of numeracy students andoffers strategies to help students with limited reading skills. "CanOrdinary People Do Real Maths?" (Joan O'Hagan) tackles the issue of"real" math through a review of historical, philosophical, andcultural perspectives on the nature of mathematics and considers theconsequences for tutors and students. "Numeracy: A Core Skill or

Not?" (George Barr) reviews the place of numeracy within recentlyestablished core skills frameworks. "Sorting Out Statistics:Confessions of a Numeracy Tutor" (Sarah Oliver) considers the placeof statistics in life and in the curriculum. "Learning Contracts in

Higher Education: Towards Confidence in Developing Numeracy Skills"(Ian Beveridge, Gordon Weller) describes the development of numeracysupport for Access (a program which attracts a lot of single parentmothers, housewives with older children, and non-mainstream groups)

and higher education students at Luton College, England. "Effective

Provision of Literacy and Numeracy Instruction for Long-TermUnemployed Persons" (Joy Cumming), an article that reviews theprovision of employment-related courses in Australia, evaluates thefeatures of good practice in both literacy and numeracy, andconsiders the most effective approaches in developing numeracyskills. (YLB)

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ViewpointsA Series Of Occasional Papers On Basic Education

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16. Numeracy

READING COMPREHENSION IN WRITTENMATHEMATICS PROBLEMS

Sue Wareham

CAN ORDINARY PEOPLE DO REAL MATHS?Joan O'Hagan

NUMERACY: A CORE SKILL OR NOT?Dr George Barr

SORTING OUT STATISTICS: CONFESSIONS OF ANUMERACY TUTOR

Sarah Oliver

LEARNING CONTRACTS IN HIGHER EDUCATION:TOWARDS CONFIDENCE IN DEVELOPINGNUMERACY SKILLS

/an Beveridge & Gordon Weller

EFFECTIVE PROVISION OF LITERACY ANDNUMERACY INSTRUCTION FOR LONG-TERMUNEMPLOYED PERSONS

Dr Joy Cumming

2

ViewpointsA Series Of Occasional Papers On Basic Education

16. Numeracy

IntroductionIn the current debate about standards of basic skills, the focus is often on the nature of

adequate literacy skills. However, the role and function of numeracy in our lives deservesequal consideration. What numeracy skills are needed in adult life, in or out of the workcontext? What linkage is there, or should there be, between the development of literacy andnumeracy skills. Is numeracy a 'core skill' and, if so, to what extent? Is nurneracy about skillsor applications? Many of the issues that are raised by these questions are tackled by theauthors in this Viewpoints.

Sue Wareham examines the levels of reading ability required of numeracy students andoffers strategies to help students with limited reading skills. Joan O'Hagan tackles the issue of'real' maths through a review of historical, philosophical and cultural perspectives on thenature of mathematics: and considers the consequences for tutors and students. Ian Beveridgeand Gordon Weller describe the development of numeracy support for Access and HigherEducat:on students at Luton College. George Barr reviews the place of numeracy withinrecently established core skills frameworks. Sarah Oliver looks behind the pie chart to c-Tisiderthe place of statistics in our lives and in the curriculum. Finally, in an article that reviews theprovision of employment related courses in Australia, Joy Cumming evaluates the features ofgood practice in both literacy and numeracy. She considers the most effective approaches indeveloping numeracy skills.

ContentsREADING COMPREHENSION IN WRITTEN MATHEMATICS PROBLEMS

Sue Wareham

CAN ORDINARY PEOPLE DO REAL MATHS?

-Joan 0 'Hagan

9

9

NUMERACY: A CORE SKILL OR NOT?

Dr George Barr 15

SORTING OUT STATISTICS: CONFESSIONS OF A NUMERACY TUTOR

Sarah Oliver 19

LEARNING CONTRACTS IN HIGHER EDUCATION: TOWARDSCONFIDENCE IN DEVELOPING NUMERACY SKILLS

Ian Beveridge & Gordon Weller 24

EFFECTIVE PROVISION OF LITERACY AND SUMERACY INSTRUCTIONFOR LONG-TERM UNEMPLOYED PERSONS

Dr -Joy Cumming 29

Thc views expressed in this publication are those of the authors alone and are notnecessarily shared by their employers or the Adult Literacy & Basic Shills Unit.

Viewpoints 16: Numeracy

Reading comprehension in written mathematics problemsSue Wareham

Sue Wareham is a part-time tutor who works in Adult Education. She teachesliteracy and numeracy at several centres in south east Essex. She is involved withthe initial training of volunteers locally and the in-service training of ABE tutors in

the county.

The ability to solve problems is at the heart of mathematics.Mathematics is only 'useful to the extent to which it can beapplied to a particular situation and it is the ability toapply mathematics to a var, ny of situations to which wegive the name 'problem solving'. However, the solution of amathematical problem cannot begin untd the problem hasbeen translated into the appropriate mathematical terms.This first and essential step presents very great difficultiesto many pupils a fact which is often too little

appreciatedParagraph 249. %Mathematics Counts*

Why do students come to classes to improve theirnumeracy skills?

In my experier ce. working in Adult Education,students come to ABE groups wishing to improve theirnumeracy skills for a wide variety reasons: tosupport the maths content of other muses such asbook keeping, navigation: to apply their new skills to ahobby such as dressmaking or woodwork: to help theirchildren with homework: to improve their job prospectsor cope with the demands of promotion: to pass a skillstest for entry into a profession such as the fireservice orpolice: and lastly but perhaps the most important ofall to improve their own self-image. This wide varietyof aims and needs poses special difficulties for the tutorin terms of class organisation and management.

Management difficultiesIn many ABE classes numeracy is taught within a

predominantly literacy group and this presents specialproblems to the tutor. There may be insufficientnumeracy students within the group to offer mutuaisupport in the form of discussion. The tutor may feel

obliged to match the student with a volunteer andprovide worksheets or textbooks for their use. In theabsence of volunteer help the student may have towork alone with the occasional support of the tutor.Indeed students who join the group with a short termspecific aim, perhaps of passing a skills test foremployment. may prefer to work alone.

NI any areas of further education basic skills.including numeracy, are taught using flexible learningmethods, perhaps with a workshop style of delivery. Itis possible that this method of organisationoccasionally places greater reliance on printed texts asteaching resources than perhaps the tutor would wish.

This special organisation and its reliance on either

published materials or those prepared by the tutorraises several points which may be worth considering.Could the selection or adaptation of numeracy textsbe an influencing factor in the students's understandingof the topic and in particular affect his skills in solvingword problems? Perhaps there are also factors outsidethe text which affect success which should beconsidered such as maths anxiety, poor experiences atschool, fear of failure and especially fear ofencountering 'trick' questions.

Why are we obliged to make use of writtenarithmetic problems?

In order to become competent with everydaynumeracy skills the student needs to meet manydifferent applications of arithmetical operations. Theapplication of some of these skills by adults ineveryday life, those of mental arithmetic, estimationand the practical use of money does not often requirethe use of pencil and paper or necessitate translatingthe problem into words. Other situations, however, dorequire the recording of the number operationsinvolved such as planning a budget or checking a bankstatement. In order to extend the students' range ofapplication of the basic concepts we offer practice ineveryday situations by using arithmetic wordproblems. However we should remember that in reallife situations we set our own problems according tothe task in hand and these problems are rarely writtendown in words or sentences. Admittedly there is nosubstitution for the real life task and in many groups itis possible to maintain a very practical approach for

example in a Plan. Shop, Cook and Eat' group.However for reasons of organisation mentioned abovein groups of students with mixed purposes and aimsthe tutor is often reliant on texts to give the studentthe necessary depth and breadth to practise his skills.It seems especially appropriate to examine thoseaspects of a text which may promote or hinder success.

Solving a word problemThe procedure for solving an arithmetic word ;

problem appears to fall into several stages. At a literallevel the reader must read, comprehend and extractthe relevant information from the text. He mustinterpret the meaning of the problem and identify thearithmetical processes needed to solve it before

proceeding.

9

Reading comprehension in written mathematics problemsI t is the early stages of this process. that of reading

comprehension. which I propose to concentrate on inthis paper.

How do we read a maths worksheet or a pagefrom a maths textbook?

When presenting students with a worksheet or apage from a maths textbook we are expecting a varietyof reading skills from them. The text may contain anexposition which could be a statement of the topic andits related concepts. There may be an explanation ofnew vocabulary, a new method or notation. Thereader must read and absorb these items. There maybe examples which the reader has to puzzle out andrelate to the topic. There will be instructions whichthe reader must interpret and act on. After reading the

! exercise the reader must formulate a plan of action forsolving the problem. He then moves outside the textto carry out his task. The above reading tasks allrequire close reading but there may be other parts ofthe text on the page. which may add points of interestor extend concepts, but which do not require suchclose reading. Nevertheless this part of the text stillhas to be read, assessed and possibly discarded as notuseful to the task in hand. The page will also containsignals, such as headings, question and page numbers.which guide the reader around the page. These are notread in the conventional way but guide the eye.perhaps by identifying or clarifying parts of the text.In reading such a text the student is making use of hisskills in reading everyday English but in addition tothis he has to recognise each word or symbol andattach meaning to it. The reader must be active, thetext informs, instructs and requires action. The readermust respond to the text by solving the problemspresented. The text may also contain data, charts anddiagrams. The reader must interpret these to build upa meaningful picture of the text.

How do we read a specific arithmetic wordproblem ?

Reading a word problem is perhaps similar toreading notices, instructions or directions. The purposeof the reading task is for a specific outcome, tocorrectly solve a problem. All the information thereader requires to solve the problem must be in the text.

In a piece of prose writing the reader makes use ofthe redundancy in the text to aid prediction. In anarithmetic word problem the redundancy in the text isreduced giving the reader less opportunity to predictand necessitating more intense reading. While readinga prose text the reader has the opportunity. by makingpredictions and using redundancy in the text, tochange direction, rethink and to decide if his previouspredictions were right or wrong. A word problem givesthe reader no opportunitx to do this because there isless surplus information in the text. Because thelanguage of the word problem is compact. applicationof the usual reading skills such as making use ofcontext cues and grammar, is severely limited, if notimpossible. The written maths problem can allow forno ambiguity of interpretation. If it does then you

may get the wrong answer. Every word is valuable tothe reader. I t must he read, interpreted and itsrelevance to the problem aszessed.

For example: 'Four brothers are left money in anaunt's will. The eldest has £200 and the others eachhave ,100. How much money did the aunt leave in herwill?

The word each is crucial to the correct solution ofthis word problem. If the word is omitted, ignored.regarded as irrelevant or just not understood, then theproblem will not be correctly comprehended.

The mathematical language of numbers is veryprecise while the verbal language of the writtenproblem is less precise. This inevitably raises problemsof interpretation of the written language into the moreprecise language of numbers.

For example: three times two plus four:3 x 2 4 OR 3 x 2H-4

How can we assess the reading difficulty inherentin an arithmetic word problem?

[sing a readability formula requires at least 100words of text. Clearly this is an inappropriate way toassess the reading difficulty presented by an individualword problem. The formula cannot be validly used inthis case. In fact, research by Katharine Perera2 hasshown that because of the terse and condensed proseused in the writing of these problems. readabilityscores, which have been carried out using mathematicstext books, suggest that these books are easier to readthan they really are.

Perhaps what is needed then is to examine whataspects of the language cause reading difficulty so thatappropriate strategies may be used in the writing of theproblems, by the tutor, in order to increase readability.

Sentence length is often used as a measure ofsentence difficulty. In a prose or narrative text it isvery often the case that a long sentence is harder toread. particularly if there are several clauses in it.Research reviewed by Perera2 indicated that:

'It does not always follow that difficult sentenceswill always be identifiable by their length: shortsentences may be harder than long ones. If lengthwere a true measure of complexity, then allsentences of the same length would be of equaldifficulty.-

The author explains in this paper that redUabilityformulae do not take into account word order.Theoretically all the words in the passage under testcould be listed in random order and the same readinglevel score would be obtained. Perhaps then problemsmay be caused to the reader by the order of words ina sentence and its grammatical complexity.

For ease of' reading the simplest sentence structure isthe basic subject, verb. object or subject, verb, adverb.

For example: Oranstes cost 12p. I bought 2 oranges.An additional advantage of this structure is the focusof the reader's attention on the subject of the sentencewhich comes first. in the word order.

Also at this level of complexity are simplearrangements of these structures into questions orinstructions.

5


For example: What is the total cost'? This basicstructure can be extended by the use of adjectives,participles. conjunctions and also by the use of thepassive tense.

For example: 2 oranges were bought. The complexityof the sentence is increased by the use of subordinateor comparative clauses.

For example: Carpet, which costs £10 per squaremetre. is to be laid in a room 10m long by 6m wide.Perera also advises caution in the use of ellipsis inconstructing sentences. Ellipsis occurs when the readerhas to mentally supply words in a gap left by thewriter. The tutor who perhaps is attempting to

simplify a lengthy problem may unwittingly increase

the reading difficulty.For example: One packet of cigarettes has five in it

and another (packet of cigarettes) has four. Howmany cigarettes are there altogether? In some cases itmay be better to recast the sentence rather thanemploy ellipsis. The writer of questions perhapsshould also aim to develop a /low of language in thesentences and avoid the technique the reader has toemploy in the following example, repeating the stem

of the sentence.A man had £6.00. He spent £4.45. How much has he

left? Give the answer a ) in pounds b ) in pence.Research carried out by Riley.' into the semantic

structure of arithmetic word problems indicates thatsentences expressed in different ways are not equaLydifficult to understand although they require the samearithmetic operation to solve them.

For example: a ) Joe had three cigarettes. ThenTom gave him five more. How many cigarettes doesJoe have now? b) Joe had some cigarettes. Then hegave five cigarettes to Torn. Now Joe has threecigarettes. How many cigarettes did Joe have in the

beginning'?Thus the ease oi solution of a problem is affected by

variables such as the length of the problem, its -grammatical complexity and the order of statementsin the problem. The syntax employed in the writing of

a problem may affect the student's comprehension butanother factor which should also be considered is thevocabulary used in the writing.

How can the reader cope with unfamiliar wordsin a written problem?

When a student reads a word problem she brings to

the text her knowledge of the world and her knowledgeof language, To make sense of a word or phrase thereader will make use of the context, syntax of the text

and her previous experience.Pimm gives the example of a student who was

asked to explain the mathematical term 'on average'.The student replied. "It's what hens lay on." Further

;investigation revealed that the student had readsomewhere. 'Hens lay on avt rage 300 eggs per year'.Indeed, the student was justified in the meaning heassigned to the word from the sentence he was relating

it to from his previous experience.Words which are infrequently used, or are perhaps

unfamiliar to the reader because they occur only in a

mathematical text, can make it difficult for the readerto extract encugh clues from the context to clarifymeaning. An additional consideration may be that if aword is familiar to a reader in one context, with onemeaning, it may be unfamiliar when used with adifferent, less common meaning. For example, the :

word 'mean* is used in a mathematical context as asynonym for average whereas in everyday Engli.,ti theword may be used differently.

For example: I don': know what you mean.He has a mean look about him.

The clarity of language, both verbal and written, is

crucial to a student's understanding of the task inhand. In a class situation it may be useful for the textto be discussed by the student and tutor but in somegroup situations this is just not possible. Perhaps then

a student's difficulty with comprehending a written I

task or problem may be aided by developing hisunderstanding of the register of mathematics, itssimilarities to and differences from ordinary English.that is written English used outside the mathematics

context.The learning and teaching of numeracy involves the

use of both everyday English terms and morespecialised mathematical words. Kane:' introducedthe terms Ordinary English and Mathematical Englishto help analyse the special nature of writtenmathematics. He says that: 'Mathematical English

and Ordinary English are sufficiently dissimilar thatthey require different skills and knowledge on the part

of readers to achieve appropriate levels of reading

comprehension.'As mentioned earlier in this article mathematics

sometimes uses words or phrases which a student mayencounter verbally or in an English text and assigns itsown specialised meaning to them. The word product.for example, is used in an everyday sense to refer toconsumer goods or the items made by industry. Inarithmetic the word has a more specialisedmeaning. A

student who is unaware of the specialised meaning of

words such as these may have difficulty comprehendingthe text if there are insufficient context clues to help

him.The Cockcroft Report' states that:

'The policy of trying to avoid reading difficultyby preparing workcards in which the use oflanguage is minimal or avoided altogether should

not be adopted. Instead the necessary languageskills should be developed through discussion and

exploration.' Paragraph 311

In a class situation where numeracy group work is

possible it may be valuable for the group to discuss thedifferent words they use to express a numerical

problem.For example: 10 - 8 take away

less thansubtractminus

A comparison could then perhaps be drawn between

these words and other expressions which also imply

subtraction.

4

Reading comprehension in written mathematics problems

For example: Bread reduced by 2p today, buy now!What does the word deductions meanon my wage slip?If I buy a new bicycle now the shopwill give me a discount.

This discussion could then be extended. ifappropriate, to examine the use of the word 'difference'.to mean subtraction in an arithmetic question and itsuse in everyday English. The answer to the question:'What is the difference between 27 and 6? ' depends onwhether the word difference is interpreted in aneveryday English sense or in its more specialisedmeaning in arithmetic. In arithmetic it meanssubtraction. In an everyday interpretation thedifference between 27 and 6 could be that one has twodigits and the other one, or that one is an even numberand the other odd.

Mathematics uses words with a high technicalcontent. Many mathematical terms have semanticroots in the classical languages, Latin and Greek.

