Numeracyat WorkSecond Edition

Lynda FownesElizabeth ThompsonJulian Evetts

Editors

CANADA

ii Introduction

National Library of Canada Cataloguing in Publication Data

Main entry under title:

Numeracy at work / Lynda Fownes, Elizabeth Thompson, Julian Evetts, editors. -- 2nd ed.

ISBN 0-9685027-6-8

1. Numeracy--Study and teaching. 2. Workplace literacy. I. Evetts, Julian, 1948- II. Fownes, Lynda, 1947- III. Thompson, Elizabeth, 1942- IV. SkillPlan (Program)QA141.15.N85 2003 513'.071'5 C2003-910729-9

Copyright © 2002Second Edition, 2003

SkillPlan – BC Construction Industry Skills Improvement CouncilAll rights reserved.

To order copies of this publication, please contact:

SkillPlan – BC Construction Industry Skills Improvement CouncilSuite 405, 3701 Hastings StreetBurnaby, BC, V5C 2H6

Order online: www.skillplan.ca

Financial assistance provided by Human Resources Development Canada.

Practitioners are encouraged to photocopy materials in this publication for educational purposes only. Pages may be reproduced, provided that each page used is reproduced in its entirety without modification, with all legal notices maintained. No part of this material may be modified, repackaged, reproduced or used for any commercial purpose or sold by any organization without permission in writing from SkillPlan.

Typography

Calgary

Numeracy at Work SkillPlan iii

AcknowledgementsNumeracy at Work is truly a collaborative work bringing workers and educators

together. We are so pleased that our colleagues across the country saw the value intelling the stories of workers in their part of Canada. Each chapter is rich with detailsand together they make this collection what it is. Thanks to the contributors who arenoted in each chapter and especially to the workers who allowed us to capture numeracyin their work lives.

We acknowledge the contribution of content experts:

• Gene Brush, ABE Mathematics Instructor, Kwantlen University College• Geoff Dean, ABE Mathematics Instructor, Kwantlen University College• Sue Grecki, Workplace Educator, SkillPlan• Marilyn Luft, Workplace Education Consultant• Bill Norton, Piping Trades Instructor, Pacific Vocational Institute• Donna Palmer, Education Consultant• Sheila Whincup, Workplace Education Consultant

We also acknowledge the professional support provided by:

• Janis Barr, Beachside Editorial Services• Anita Rathje, Administrative Services• Anne-Marie Ventura, Translation Services• Hallographix Design & Multimedia

A sincere thank you is extended to those who allowed us to use their authentic workplacematerials and photographs.

A national advisory committee, whose names are listed in Appendix 6, guided this projectfrom the start. We extend thanks to them for their suggestions and support. Thanks to HumanResources Development Canada for financial assistance, and in particular to Debra Mair forher dedication to the promotion of essential skills.

iv Introduction

About SkillPlanSkillPlan, a joint labour and management

initiative of the BC construction industry, wasestablished as a not-for-profit society inMarch, 1991. The Council is the industry’sresponse to an evolving workplace thatrequires greater reading, writing, numeracy,problem solving and oral communicationskills. SkillPlan believes that a solidfoundation of these skills is the VelcroTM towhich all other skills stick. We recognize thatthese skills are part of an adult’s life not onlyat work but in their unions and the widercommunity.

Numeracy skills are an integral part ofsuccess in trades. Heat and Frost Insulators andsheet metal workers use geometry, trigonometryand triangulation to draw patterns of both regularand irregular shapes. Electricians use formulas tocarry out a number of different calculationsincluding energy loss, voltage drop, and actualload on a service. Boilermakers measure and cutmaterials using metric and Imperialmeasurements. Refrigeration workerscalculate refrigeration requirements bydetermining the transmission of heat gaintaking into consideration such factors asconduction, type and thickness of insulation,external architecture and ‘U’ factors. The listis endless.

