Vocational Mathematics

Numbers and mathematics in AVT______

Tine Wedege

Translated by Gail FitzSimons and Anna Folke Larsen

Contents

Vocational Mathematics Preface 3 Tine Wedege Introduction 4 Malmö University, Malmö Chapter 1 8 © 2007 The author Numbers and technology Translation: Gail FitzSimons Chapter 2 14 Functional mathematics and and Anna Folke Larsen reading skills Chapter 3 20 Photos: Mathematics in work and in school 2Maj/Mira cover Chapter 4 26 Adults’ blocks towards learning Dreamstime p. 14, 26, 44 mathematics Apart from these: Tine Chapter 5 30 Wedege Numeracy in everyday life and in education Drawings: Anders Folke Chapter 6 38 Larsen Numbers and vocational mathematics First edition 1998: in semi-skilled jobs Fagmat – tal og matematik i Chapter 7 44 Numbers and vocational mathematics AMU. in AVT teaching The Danish Labour Market Chapter 8 50 Authority, Copenhagen Relevance and visibility of mathematics Literature 54

Preface It has been argued internationally in policy reports that numeracy as well as literacy are both necessary for work and for citizenship. Accordingly, there has been a focus on the lack of basic reading, writing and arithmetic skills of many adults. In the International Adult Literacy Surveys (IALS) from the 1990s, the focus was on quantitative literacy, and in the Adult Literacy and Life skills survey (ALL) from 2003 it was on numeracy. Following an initial survey, the Danish Labour Market Authority initiated the analytical project “Adult Vocational Mathematics” (Danish: Fagmat) on numbers and mathematics in semi-skilled job functions and in Adult Vocational Training (AVT). In this project, the focus was not solely on the adults’ problems with mathematics but also on the problems caused by mathematics education itself. As a senior adviser with the Authority at the time, I led this project from 1995- 1998. Together with Lena Lindenskov (then at Roskilde University), I organized four surveys on AVT-teaching, on AVT-students and on workplaces. The following people participated in the project for shorter or longer periods of time: Susan Møller, Bruno Clematide, Lothar Holek, Kim Foss Hansen, Tage Munch-Hansen, Dennis Karlsson, Nina Skov-Hansen, Per Gregersen and Tomas Højgaard Jensen. A series of AVT-centres, Vocational Schools, workplaces, employees and AVT- students participated in the empirical investigations. Four industry representatives from education committees participated in the reference group: Jan Mogensen (construction), Knud Madsen (commerce and office work), Gorm Holsteen Jessen (metal) and Jørgen Abildgaard Nielsen (transport). I thank all these people and institutions for engaging in the project. Also many thanks Anna Folke Larsen and Gail FitzSimons who did an excellent job translating FAGMAT from Danish into English. Finally, I want to thank the Danish Ministry of Education and the European Network for Motivational Mathematics for Adults (EMMA) who made this publication possible. I hope that this publication might lead to a further development of the mathematics (containing) instruction by teachers and educators in adult and vocational training. Tine Wedege Malmö University, September 2007

In the European labour market there is a need for functional understanding of numbers and mathematical skills just like there is a need for skills in reading, writing and use of information technology. The needs are found both among the workers themselves and in workplaces where new technology with its changing techniques and work organisation places new requirements on the competences.

Introduction

Why the project “Vocational One could be led to ask: “But aren’t Mathematics”? we well of without these competen- At Grundfos, one of the world's lea- ces?” Yes, it is accepted that many ding pump manufacturers, the semi- adults are immensely competent in skilled workers receive a special offer their work function after many years to learn mathematics (at the AVT- of experience, even though they do not centre or the Adult Educational have formal qualifications in mathe- Centre). One aim is to qualify the em- matics. If you do not feel comfortable ployees for education and work; a with numbers there are many strate- second aim is that a number of em- gies for avoiding them in the everyday ployees with blocks in the subject can life, but it will constrain your flexibi- move on. The Head of education at lity. Furthermore, quality certification Grundfos has not been in doubt that in the workplace implies rigid require- the basic qualifications in mathema- ments on the employees’ literacy and tics are necessary in the labour arithmetic skills. market today: “They are necessary in The teaching in most AVT-education order to enter work. That is one contains some calculation and voca- hundred per cent sure, as calculation tional mathematics of some kind or has not been automatised: the em- other - either as general arithmetic in ployee has to decide upon the special modules or fitted into the voca- numbers coming out of the machine.” tional teaching, or as vocational arith- In the 1990s, large national and metic integrated into the technical- international surveys of adults’ literacy vocational teaching. took place. Corresponding surveys did Within a number of areas during the not exist in the area of calculation and past 15-20 years, there have been mathematics, but it was striking that new/stronger requirements for the some of the reading tasks causing participants to have mathematics- especial difficulties among both semi- containing competences in the Danish skilled workers and others AVT-education. For example, this presupposed that the reader had an applies to the courses in sewerage, understanding of numbers and the CNC-turning, techniques of measure- skills of reading maps and tables. ment, and logistics and co-operation. 6