These words can cause particular difficulty tostudents as they may only be encountered in themathematical text and often it is not easy to deducemeaning from them. Introducing the roots of wordsnew to the student may be a productive way ofgenerating interest in the topic by relating it to history.

An interesting example of the links the mathematicalword may have with ordinary English may stimulatethe memory for later recall.

For example: bisect'bi' means two 'sect' means cutbicycle two wheels sectionbiceps dissectbigamy sectional

How can the tutor help the student predict themeaning of unknown words?

As previously discussed, the brevity of the languageused in the writing of a word problem can causedifficulty for students if they encounter unfamiliarwords. Earp" found that care must be taken with eventhe simplest words when writing problems. Hepresented secondary school children with a simplehomemade worksheet on fractions. Pupils were askedto complete both phrases

parts shaded

equal parts

Most students correctly completed '3 parts shaded'but several alternative answers were offered for

equal parts'. Some students suggested 2.presumably referring to the unshaded sections. Otherssu ggested 1 three parts are shaded and these are threeequal portions. This is thus a valid answer. Somesuggested five, referring to the whole figure. Obviouslythe intention of the writer in the phrase 'equal parts'was not clear to the reader.

In reading a prose text a competent reader will

usually determine the plain sense, that is get the gist,of the passage fairly quickly. Then, more slowly, byasking questions of the text and drawing inferencesthe reader begins to comprehend the implications ofthe message. He gradually builds up a meaningfulwhole and makes sense of the passage. It is thismeaningful message which tells us that, for instance.the word read should sometimes rhyme with beadbut that in another context it should rhyme with bed.It is not what our eyes see that tells us which is whichbut the message in our heads.

'The brain tells the eye more than the eye tells thebrain.'

Frank Smith 7

If an ambiguity of meaning arose in a prose text thereader would search for clues to clarify hisinterpretation by re-reading, looking at the contextand perhaps analysing the syntax. In these very terseexpressions parts shaded" equalparts', not only is the reader expected to reactpromptly to the text by filling in the answers on thesheet but he has to do this without a richer context tohelp him. The exact comprehension of the text, as inall arithmetic problems. is vital for obtaining anaccurate answer. Earp suggests providing a richercontext by writing:

'The shape has been cut into equal partsand of the parts have been shaded.'

An example offered as an introduction to a worksheetsuch as this would also aid comprehension.

Thus it may be helpful if the tutor extends thecontext of the question to give the student moreopportunity to predict the meaning of unknown words.

For example: A loaf cost 65p yesterday. Today it isre -Weed by 2p. What will I pay?

If the student ; unfamiliar with the word reducedthere are no clues in the sentence to help him. Perhapsthe question could be extPnded to:

A loaf cost 65p yesterday. Today it ischeaper. It is reduced by 2p. I willpay 2p less than I did yesterday.What will I pay ?

The meaning of the unfamiliar word reduced canperhaps now be predicted by using the clues ofcheaper and less than.

Does the context in which a problem is set affectcomprehension?

Research by Rees and Bare concluded that using afamiliar context in arithmetic questions aided adultstudents. They posed two questions to a group of i

students whose ages ranged from 13 years to 57 yearsold. The first question was a pure arithmetic one, notset in context.Pure question Find the average of 15. 17, 18.25

and 15.75The second question required the same arithmetic

operation to solve it but the question was set in context.Context question The weekly wage of four teenagers

is £15. 4:17. .i!.1 8.25 and £15.75.What, is t he average weekly wage?

The results ist.)wn on the next page ) indicated that

ON, 5


having a question set in context improved theperformance of all students in the study. However, theimprovement was most noticeable in adult students.where there was a 14°,, increase in the success rate.

Results of the study:"0 Correct I

PURE 49°0 24°0 56° o

CONTEXT 54°0 31°0

AGE 13-14 13-14 19 - 57 I

mixedability

slowlearners

TOPS I

trainees

These results suggest that having a purpose.preferably a familiar one which the student can relateto the real world, improves his performance. Thisrelevance of purpose also, of course, applies to readingcomprehension which is improved if the student cansee that the reading task is purposeful. We should alsoconsider perhaps that an arithmetic problem which isdone for its own sake to be marked by the student ortutor as right or wrong may induce feelings ofpointlessness, anxiety and fear of failure. In the realworld we are less exposed to right and wrong situations.there are more grey areas. The tutor mav thus wish toconsider the motivation of the student when seekingrelevant contexts for questions. Students may wish touse their numeracy skills for employment purposes.such as ordering materials, checking invoices andcalculating VAT. Others may need help to extend theirnumeracy skills to enable them to understand recipes.instructions for beer making or other hobbies, as wellas the everyday tasks of shopping and moneymanagement. Whatever context the question is set inthe knowledge of the arithmetic principles needed tosolve it must be sound. The ability to transfer skillslearned to be used in many and varied contexts isperhaps what the tutor and student would togetherwish to develop.

How can the tutor help?Working with adults, we build on the student's

present knowledge and past experience. If a studentcan do a sum such as 8 + 3 this could be used as a basisof a problem construction task. If the answer to theproblem is known there is less anxiety in the situation.In reversing the problem solving situation, by askingthe student to actually compose the problem, thestudent will feel more secure. Moving from a wordproblem to a solution is a step into the unknown.fraught at every stage by the possibility of error.Asking the student to compose his own problem mayhelp to remove this fear of the unknown.

TASK 8 + 3Find a problem to match the sum. Think about:

getting a wage increasefood prices risingputting on weightbuying extra milk.

TASK 14 - 2Find a problem to match the sum. Think about:

losing weight

bus tares going downsale prices, everything reducedspending money, having less in your purse orwalletsmoking fewer cigarettes.

A conscious use of vocabulary which implies additionor subtraction may provide a useful discussion point.

If this strategy is used, the student, as the author ofthe problem, is then in the best possible position toread and interpret it. Writing texts aids ourunderstanding and analysis of the texts we read.Indeed at a fundamental literacy level a tutor mavmake use of this language experience approach in theteaching of reading. The method seems to lend itselfalso to composing, analysing and solving arithmeticproblems.

Pages of sums. although providing reinforcement ofnewly learned concepts. do not aid in the applicationof skills. A student who perhaps needs support withreading may be helped, after the initial concept hasbeen introduced and reinforced using numericalexamples only, by offering a written problem withlimited words. The problem could be presented in such I

a form that inaccurate reading will not affect thecorrect solution of the problem.

For example: student and tutor play the card gamepontoon. The numerical skills needed are addition andsubtraction. The purpose of the game is to have a handof cards totalling as near to 21 as possible. When thetutor is certain that the student is competent with thenumerical skills needed, a written worksheet of theformat below could be introduced.

WorksheetThese are the cards in my hand.What is my score? 8 3 4

Inaccuracy in reading the problem should not affectthe correct solving of it.

For the student who can read well but has difficultycomprehending the appropriate numerical skillsrequired to solve the problem other strategies may be

useful. A range of problems needing the samearithmetic skill to solve them, but expressed in avariety of ways, could be presented to the student.The student could then decide if. for example, additionor subtraction is needed without actually doing theproblem.

For example: To find an answer what do I need todo'?My bus fare costs 30p. It goes up by5p.

Talking through the problem with a tutor or anotherstudent may clarify the course of action necessary.The tutor may then wish to vary the approach andoffer questions with a choice of two skills.

For example: To find an answer do I need to add orsubtract?The petrol tank in my car holds 6gallons of petrol.It has 3 gallons in it now.How many gallons do I need to fill itup?

The tutor's role in developing and using techniques

6

Reading comprehension in written mathematics problems

to aid the student in interpreting the text and solvingthe problem seems to be crucial. A judicious use ofhobbies and life skills matched to the student'sinterests and needs can be invaluable, in thedevelopment of understanding and the application ofskills learnt, to the student's life.

When preparing worksheets, the tutor may alsowish to consider the effect which the layout of the text.the size of print and the position of diagrams may haveon comprehension. It is also clear that discussion andoral work should play a prominent part in establishingand reinforcing any concepts formed before they areapplied to a suitable and relevant context. The adultstudent will find it easier to solve a problem which hecan relate to his own experiences. It is also importantthat the mathematical concepts should be presentedin varying contexts to extend the versatility of thestudent's skills in problem solving.

The degree to which the syntax and vocabulary ofthe question can influence the problem outcomeshould also be considered when writing questions forstudents. The simplest, most straightforward form of

expression should be chosen and where possiblestudents should be encouraged to compose their ownq uestions. The differences between reading amathematical text and an English text are considerablein the extra demands the mathematical text makes onthe reader. The student's reading task is made moredifficult by the economy of writing in the text whichmakes prediction of unfamiliar words difficult. Themathematical text allows for no ambiguity anddemands an instant response from the reader in theform of solving the problem. We should be aware thatwe are asking so much more of a student than just thearithmetic calculation. Indeed once the problem isunderstood, its translation into the appropriatemathematical symbols and working out a solutionmay itself not be so straightforward.

For example: I have 60p. I give John half of it. Howmuch do we each have?

60

11.anslating2 60 ! of 60 60 ÷ 2

the written word into arithmeticsymbols is vet another major step for the student.

SummaryStudent DifficultyUnfamiliar wordsAmbiguity of meaning

:Tutor StrategyProvide stronger context cuesProvide stronger context cues

IDevelop understanding of mathematical registerDiscuss use of words in arithmetic problems

Technical words I Controlled introduction of new wordsMake a glossary of terms

I Repeat definitions; Verbal reminders! Marginal comments on worksheets! Examine semantic roots' Use new word in severai contexts

Confusion of word meaning ! Discuss compare everynay words with those words used in a special way in arithmeticChoose oral language carefully

Length of sentence Several short sentences preferable to one of high concept density

Order of words in a sentence

Grammatical complexity

I Begin the sentence with the subjectI Use the present tense if possibleI Avoid inferential statementsIAvoid use of subordinate and comparative clausesI Choose simple sentence structure

Lines of print too long I Line break the textInadequate signals for eye I Bolder question numbersmovements Use of colour, simple diagrams

I Diagrams positioned near to relevant textInadequate legibility I Increase size of type:writing

Increase spacing between words['se lower case where possible

('onfusion between typed digits Handwrite 1 's and l'sand lettersToo much print on the page : Vrite individual questions on cardsUnfamiliar:Inappropriate context I Give students initiative in choosing context inaxinuse motivation

i Give opportunity for practice in different contextsEnsure context relates to familiar everyday tasksEncourage students to write their own problems

Fear of failure Re-inforce understanding of numerical conceptIncrease confidence by using group work

I; Maximise enjoymentCreate a supportive. relaxed atmosphereUse anv mistakes constructively

I Build in self-checking devices


Bibliography1 Cockcroft. W.H. 1982'1 'Mathematics Counts'

H.M.S.O.

2 Perera. K. 198W 'The Assessment of LinguisticDifficulty in Reading Material' Education Review.32.2.

3 Riley. M.S. 1983'. 'The Development ofMathematical Thinking Ed. Ginsburg Academicl+ess.

4 Pimrn. D. .1987! 'Speaking Mathematically'Rout ledge, Kegan, Paul.

5 Kane, R.B. 197W 'Readability of MathematicsTextbooks Revisited' The Mathematics Thacher 63.579-81.

6 Earp ;1971. t1)roblems of Reading in School Maths'School Science and Mathematics.

7 Smith. Frank .1975.: 'Comprehension andLearning' Holt. Rinehart and Winston.

8 Rees 8.z Barr 1984 'Diagnosis and Prescription'Harper and Row.

8 1 0

Can ordinary people do real maths?

Can ordinary people do real maths?Joan O'Hagan

Joan O'Hagan has taught mathematics and numeracy in schools (very briefly) andin Further and Adult Education. She now teaches in Fircroft College of AdultEducation, Birmingham. Some students on the one year residential course aremaihs-phobic, and some need to reach GCSE-equivalence. Sometimes it's the sameperson. . . .

This question matters to me for several reasons:Queries are still raised both inside and outsideABE about the academic legitimacy of ABEmaths/numeracy ( Is working out what train tocatch to get to an ALBS': training event for 11amreal maths?)'Motors, perhaps especially those who have cometo adult numerac. .aths work through adultliteracy, }lave difficulty in reconciling what theysee as the need for mathematical rigour with thekind of adult-student-centred approach theytend to adopt How do you build on adults'previous learning experience when a great deal ofit has been negative? How do you build on theirprevious knowledge when a great deal of it isconfused? How do you avoid the trap ofconstantly telling students they've "got itwrong "? .

Some tutors, perhaps themselves unconfidentabout their own mathematical ability, areconcerned that they may be short-changingstudents by giving them access only to asecond-rate, hand-me-down type of mathematics- and these tutors may be painfully aware thatthese students may already have been and feltshort-changed by educators before in ways whichrelate to their gender, ethnic origin, or class.Some ABE students want to follow up theirinitial, generally numeracy-based work with abroader, richer mathematical diet - can we helpthem prepare for this by encouraging them to do"real" maths early on'? And some students want,from their first contact with adult provision, tohave this rich mathematical experience.

These issues arise in a climate where the debate inthe seventies and eighties about adult-centred processhas been largely overtaken by a debate aboutmeasurable outcomes: against a background whereincreasingly, nothing moves unless it's assessable -i.e. nothing gets funded unless it's assessable), in a

situation where ALBS(' has taken a strong lead onaccreditation with a scheme ( Numberpowen whichalso operates as a curriculum driver. We all need to beclear about what constitutes "good" or "real" or even"really useful" mathematics.

In this article I'm describing a line of thought inwhich I am tryine to weave together into a coherent

position what I've read about philosophy ofmathematics with what I've seen and heard whenadult students including myself) are learning anddoing mathematics. I am logging a personal learningjourney which I made in an ad hoc. fairly unstructuredmanner - I drifted into philosophy of mathematics. Istumbled across Lakatos, and then thought I could seeconnections with multiculturalist and feministepistemology. I know now that school teachers havebeen thinking along somewhat similar lines - PaulErnest (1991 and 1992 ) has written at length aboutsimilar questions as they arise there. The journey foradult tutors has, however, different signposts andrequires analysis of different material: so it has beenwell worth doing; but the process has reminded methat adult tutors are still too isolated from each otherand from school-teachers. I am very glad therefore, tobe able to write this article. and I'd like to echo DianaCoben's . 1992) call for a forum for adult numeracy/maths research.

I'm also aware that I'm not dealing directly or fullyhere with questions about how "basic- adult studentslearn maths. about :he psychology of the process. I ammainly interested here in whether their processes areto be taken seriously as potential sources of "real"mathematics.

Nor am I trying to have the last word on whetherstudents should be encouraged to stop doing"numeracy" and do "real" maths. I want to increasestudents' options, not curtail them. And anyway.maybe it's possible, perhaps even preferable, to use"real" mathematical thinking in a nurneracycurriculum.I'm reminded of a story told to me by my mother. Shereported a lively discussion among her friends aboutvalue for money in washing-up liquids (a classicNumberpower situation ). They were all willing toaccept that super-concentrated liquids wash moredishes than the cheaper ( per ml) varieties, and thatthe cost per dish worked out less with thesuper-concentrated ones. Therefore, said her friendstriumphantly, we should buy the super-concentratedones. Full Numberpower marks to them. My motherpointed out however that the way in which they, all ofthem doing small amounts of dish-washing at a time,consumed the liquids. was crucially different from theTy advert household. She sugge3ted that there is a

1 19


minimum amount of liquid which you use each time -function of the size of the hole in the lid and of thesqueeziness of the plastic bottle and she suspectedthat this minimum amount of the super-concentratedstuff was far more than was needed at each washing-upsession, whereas the minimum quantity of the ordinaryones was sufficient. Therefore, she hazarded, you were

; wasting the advantage of the super-concentratedliquids by in effect under-using them. Super-concentrated manufacturers !like mustard makers Iwere making money from the stuff you threw away. Itseems to me that she, without a number in sight. hadembarked on a good piece of mathematics: her friendswere doing good Numberpower numeracy. And clearlythe key consumer insight was the mathematical, notthe numeracy one

So what is "real" maths?Classical views (for thoughtful analyses of these

views see, for example. Davis and Hersh 1980 orErnest 19911 of the nature of maths tend to be veryprescriptive and absolutist, centring on the idea offormal proof as the key to both describing and doinglegitimate mathematics. Proofs are to be built up intounassailable fortress-like structures: the legitimatepublic language of mathematicians is about attackingand defending positions: and there are absolute truthsof mathematics waiting to be either discovered orinvented. And accounts of mathematics educationallied to these positions tend to be concerned witheducation as authority-based transmission of culture.To the ABE tutor, accustomed to encouraging studentsto build on. and to analyse and generalise from, theirshared experiences, these models offer little.

7)-uth through proof?A brief look at these absolutist positions highlights

the importance of the issue of proof as a test used byprofessionals of whether they are doing mathematicsat all or doing it well: and therefore as an issue whichadult students and educators need to look at.

A Platonist view holds that mathematical objectsare real - as real as any other ideas. They existindependently of the efforts of people to know aboutthem. The task of the mathematician is to know theseobjects: to understand for example the nature of space.