SkillPlan services to assist members includeproviding tutors, study groups and classroompresentations. A third-year carpentry apprenticemay need to refresh trigonometry skills. Anothermember may be at a loss when it comes tofractions. For yet another, the need may be tolearn about programming a scientific calculator.Requirements vary and so do the responses. Theupgrading of essential skills supports individualswho want to be able to make choices and besuccessful. Individuals make up the pool ofresources available to unions and to the industry.Everyone in the industry benefits from the highestpossible skill level of each individual. For moreinformation, visit SkillPlan’s internet website atwww.skillplan.ca

Numeracy is the‘nuts and bolts’ in

construction

Numeracy at Work SkillPlan v

Table of ContentsAcknowledgements .................................................................................... iiiAbout SkillPlan .......................................................................................... ivTable of Contents ....................................................................................... vPreface ................................................................................................... viNumeracy: A New Understanding .............................................................. viiNumeracy Matters ..................................................................................... ixA Workplace Education Perspective ........................................................... xThe Content Framework ............................................................................ xiiInstructional Settings .................................................................................. xiiiNavigating the Collection ...........................................................................xivMoney Math .............................................................................................. 1

1. Financing New Wheels ................................................................. 32. Tending Bar ................................................................................. 193. Invoicing in the Rockies ............................................................... 354. Pricing Hair Care Products .......................................................... 475. Managing Furniture Sales ............................................................. 61

Scheduling or Budgeting and Accounting .................................................... 756. Valuing a New Contract ............................................................... 777. Reconciling Petty Cash ................................................................ 998. Harvesting Fish for Profit ............................................................ 1179. Comparing Shipping Costs .......................................................... 13310. Planning Catering Service ............................................................ 147

Measurement and Calculation ................................................................... 17111. Administering Medication ............................................................ 17312. Calculating Offsets ...................................................................... 18713. Measuring for Production ............................................................ 20514. Building Brew Tanks ................................................................... 21915. Applying Anhydrous Ammonia .................................................... 235

Data Analysis ............................................................................................ 24916. Analyzing Materials for Quality .................................................... 25117. Reaching Northern Hiring Targets ................................................ 26518. Collecting Data for Safety ........................................................... 28319. Tracking Equipment Downtime .................................................... 29920. Producing ‘Winner’ Conifer Seeds .............................................. 311

Numerical Estimation ................................................................................ 327Appendices .............................................................................................. 347

Appendix 1 – Numeracy Tasks Described in Each Chapter .................... 347Appendix 2 – Math Foundations by Chapter .......................................... 350Appendix 3 – Mathematical Foundations ................................................ 351Appendix 4 – Complexity Levels for Numeracy Tasks ............................ 359Appendix 5 – Reading Text and Document Use ...................................... 360Appendix 6 – National Advisory Committee ........................................... 361

vi Introduction

Numeracy at Work is a rich collection ofexamples of how numbers are used in theworklife of a selected sample of Canadians. Weat SkillPlan have had the privilege of bringing thiscollaborative work together to make a statementabout a view or concept of what it means to benumerate.

Over the years, our viewpoint as workplaceeducators has evolved through our experiencescoaching working adults to acquire or renewnumeracy skills. The impetus to share thisinformation became clearer in March 2000. Atthat time, invited delegates met at a nationalround table, Numeracy in 21st CenturyCanada, at Queen’s University in Kingston,Ontario. First on the agenda was discussion andwork around the concept of numeracy and theattempts to develop workable definitions of it.Since that time symposia, conferences, projects,publications and classroom experience havebeen bringing about a common view ofnumeracy in Canada and internationally.

Our mandate and interest at SkillPlan is inthe broad range of essential skills. Theindividuals, groups and organizations we serveare dedicated, like we are, to success at work.We have made a place in the community ofwork. Our experience leads us to recognizeworkplace education as important and worthy ofdedicated research and development. Along theway, we acknowledge and respect the broaderfield of literacy education and the diversity ofcontributions made by our colleagues in manyareas. When the people we all serve do well, theexcitement and successes extend throughout theworkplace, family and community.

Preface

Preface to the Second EditionAs editors we are pleased with the response

to Numeracy at Work, and to the opportunity totake another look at the stories, descriptions andactivities contributed by so many people acrossCanada. We have taken advantage of thisopportunity to review the chapters and havemade minor changes that clarify and correct.

An important addition to this edition is adescriptive list of numeracy tasks, making themsimilar in form to the reading text and documentuse tasks listed for each chapter. The concept ofworkplace numeracy encompasses mathematicalfoundations but extends beyond mathematics toinclude the observation, critical thinking andproblem solving that a worker uses to completetasks. The inclusion of Numeracy TasksDescribed in This Chapter, in the introductionto each chapter and in Appendix 1, gives a morecomplete picture.