This can lead to some vocational and figure it out. I’ve never been good in educational problems in teaching; but maths.” also in other places with no new Many adults have a frozen attitude requirements, problems can arise in towards mathematics. Some people the classroom. A vocational teacher call it mathematics anxiety, while expressed it this way: others talk about blocks towards “Calculations and mathematics are numbers. considered as a problem by incredib- What is project “Vocational ly many students. “I’m simply not Mathematics”? able to do anything with numbers.” Vocational Mathematics is an analysis They give up beforehand. Maybe and development project about num- they have had bad experiences with bers and vocational mathematics in mathematics... It is women my age Adult Vocational training. We have who left school as thirteen-year-olds. posed three main questions in They don’t have faith in themselves Vocational Mathematics: and they are left out. Here is a specific example illustrating the problem: • What skills and understandings in A pattern for a skirt is to be made calculation and mathematics are with a waist of 100cm. The 100cm needed in the semi-skilled jobs has to be divided by four. A typical compared with the requirements in student’s reaction is: “No, I can’t the AVT-teaching?

• What are the difficulties in (3) The workplace survey on the use numbers and mathematics that the of numbers, charts and formulas in AVT-students encounter, why do semi-skilled job functions in a number the problems arise, and what are of selected workplaces. The survey the implications for those returning consists of observations and short to vocational education. interviews with nine core employees and an examination of seven existing • How can the mathematics- qualification analyses. containing AVT-teaching be arranged such that it supports the In the surveys we were interested in students and provides them with numbers and vocational mathematics opportunities to exploit their within the following four areas: potentials? Construction, commerce and office In order to shed light on these ques- work, metal and transport. tions, project Vocational Mathema- Furthermore, our starting point was a tics consists of three surveys: questionnaire survey among vocatio- (1) The teaching survey on numbers nal teachers at the AVT-centres. The and vocational mathematics in AVT- survey was carried out in Spring 1995 teaching in selected education pro- in the framework of the cross-sectoral grams for semi-skilled workers. The development project “Profile in survey consists of observations of the Mathematics of Adults,” where the teaching in AVT-centres and exami- teachers were asked to evaluate the nation of educational documents. participants’ arithmetic and mathematical skills in relation to the vocational (2) The student survey on the AVT- teaching. students’ general understanding of numbers, arithmetic and mathemati- This publication is not a report on the cal skills, their experienced needs for three surveys but it is based upon the using these skills in the workplace, findings and on international research and their attitudes towards numbers on adults, mathematics and work. By and mathematics. The survey consists the translation in 2007, we have up- of qualitative interviews with 45 stu- dated the text and generalised it to the dents at AVT-centres and structured broader European context whereever interviews with 160 students at a possible. commercial school and four AVT- centres.

Technological development is about new techniques/machinery, work organisation and qualifications/ competences. Workplaces are swarming with numbers, but mathematics is hidden in the technology and it is a widespread conception among adults that “mathematics is important, but not for me”.