Euclid described his discoveries about one of thesereal mathematical entities - space as a process ofrelentless deduction of theorems from a number ofaxioms about the real world which he felt were soself-evident that they did not need to be justified.When exploration of alternatives to one of theseaxioms generated non-Euclidean geometries throughthe Euclidean method, the link between the Euclideanmethod and the truth value of its proven theoremswas bro'.en. Some mathematicians moved away frombelief in geometrical axioms as the basis for allmathematics, whilst holding onto the hope thatEuclidean method could be applied from a differentstarting point to the problem of underpinningmathematical thinking. Number was chosen as thenew basis, but this in turn proved to he problematic

when it became necessary to introduce irreconcilableideas about infini te sets. Frege tried to basemathematics on logic alone, but ran aground onRussell's paradoxes about self-referential sets. Russellthen worked with Whitehead to try to reconstitute settheory in such a way that it did not lead to suchparadoxes. Unfortunately, the system became sotortuous that Russell gave the project up "Havingconstructed an elephant upon which the mathematicalworld could rest. I found the elephant tottering, andproceeded to construct a tortoise to keep the elephantfrom falling. But the tortoise was no more secure thanthe elephant. and after some twenty years of veryarduous toil. I came to the conclusion that there wasnothing more that I could do in the way of makingmathematical knowledge indubitable.- !Russell.Portraits from Memory.. Yet another attempt to putmaths back onto a respectable basis was made byconstructivists. who started from the belief thatnatural numbers are given to us by fundamentalintuitions: and who say that real maths consists inusing only something called the finitary method onthese objects to create other objects no objects whichcannot be created by this process exist as validmathematical objects. This solves the problem of theinfinities more or less by pushing it off the stage.Hilbert's formalism. based on constructivism, alsotried to resolve the Russell antimonies by sidesteppingthe Platonism question with a declaration that mathswasn't about anything in particular. it was to bedefined as a method of manipulating ideas.Unfortunately Godel's theorem wrecked Hilbert'sprogramme by establishing that it was impossible toprove consistency within arithmetic without recourseto methods from outside the system.

For the non-professional mathematician, theseabsolutist positions give very little hope of admissionto the fraternity. t best, mathematics seems to bedo-able only by people with long rigorous years oftraining: at worst it is actually impossible. The adulteducator working with students at a basic stage, feelsdisinclined to encourage them to embark on activitywhich would lead, after twenty years of very arduoustoil, to disappointment. Perhaps we should, after all,encourage ABE students to narrow their horizons anddo "numeracy- instead of "maths"?

The failure so far of absolutist stances might on theother hand give the ABE worker grounds for optimism:at least she doesn't have to try to attain theunattainable absolute: though the optimism isshort- Nved if she is thus sucked into a suffocatingepisternological vacuum.

Are alternatives available'?

Some alternative epistemologies.

LakatosImre Lakatos's Lakatos 19761 attempt to resolve

the problems generated by adherence to absolutistviews about the nature of mathematics and of proofstarts from an acceptance that mathematics may notbe preservable as an infallible system, even in principle.

10 12

Can ordinary people do real maths?

and that this aim of philosophers of mathematicsis best abandoned. He doesn't feel though that this isa position which is limiting in any sense - on thecontrary he dislikes formalism because it "deniesthe status of mathematics to most of what has beencommonly understood to be mathematics, and cansay nothing of its growth" ; Lakatos, op.cit. p21.He was interested in recognising mathematics as acreative activity, engaged in by fallible people whomake observations about real or mathematicalobjects, guess a generalisation. work with otherpeople to try to falsify it in a Popperian sense, andbuild a proof G, the conjecture. He uses "proof-in a pre-Euclidean sense, as meaning a "thoughtexperiment which suggests a decomposition of theoriginal conjecture into subconjectures or lemmas,thus embedding it in a possibly quite distant bodyof knowledge- 'Lakatos, op.cit. p91 and suggeststhat a major function of attempts at proof-buildingis to help improve a conjecture even where, in factespecially where. no Euclidean "proof- of thatconjecture is found. He is prepared to give up theidea that proofs ultimately prove anything in favourof this idea that the process of proof-buildingimproves thinking. He maintains that by this processof publication and analysis the mathematicalknowledge geni.,-ated in the subjective domain of anindividual's im transforms and becomes objectiveknowledge.

Proof-building - style and functionThe process of proof-building involves the generation

of sub-conjectures or lemmas, and the subjection ofthem to a process of attempted refutation. But thisattempted Popperian falsification is a positive process:there is always the strong possibility that whilstsub-conjectures or lemmas may be refuted by localcounterexamples. and the main conjecture by globalcounterexamples. this may lead to the creation of newmathematics, or at least to a much deeper under-standing of the original conjecture. After all."Columbus did not reach India but he discoveredsomething interesting- 'Lakatos. op.cit. p141.Although the cultural relativists referred to later

might have something to say about the Euro-centricity of that remark ). An apparently fatalcounter-example may be seen either as a malignantmonstrosity, or it may be a wonderful opportunity toclarify and define the scope of the earlier thinking, tothink laterally, to open up new areas and ideas, tomove on from the first conjecture or problem to a moreenlightening one. Defensive formalist or logicistthinking on the other hand may lead the mathema-tician into retreating into an unnecessarily safeposition, throwing out the baby in an attempt to resistcontamination from polluted bathwater: or may leadto the sort of endless revision and refinement whichgets the thinker no closer to certainty, and behindwhich lies the realist nightmare that mathematicalobjects may be infinitely complex and thereforeincapable of description by finite minds working withfinite language.

Informal mathematicsLakatos' ideas about informal mathematics are

crucial here. He asserts that informal proofs are valid,that they are not just "formal proofs with gaps-: heshows us, ;Lakatos 1976) and comments extensivelyon. the genesis of an informal proof of Euler's [ V - E +F = 21theorem for polyhedra. This proof is arrived atby a group of students who take an initial conjectureand subject it to a process which allows them to usetheir intuition, which values and uses the dynamics ofidea generation within a group, which positivelyencourages an atmosphere in which speculations arefreely made, which uses peer scrutiny to deepen thegroup's understanding of the original conjecture andwhich expects and values "unexpected" outcomes inthat it generates new mathematics. This is a kind ofproof, a kind of activity, which, in my judgement.adult students find satisfying, because the prover isengaged in a process of the creation for herself of newknowledge and maintains ownership of the knowledgethroughout. This definition of mathematics, and thisheuristic, have a great deal to offer to adults, weariedas they often are by mathematical experiences whichdemand that they subscribe to "truths" of which theyfeel no ownership. And if this is mathematics, I wouldwish to encourage students to do it. and I would wishthem to gain credit for doing it.

Lakatos however doesn't give us much guidanceabout where the initial conjectures come from: I turnnow to related epistemologies which comment on thenature of mathematics and of proof from a sociologicalperspective.

Multicultural considerations

Proof in Vedic MathematicsAnother way of thinking about tests of the

acceptability of mathematical conclusions lies for mein my experience of the techniques of Vedicmathematics. Vedic mathematics is a system basedon the Vedas. Indian manuscripts dating from around1000 BC, and has been made accessible in this centuryby Sri Bharati Krsna Tirthaji (1965). I was learningin an ILEA adult education evening class taught by

Andrew Wright at The Polytechnic Of North London)Vedic algorithms for whole number multiplication anddivision, and I found after some practice that I wasseeing the answers to problems before I'd workedthrough the Vedic algorithm; as I checked my answersby going through the algorithm I came to rely moreand more on my initial judgement and less on thealgorithm, which became sometimes merely a secondway of checking my first. confident answer. i felt I wasbecoming intuitive about the answers to numericalproblems, and was surprised at this. My teacheragreed that surprise was a fairly common experienceamong Western trained people. I then found thatPratyagatmananda Saraswait lin his foreword toTirthaji's 1965, p.131 talks about conscious orunconscious mathematical thinking, and allows that"even some species of lower animals are by instinctgifted mathematicians: for example the migratory

1311


I bird which flies thousands of miles off from itsj nest-home, and after a period. unerringly returns". He

Ilater insists that when you are called upon to prove.j you must have recourse to logic, but he has

1

distinguished clearly between doing mathematics andproving mathematics it is legitimate to claim to

I know before you have proved, and indeed. "proving"i .I is not a necessary part of the mathematics; it's merelyi something that you may (or may not) be -called upon! to do". The suggestion is that this style of proof is

Iembarked on as a way of convincing sceptical western1

mathematicians of the validity of the methods,Ialthough the proponents do not feel this is always

, necessary for them.I I have found direct echoes of this issue when working

on Vedic mathematics with maths phobic adults. Theintroduction of one of the simplest Vedic algorithms hasinspired students to work together to create ideas abouthow to extend and develop the algorithm, to expressand test the ideas, to see mathematics as a human

I

artefact rather than as a set of rules which rain down1 on unfortunate students from some great epistemologi-

I

cal height, and to think about whether/when they hveI created or proved mathematical ideas.!

I

i RelativismThis perspective on cultural translation of ideas is

echoed by George Joseph (1992) where, tracing thedevelopment of Greek and Western mathematics fromIndian and Chinese origins, he talks about the claim

1

that mathematical proof began with the Greeks and!

that earlier maths can be dismissed. as Kline does, asI "scrawlings of children just learning to read" Kline.I

11962. quoted in Joseph op.cit.). He points to a

Idifferent kind of (inductive ) proof in Egyptian and

: Mesopotamian mathematics he argues that this. method of piling example on example was culturallyI more relevant and acceptable to those mathematiciansj and to the communities for whom they were creating1

, mathematics than a Western style deductive proof1

I would have been. Joseph, and later Bishop [1990. p51I,!

also suggest that mathematical ideas may be createdand should be understood within that context, andthat we should resist the easy temptation to getinvolved in crude comparisons across cultures andoppositional ways of deciding between ideas. Bishop,for example, talks of the cultural dependency of 1

Euclidean ideas on an atomistic and object-orientedview of space, ( points. lines, planes, sohds) in contrastto a Navajo idea of space as neither subdivided norobjectified and where everything is in motion. Thiscould be expected to produce a different kind ofgeometrical ideas. This kind of relativism, and thislack of trust in the Euclidean method opens up ways ofviewing mathematical ideas and methods which haveuntil recently been seen as outside the Westernparadigm.

Feminist epistemologiesMore strands in this piece of epistemological weaving

comes from feminist ideas which argue in various waysfor a form of relativism which is not however weakly

subjective. Griffiths j Griffiths and Whitford 1988, p9]argues that to separate feeling from knowing is to runthe risk of distorting scientific knowledge. I amreminded of how often adults are blocked becausealthough they may see, even subscribe to a classicalproof of an idea, they can't reconcile that with theirfeelings about it and so they don't really know orbelieve the idea. If the conflict is not resolved they arelikely to operate on their gut feelings rather than ontheir (to them) empty and unconvincing intellectualconclusions; and mathematical knowledge is thusdistorted. Seller (1988) suggests that an epistemologywhich has some democratic features offers the prospectto ordinary people of arriving at valid propositions ina finite time. She links this with an argument lop.cit.p182I that "the ultimate test of a realist's view is theiracceptance by a community", and maintains that "itis not that every individual's unexamined andundiscussed experience is true, much less her opinions,but it is only through examining and discussingindividuals' experiences that we can do what therealist called finding truth, what the relativist callscontributing to the structure of reality". Walkerdine1989, p37 ) suggests that the domination by male

thinkers of the mathematical stage is related to a viewof and a need for unassailable truth through proof;and that this is an unnecessary restriction. Theseinsights are useful in clarifying further the nature ofknowledge as something generated from individuals'subjective realities, which when mediated throughdiscussion, become objective knowledge.

There is another aspect to this discussion aboutgender which relates to the discussion about differentmathematical truths arising from different cultures.Work by Mary Harris (e.g. the Common Threadsexhibition ) and Claudia Zaslavsky ( 19731 suggestthat women and men arrive at the same mathematicaltruths, although by different, gendered. routes. This isquite different from what has been suggested earlierabout for example the nature of ideas about space indifferent cultures: and perhaps the question shouldremain open here. What the two perspectives have incommon is the idea that very differing experiencesmay lead by differing ways to mathematical truths.and that an appropriate learning style includes theexamination of all these experiences.

Where does this leave basic maths students?Students are Lakatosian when they are adven-turously engaged in investigational mathematics.They are Platonists when they are defending thestudy of maths against people who are moreinterested in. for example, literature - they saythat they are finding out true facts about theunive, se, which is more than can be said aboutstory-tellers or literary critics.They also resort to Platonism when the going getsrough - when they find abstraction difficult theyoften ask me what it's all for, insisting that it mustrelate to some piece of real knowledge about realthings in the real world, although they activelyenjoy abstraction when they're being successful at it.

1214

Can ordinary people do real maths?They are enthusiastic cultural relativists whenthey are working on Vedic mathematics, or Mayanor Igbo counting systems, or embroidery patterns.To the extent that they are confident of their ownjudgement, they are happy conventionalists -although they are often surprised to find themselvesin this state - and indeed I think of this as anindicator of confidence and of mathematicalmaturity.They sometimes "succumb" to formalism when, inthe first flush of relief at being able to handle "x"they don't want to use "x" for anything, they justwant to push it around a lot.Some unconfident students are heartily opposed tothe adversary method, feeling that they will neverbe able to venture any thoughts if they are to beimmediately criticised, however gently.Lakatosian-style discussion however, with itsrespect for everybody's ideas and its delight inunexpected outcomes, is a much more attractiveproposition. (Other students with about the sameamount of confidence welcome the security ofknowing that someone else, but it must be someonewhose academic authority they respect, willimmediately "put them right" if they have strayedfrom truth. I think this split happens in most adultlearning, and I feel that significant learning growthhas happened when students can choosecomfortably to operate in different styles atdifferent times).ABE students deciding in learning groups whetherto add or subtract numbers to solve a problem areconventionalist. mathematicians. They arefollowing a Lakatosian schema in which, by the"incessant improvement of guesses by speculationand criticism, by the logic of proofs and refutations".op.cit. p51. their mathematics is validated byagreement among the learning and practisingcommunity. By the same argument, students beingtrained by us to -do" word problems or to fill outcheque stubs are not. Study circles such as I haveseen operating at Walk In Numeracy and atFircroft, where students often carried outmathematical projects without appealing to outsideauthorities ( teacher or textbook ) for proof-approval, were mathematizing. I have often beencalled in by a group to discuss the second stage of apiece of work, where they were quite sure they haddealt with the first part satisfactorily although Ihad provided no solutions, they had not found anyfrom outside the group and the mathematics wasnew to them. If it be objected that they are tooinexperienced a group to legitimise their ownbehaviour. I need only point to the dilemma ofprofessional mathematicians who may be workingwithin equally limited groups-John Von Neumann;quoted in Davis and Hersh. 1980. p181 estimatedin the 1940s that a skilled mathematician mightknow ten percent of what was available. If we allowthat the professionals are mathematizing, we mustallow that ABE students are too.Lakatos and the relativists legitimize a practice

which can start with naive guessing althoughinductive guesses may be more fruitful. This meansthat a legitimate and very fruitful starting pointfor students can be their own guesses aboutmathematical objects. Better still is a combinationof such naive guesses with guesses which arise fromeven limited observations of special cases.Androgogical practice, where the experiences ofindividuals can legitimately be taken as startingpoints, and where part of the aim of the exercise isto take that experience, explore its meaning,generalise from it. and test that generalisation bycomparison with the experience of other people isthus validated.Students can feel entitled to move around a topic,following promising lines of thinking without feelingthat they ought to know where they're going beforethey get there ( this last is the stultifying trapwaiting for the Euclidean mathematician, who isrelatively uninterested in theorems which ariseincidentally on the way to the proof of a theoremwhich has been ascribed prior value). Students arespared the long and linear march tow? apre-ordained truth, since to mathematicize well, itis positively important to look down all theinteresting by-ways which appear. Mathematicalthinking can thus be student-led, can reflect theirinterests, their culture and their experience; andcan, by application of the creative process of proofsand refutations. improve the skills of the thinker.

And how do things look for the tutors?The tutor has a clear role in helping students toarrive at initial conjectures by drawing on previousexperience; the tutor is not put in the uncomfortableposition of constantly carping critically at thisexperience, since the relativistic epistemologiessuggest that knowledge and understanding willemerge from critical evaluation by students of thisexperience. The tutor's role here is to encouragestudents to express their current understanding ofideas, even if she feels that this understanding isincoherent, internally contradictory and just plain"wrong". Given a clear agreement in the learninggroup that all ideas are to be critically evaluated.tutors can escape from the pressure to follow theexpression of an "incorrect" idea by an immediatedenial or correction.The tutor's role then becomes that of the teacher inProofs and Refutations to chair the discussion, tobe willing to contribute insights from her ownexperience, to help students keep track of thediscussion, to be sufficiently experienced to makeconnections which students are not yet ready tosee, to be skilful in deciding when to suggest theseconnections to students: and very importantly, toprovide from that wider experience. Popperianfalsifiers against which students can test theirconjectures in a finite time scale.Ilitors cannot always of course hope to chair

high-minded or even entertaining( Lakatosiandiscussions, we often resort to absolutist heresies

1 513


themselves ( we get it absolutely wrong! ), and we haveto keep at least half an eye on the sumrnativeassessment around the corner. Students cannot alwayscourageously express and scrutinise ideas. So chat?Real mathematicians aren't consistent about theirmethods either being, according to Cohen andDieudonne [Davis and Hert'i 19801 realists onweekdays and formalists on Sundays.

For me as an adult numeracy/maths tutor, trying tomake a synthesis/analysis of the varied strands of myexperience, this initial exploration of a fallibilistphilosophy of mathematics and of mathematicseducation has been a foray into fertile territory.Cultural relativism, validation of informalmathematics, accreditation and sympathetic analysisof adults' prior learning, seeing students createmathematics before my very eyes what more could Iask for? And now I know it's real maths too?

References

! Bishop. Alan Western Mathematics: The Secret Weapon

of Cultural Imperialism. Race and Class. Volume 32.October-December. 1990, London. Institute of RaceRelations.