Interest in a second edition of Numeracy atWork reflects the increasing demand forresources that support the work of instructorsand students who seek to incorporate actualworkplace tasks into their educational settings.Once again, Human Resources DevelopmentCanada has responded with financial assistancefor this publication. We appreciate theirleadership in this field.

Numeracy at Work SkillPlan vii

Numeracy: A New Understanding

Use of the word numeracy suggests a newparadigm that is not limited to arithmetic,that is different from mathematics and that isuseful when discussing what goes on at work.Mathematician Richard Noss in his paper,New Numeracies for a Technological Culture,presented at the Institute of Education,University of London, England in 1997,provides a historical starting point for adiscussion on numeracy. He cites theCrowther Report of 1959 which first coinedthe word numeracy as:

“An understanding of the scientificapproach to the study of phenomena–obser-vation, hypothesis, experiment, verification [. . . ] ; the need in the modern world to thinkquantitatively, to realize how far ourproblems are problems of degree even whenthey appear as problems of kind.” (1)

Noss goes on to describe the Crockcroftreport of 1982, a report on the teaching ofmathematics in schools as it relates to furthereducation and employment. Noss indicates aconcern about ‘culture of utility’ that is seenin the UK and elsewhere. He quotes from theCrockcroft (2) report, “Mathematics shouldbe taught to the extent that it is useful in theworkplace, and in adult life.” and furtherdescribes findings of the report that indicatemany adults, in deskilled jobs, did not usemathematics at work. Education policyresulting from this report, and the climatefollowing, served to narrow and simplify theteaching of mathematics as it applied to adultlife.

Noss makes a strong case for teachingbeyond isolated skills toward a moretheoretical mathematics. He speaks of “newintellectual tools designed to make visible themathematics which lies beneath our socialand working lives.” (3)

Participants at Numeracy in 21st CenturyCanada at Queen’s brought out definitions ofnumeracy that fit somewhere in between roteskills or number recognition and ‘higher’mathematics. Two significant initiatives haveinfluenced this understanding, theInternational Adult Literacy Survey and theHuman Resources Development Canada,Essential Skills Research Project.

The 1995 report of the International AdultLiteracy Survey (IALS) defined QuantitativeLiteracy as:

“The knowledge and skills required toapply arithmetic operations, either alone orsequentially, to numbers embedded in printedmaterials, such as balancing a chequebook,figuring out a tip, completing an order formor determining the amount of interest on aloan from an advertisement.” (4)

In 1997, HRDC (Human ResourcesDevelopment Canada) in the Essential SkillsResearch Project chose to blend labels andconcepts:

“Numeracy refers to the use of numbersby workers in the occupational groups andtheir requirement to think in quantitativeterms in order to complete tasks.” (5)

Numeracy at Work defines numeracy as:“the quantitative skills needed to completetasks, acknowledging that in the work life ofCanadians these quantitative skills areexecuted in concert with other essentialskills.”

Other definitions and constructs ofnumeracy in the workplace continue toevolve.

viii Introduction

International Adult Literacy SurveyResults by Country - Quantitative Scale

Percentage of Canadian adultsperforming at Level 1 - 16.9%The individual performs a single,relatively simple operation(usually addition) for whicheither the numbers are alreadyentered onto the given documentand the operation is stipulated,or the numbers are provided.Complexity moves to Level 2 fora subtraction calculation thatinvolves regrouping numbers.

Percentage of Canadian adultsperforming at Level 2 - 26.1%Tasks in this level typically requireindividuals to perform a singlearithmetic operation (frequentlyaddition or subtraction) using numbersthat are easily located in the text ordocument. The operation to beperformed may be directly inferredfrom the wording of the question or theformat of the material (for example, abank deposit or an order form).

Figure 1 - Distribution of Numeracy by Country. IALS, 1996

CanadaGermany

NetherlandsPoland

SwedenSwitzerland (Fr.)

Switzerland (Ger.)United States

0 20 40 60 80 100

Level 1Level 2Levels 3-5

Numeracy at Work SkillPlan ix

Numeracy MattersThe first and subsequent International

Adult Literacy Surveys (IALS) providesomewhat startling evidence that numeracymatters both to individuals ages 16 to 65 andto the economic well-being of the countriesthey live in.