Coben, Diana ( 1992) What do we need to know! Issues innumeracy research, Adults Learning, N IACE. September

1992.

Davis. Philip J.. and Hersh. Reuben, The MathematwalExperience. Pelican, 1980.

Ernest, Paul (1991) The Philosophyof Mathematics Education,Basingstoke, Falmer Press.

Griffiths and Whitford ieds) {19881 Feminist Perspectwes inPhilosophy, Basingstoke. Macmillan Press.

Joseph, George Gheverghese, ,19921 The Crest Of ThePeacock, Penguin.

Kline. Morris. ,1962) Mathematics, A Cultural Approach,Addison Wesley.

Lakatos, Imre. .197(3) Proofs and Refutations The Logic of

Mathematical Discovery, Cambridge, Cambridge UniversityPress.

Anne Realism versus Relativism: Towards a PoliticallyAdequate Epistemology, Griffiths and Whitrod teds) 1988.

Tirthaji, Sri Bharati Krsna 1965. Vedic Mathematics. Delhi.Motilal Banarsidass.

\Valkerdine. Valerie 'compiler,. with The Girls and MathsUnit, The Institute Of Education. Counting Girls Out.

London, Virago.

Zaslavsky, Claudia. 19731 Africa Counts, Lawrence Hill.

I 4 I fi

Numeracy: A core skill or not?

Numeracy: A core skill or not?Dr George Barr

Dr George Barr is currently Assessment Quality Manager at City and Guilds. Hespent nine years at Brunel University as a Research Fellow researching thedifficulties students experience in learning mathematics, before being appointedto City and Guilds as Senior Research Officer to work on assessment developmentissues for NVQs, as well as curriculum development and assessment in numeracyand mathematics schemes.

Introduction

This paper addresses some of the issues involvedwhen looking at numeracy and trying to decidewhether it is a core skill or not. The issues that are seenas critical in making a decision on this question are thedefinition and specification of numeracy and thepotential for transfer of skill from one context toanother.

Numeracy together with communication skills, hasalways been seen as an essential ingredient in a coreskill framework. Many different core skill frameworkshave come and gone in the past few decades. TimOates (1992) provides a brief background andchronology to the current developments in core skillsin post 16 education and training and his previouswork gives an even more detailed history of core skilldevelopments (Oates, 1990). Historically numeracy isseen as an essential core skill. but of late it has beenrelegated to group two by the National CurriculumCouncil ( NC('. 1990 ) although more recently it hasregained its central status from the National Councilfor Vocational Qualifications NCVIQ

Numeracy is frequently seen as an area that peoplelack confidence in and find difficult to master.Employers frequently report that their employees areinnumerate and that education has failed the workforcein not preparing it for industry and commerce. Testingon adults frequently shows that some aspects ofnumber are found to be difficult by the population as a

I whole.The Cockcroft Report 1982). however, which

looked at, among other matters, the mathematicsrequired in further and higher education, employmentand adult life generally. stated:

'We have found little evidence that employersfind difficulty in recruiting young people whosemathematical capabilities are adequate.'para 46')

There is clearly a perception problem here.Employers may not find it difficult to recruit youngpeople who are mathematically capable ( presumablythis is interpreted as having the potential ), but thereare frequent complaints from employers about the

skills that young people have ) ie what they can do).Hence the core skills movement! The Cockcroft reportalso states that:

'Few subjects in the school curriculum are asimportant to the future of the nation asmathematics ...'

Numeracy is seen to relate to mathematics sincemany of the concepts and skills are part of the subjectand yet it is also seen as being something more thanmathematics. But how is numeracy defined?

Definitions of Numeracy

Numeracy has often been defined by a list ofsignificantly computational skills such as add andsubtract whole numbers. Sometimes numeracy is definedaround basic life skills as by the Adult Literacy andBasic Skills Unit ALBSU. 1993) in the numeracystandards for Numberpower. ALBSU define four skillareas, as:

handle cash or other financial transactionsr.ccurately using till, calculator or ready reckoneras necessarykeep records in numerical or graphical formmake and monitor schedules or budgets in orderto plan the use of time or moneycalculate lengths. areas, weights or volumesaccurately using appropriate tools. eg rulers,calculators. etc.

Sometimes numeracy is defined more qualitativelyas by the CBI (1989):

'Application of Numeracy understand, interpretand use effectively numerical information whetherin written or printed form: identify which form ofnumerical communication is most effective ( maps,flowcharts, models for a given situationdepending upon recipients: use numerical basedapproaches to solve problems.'

The National Curriculum Council 1990 ) has asimilar definition: they say:

'The following outline descriptions provide abasis for further development....

I 7 15


Numeracy includes the ability to:understand and interpret numerical datapresented in a range of forms:present numerical data in a range of formsappropriate to different purposes and audiences:select and apply numerical methods to problemsolving.

Students should be able to use numericalinformation, tackle quantitative problems anduse statistical analysis and presentation.' CoreSkills 16-19: A response to the Secretary of State,page 10).

The National Curriculum Council :1990) explains:'Definitions of core skills must be flexible toaccommodate changing needs.'

Building on the above we need to remember that theword 'numeracy' was introduced to partner the wordliteracy'. Conceptually both literacy and numeracyare subject to change over time. Further, both wordshave social definitions which wiil be based on people'sperceptions of the numeracy they think they use andwhat they think people should know.

There is significant research data to show thatpeople are not always aware of the numera.:v conceptsand skills that they employ daily leg the CockcroftReport. 1982). Consequently views on what numeracyconcepts and skills should be known are likely toexclude some things which are probably essential.

Investigations have been carried out on themathematical requirements of employment since themid-seventies using a number of different techniquessuch as job, skill and task analysis: observation ofwhat. employees do at work: and employers' trainingprogrammes. Clearly different aspects of numeracyunderpin performance in different areas of work.Further, the NC(' '1990) in its analysis of post 16requirements for A and As levels points out:

'Numeracy. ... cannot be fully developed inevery subject ... Core Skills 16-19: A responseto the Secretary of State. page 11: .

This is why the NCC put it in the second group ofcore skills, rather tnan the first. Jessup 1990 developsthe NCC argument further and goes on to explain:

'These /numeracy is included) are considered tobe of a different order in that they do notunderpin performance almost universally. ... Thisset of core skills could be described as highlyvaluable techniques or methods that have wideapplication and which it is desirable that allpeople should achieve a certain level ofcompetence.' Common Learning Outcomes: CoreSkills in A/AS levels and NVQ5, page 20).

So it becomes clear that is it not a simple task tocreate a precise definition of numeracy acceptable toeverybody. At the extreme there are two schools ofthought. One school of thought focuses on the basicskills of adult life, whilst the other focuses on theperceived needs of industry.

The N('vQ 11992) has created the core skill area'Numeracy applications of number' as part of thecore skills for National X'ocational QualificationsNvQs) and General Vocational Qualifications

,GNVQs and defined it by identifying the key themes )

of:

gathering and processing datarepresenting and tackling problemsinterpreting and presenting data.

They have then defined five levels around the threethemes and say that progression through the levels ismarked by increasing complexity in the nature of themathematical operations performed.

In this brief overview of definitions it is possibleto see that some are focused on daily activities tegALBSU . whereas others are more process oriented,eg NCVQ).

From these definitions assessable levels ofperformance have been specified.

Specifications of Numeracy standards

There is a wide variety of models that can be use fordescribing and specifying nurneracy skills. The FEL:

1979) adopted the following model:'We have decided to describe the curriculum ofthe common core we recommend in terms of achecklist of:

a those experiences from which we thinkstudents should have had the opportunityto learn

b the nature and level of performance wethink students should be expected toachieve. A Basis for Choice. para 51).

They went on to suggests:'In some areas of learning, mathematics forexample. it will doubtless be desirable to specifyan assessable level of performance which it isintended the student should reach and theexistence of such a specification will help towardspurposeful course planning and the currency ofthe qualification.' ) A Basis for Choice. para 551.

However, this still begs the question: how do youspecify an assessable level? Research at BrunelUniversity in the late seventies and eighties looked atthe mathematical requirements of people aged 14 +with a view to creating a hierarchy of concepts andskills based on qualitative and quantitative researchinto the difficulties students experience in learningmathematics. Such a hierarchy could provide aspecification fcr assessable levels. There are clearlymany ways of creating conceptual hierarchies:consequently the Brunel model is one of many. One ofthe aims of the Brunel research was to provide a modelfor specifying a structure far the numeracyrequirements in employment and life in general.

One of the products of the research at Brunel was aMathematics Curriculum Framework (see Barr. 19891which was designed to help curriculum developersappreciate the mathematical demands made onstudents when certain mathematical concepts andskills were specified in pre-vocational and vocationalsyllabuses. This framework was used, for example, inthe ,,12/. elopment of the City and Guilds Numeracyscheme from 1978 onwards) and the NI.13Sl

16 . Is

Numeracy: A core skill or not?

Numberpower standards in1989. In presenting Number-power a competence-based model was adopted and anNVQ style of presentation was used. An example of thispresentation taken from a stage one unit is given below:

UNIT 3 PLANNING THE USE OF TIME ANDMONEY IN EVERYDAY SITUATIONS

ELEMENT 1 Planning the use of available moneyRangetypes of money: pounds and pence (sterling;planning constraints: the plan should include at least

five but not more than twentyitems: the names of the items willbe given (eg food. gas. electricity.telephone. rent.... : total moneyavailable will be a fixed amountbetween £20 and S:2000weekly spending; a party: works'outing; wedding; holiday spend-ing money; petty cashcalculator, ready reckoner

example contexts:

optional aids:ASSESSMENT SCHEDULEIn at least two real or simulated situations of the typeillustrated in the range the candidate must demonstratethat he/she canPERFORMANCE CRITERIA1.1 perform all calculations accurately1.2 include all essential items1.3 keep within the fixed amount of money.

The NCVQ in developing its five levels for the numeracyunits based them around the three themes ( see above !and also wanted a close alignment with the NationalCurriculum. All the numeracy level 1 techniques wereextracted from National Curriculum statements ofattainment at levels 1-4: level 2 techniques are fromNational Curriculum level 5 & 6: level 3 techniquesare from level 7; level 4 techniques from levels 8 & 9and level 5 techniques from National Curriculum level10. The National Curriculum for mathematics is ahierarchical structure of concepts and skills brokenclown into ten levels. The NCVQ development alsobuilt on previous numeracy specifications.

The NCVQ model for a specification of numeracy hasbeen presented in an NVQ style of units, elements.performance criteria and range. It should also benoted that it is a 'criterion-referenced, outcome based'model, that is it presents a statement of a level ofcompetence to be demonstrated, it does not present ateaching or learning programme. An example of thestyle of presentation is given below.

Application of number level 22.1 Gather and process data using group 1 & 2

mathematical techniques1 techniques appropriate to the task are selected and

used by the individual2 activities required by the techniques are performed

to appropriate levels of precision and in the correct:4equence

3 data recorded in the required units4 data are recorded in the required format5 records are accurate and completeRangeGroup 1 techniques: converting between differentunits of measurement using tables, graphs and scales:making estimates based on familiar units of measurement.checking results: conducting a surveyGroup 2 techniques: drawing and constructing 31)objects; designing and using an observation sheet to collectdata: designing and using a questionnaire to survey opinion

Early feedback suggests that there are concernsabout the need for specialist tutors for both teachingand assessment purposes. It seems that non-numberspecialists lack confidence in assessing the skills. It istempting to ask how these skills can be 'essential toand embedded within performance by an individual'

, Oates 1992). page 34) when the assessors lackconfidence in assessing these skills? Of course the nextquestion to iollow is: Is numeracy a core skill when theassessors do not have the confidence to assess it? Oates(1992) suggests that the problem derives from the'state of the world' rather than a probtEm associatedexclusively with the units.

The two examples presented show differing views onthe presentation and specification of numeracy as acore skill. The first presentation is a 'life skill' modeland was based on a 'functional analysis' of thenumeracy used in work and non-work activities.whilst the second is a 'generic unit' model anchored tothe National Curriculum.

It is interesting to note that ALBSU (1993) hasbroadly mapped the Numberpower standards withthe National Curriculum ( NC and the NCVQ CoreSkill Units as:

Numberpower 1 NC Mathematics NCVQ Numeracy

Foundation 1 Levels 3 & 4 Level 1

Stage 1 Levels 3. 4, 5, 6 Levels 1 & 2

Stage 2 1 Levels 4. 5, 6. 7 Levels 2 & 3

The common aims of these specifications would beto prepare people for work and create opportunitiesfor personal development. Again it would appear thatit is possible to identify two schools of thought for thenurneracy specifications, one focusing on basic skillsand progressing from them ( a bottom up strategy),whilst the other appears to focus on a 'generic unit'which attempts to 'bridge the academic/vocationaldivide' a top down strategy).

It is anticipated that by providing a clear !

specification people are able to recognise their ownskills and prepare for future learning. However, islearning numeracy so simple?

Difficulties of learning numeracyconcepts and skills

There is a substantial body of research on thelearning difficulties. Ideally the specification should n

not include concepts and skills which are overburden-some for the people they are intended for. Further, theprogression through the specification should beorganised by increasing complexity in the nature ofthe numeracy concepts and skills through a series oflevels. This approach should help build peoples'confidence in their own ability and help preventconceptual blockages occurring.

Placing numeracy in contexts is frequently seen as ageneral panacea. The relevance of the numeracy to belearned is important but must not be regarded toosuperficially. An applicable context will be anappropriate context for some because it appeals to the

17


imagination ,such as. how would you invest/spend.£1 000 000?). whereas for others it may be a familiareveryday context (such as. working out how muchchange you should receive from 10). Whatever thecontext the learners' knowledge of numeracy shouldbe sound in order to be able to solve problems and alsoto learn further skills and techniques. There is a needto be aware both of the level of difficulty at which thenumeracy is 'pitched' and the level of familiarity/relevance that the context presents. This hasimplications for teaching, learning and the specificationof numeracy standards. In some senses this issueconcerns transfer.

Transfer

'11-ansfer is seen as the ability to apply concepts andskills learned in one context to another context.Research suggests that making the core skills explicithelps learners both to learn and to transfer their skillsmore readily. However. Oates 1992) points out that:

'Core skills offer the potential for enhancingtransfer of learning to new contexts, not least bymaking people more aware of the skills theypossess and of the skills required in differentcontexts. But there has been difficulty intranslating this widely held belief into effectivepractice' page 11).

In spite of evidence against it, the viability of theconcept of transfer remains largely unquestioned.From an assessment standpoint it is possible to askwhat appears to be the same numeracy question indifferent contexts which generate markedly differentperformance from candidates. Transfer is somethingnot to be assumed.

Conclusion

Some aspects of numeracy underpin our daily livesand on that basis it is fair to argue that numeracy is acore skill. Further, if the underlying numeracyconcepts and skills can be transferred from one

,context to anothel then the argument isstrengthened.

The main problem with nurneracy as a core skill isits specification and perceived relevance. It is easy toargue a case for the Nurnberpower model since thespecification is so clearly life skills related. It is not soeasy to argue a case for generic units [especially at thehigh levels) since they are clearly not relevant to a

significant proportion of the population. Further, if itis possible to sl-pw that people can satisfy theperformance criteria in the NVQ numeracy standards ,

at level 3, say, it is unlikely that if these people do notuse these skills and techniques on a regular basis thatthey will retain them on the future occasion that theydo need them which is clearly one of the desiredgoals. It is generally accepted that if skills are not usedon a regular basis then they deteriorate.

The middle ground in this argument would be thatthere should be an agreed minimal level of numeracy,which is part of the core, with free-standing genericunits for the occupational areas that demand a higherlevel than the accepted minimum.

References

.ALBst 1993) Basic skills support in colleges: assessing ,

the need', Adult Literacy and Basic Skills UnitALBS(' London

Barr G 1989) 'The development and application of a :

mathematics curriculum framework for studentsage 142'. PhD thesis. Brunel University

('BI (19891 'Towards a Skills Revolution', Confeder-ation of British Industry: London

Cockcroft Report 1982) 'Mathematics Counts' HerMajesty's Stationery Office. London

FEU .1979 ) 'A Basis for Choice', FEU, London

Jessup G 1990) 'Common Learning Outcomes: CoreSkills in A:AS levels and N NCVQ, London

N cc 1990) 'Core Skills 16-19: A response to theSecretary of State', N(T, York

Oates T :1990 ) 'General Competences, parts 1 - 4', ED,Sheffield

Oates T 1992) 'Developing and piloting the NCVQCore Skill Units' Report 16. NeVQ, London

The views expressed in this paper are the views ofDr George Barr and do not necessarily represent theviews of City and Guilds.

18

4

Sorting out statistics: Confessions of a numeracy tutor

Sorting out statistics: Confessions of a numeracy tutorSarah Oliver

Sarah Oliver is Basic Skills tutor at Raining for Skills, Kensington & Chelsea'sY.T. scheme. In the past she has been a numeracy, maths and IT tutor andco-ordinator in adult education. She moved into youth work after becomingincreasingly disenchanted with the prevailing ethos in education.

"Nos numerus sumus et fruges consumere nati.""We are just statistics born to consume resources." (Horace 65 8BC)

Statistics are inextricably associated with tedioustabulations and little grey men telling lies on television.The soullessness of our society in general and oureducation curricula in particular can be blamed largelyon the "stochastization of the world" L a world inwhich computers and qualifications have been raisedto the status of demigods. in which the word 'value'has been denuded of all ethical significance and inwhich the misconceived notion of 'norms' is theultimate authority dominating our lives.

J. do not intend in this paper to pretend that I findstatistics exciting. And that is the crux of my problem.If I were on the wavelength of Marilyn Frankenstein.if I were a radical mathematician. I wouldn't have aproblem. I would realise that although "statistics areoften mystifying, they can be empowering and criticaltools." But I neither feel empowered by nor endowedwith greater critical faculties as a result of my readingand interpretation of statistics. On the contrary. themore information that I assimilate, the more I amoverwhelmed by the enormity of information to beassimilated. How, then. do I set about teaching thissubject with the enthusiasm and integrity that is theessence of all successful communication?

This paper is written in an attempt to grapple withthe problem of how a teacher can approach a subjectfor which she initially feels little affinity but which isan obligatory part of the syllabus. And rightly so: forhowever much I might wish it were otherwise, anunderstanding of statistics is rapidly becoming thelynchpin upon which so-called 'numeracy' rests.

Changing numeracy needs in the ageof information

Although I have been teaching numeracy for adecade. I have to admit t at I have never been verycertain as to what exactly I was teaching, I am evenless clear today than I was ten years ago. A blindinglyobvious fact has only recently dawned on me: I don'tknow what I'm teaching because what I'm supposedto be teaching changes.

I t is inevitable that numeracy will always be animpossible subject to define for the simple reason thatdefinitions of numeracy vary according to the degree

of stochastization of society. In the sixteenth centuryarithmetic was a university degree subject. Today wejudge a person to be seriously impaired on thenumeracy front if he cannot perform the four rules.Perhaps we have got cleverer in the intervening fivehundred years? Given the condition of today's societythis is hardly a convincing explanation: more informedcertainly, but information is not at all the same thingas cleverness. It seems far more likely that numeracyneeds fluctuate and that, in an age in which the rate ofchange appears to have exploded exponentially,blitzing us with the future before we have assimilatedthe past or breathed in the present, the basic mathscurriculum and its exponent) suffers accordingly.Tomorrow's numeracy is not today's and yesterday'sworksheets had better be binned because they won'tbe any ui next week.

We live 'The Age of Information, a Golden Age ofDigits'. Our information is force-fed to us viacomputers. And computers, being less bright thancommonly supposed. can only handle digitalised data.It is hardly surprising that the information resultingfrom that data should have a distinctly numericflavour. Our brains are continuously bombarded witha "statistical blizzard that numbs the attention."Society. we are reliably informed by the pundits, isbetter informed than at any time in human history.These same pundits also inform us that educationstandards are sadly low and that we must all becomeeven better informed if we are to keep pace with themarch of mankind. Hence we cannot open a newspaperwithout being regaled by column inches of figures, wecannot select a school for our child without firststudying the latest league table of exam results and wecannot have a peculiar thought in our head withoutwondering whether we are more than a standarddeviation from the norm in urgent need of psychiatrichelp.

Statistical thinking pervades our society leaving usindividually isolated and impotent. The ever-increasing flow of information communicates less andless on a personal level. Even the statisticians havedrowned in their own mania for data collection. Thedeterioration of the ozone layer could have beendetected ten years earlier from existing data if anybody

21 19


had bothered to analyse it. But nobody did. Theywere too busy collecting more data. The overload ofdata has replaced ideas as the foundations upon whichour thinking rests and information masquerades asknowledge.

The tutor's problemThe numeracy tutor, along with rest of society, has

inevitably been swept along in the wake of periormanceindicators and market forces. Statistics dominate herprofessional life: her students are statistics, her examresults are statistics, her very existence in adulteducation is a statistic. And, as if that weren't enoughstatistics for anyone, she must be a purveyor oftabulations and pie charts to consumers who havelong since been "bludgeonedi into a dumbacquiescence" by the overwhelming authorityinvested in numbers.

I have with some diffic ilty come to accept that welive in a digitalised world and that we need to benumerate to cope with it. I am also aware that it is myjob tu communicate to my students that, far frombeing baffling rows of figures and slices of cake withpercentage signs in them, statistics are vital andempowering sources of information. I have thereforedutifully followed the rules of good practice as laid outin every '.utor's basic education guidebook. Surely.despite my negative feelings. if I follow the rules Ican't go wrong?

All I have to do is help the students pick a topic ofinterest to them: local authority spending plans. say,or the relative earnings of men and women. I canprovide relevant worksheets, instigate a healthy classdiscussion, dissect the figures with the aid of a piechart and enable the students to draw another piechart more to their liking. Hey presto. I haveintroduced my students to critical thinking,empowered them and ensured that they havecompleted one more element of their City & GuildsNumberpower unit all in one blow.

This is where my problem starts. Have the studentsreally achieved any of the intended goals? Or have I.rather, presented them with an inherently boringtask, given them the illusion of empowerment bypermitting them to air their views in a protected spaceand a tick in their Numberpower book with no moreguarantee that they will remember how to draw a piechart in six months time than they were able to recallhow to do fractions after they had left school? Have Inot sent them out onto the street to face theunpalatable truth that they are just as impotentagainst local authority spending cuts or inequitablepay scales as they were two hours pre Jiously? I cannothelp feeling that the student is no more empowered, nomore educated, and more likely than not reduced to astate of paralysed helplessness in the face of theoverwhelming odds against obtaining a job even witha Numberpower certificate pinned to the wall. Theonly conceivable result of those two hours in theclassroom is that a student's awareness of anunalterable reality has been raised. It is not always

sensible to ram reality down people's throats unlessyou can also equip them with the necessary tools forhandling that reality.

The root of my problem lies in the nature ofstatistics. Statistics are like electricity bills. They maybe frightfully relevant but they are also intrinsicallyboring and frighteningly unalterable. The truth aboutboth statistics and electricity bills is that the numerateindividual does not bother unduly about them. Shedoes what everyone else does pays her electricity ifshe can and complained if she can't. She does not readthe latest trade figures nor write to the local authoritybecause she disapproves of their pie chart breakdowns.Her empowerment comes from the knowledge that shecould interpret the figures if she so wished not from theinformation provided by the figures. And she did notget that sense of empowerment by studying statisticsor reading electricity bills. She got it from entirelydifferent sources: from learning a way of thinking,from connecting with ideas not data, from beingeducated in maths not informed by numbers.

20

The solution to the problemWhenever I have a problem I invariably turn to

George Polya's 'Mathematical Discovery' in the firmbelief that any problem, mathematical or otherwise.can either be solved with his guidance or will haveevaporated by the time I put. the book down. Polyalists 'Ten Commandments For Teachers'. His firstcommandment is: Be Interested In Your Subject."There is just one infallible teaching method; if theteacher is bored by his subject, his class will beinfallibly bored by it. If a subject has no interest foryou do not teach it. because you will not be able toteach it acceptably." In other words, since I have gotto teach statistics. I have got to get interested instatistics. This is no easy task for me. I have a seriousattitude problem towards statistics. I find them eithertoo dull or too frightening to spend much timecontemplating them. But I did once read an interestingstatistic: "Homosapiens took around 3.8 billion yearsto evolve. A further 700 million years was enough toinvent radio and 57 years later came nuclear weaponsand potential annihilation" (Independent Sep. 1990. )

These figures made a deep impression on me,although I'm not sure why. For some reason theyappeared to impress my students as well. It's abeginning. At least it has sparked my interest instatistics.

Polya's second commandment is: Know YourSubject. "No amount of interest or teaching methods,or whatever else will enable you to explain clearly apoint to your students that you do not understandclearly yourself."" This commandment seems at firstsight easy enough to obey. I am numerate: of course Iunderstand basic statistics. It's true that I get a littlelost when it comes to chi squares. distributions andstandard deviations but, fortunately. these are not onmy syllabus. However, when I examine myunderstanding more closely, I realise that, althoughon one level I understand basic statistics in that I

4.so


know how to 'do' them, on a deeper level I have no realunderstanding of the underlying ideas. I do not knowmy subject. I have the requisite information but Idon't have a firm grasp of the ideas that are thebedrock of that information.

I must make it clear at this point that I am notpropounding an argument for teaching these ideas tostudents. Ideas do not get taught; they get caught.And they only get caught by connecting withpre-existing ideas in the learner's head. But if theteacher does not have any ideas about her subject thenthe students have nothing to catch. They may wellleave the classroom a veritable mineful of information,an apparently desirable attribute nowadays, but theywill not leave any better equipped to think. Surely,even in the Information Age, the primary purpose ofall education must still be to teach people to think?

The solution to my problem is becoming clear. Myinterest in statistics has been stimulated. Now I mustdelve into some statistical ideas in the hope that theywill connect with what is already in my head andenable me to get to know my subject.

Big numbers"That which everywhere oppresses the practical

man is the greater number of things and events whichpass ceaselessly before him and the flow of which hecannot arrest. What he requires is the grasp of largenumbers" (Theodore Merz

My only interesting piece of data immediatelystrikes the reader with the enormity of number.However, if the student has no grasp of numberslarger than thousands and the tutor has no realunderstanding if the relative sizes of millions andbillions, the full implication of the figures is lost.

All of us. presumably, have taught big numbersthat is to say, we have enabled our students to readand write numbers of more than four figures. But howmany of us have thought about why we are teachingbig numbers, other than the obvious fact that they areon the syllabus and they are useful? And when have wewasted any time wondering whether he has a feel forthem?

I have been able to read and write big numbers formost of my life. I have only recently (thanks to J.APaulos learnt to get a feel for the relative size of amillion and a billion. ( A million seconds isapproximately 11 I days: a Hlion seconds is 32 years>.If I don't have any idea of the relationship betweennumbers, how can I expect my students to?

An understanding of big numbers is not merelyuseful, they have a relevance for us far beyond the factthat international debt, currency dealing and tradefigures are counted in billions. A grasp of big numbersis essential to our psychological wellbeing. In a worldin which we are surrounded by billions of beings verysimilar to ourselves, we need to be able to visualiseourselves in relation to a large number. Just as a graspof the relationship between 57,700 million and 3.8billion is essential to the understanding of my statistic,so a grasp of the relationship with the world. If anumber is outside our concept of number then it has no

meaning for us. Only when we have the concept ofnumber can we make the necessary adaptation to it:although we are very small (one actually), we are partof the whole. Rather than being squashed by size, wecan be aware of our own small but significant part. Wecan flit comfortably back and forth between the ideaof one and the idea of a billion, relating to both withsense of connectedness instead of alienation.

Exponentiality and rates of change"The rate at which the taste of Juicy Fruit gum

diminishes is exponential" (Paulos).A second idea implicit in my statistic is that of

exponential growth or, in this case, shrinkage.Exponential notation is not included in the basicmaths syllabus. It should be. It is not difficult to teachindex notation and once students have got to gripswith the chessboard problem ( Q: if you put two 2mmcoins in a pile on the first square, four on the second,eight on the third etc., how tall will your pile be on thesixty fourth square? A: reaching beyond the sun andalmost to the nearest stan . It puts social issues such asAids and population growth into perspective. We'reright to be a bit concerned. And if statisticians were tofind that the ozone hole was growing exponentially, itwould be time to stop getting informed and start to dosome thinking. People automatically tend to think ofgrowth and shrinkage happening at a constant or linearrate. I have no idea whether or not the banner ofprogress encourages mankind to march to an exponentialtune, but my statistic looks as though it may well be.

Averages"As I was going up the stairI met a man who wasn't thereHe wasn't there again todayI wish, I wish he'd stay away" . Hugh Mearnsi.Far too much of the statistical information presented

to us is based on averages. Social scientists, in theireagerness to achieve scientific respectability and reducehuman behaviour to quantities, seized on averages asa suitable mathematical device for their purposes.They then leapt from the maths of averaging to theidea of norms. The 'normal person' has become anarchetype towards which we are all meant to aspire.This archetype is a myth conjured up by statistics. Itis a dangerous myth. It encourages us to deny ourindividuality in a vain attempt to find security byfitting into a data-designed mould.

Averages are very useful for governments andbreakfast cereal manufacturers. Neither of theseparties is remotely interested in the individual. In theone case they are interested in crowd control and inthe other in how they can get that crowd to consumeas much as possible. I do not deny the usefulness ofaverages. "When many events are averaged, onearrives at stability, order and lawful behaviour" 7. I dodispute the relevance of the way they are presented onthe basic maths syllabus. I find it hard to believe thataverage citizens sit around working out their averageannual petrol consumption or their average householdexpenditure.

01.

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The interesting thing about averages is that theresulting figure is illusory. This is what makes themdangerous and it might be helpful for all of us if wewere a little more able to distinguish between anaverage and reality. There is no such thing as thenormal person therefore there is no point wishing to benormal. A norm is defined as an 'authoritativestandard'. But whose authority? "The CriterionMakers tell us that society should move so that suchand such a norm is optimised and they base policy onthis, but no one can say why the criterion itself isappropriate"8. Our twentieth century idea ofadjustment to the norm is governed only by numbers.Numbers are neutral symbols thus, at rock bottom,the idea of a norm is a sterile absurdity.

Probability"Probability is not just for mathematicians anymore

it permeates our lives" (Pau los).Probability crops up everywhere. There is an

advertisement around at the moment which states:"There's a 1 in 5 chance it could be you" I presumethis advert is designed to scare us into donating moneyto cancer research in the hope that a cure will be foundbefore we become that 1 in 5. It is objectionable onboth ethical and mathematical grounds. Firstly.because it plays on our anxieties rather than ourcompassion, our motive for donating would be fear notcharity. Secondly, in statistical terms, it is nonsense;quite apart from the fact that it should give usgrounds for rejoicing that we have an 800 0 chance ofnot dying of cancer good odds against. The figure hasbeen arrived at by blanket averaging without anyallowance for age, smoking habits, hereditary factorsetc. It is a meaningless statistic and is a classicexample of the abuse of figures in the manipulation ofthe innumerate.

A realistic understanding of probability can help usin our everyday lives. It can help us to rationalise ourirrational fears and take proper precautions when weare at high risk. If we examine the statistics we can seethat women's fear of being raped by a stranger, is outof all proportion to the actual risk of being raped by astranger, and their sense of false security whentravelling by car is astoundingly foolhardy. Theinnumerate person will take umbrage at the fact thattheir fear of rape is statistically uncalled for. They willsay, "Yes, but what if it does happen to me? Whatabout the Yorkshire Ripper?" One cannot helpsuspecting that many people satisfy their need to feelfear in such a way that it does not interfere with theirconvenience. Cars are much too convenient to beoverly concerned by accident statistics whereaswalking on a cold windy night is distinctlyuncomfortable. A trivial but true consequence of thisattitude is that it substantially increases the chancesof dying of circulatory disease through lack of exercise.

Every day of our lives we are confronted byuncertainty and governed by the laws of chance. Yetall too often, as a result of information overload, wecan be made unnecessarily fearful because of ourinability to determine rationally the risk of what we

do. It is almost unbelievable that something so centralto our lives should be missing from the basic mathssyllabus, that the arbiters of numeracy should havedecided that pie charts take precedence overprobability. Even ak GCSE level problems ofprobability are confined to absurd questions aboutcoloured balls in bags. There has been a lot of fine talkover the past decade about relevant and meaningfulmathematical activity. It is time for the talk to bereflected in the curriculum. What could be morerelevant than the maths of risk?

Generality"A single death is a tragedy; a million is a statistic"

(Stalin).The ability to generalise while retaining meaning is

at the heart of mathematical thinking. When wegeneralise we are either looking for connecting patternsin apparently shapeless heaps of facts or we areattempting to create a pattern out of very few facts. Itis through the process of generalisation that data canbe used to illuminate an idea or can help us predict anoutcome.

Innumerate people tend to feel ill at ease withgeneralisation. They fear that their individual treewill be suffocated by the wood. Numerate people, onthe other hand, either innately or through soundrnathemati :al education, learn to leap from aparticular tree. These mental gymnastics, the art ofspecialising and generalising, are involved in allmathematical activity. The more consciousness ofthis that there is in the classroom the better. Statisticalinformation is an ideal tool for discussing ideas ofgenerality in a real context, since statistics, by theirnature, lead to generalisations.

The art of differentiating between personalisationand generalisation is not only an essentialmathematical skill, it is psychologically important ina mass culture. - Innumerate people have a strongtendency to personalise - to be misled by their ownexperiences, or by the media's focus on individuals ordrama"9 with the concomitant danger that they cantake a tragic view of their own life while maintaining aStalinist indifference towards mass disaster. Aneducation in generality is essential if we are tomaintain any vestige of humanity on our over-populated planet. In a world in which, increasingly,any of us could be anybody, a grasp of the generality ofstatistics in connection with the particularity ofindividual behaviour is vital to our self understanding.

Ideas not informationIn the process of writing this paper I have become

aware that I have a lot to learn about statistics. I needa deeper feel for the difference between arithmeticaland geometric progressions in order to understandexponential growth; I need to understand distributioncurves in order to clarify the process of averaging; Ineed to be aware of variables and false correlationsand the difference between practical and statisticalsignificance. I need all this in order to teach students

22 24


how to interpret a pie chart. Only when I haveempowered myself with ideas will I be able to empowermy students. The information that I have at mydisposal is only meaningful if it backs up an idea.Students are not sausages to be stuffed with a syllabusand turned out as a satisfactory product. They canonly learn successfully if the information they receiveis based on ideas that make connections. And now Ibegin to feel in tune with Marilyn Frankenstein:"There are many levels on which we have to fight tomake our individual and ,:ollective lives meaning-ful knowledge of basic numerical and statisticalconcepts is important in these struggles."'"