What seems surprising is an apparentdearth of numeracy skills. According to thesurvey, 43% of Canadians between the agesof 16 and 65 have an 80% probability ofperforming tasks correctly at Levels 1 and 2.(See Figure 1, Distribution of Numeracy byCountry) These are tasks that require a singlearithmetic operation, such as addition orsubtraction, using numbers that are easilylocated in the text or document.

The majority of Canadians are less likelyto solve quantitative problems involvingmultiple steps, inferring the operation neededor identifying the correct quantities to use inthe calculation. Levels 4 and 5 at the otherend of the scale are only achieved by 4.7% ofCanadians. Questions at this level requiredindividuals to look, for example, at a tableproviding nutritional analysis of food andthen, using the information given, determinethe percentage of calories in a Big Mac® thatcomes from the fat content.

What is not surprising is the link betweenliteracy, numeracy and employment. Bycomparing how individuals score in terms oflevel with the number of weeks employedand income, there is a clear relationshipbetween skills and achieving and maintainingemployment. According to Scott Murray,Statistics Canada:

“Literacy is strongly correlated with lifechances and use of opportunities. While theprocesses that lead to this result are certainlycomplex, there can be no doubt about itsimportance to employment stability, theincidence of unemployment, and income.

Moreover, in most countries the structuraladjustment that is reducing the economicprospects of adults with low literacy levels isfar from complete. Therefore, those with lowliteracy levels will have even feweropportunities in the future.” (6)

For workers, lack of numeracy haseconomic consequences. For Canada, ourcollective numeracy matters. Our contrib-ution is this resource but it is only a startingplace. We invite educators at every level touse these materials to promote a morenumerate Canada.

References

1 Ministry of Education, UK 1959, p 2702 Learning of Mathematics 18 (2) FLM

Publishing Association, KingstonOntario, Canada, p. 2

3 Noss, Richard, New numeracies for atechnological culture, from the Learningof Mathematics, 18 (2), FLM PublishingAssociation, Kingston, Ontario

4 Literacy, Economy and Society: Results ofan International Survey, StatisticsCanada, 1995

5 HRDC, Reader’s Guide to the EssentialSkills Profiles www.hrdc-drhc.gc.ca/essentialskills

6 Statistics Canada, Literacy, Economy andSociety: Results of an InternationalSurvey,. p. 116

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x Introduction

After ten years of tutoring hundreds ofbuilding trades members and participating inextensive workplace research, SkillPlanworkplace educators are beginning tounderstand more about numeracy; what theelements are and what we can do to facilitatebetter learning strategies.

Many of the individuals who seek tutoringassistance are faced with an apprenticeship examin any one of 16 building trades areas. Examquestions require understanding of mathfundamentals that vary from decimal placementto using a formula to find the lateral surface areaof a frustum. Repeatedly, we observe thestruggle in moving from theory to application. Athird year apprentice carpenter may have somestored memory of trigonometry from high schooldays, usually some ten years in the past, butrecalling the operations is not nearly thechallenge that applying the operation presents.Many adults faced with solving workplacenumber problems don’t know where to start,which numbers to use and what a reasonableanswer might be. Even operations such ascalculating the area of surfaces can beproblematic, not because the answers areincorrect, but because the method of arriving atthe answer is time consuming. In other words,there does not seem to be a cognitive awarenessof efficiencies of method. Converting andnavigating between measurement systems is anongoing challenge in Canada. Materials may beordered using one system and cut, formed andinstalled using the other. As every educatorknows, actual mathematical operations skillsare only part of the picture and the processesof mathematics are rarely neat and tidy. Mostworkers face tasks using numbers embeddedin the contextual texts of their work environment.Within the cultures of various workplaces,strategies have been developed that make quick

and accurate applications of skills possible. Inreviewing the contributions for Numeracy atWork, we were again impressed by the ingenuityand practicality of these strategies. We hope thechapters reflect the varied context of workplacesand the holistic nature of numeracy.

While our context began and continues in theconstruction industry, we have been fortunate toadd a research perspective that is expanding ourexperience to include hundreds of occupationsspanning nearly every sector.