References

Philip J. Davis and Reuben Hersh (1986) DescartesDream, 1,7,8.

Marilyn Frankenstein (1989) Relearning Mathematics.A Different Third R Radical Maths, 2.10.

Theodore Roszak(1986) The Cult Of Information, 3.

John Allen Pau los (1988) Innmeracy. MathematicalIlliteracy and its Consequences, 4.9.

George Polya (1962) Mathematical Discovery. OnUnderstanding, Learning and Teaching Problem Solving,5,6.

Other SourcesJohn Allen Pau los (1991) Beyond Numeracy

Morris Kline (1953) Mathematics In Western Culture

2523


Learning Contracts in Higher Education: Towards confidencein developing numeracy skillsIan Beveridge & Gordon Weller

Ian Beveridge is based at the Centre for Education & Development at Luton Collegeof Higher Education where he is responsible for organising the Maths LearningCentre. He has worked at the University of Minnesota where he was involved witha developmental maths project and the running of the Maths Learning Centre.Gordon Weller is based in the Enterprise Una at Luton College of HigherEducation and is engaged in a college-wide introduction of the Ente:riseLearning Contract. He has experience of working as a tutor-counsellor with theOpen University, Faculty of Technology.

Introduction and Background

In this report we intend to explain how newdevelopments in numeracy-related teaching supportat Luton College of Higher Education can break downtraditional barriers to developing numeracy skills.This study is currently in its preliminary stages, andbase line data that has been used in this paper, willform the basis for more in depth research using dataaccumulated throughout the first full year of operation.

Over the past decade, higher education has slowlyopened its doors to those who have traditionally beendisadvantaged through lack of G.C.E. qualifications.Today, many Higher Education Institutions areencouraging adult learners to prepare for entry tovarious degree level courses. It is possible for anyonewith little or no formal qualification, but who has pastexperience which can be taken into consideration, tobe enrolled on to a one year Access course, or indeed.directly on to a degree course ( Handy 1985).

Luton College of Higher Education has, over theyears. developed a reputation for being a good 'accesscollege'. A large proportion of 'mature' students areenrolled on to 'access' courses, and are encouraged topursue higher education to degree level. Theassessment for enrolment is dependent upon the pastacademic and work experience background of thestudent. The student has the opportunity to questiona course manager on the content and level of thecourse which is of interest to them. In many cases'mature' students, particularly women, retainunpleasant memories of their last period of formaleducation at school, up to the age of 15 or 16Stanworth 1984).

Enterprise in Higher EducationThe needs of adults returning to education may be

very different from students who enter higher

education straight from school or college. However.there is a common need for ongoing motivation andself development. In this respect Luton has recognisedthe vocational nature of most of the courses offeredwithin the college, and has sought the help of theEmployment Department and support from local andnational employers. This aid came in the form of theEmployment Department funded Enterprise in HigherEducation EHE) programme. which Luton wasawarded in July 1991.

From the start of the programme employers havecontributed in cash and kind to add to E -nploymentDepartment funding to achieve the EHE objectives ofLuton College of Higher Education, which are brieflyoutlined below:

1. To give students the opportunity to participatein a module to develop job seeking skills. Thiscovers self awareness, sources of information andthe employer market. CY's and application forms.interviews and methods of selection.

9. Changes in the curriculum to foster self-managedand student centred learning, for examplethrough the use of case studies or projects.

3. Closer involvement with employers in coursedesign and development.

4. A commitment to providing some real worldexperience for all students while at Luton Co:legeof Higher Education. This may be through workplacement, work shadowing, work based projectsor structured company visits.

5. A commitment to staff development throughemployer led workshops and seminars, staffwork placement for updating and practicalexperience and consultancy.

6. A Personal Development Portfolio for students.in which they record and assess their skillsdevelopment with a view to promoting continuousself development. This portfolio will also formthe basis for a comprehensive CY which can be

24 26

Learning contracts in higher education: Towards confidence in developing numeracy skills

supported by evidence of skills if necessary.7. A Student Enterprise Committee, organised in

collaboration with the Student Union in order topromote an awareness of skills development bothwithin and outside college based activities.

8. Students are encouraged to undertake anEnterprise Learning Contract for skillsdevelopment. They assess their own skills andnegotiate with tutors on how they will developand improve in target areas. Where studentsparticipate in a wnrk placement the negotiationis three way; tutor, employer and student.

(Crystal 1992)

A Learning Contract for NumeracyConfidence

It is this last objective of Enterprise in HigherEducation that is being used as a change agent forLuton students to develop transferable skills. One ofthe most important transferable skills recommendedby employers, as necessary in. virtually all aspects ofemployment, is competence in numeracy. Indeed arecent survey of 220 new graduates by employersfound that "a sizable minority have difficulties withand are nervous about numeracy tests set byrecruiters" (Cornelius 1992).

With this in mind every effort is now being made toencourage all students to target numeracy skills usingthe Enterprise Learning Contract see Appendix 1).This contract is owned by the student and is personalto them. .t is not marked or graded. and is intended tobe a genuine opportunity for active participation inskills development. The student works with a tutor innegotiating how s/he will achieve a certain skill, andhow evidence of this will be provided. The timescale isalso negotiated and, where skills development involvescontact with an outside organisation. the student isencouraged to discuss their skill targets with theemployer.

Learning Contracts can be understood as the efficientand effective use of tutor and student time, whichresults in clear student objectives being set, thusgiving a framework for discussion. Earwaker t 1992 a'notes the importance of effective tutoring in educationinstitutions with ever increasing student numbers.

"Our students need our help and in future theyare going to need it even more. We can respond byincreasing the quality if not the quantity oftutorial time. The mark of that quality will be inthe businesslike way we establish the kind ofproductive, working relationships they areentitled to expect" (p.18 Earwaker 1992 b

There are many books and manuals available totutors suggesting ways in which students can developtransferable skills doing useful and practical tasks.Paul Kearney's Mauling Through Enterprise t 1991) is agood example. Further supporting information forstudents, tutors, and employers, on developingtransferable skills using the Enterprise LearningContract, has also been produced by the EnterpriseUnit at Luton College of Higher Education (Crystal

et al 1992). The concept of the Learning Contract isnot new, although when initially conceived in theU.S.A, it was intended to help research students innegotiating research objectives with their supervisor(Knowles 1978). Currently many Higher Educationinstitutions, and some U.K. employers are promotingthe benefits of Learning C ontracts in skillsdevelopment (Higher Education Developments 1991).

The Enterprise Learning Contract (ELC), asdeveloped at Luton College of Higher Education, isparticularly useful for nurneracy skills development.The student can use the ELC for targeting any numberof transferable skills, although it is recommended thatbetween 3 5 skills are worked on for each year atcollege. This program is open to all Luton students infull or part-time education and was first introduced inJanuary 1992, involving some three hundred studentsacross the college. All students who complete one ormore skill objectives are eligible for the Enterprise inHigher Education Certificate of Achievement which isendorsed by employer partners of EHE at Luton.

Support for students who wish to target anynumeracy related skill as part of their ELC will beprovided by a new Maths Learning Centre, which willinitially be staffed by tutors. The eventual aim is totrain and develop student tutors who will be able towork in the Maths Learning Centre with clientstudents and negotiate numeracy skills achievementusing the ELC. A similar system has been developed atthe University of Minnesota. General College, wheremore advanced students were trained as numeracyinstructors (Beveridge 1992, unpublished).

The Job of a Maths Tutor (student tutor)

A maths tutor is someone with respectively goodlistening, counselling and, eventually, numeracy skills.The customers of the Maths Learning Centre (mLC)will include students with considerable anxiety aboutmathematics and statistics. It is not the job of mathstutors to do the work for students who come to theMLC for help. It is their job to listen and understandthe problem of the students at its most particular leveland then to advance the student forward in his or hertask and then return to find out how muchprogress has been made. There will be peak loadsituations around exam times which will requiretutors to increase their pace and make decisions toservice all the extra students. Ilitors will also ensureall users of the MLC are signed in and assign exams andother work the MLC is keeping for particuler mathsclasses. Where arranged, they will grade and givefeedback immediately after testing.

The skills cited in the Enterprise Learning Contractthat are exercised in the job of Maths tutor includeskills in all six areas: presentation skills includeself-presentation, self-management skills includetime-management, study skills, assertiveness.awareness of resources around the college andprioritising. Communication skills will be of greatimportance as development of good oral communica-tion skills can increase effectiveness, and put the client

25

27


student at ..ase with the student tutor. ComputerAided Learning mathematics study packages arebeing introduced and will entail computer literacyskills on the part of the student tutor, and the clientstudent. Interpersonal skills development is a highpriority and weekly review sessions for student tutorswill focus on this vital aspect of the work. Numeracyskills will sharpen both as students think aboutmathematical problems in their tutoring time and asthey realise the patterns of errors do not change fromsemester to semester. The general skills we are mostconcerned about as we satisfy many Access students isa sensitivity to Equal Opportunities. Customer care,graphics, assertiveness and interviewing skills will beworked on continuously and in weekly review sessionswith the MLC manager.

Maths Learning Centres - Catalysts formore Equal Opportunities?

Maths Anxiety was a term coined by Sheila Tobiasin her workshops for women students during the early1970's in the New York area (Tobias 1978. ) It refers tothe stress-induced behaviours which result from yearsof feeling inadequate during maths at school. Inparticular it affected women and the various ethnicgroups that live in the inner cities where the mostoppressive teaching of maths occurred. Theperformance of the same groups were increasinglydivergent from that of the white male population . Forthe U.K. we need only look at the absence of womenengineering students to know the same situationapplies here. When examining the causes of thissituation, we enter the debate of "nature versusnurture- lying behind our revealed abilities. ElizabethFennema, among others, has documented theworsening relative performance of women atter theage of 15. She suggests there is more acceptance ofwomen dropping maths as they are not expected toneed it much in later life Fennerna 1976. )

The Access programme attracts a lot of singleparent mothers, housewives with older children andnon-mainstream groups. Dealing with maths is oftena key issue in surviving higher education for manystudents. There is a need for a fresh approach inlearning mathematics, one which does not restimulateold fears and insecurities. Maths courses must makesense, allow students to work more at their own pace.and pursue personal interests and needs. However,students cannot be expected to become independentlearners of maths overnight since the skills involvedare not widely knowr .

The Maths Learning CentreRenaissance in student numeracysupport

A Mathematics Learning Centre is a place wherestudents can go and get help in learning maths in asupportive environment and where students are dealtwith on a one-to-one basis. Its primary resources aretutors who are non-judgmental and expert problem

solvers. The training for such tutors focuses mainly on i

the counselling techniques that make it safe forstudents to reveal n .ore of their own good thinking.Students' ideas are never "wrong" and all ideas aredevelopmental. Any idea has the possibility of beingdeveloped. Naive ideas can be tested against situationswhere they must be redefined rather than taken awayand replaced by someone else's ideas. The adage,"From what to what", is the developmental goal ofthe tutors. They must find out how students think ;

about the problem as the starting point of anydialogue. Handling feelings is basic in problem-solving ;

and tutors have to give students room to allow their i

feelings to surface. The maths learning centre is notlike a regular classroom, nor should it be.

Higher education will have expanded from 70,3 ofthe 18 year old population in 1980 to 30°,3 of thepopulation by the year 2000. official targets are met.Currently, over 200 of the 18 year old population is inhigher education and we are likely to meet the targetbefore t he year 2000. Past assumptions of homogeneousgroups taught in similar ways at -0" and "A" Levelare now irrelevant. More students need maths thanbefore and the standards are increasingly diverse asare the students. The learning style of lecture andtutorial is both expensive and inefficient. It is time tomake radical changes in the way maths is taught andwe must effect these changes with existing teachersexcited about them.

Maths learning centres can catalyse such changes.They support the work of existing teachers by theirremedial work. They offer students the chance ofindependent but supported study to complete mathsrequirements. They offer good lecturers a resource tosupport students through more interesting open-endedquestions to continue the discovery maths of the(;csE's best aspect, its project work.

The central quality of this agent for change is thatall tutors are charged with listening to students andunderstanding their thinking on a one-to-one basis.Emotional needs are recognised. 'Ilitors work hard atlearning how a student thinks about a problem inorder to advance the situation in the directiondetermined by the student. This is respectful, studentcentred, independent learning but students aresupported. Many problems will not get "solved" thisway, but students are more likely to develop theirthinking when not "told the answer".

Learning to think Mathematically:some examples

Promotion of the Maths Learning Centre t MLC 1 hasinvolved short presentations to various groups ofstudents on courses throughout the college. Theobjective of these presentations was to encouragestudents to discuss mathematical problems, and offerthe opportunity to further their interest at the mu'.The following is an account of one such presentation:

Take 1 minute with a pencil and paper and consider thisquestion: what concentration of an acid solution resultswhen .ou mix a 4000 with a 10" 0 solution? After you have

26 9 S

Learning contracts in higher education: Towards confidence in developing numeracy skills

written down what you consider to be a believable answer,take 5 minutes to tell each person sitting next to you what itis that makes your answer reasonable. We are anxious thatyou and your neighbour can agree on an answer or, (ailing.that, that you are able to explain your neighbour'sreasoning to the rest of us here.

About one-half of all students decided to agree thata 5000 solution resulted with the mixture, one thirdthought a 30°0 solution was produced while theremaining one-sixth mostly agreed on a 250 0 solution.

Take another minute with another ptece of paper andconsider what temperature results in your bath when youmix 40°C hot water with 10°C cold water? Once again wewant you to agree on a reasonable answer with yourneighbours in the following 5 minutes. Alternatively youmay choose to show respect for your neighbour by learningthis other explanation well enough to explain it clearly tothe rest of us.

A resiliant few students thought a 50°C bathresulted and one-half decided the bath temperaturewould be 30°C. Another third chose 25°C as theiranswer. The remaining sixth were perplexed andtentatively wondered how we could know thetemperature, not knowing if more hot or more coldwater was used in the bath.

Each group who were asked this pair of questionswere equally divided as to whether they were the sameor different questions. There was also division as towhether the bath temperature or the acid mixing waseasier to think about.

They were asked what was similar between them.The numbers were the same and there were twoquantities in each problem. Both problems involvedthe same process of mixing similar things. Both askedto find a result. What was different about these twoproblems? They involved different things, acid andbath-water, percents and temperatures. Bath-water isless than 100°C and acid can be 100°0 ;and somethought more than 100°0). One problem is unfamiliarand another is quite familiar.

.1 large ice-cube of 0°C is dropped into a cup of stronghot cof fee of 40°C. What is the likely temperature and whatis the strength of the coffee by the tune the ice-cube is finallymelted? Take one mutute and write down your thoughts asto the consequences of the we-cube for the coffee. Onceagain get an agreement with your neighbour if you can.Alternatwely make sure you understand a different answerand get the neighbour to understand your answer wellenough to explain 11 sensibly to the rest of us. Practicesummarising your netghbour's thinking until it is agreedthat it is accurate.

Just a few students held on to the strategy ofsubtracting the numbers to get a result. Thediscussions began to get longer. Demands to know"the answers" to the questions began to grow. Aconsiderable amount of irritability was expressed atthe back of the auditorium at the vagueness of thequestions and the lack of "answers" as feedback.Those students who decided the questions lackedsufficient information, in particular about thequantiti.:s of the ingredients of the mixes grew to

about one-third in some groups. Along with the 1

irritability, there were animated discussionsthroughout the adjoining rooms.

Analysis of students visiting theMaths Learning Centre during its firstfull week

Currently the Maths Learning Centre at LutonCollege of Higher Education has had an impressiveresponse from students, and wide acceptance of theobjectives from college lecturing staff. Statistics fromthe first week of operation are shown below.

The Maths Learning Centre is open Thesday,Wednesday and Thursday from 12:00 noon to7:00 pm. Students were asked to sign in as theyentered the Study Centre. They wrote down theirnames, their course and year. the time they enteredthe room and the help they needed. They were askedto sign out too but most forgot to do this. The data onthe year of the students was also incomplete.

Student Visits Once TwiceNo. Students 61 10

Student Backgrounds Year 0(Access)

No. Students 31

CoursesServiced Health/ Design/Soc. Tech

3 times 4 times2 1

Year 1 + Year 1 +(HND) (BA.BSc)

19 26

App Business AccessSt i. (various)

No. Students 11 31 3 11 31

Time of Entry 10-12 12-1 1-2 2-3 3-4 4-5 5-7No. Students 5 17 20 0 13 9 11

Type of Help Required

No. Students

General Study Specific MathsSkills Skills

8 67

The future of this studyThe above figures give some credence to the need for

such student support in the general area of numeracyand, with increased promotion of this facility, it willbe possible to monitor the type and frequency oftypical numeracy problems. This information canthen be made available to tutors, and may result inchailges of teaching material or style. Only throughongoing evaluation can a combination of studentLearning Contracts and the Maths Learning Centrebe assessed in terms of effectiveness.

Note:If you would like to know more about EHE orparticipate in developments, please contact theEnterprise Unit at Luton College of Higher Education,Park Square, Luton ,05821 489246.

References

Beveridge, I. 1992 The M innesota Model ofDevelopmental faths. Unpublished Paper.

Cornelius. M. ;1991 . Unwersity Numeracy Project.University of Durham.

2729


Crystal, L. (1992). Enterprise in Higher Education.Local Economy Quarterly, vol 1, issue 3. October1992.

Crystal, L. et al (1992). Enterprise Learning ContractHandbook. Luton College of Higher Education.

Earwaker, J. (1992 a). Help ing and Supporting Students.Open University Press.