Lessons Learned fromWorkplace Research

Essential Skills Research conductedbetween 1996 and 1998 examined thenumeracy tasks of workers through extensiveinterviews with job incumbents. Over 6000workers were interviewed in 200 occupationclassifications. From a researcher’s point ofview, the experience of speaking directly tojob incumbents and observing their workactivities has led to some interestingobservations that should influence how wedesign effective instruction.

• Most workers did not always see the mathwithin a task. The focus was on gettingthe job done. Tasks required anintegration of skills.

• Data analysis was used by many workerssimply because it was available.Computers track and report in everyconceivable work setting.

• Measuring ranges from rudimentary tosophisticated and involves using anexpansive list of measuring tools. Did youknow that inspectors use ultrasound graphsto measure flaws in aluminium ingots?

A Workplace Education Perspective

Numeracy at Work SkillPlan xi

• Accuracy matters. Mistakes affectproduction and in some cases, can causeaccidents.

• There is a time limit. It is not enough tojust get the right answer to a calculation.The method used to arrive at the rightanswer is scrutinized for efficiency. Thepressure to produce more with increasingattention to quality is a fact of businesseconomics.

• Workers actively incorporate mechanismsso that they won’t have to calculate.Development of job aids or estimation isa practical response to demands of timeand accuracy.

Many of the observations and perceptionshave been born out during the development ofNumeracy at Work. The publication reflects asmall part of the work lives of twenty people.These workers were willing to separate outessential numeracy skills from the complexity oftheir daily tasks and describe them for thecontributors. Many workers, however,mentioned that these were just examples, thatnot only do the numbers change in differentcircumstances, but the processes do too. Theynoted a multitude of factors and variables thatinfluence the numeracy tasks and make itimperative that they have an understandingbeyond the basic foundation skills they use eachday.

Essential skills research has provided aframework for this collection. Theexperiences of the contributors, the workersand workplace educators, give an insight intoreal life numeracy situations. In ourexperience, adult educators and teachers areenthusiastic about providing transferableemployability skills for their students. We hopethat this collection will provide a resource that isunique and interesting for professionals andlearners alike.

xii Introduction

The Content Framework

During the Essential Skills ResearchProject, workers were asked about how theyused numerical calculation and theirresponses were categorized and rated in fivethemes or application settings. Tasks havebeen given complexity levels ranging from 1,least difficult, to 5, most complex. SeeAppendix 4 for more information aboutcomplexity levels. While Numeracy at Workpresents a range of complexity levels, it hasnot been designed to address occupationalrequirements or to parallel sequentialcurricula. It is instead presented with aids toassist teachers and instructors who areinterested in using these applied workactivities to enrich what they do.

The editors have selected chapters that bestfit each application setting. Like so much of whathappens at work, skills are integrated. Someapplications may overlap. Our emphasis is onrepresenting the multiplicity of work tasks ratherthan risking misrepresentation through isolationor over classification of math skills.

As workplace educators, we offer thiscollection as a conduit to the workplace.

CANADA

Money MathThese are tasks that involvefinancial transactions such ashandling cash, preparing billsor making payments.

Scheduling or Budgetingand Accounting MathThese are tasks thatinvolve managing timeand money as resources,planning and monitoringtheir use, assessing bestvalue or reducing waste.

Measurement andCalculation MathThese are tasks thatinvolve measuring anddescribing the physicalworld.

Data Analysis MathThese tasks involve theanalysis of numerical data.

Numerical EstimationThese tasks involve anyestimation that results in anumber.

Numeracy at Work SkillPlan xiii

How can Numeracy at Work be used ininstructional settings? Here are some ideas thathave been suggested to us by K to 12 teachers,ABE and technical instructors.

Numeracy at Work provides:• practice opportunities as a supplement to

curriculum. With practice, comes skillgrowth. Numeracy at Work providesmaterials for an applied approach to supportcurriculum concepts.

• reinforcement and support for curriculumtopics. Real world examples reinforce theimportance of what is being taught ininstructional settings. Learners may have littleexperience in which to relate math conceptsto applications.

• evaluation of transfer of learning. To movefrom theoretical concept to applicationinvolves the integration of skills and requiresthe confidence that comes from practice.

• challenge assignments for high achievers. Ingroup instructional situations, high achieverscan be assigned chapters to enhance corecurriculum.