Earwaker, J. (1992 b). A Relationship That Works: onthe importance of the tutoring role. The Times Higher.30th October 1992, p.18.

Employment Department Group k1990). HigherEducation Developments The Skills Link. CrownCopyright, November 1990.

Fennema, E. (1976). Influences of Selected Cognitive..4ffective, and Educational Variables on Sex-RelatedDifferences in Mathematics and Learning and Studying.A report prepared with the assistance of Mary AnnKonsin for the National Institute of EducationU.S.A.). Grant P-76-0274. October 1976.

Handy, C. i1985). The Future of Work. Basil BlackwellLtd.

Kearney, P. (1991). 7)-aining Through Enterprise: APractical introduction using Enterprise Briefs. ArtemisPublishing.

Knowles, M. (1976). Using Learning Contracts.Jossey-Bass Publishing.

Stanworth. M. (1984). Girls on the Margins: A Study ofGender Divisions in the Classroom. In Hargreaves, AWoods, P. (Eds), Classrooms Staffrooms: The Sociologyof Teachers Teaching. Open University Press.

Tobias, S. (1978). Overcoming Math Anxiety. WW.Norton Co.

Note:If you would like a copy of the EnterpriseLearning Contract Handbook and the unpub-lished paper by Ian Beveridge, please contactthe Enterprise Unit on (0582) 489246, a charge of/5 will be made to cover printing and postage.

28 3(1

Effective provision of literacy and numeracy instruction for long-term unemployed persons

Effective provision of literacy and numeracy instruction forlong-term unemployed personsDr Joy Cumming

Joy Cumming is a Senior Lecturer in the Faculty of Education, Griffith Universityof Australia. A former secondary teacher of English and Mathematics andlongtime researcher, she has undertaken research projects on adult literacy andnumeracy at the local, state and national levels.

Introduction

In Australia, as in many countries around theworld, funding for provision of adult literacy andnumeracy programs has been considerably increasedin the last few years. Much of this increase is due to thesuccess of public awareness campaigns whichculminated in International Literacy Year in 1991.However, much of the increase is also due to theeconomic rationalism and human capital models whichhave dominated government planning since the 1980's.Funding for programs in adult literacy and numeracyis seen to be desirable not from the perspective ofenhancing people's quality of life and empowermentbut for the purpose of enhancing the productivity ofthe nation. The changing nature of employment isbelieved to place more demand on literacy andnumeracy skills of workers and the ability to bemultiskilled in work performed. In Australia. themost often quoted figure for the loss of productivitydue to poor literacy and numeracy is 83.2 billionyearly, a substantial amount relative to our grossnational product.

At the same time, the increase in unemploymentlevels in Australia, as in many other nations, hasmeant an increase in expenditure on training programsfor the unemployed. In 1989-90. when the followingproject took place, the Federal Government alreadyexpended $100 million dollars on training programsfor the unemployed through a scheme known asJobTl'ain alone. Again, the government perceived thatnational productivity and employability would beincreased by the improvement of literacy andnurneracy skills of those who were unemployed,particularly the long-term unemployed.

It is important for all stakeholders in these programsparticipants, staff, employers, government and

taxpayers that such programs are as effective aspossible. This paper presents the findings of researchwhich has looked at best practices in literacy andnumeracy provision within the context of suchgovernment commitment and expenditure and withinthe constraints of the types of programs being funded.The discussion includes a brief analysis of criteria of

best practice which emerge from the literature onadult literacy and numeracy provision in general andfor unemployed persons in particular.

Best practice in literacy and numeracyprovision for unemployed persons:previous research and writing

Most definitions of literacy and numeracy are nowcomplex, involving not only skills of reading, writing,speaking, listening and mathematical computationand space manipulation, but also critical thinking,problem solving, questioning, and the appropriate useof language and mathematics. In conjunction withthis, research has shown that there is a multiplicity ofneeds and contexts in literacy and numeracy (forexample, Guthrie & Kirsch. 1985; Levine, 1982; Moss,1984; Sticht. 1988; Szwed, 1982).

In practice, therefore, this indicates that there is aneed for diversity in the provision of programs foradults and for long-term unemployed persons. Onetype of program is not going to suit all. Literacytraining with an overall vocational perspective maytake several forms depending on the level of literacy ornumeracy of the long-term unemployed person.Provision for people with very little literacy ornumeracy may be directed at basic education; peoplewho are moderately literate may develop skills quicklyin the general context of job demands; while provisionof integrated skills and vocational training in specificjobs may be most suited for people with reasonablywell-developed literacy or numeracy skills ( M ikulecky,1990).

Both literacy and numeracy are now seen torepresent a range of performance with literacydemands and performance context-based ( Kirsch &Jungblut. 1986. ) Today's models of literacy andnumeracy instruction emphasise that learning shouldbe based on meaningful, contextual and purposivepractices ( M ikulecky, 1989). These models acknowledgethat in learning to read or be successful in mathematics,background knowledge, familiarity with a topic andrelevant experiences will influence the acquisition,understanding and application of new knowledge.

3 129


A critical question therefore for adult literacy andnumeracy programs with a vocational expectation.Such as those for long-term unemployed persons, ishow to choose a context on which to focus the literacyand numeracy instruction. Conversely, relatively littleis still known about the literacy and numeracydemands and activities which do take place in manyworkplaces. Both the duration of literacy andnumeracy involvement and the level of literacy andnumeracy activities at work are often underestimatedby workers and employers )Mikulecky. 1989).

Choice of context is not the only issue in provision ofliteracy and numeracy instruction for unemployedpeople. Even when a vocational context is chosen asthe focus for the instruction, the degree to whichauthenticity of the workplace is maintained must beconsidered (Mikulecky, 1989). Literacy and numeracytasks within the workplace have been shown to differfrom most academic-based learning tasks. Workplacetasks have generally problem-solving orientations.involve communication within social settings andoften take place in an informal way Brinkworth.1985; Mikulecky & Ehlinger, 1987; Sticht. 1983:

Willis, 1984). The development of metacognitive andproblem-solving strategies are seen as desirableoutcomes of training programs in literacy and numeracyfor the workplace (Fingeret. 1984: Mikulecky, 1989).

Choice of a specific vocational context over ageneric vocational context or a basic education contextis also problematic. Sticht's :1982) research foundthat general training in literacy did not transfer to jobtasks, that the effect of any such training may be lostwithin eight weeks and that it took between 80 to 120hours of instruction for adults to improve a grade levelin general reading. The complexity of mathematics indifferent contexts means that such estimates areimpossible.

Programs for adults should be designed to meetindividual learner's needs and goals. In addition, theseprograms should emphasise learning rather thanteaching ALBSU, 1987; Heaney, 1983 Staff workingin programs for adults should be well-trained Draper.1986; Ulmer. 1980). They need to be mnItiskilled bothin teaching strategies and in their content area as well

as very confident in order to be able to meet theseindividualised goals. The ways in which this can bedone include flexibility both in program design andavailability and use of alternative instructionalapproaches and sharing of goals of instructors andlearners. Teachers should avoid isolation of the adultlearner and make use of group work and work settingsto support learning development. The teacher becomesthe facilitator t Morrison, 1986).

Research has shown again and again that literacy istied to social context and environment and these mustbe considered in addressing literacy and numeracyprovision. A recent study Wickert. 1989 )demonstrated the need for adult literacy and numeracyprovision in Australia for English-speakingbackground people as well as for those fromnon-English speaking backgrounds. Results from thestudy showed that people who were unemployed or in

unskilled employment had the poorest performanceon the three dimensions of literacy (document, proseand quantitative) assessed.

As most practitioners know, programs for thelong-term unemployed in particular may need toaddress other factors which can coexist with literacyand numeracy problems such as self-concept, workattitudes and life skills. Programs need to emphasisethe personal responsibility and empowerment of thelearner through negotiation of goals and programcontent (Adams, 1982).

Previous writers have also alluded to the alienationwhich adult literacy and numeracy learners can feel ininstitutional settings ( ALBSU, 1984; Simpson, 1988, )since many may have had very negative experiencesfor most of their formal schooling. This, combinedwith the sensitivity of publicly admitting to problemswith literacy and numeracy (although numeracyproblems are more socially acceptable) means that itcan be difficult attracting those people with greatestneeds to courses Lane, 1983; Munday & Munday,1983; Reubens. 19861.

Finally assessment is still an area where much workstill needs to be undertaken. Assessment here isdefined as meaning processes by which judgementscan be made about learner's needs and progress, notnecessarily traditional or formalised testing. If literacyand numeracy are perceived to be culturally andcontext s pecific. the development of assessmentstrategies which can be applied across a range ofprograms is problematic. Difficulties therefore arisewhen literacy and numeracy instructors try to gaugethe progress of their students. Common practices inadult literacy and numeracy include using studentself-assessment, the setting of small, reachable goalsand certificates of completion. The compilation offolios of work over the duration of study is a simpletechnique through which both students and teacherscan realise the progress made.

Effective provision of literacy andnumeracy instruction in programs forlong-term unemployed persons:observation of programs in Australia

ContextIn 1989-90. Bert Morris and I undertook a national

research project funded by the CommonwealthDepartment of Employment. Education and Trainingt DEET), under the auspices of the National Policy ofLanguages, to investigate alternative models ofvocational literacy and numeracy provision for long-term unemployed persons :Cumming & Morris, 1991).

The two models to be investigated were classified as:

pre-vocational programs with main emphasis onthe acquisition of literacy and numeracy skills;andvocational programs with main emphasis (over60°0 of the content ) concerned with training invocational areas and wi th a literacy and numeracycomponent.

30 :3 2


Although the original brief called for a contrastiveevaluation, we saw as our goal the identification ofwhat was good practice in literacy and numeracyprovision in both pre-vocational and vocationalprograms, and in what context were the differenttypes of programs most beneficial.

In order to do this, we examined 32 exemplaryprograms across all states and territories of Australia,from inner city areas in Sydney to the outskirts ofAlice Springs. To be included in the study a programhad to be identified as exemplary by at least twopeople from different sectors who had direct knowledgeof the program but were not involved in the teachingof the program. For example, the program might beknown favourably to a funder such as a Common-wealth Employment Service (CES) officer and to anadministrator in a TAFE college. Not only were theseprograms usually seen to involve exemplaryinstruction and structure, they usually had highretention rates, placement in work or training andremarkable learning and affective outcomesconsidering their duration.

One immediate finding of the study was that in 1989very few programs which could be classified aspre-vocational, or in other words access basiceducation, were being funded externally or internallythrough the adult education sectors in Australia. Infact programs for the unemployed which were fundedby CES were required to have at least 6000 vocationalcontent. Programs which deviated from this to have abasic education focus did so only because of anindividual relationship between a provider and a morevisionary CES officer.

A second immediate finding of the project was thatmost courses which were operating, either vocationalor basic education in nature, were of very shortduration, usually 3 to 12 weeks. Repeats of courseswere always dependent on funding availability andneed. A technical reality of doing the study wasidentifying courses for visiting before they werefinished.

A case study approach was used. Because of therequirement to travel large distances, and the usualrestrictions on budget, only short visits could bearranged to each site. Data were collected throughinterviews with funders, administrators, teachers andstudents, collection of work samples and studentrecords, as well as observation of the literacy andnumeracy practices in these programs. We looked forrecurrent and idiosyncratic themes emerging from ourdata which reinforced the criteria of good practicederived from the research literature as well as addingto these.

Characteristics of effective programsThe general finding of the study was that the basic

education ( pre-vocational) or vocational focus of aproject was, of itself, not a limitation on the quality ofliteracy and numeracy instruction provided. Excellentexamples of both types of program were found. Whatwas found to be important was the provision ofappropriate instruction oriented towards achievement

of the aims of the program and the needs of theindividual participants.

From the case studies undertaken in the project,general characteristics of effective provision wereidentified. For the most part these were congruentwith those identified in the research literature.although there were also some characteristicsidiosyncratic to programs. The criteria can be looselyconsidered in three categories: content and teachingstrategies within programs; the qualities of theteachers in the programs: and organisational factorsassociated with the programs. These categories areobviously not discrete but provide a simple frameworkfor consideration of the nature of these effectiveprograms.

Content and teaching strategiesIndividualisation of instruction. As identified

in the literature, individualisation of curriculum andinstruction occurred in the exemplary sites. Teachersmade judgments about participants' needs in a numberof ways, usually informally. An outstanding andrecurrent feature of the teachers in these programswas their ability to respond flexibly to different needswithin their group. Students commented that lesscompetent teachers lacked the same insights intoindividual needs and learning styles and were far morerigid in their teaching. For many students theseteachers recalled memories of school days with teachersmore concerned with teaching curriculum than thechild.

Of course not all teaching observed was as desirableas the above practices. Many teachers, particularly inmathematics, used workbook activities based onprimary or secondary school textbooks. 'busy' tasksand whole-class teaching regardless of the individualperformance ability of the students. But effectiveteachers were able to individualise their programsand the students were appreciative of their efforts andmade noticeable improvements.

Teaching strategies. Teaching of literacy andnumeracy in effective programs was alwayscontextualised, whether in vocational context or lifeskills. Worksheets which had no transferability norpurpose for the students were not used. In addition.staff tended to make use of peer tutoring, group-basedapproaches and have a problem-solving orientation toany task. Work centred around the development ofmetacognitive strategies as much as contentknowledge.

Another common approach in both literacy andnumeracy instruction in the most effective programswas the emphasis on oral communication in thelearning process. Most students have strong oralability and a capacity to verbalise problems andsolutions without knowing the formal written forms ofliteracy or numeracy. Thus the use of an attainedstrength provides a firm basis for the development ofnew abilities. Not only did the use of oral skillsfacilitate group problem-solving activities andparticipants working together and helping each other.it simultaneously provided an environment more

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1

authentically related to the workplace. At the sametime this removed students' preconceptions of whatan education setting should be. In fact, in manyprograms providing for people who have obviouslyhad very negative previous learning experiences, staffhave to allow students to undergo a form of re-birthingto allow them to divulge themselves of thesemisconceptions of how learning should take place.This was particularly true in modules based onmathematics, where many students who have failed inmathematics enter with the notion of correctness ofanswers, that only a correct answer is indicative ofsuccess, and that there is only one was to solve aproblem.

Two other noticeable features of instruction relateto the numeracy provision in the effective programs.Firstly, because of the predominantly oral approachtaken it was often difficult to distinguish betweenwhat was actually literacy, numeracy or vocational orbasic life skills content. Therefore. the better theprogram, the more blurred the lines between theseareas.

The second feature related to the content of thenumeracy in the effective programs. Lacking were theworksheets and emphasis on basic facts and operationsworksheets which are often regarded as numeracy.These were still regarded as important needs,particularly for the workplace. but were approachedin different ways. Numeracy was much broader andgenerally included many dimensions of mathematicsfrom the number system through games to spatialawareness. Large physical objects such as tables andchairs were manipulated in the classrooms to deal withproblems ranging from percentage and discount tovolume and area. Estimation occurred as a physicalactivity such as standing beside a window and decidinghow low it was, rather than the topic approach ofestimation in school curricula. Money was used as avaluable tool in aiding understanding of place value.Through the use of varied approaches and content innumeracy, students not only acquired new knowledgeand problem-solving strategies, they also enhancedfrom a different perspective the imperfect mathemat-ical knowledge and concepts they usually brought to aprogram.

Adult learning principles. The features of adultlearning which were mentioned by staff in effectiveprograms were negotiation and responsibility forlearning. Students were encouraged to set or help settheir own learning goals and how best they might

I achieve them. The flexibility of the programs and theteachers meant that meeting such individual needswas achievable. In some programs the nature ofmodules to be studied or the content of individualsessions was negotiated between students and teachers.

: It is important to note however, that this was not anattitude of lazssez-ia ire. for when necessary teacherswould help provide a structure and knowledge basewithin which students could make an informed choice.Often the adult learner with basic literacy andnumeracy needs is not in a position to he able toidentify their needs.

A theme that emerged constantly within the bestprograms related to group dynamics. Staff believedthat the development of group dynamics was vital tothe success of a program, particularly in assistingmany long-term unemployed people in regaining oracquiring social skills. Positive encouragement by agroup of peers is one of the strongest motivators anindividual can have. In several programs deliberatestrategies were employed at the beginning to assist inthe formation of group dynamics including socialoutings or the formation of a tea club on the first daywith students having to take responsibility for differentroles.

Integration of literacy and numeracy acrossthe program. Vocational programs usuallyincorporated a number of modules including literacyor communication skills). numeracy (or workplace

mathematics . life skills and vocational content.Full-time programs with a basic education orientationusually had a similar format with other modulesinstead of vocational content. These additionalmodules ranged from computer skills to study of thecommunity.

In both types of programs, effective programsincorporated and integrated literacy and numeracyinstruction across all modules. The orientation ofmost of these programs was that literacy and numeracydevelopment were the major need and focus of theprograms. Therefore, every module had a literacyand/or numeracy focus. This was undertaken in arange of ways such as the of instructional materials,manuals and activities written at appropriate levelsand team teaching by vocational and basic educationteachers in vocational modules. In actuality, some ofthe better basic education and vocational programsshowed only marginal differences in orientation, butwere identified as pre-vocational or vocational by theproject brief definition.