• career exploration for identified goals. Manyadults and some youth have career pathsclearly in focus. The occupations describedmay challenge or confirm appropriate skilllevels that will be needed for success in thesechosen occupations.

• opportunities to build confidence. Someadults in learning programs have workexperience or aspire to the occupationsdescribed. Building on the strengths of adultsis fundamental to learner-centered education.

• career research. While numeracyapplications are the subject of each chapter,the activities are typical of workplace tasksassociated with specific occupations.Learners have an opportunity to simulatework experiences.

• group projects for building teamwork andmathematical thinking. The activities andthe suggestions found in ExploringNumeracy Further are designed togenerate discussion about the thinkinginvolved in completing workplace tasks.

Instructional Settings

xiv Introduction

Appendix 2, Math Foundations byChapter, and Appendix 3, MathematicalFoundations are important resources forteachers and instructors who are fitting activitiesinto prescribed curriculum.

Appendix 2 is a composite of the mathfoundations compiled from the clipboardfound in each of the twenty chapters. Listingsallow teachers, instructors and individuals toidentify and compare chapters where thesame skills are used.

Appendix 3 more fully describes what isincluded in the topic headings used inAppendix 2. This lengthier appendix is takenfrom the Readers’ Guide to the Essential SkillsProfiles which can be downloaded from HumanResources Development Canada’s websitedeveloped for Essential Skills.

Chapter Features1. About this Chapter

Each chapter begins with an overview of theworkplace setting, tasks and numeracyactivities that are highlighted in the chapter.

2. Numeracy Tasks Described in ThisChapterThe concept of numeracy often goes farbeyond the skills of mathematics. A short listof numeracy tasks as they are described inthe chapter focuses attention on the worker’sperspective.

3. ContributorsA note about the contributor acknowledgesthe individuals involved in developing thischapter.

4. Clipboard - Skills used in the ChapterMath Foundations are categorized andcross referenced in Appendix 2. Mathfoundations may be described in the storybut are not listed on the clipboard unless thelearner uses the skill to complete theactivities. Appendix 3 contains an extensivedescription of Mathematical Foundationswith examples of applications in workplacesReading Text and Document Use identifyelements described in Appendix 5. Theactivities in every chapter require theintegration of essential skills. Theclipboard list includes document use andreading text skills featured in the chapterand used to complete the activities. Theselists describe the features that contributeto the difficulty of finding and usinginformation. Refer to the Readers’ Guide(www.hrdc-drhc.gc.ca/essentialskills) formore detailed information.

5. The “Story”This section of the chapter is not the wholestory but has been tailored to feature thenumeracy aspect of workplace tasks. Whilethe objective of demonstrating applicationshas been met, a disclaimer of sorts needs tobe stated. Jobs involve many more skills thancan be described here. Situations too aremore complex than they appear and may notdescribe all of the economic or social factorsthat make tasks more or less difficult.This part of the chapter is intended to beauthentic to the person telling the story. It isnot necessarily representative of others in theoccupation, company or industry. Theworker explains the workplace task, whythey need to solve a problem and how theydo it.

Navigating the Collection

Numeracy at Work SkillPlan xv

6. ActivitiesIn Activities, the work task described ismimicked. This format allows the learneror teacher to ‘talk through’ the problemusing parallel reasoning.

7. Exploring Numeracy FurtherThis section may include three types ofinformation. One type is more informationabout this work setting which adds to therichness of the application. Some chaptersinclude a review of some skills used in thechapter. These brief explanations areintended to be prompts for memory or

learning aids but do not replace a sequentialcurriculum. A third type of information thatmay be included in this section is suggestionsand ideas for related activities.

8. Answer KeyThe Answer Key includes answers to theActivities and are boxed for quickidentification. The key also provides abreakdown of the task into steps. The stepsare intended to review the thinking processand may be used to correct answers and,more importantly, to encourage discussionabout the thinking process.

As professionals, you will know how to best use the storiesand activities included in Numeracy at Work. The loose leafbinder format is an invitation to photocopy. We believe that realworkers doing real tasks will reinforce the value of skills taughtin math classes, allowing learners to test their quantitativeunderstanding, and helping them to learn about how numeracyskills contribute to employment.