Balancing specific and transferable literacyand numeracy skill development. Most of theprograms we visited were aimed at low occupationalskill levels because generally these are the jobsundertaken by people with much lower literacy/numeracy skills than the general community. If aprogram focused only on the skills needed to do ajob such as factory hand or sewing machinist, thestudents would have little opportunity of improvingtheir general literacy or numeracy abilities. Wefound that in effective vocationally-oriented programsstaff tried to balance the development of skillswhich were specific to undertaking a job with themore general literacy and numeracy needs of theparticipants. Work with money might help inoperating a cash register but was also incorporated inapplied areas such as personal budgeting and foodshopping, while at the same time developingtransferable knowledge of mathematics such as placevalue and basic operations. In this way, staff knewthat they were not only meeting the immediate workand life needs of the participants. but were alsoaffecting their capacity for better skilled work ortraining.

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Identifying vocational literacy and numeracycompetencies. The effective vocational programs,which usually offered skill development for up to threejobs, had specific curriculum orientations to pursue.In the most effective vocational programs, staff devisedtheir curriculum by undertaking skills or competencyaudits of those jobs. For example, one programprovider had identified that there was a local need forclerical hospital staff to help organise admissioninformation forms. The nature of the work to beundertaken was audited and appropriate curriculumdeveloped. In fact, this provider operated outside thenormal guidelines by spending a few monthsdeveloping appropriate instructional materials for arange of jobs before taking clients, again incorporatingbasic literacy and numeracy development throughoutall content. Needless to say, these materials were thengreatly sought after by other providers.

The practice of audits and identification of skillsand competencies facilitated individualisation ofinstruction. The staff were able to examine whatincoming participants were able to do against whatwas necessary in the job. This gave a clearer picture ofwhat areas of literacy and numeracy developmentneeded most focus.

On completion of the program, students were givena certificate on which was listed the skills orcompetencies which they had either 'achieved' or'attended for', out of the total required for the job.Thus all students could be shown to have achieved in apositive way. The employmeni. placement of thesestudents was very high because the providers had firstidentified an employment demand in the region andthen developed the students' skills to suit.

Teacher qualitiesTeachers' qualifications and expertise. There

can be great differences in the backgrounds of staffemployed in basic education and vocational programsfor the unemployed, according to the location of theprogram. In general the teaching staff in basiceducation programs are more likely to be qualifiedteachers than those in vocational programs. However,the generally poor employment conditions providedfor staff in these programs, low pay and poor ornonexistent job security, can make it difficult toemploy qualified staff. rhis was particularly true inrural Australia where such staff would have to leavethe region to find employment elsewhere. Shouldanother program suddenly be funded, qualified staffsimply would not be available.

Within such programs, however, it is not always thecase that formal qualifications make for the betterteacher in general. Adults with trade or clericalexperience who related positively to long-termunemployed people and who provided the flexible andindividualised attention needed in effective provisionwere also found to be appreciated as staff in theseprograms. Among the qualified teaching staff, thereappeared to be no clear pattern that primary trainedteachers were more suited to these programs thansecondary trained teachers, or vice versa.

Teachers who had had formal training in adult basiceducation were found to be among the best teachers inprograms. Teaching literacy and numeracy to adultsin these types of program involves skills which areextensions of and often different from those used byteachers in mainstream education. In Australia untilrecently very few training programs for teachers havebeen offered in this area, and hence the number ofqualified adult basic education teachers are very few.Again, more programs are being offered to meet thisneed but there are not enough qualified people to meetthe growing number of literacy and numeracy teachingpositions being advertised nationally. This is extremelycritical at this point. Literacy and numeracy instructioncannot be left to untrained or inexperienced staff.Should this occur, and the programs are not effectiN ein improving literacy and numeracy and hence deemedunsuccessful, the present emphasis on meeting theseneeds and associated funding may disappear.

What we often found in programs was that the mosteffective qualified and non-qualified teachers had'been around'. While not identifying with theparticipants in a depressing manner, these teachershad often had a range of occupations from professionalto menial, been unemployed, and undergone personaltraumas common to our society such as divorce. Thisis not to paint a picture that the only good staff forprograms for long-term unemployed persons have hadto undergo all these negative experiences first, but tosay that the most effective teachers were those whounderstood some of the issues confronting their clients,who could maintain respect for people from whommost others would turn, and who could be positive andencouraging throughout.

Team spirit with respect to other staff andstudents. In the most effective programs, staff workedtogether as a team and deliberately fostered acooperative working atmosphere. Communicationabout the program and participants was very good,with regular, often daily meetings for planning,program organisation and discussion. In one case thisoccurred with staff who had worked together for tenyears. This cohesion is both more important and moredifficult to achieve when programs employ a largenumber of part-time staff.

Team work was also important within the teachingcontext. With regard to literacy and numeracyemphases, this ranged from the literacy and numeracystaff assisting in the development of materials for allcontent areas and teaching strategies to the teamteaching by vocational and basic education teachers.

In most programs, staff were carefully selected ontheir ability to work with long-term unemployedpeople who had many learning and social problems.Staff in the effective programs were always positiveabout their students, often with humour, butemphasised respect for the students as adults andhonesty as important ingredients of successfulprovision. The staff had the capacity to maintaincontrol of the situation but were never patronising tothe participants. In general, the staff enjoyed theirwork despite its demanding nature.

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Organisational factorsFlexibility of programs. Just as the teachers

could be flexible, so too the most effective programswere in themselves flexible. Although operating withinbasic constraints of starting and finishing times andnumber of contact hours per week, the best programswere continually adjusted to meet the needs of eachintake group.

Expected outcomes. Staff were usually realisticabout the outcomes they could expect from programsof short duration. They recognised that forparticipants with major literacy and numeracy needslittle could be achieved but to set them on a pathwayof instruction and learning. For some students.recognition that they had a literacy and numeracyproblem which needed to be addressed was a criticaloutcome.

While the success of the programs in funding termswas measured in terms of placement of participants infurther training or employment, staff saw these as notthe only outcomes. Affective areas such as self-esteemwere always targeted in effective programs. Modulessuch as life skills, hygiene, personal development andconflict resolution :explained by one teacher as 'youdon't hit the boss': were included in most programs.Often physical activities such as walking or basketballwould be included to help students attain a healthylifestyle. Self esteem and confidence-building weremost important in programs where clientele weremature-age women with reasonable literacy andnumeracy skills who were returning to the workforceafter twenty years' absence.

Selection of participants. Although it may beobvious that a program operates best if the range ofstudent ability is reasonably constrained andappropriate to the level of a program, the opposite didoccur in several ,programs. This was mainly due toinappropriate placement in programs being made byEmployment Officers of the government. Fo, 2xampie.within one small vocational program we saw anEnglish-speaking background student who was unableto write his name, and a non-English-speaking studentwith a Ph.D. in a technological area. However, in thebetter programs, participants were generally matchedwith an appropriate level basic or vocation-ally-oriented course.

Often participants had to successfully complete ascreening task or tasks set by the provider. In thesecases the provider had established working relationswith the local Employment Officers which facilitatedthis process and conveyed the importance ofappropriate placement. In Australia today, screeningand placement is undertaken by selected consultantsoutside of the Commonwealth Employment Serviceand recommendations are made as to the general levelof literacy and numeracy needs of the individual aswell as to whether they are best suited to a basiceducation or vocationally-oriented program.

As an outcome of our project we recommended thatunemployed people with extreme literacy andnumeracy needs should be funded to undertake basiceducation programs prior to vocational training. This

is now happening. We felt that students who had needof intensive instruction in general literacy andnumeracy were disadvantaged by being placed in avocational prcgram. However, at the same time.many such students are either strongly targetedtowards getting a job or reluctant to face their literacyand numeracy difficulties. Initial involvement in ahands-on vocational program is often a procedure bywhich these reluctant literacy and numeracy learnersdecide that they have a need in this area and forarrangements to be made for further instruction,either individually or in a group class.

Assessment and certificationEffective literacy and numeracy programs were

individualised to meet both program goals andindividual needs. This places great demand onteachers skills in identifying those needs. In someprograms staff made use of standardisednorm-referenced reading and mathematics tests whichwere theoretically suspect and contextuallyinappropriate.

However, in the effective programs we saw, teachershad developed expertise in using informal testing.observation, examples of work and interviews forassessment. Either through knowledge of adult literacyliterature or naive knowledge of their clients, theteachers were able to gauge how far they could stressstudents, who were often easily stressed in formaltesting situations, to assess their literacy and numeracyperformance ability. The most common practice inmonitoring progress was for teachers to have ongoingdiscussions with individual participants about theirneeds and to observe closely the work which wascarried out.

In 1992 and the future, programs are findingthemselves lacing stronger accountability require-ments and the need to demonstrate in tangible waysthat students are making progress. Staff acrossAustralia are looking at certification procedures similarto those developed by ALBSt- and at the same time anumber of research projects for 1992 and 1993 arecontinuing to investigate the issues and means ofassessment in adult literacy and numeracy. This stillremains the most problematic area of adult literacyand numeracy provision.

Finally, most effective programs provided someform of certification on completion of the program,usually with a moderately formal ceremony andpresentations with family present. This was obviouslya very motivating and positive activity and for many.the first feeling of success in their lives. With theintroduction of standards frameworks and competencymodules in Australia as in the United Kingdom, suchcertification can eventually be made more meaningfuland can contribute to further learning pathways.

Structure. Despite the often disparate nature ofthe participants in these programs and manyparticipants' lack of life skills such as self-management,staff in effective programs provided a strong structurein content as well as daily organisation. There wasalways an expectation that students would arrive on

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time, participate fully and attend regularly. In someprograms students would complete a personal contractwith the staff regarding not only their learning andpersonal goals but also behaviour on site. In mosteffective programs, behaviour control was exercisedby the student group themselves who would establiShrules of conduct and undertake their own 'enforce-ment'. Staff found that not only were the participantsundertaking responsible behaviour but that theywould generally be much harder on each other thanthe teachers would be. In one site, students haddecided that latecomers would not be admitted andhad themselves once locked out a student.

Program structures. Programs varied consider-ably in duration and intensity. Some programs wereonly three weeks long and focused on life skills and selfesteem, others in Thchnical and Further EducationColleges were from six months to twelve months longand intended to raise participants to a Year 9 standardof education.

One successful and different structure was acontinuous entry model. The program started at thebeginning of the year with between 30 and 45participants, participants stayed up to 8 weeks, andnew participants could enter monthly into vacantplaces. Advantages of this model included the capacityto employ good staff consistently, small numbers ofnew students entering for whom instruction couldthen be tailored and, overall, the atmosphere ofsuccess generated when students could see othersbeing successful in getting jobs.

Most vocational programs offered or wanted tooffer work experience as part of the program. In somecases this was being hampered by appropriatelegislation to cover areas such as liability for workerinjury. However, by 1992, work experience had becomea part of most vocational training programs. In manycases students are now expected to find their ownplacement with a contract then being made betweenthe employer and the training provider.

No clear pattern emerged from the project aboutthe desirable length of programs. Short durationprograms were clearly limited in the learning theycould bring about. At the same time many long-termunemployed people are anxious to obtain work anddaunted and disillusioned by the prospect of longinstructional programs. Conversely, once immersed ina program they often considered that programs shouldhave been longer, and many arranged to continuefurther studies themselves. An important outcome ofthis is that articulation from the short vocational andbasic education into other programs is an importantconsideration for the training of the workforce ingeneral. Participants who have a long time lapse fromcompletion of their program to enrolment in a furthertraining or education program often lose both theskills and motivation just developed.

Location. The programs we visited were arrayedin a variety of habitats from sleekly modern officeaccommodation to extremely run down houses. Whilethe warmth and quality of the teaching staff was themajor contributor to the effectiveness of a program,

staff were also concerned about the self-image ofparticipants if their environment was really poor andhence their training appearing to be devalued.Therefore it is desirable that all training programsshould be in accommodation of a standard similar tothat provided for any other learning environment.

Just as it was not clear from the visits to differentprograms as to what was the most desirable length of aprogram. so the effect of location of the program on itseffectiveness was ambivalent. It is often stated thatadult learners who have had unsuccessful or negativelearning experiences at school will fmd an institutionalsetting alienating. In fact, many adults feel that theirtraining program is more prestigious and respectableif it is held in an environment such as office space or aTechnical and Further Education College. What ismore important, it appeared, is for staff and studentsto be in reasonable proximity with a home room forthe students. The more isolated the staff offices fromthe teaching spaces, the more difficult it was toencourage team work and cohesion amongst thestudents. The students themselves may not bealienated from a formal training setting but it is lesslikely that they will engage with regular campusactivities and liaise with regular students.

A final comment with regard to the effect of locationof the program and alienation due to school experiencesis the presumption that all participants have hadnegative experiences or in fact even attended school.Many participants had quite happy times at school,they just didn't learn. Others even in these times haveattended only one to two years. if at all.

ConclusionsThe major conclusion drawn from the study was

that both basic education and vocational trainingprograms incorporating literacy and numeracy shouldbe available to the unemployed. Since 1989 there hasbeen a dramatic change in the recognition thatliteracy and numeracy ability can be a major inhibitorto people obtaining work and the provision of fundingand programs to address this. However, on conclusionof our project, we recommended a model of provisionof training for long-term unemployed people which wefelt would meet most needs and be most effective. Tosome extent the concept of the model has beenintroduced in the types of programs now being offered,although the fully basic education programs are seento be the domain of further education providers ratherthan the Commonwealth Department of Employment.Education and Thaining.

We suggested that running parallel should be 10week intensive basic education modules and the samelength or longer vocational programs. Followingscreening, participants should enter either programand movement between programs should befacilitated.

We also suggested that smaller centres which couldnot support both types of program should offer acontinuous entry model. The content of the programcould be adjusted to suit vocational vacancies in the

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district with the advantage of being able to keepqualified and experienced staff employed.

In addition to these programs, we recognised thatthere would always be a role for community andindividual tutoring to assist those with the greatestliteracy and numeracy needs.

Finally, we recommended that all such programsshould try to address the criteria for effective practicewhich we had identified from the literatufe and fromour own research, while at the same time theyaddressed 'the needs of the whole person in aconstructive and deliberate manner' ) Cumming &Morris, 1991, p.481.

References

Adams. R.J. 1982. The functionally illiterate workerand public policy. TESL Talk Falk 9-16.

Adult Literacy and Basic Skills Unit. 1984.Developments in Adult Literacy and Basic Skills.England: ALBSC.

Adult Literacy and Basic Skills Unit. 1987. BasicSkills and Unemployed Adults. A Report on ..t LBSI''AISCDemonstration Projects. England: London ManpowerServices Commission.

Brinkworth. P. 1985. Numeracy: What, why. how'?Australian Mathematics Teacher 41: 8-10.

Cumming. J.J. & Morris. A. 1991. Working Smarter.Canberra. Australia: Department of Employment.Education and llaining.

Draper. J.A. 1986. Re-thinking Adult Literacy. Ontario:Toronto.

Fingeret. A. 1984. Adult Literacy Education: Currentand Future Directions. Ohio: Columbus.

Guthrie. J. & Kirsch, I. 1985. What is literacy in theUnited States? New Directions for ContinuingEducation 20: 53-63.

Heaney, TW. 1983. 'Hanging on' or 'gaining ground':Educating marginal adults. New Directions forContinuing Education 20.

Kirsch, I. & ungblut. A. 1986. Literacy Profiles olAmerica's Mung Adults. Princeton, NJ: NationalAssessment of Education Progress at EducationTesting Service.

Lane. R.W. 1983. Making adult basic educationwork: A change in emphasis. .v.ISSP Bulletin 67:89-91.

Levine. K. 1982. Functional literacy: Fond illusionsand false economies. Harvard Educational Review52: 249-266.

Mikulecky, L. 1989. Real-world literacy demands:How they've changed and what teachers can do.From Lapp, D., Flood. J. & Franan. N. Content areaReading and Learning: Instructional Strategies.Englewood Cliffs. NJ: Prentice-Hall.

Mikulecky, L. 1990. Basic skills impediment tocommunication between management and hourlyemployees. Management Communication Quarterly 3:452-473.

Mikulecky, L. & Ehrlinger, J. 1987. Training for JobLiteracy Demands: What Research Applies to Practice.Pennsylvania State University: Institute for theStudy of Adult Literacy.

Morrison, A. 1986. "Ritor training for teachers ofadults: A flexible approach. Australian Journal ofAdult Education 26: 15-17.

Moss. M. 1984. The language of numeracy. I'iewpoints1: 15-18.

Monday, J. & Monday. S. 1983. What the unemployedwant'? Adult Education 55: 334-342.

Reubens. B.G. 1986. Adult Education and Thaining inWestern European Countries. Washington DC:National Commission for Employment Policy.

Simpson, N. 1988. Opening doors: Learning to readand write as an adult. Vox 1: 40-43.

Sticht, T.G. 1982. Basic Skills for Defense. Alexandria.VA: Human Resources Research Organization.

Sticht, T.G. 1983. Literacy and Human ResourcesDevelopment at Work: Investing in the Education ofAdults to Improve the Educability of Children.Alexandria, VA: Human Resources Organisation,

Sticht. T.G. 1988. Adult literacy education. Review ofResearch in Education 15: 59-96.

Szwed. J.F. 1982. The ethnography of literacy. InM.F. Whiteman Ed. ), Variation tn Writing:Functional and Linguistic-cultural differences. Writing:The Nature. Development and Teaching of WrittenCommunication Series. vol. 1. Hillsdale, NJ:Erlbaum.

Ulmer, C. 1980. Responsive teaching in adult basiceducation. .Vew Directions for Continuing Education6: 6-15.

Wickert. R. 1989. No Single Measure: A Survey ofAustralian Adult Literacy. Canberra: CommonwealthDepartment of Employment. Education and

Willis. J. 1984. Who are these people with numeracyproblems? iewpoints: 19-21.

36 S

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