Numbers and technology

Everyone in the labour market experiences and participates in technological development. While some people experience that they are in control of their situation at work, others do not. Many people speculate about development – politicians, philosophers and educational planners. The great question for philosophers is the possibility of humans being able to control The tools are a setsquare, ruler, technological development. compass and pocket calculator But, how is the teacher to judge what in the small workshop. are the relevant objectives from the students’ personal learning perspectives? There are a lot of calculations in a course on vocational cleaning. The vocational teacher explains that many of the students cannot cope with these and she adds: “But that’s alright because they always help one another.” Thus, the teacher is content with the fact that quite a few students do not learn how to calculate the area of a floor or how The most important tools for controlling to price a job – because “they don’t the storeroom of the machine shop need that in the typical cleaning job”, of a big firm. as she says. 10 Numbers and technology

The extent to which the education of them taking an active part in their job system should react exclusively to the reorganisation, at some time in the future? demand for qualifications or whether it 600 years ago the first mechanical also should have an active function is a clock was constructed. Up to that point political question. One of the central all technical instruments had been an extension education questions is about the need for of the human arm - assisting or replacing human qualifications. In Adult Vocational labour (e.g. the plough, the mill, the weaving Training, the aim is to qualify the loom). The clock was the first mechanical device. workforce to meet the needs of both the With this invention, time was divided into labour market and the individual, in line random units (hours, minutes, seconds), and in with technological development. Europe time gradually came to be perceived as The participants might achieve the the sum of these units. The mechanical clock formulated objectives of the course, but extends the area for measurement and quantifi- what about their personal needs and cation: space, weight and time. Precise measure- further prospects for work and ment of time became one of the central elements education? What about the possibility in the organisation of social and working life.

Numbers and codes are to be controlled, and all on time. 11

Calculation in Midland Bank 1929.

Although it would be several hundred mathematical ideas and techniques has also years before the clock became an changed. everyday possession, it gradually While in its time the clock introduced and changed the relationship between human made visible figures for time, mathematics beings and reality. It introduced has now been made invisible by the new objective measurements and made information technology. When a spread- possible objective mathematical rules for sheet is first set up, the formula and dealing with time. Time became an calculations are hidden. It is only when authority. reorganisations are necessary that formulas Twenty five years ago IBM released the and mathematic-cal knowledge become first personal computers [PCs] onto the visible. It is only when the daily market. Today there is a PC in more than routine is broken by a new 50% of European homes and a simple problem that the worker becomes conscious of his/her use of pocket calculator is often given away mathematical ideas or when one buys other electronic products. techniques. In the space of less than a quarter of a century an infinite volume of calculation Changing technology and organisation of has been taken over by computers – at labour are important for the demands made the same time, adults’ need to use on mathematics-containing qualifications. We will look further into this in chapters 5

and 6. 12 Numbers and technology

Mathematics is hidden in the technology level of service may not be so high if In an airport, different functional the balance factor is out. The fact is groups involved in the handling of that safety considerations can mean luggage co-operate by means of that all the planned cargo has to be computer and telephone. When the loaded even though it cannot be done aircraft is being loaded and on time. It is within the foreman’s unloaded, the loading group and the competence to release the aircraft, load planner are in constant but he has to consult the loading computer contact. In the loading planner before taking any decision instructions, the planner has placed not to pursue the original baggage, cargo, and mail in the four instructions. cargo compartments in front of and Many years of experience give the behind the wings. The ideal balance foreman a background for judging factor (38.0) and the limits weight and weight distribution, but in (5.9/51.6) also appear on these order to gain the formal competence instructions. The loading group reads – and maybe the real responsibility in the balance factor of the aircraft on a reorganisation of work – knowing the screen during loading: In the the formula and being able to handle loading report the actual balance it is essential. This is one example factor for this specific aircraft is where the necessary qualifications 28.2. Entering the distribution of cannot solely be found in the weight between the four cargo technique. On the computer one can compartments into the formula for find the balance factor of the actual the balance factor is not required. load. This means that without This figure is automatically knowing the formula the foreman has calculated when entering the cargo a measure for what will happen if and the weight on the computer they leave out freight which is during loading. included in the instructions. In When decisions are being made at principle, he could take this type of the airport about loading an aircraft, decision, but if extra freight that is the priorities are: 1) safety, 2) not mentioned appears, he cannot keeping to the timetable, and 3) forecast precisely what will happen to service. Time is often scarce when the balance factor – if the freight is loading and unloading an aircraft, loaded in compartment 4, for and keeping to the timetable means instance. The foreman would have to that some cargo may have to be know and be able to use the formula sent on a later plane. Meanwhile, for the balance factor, if – with the giving safety first priority can lead to same technique – he was also the the flight being delayed, and the decision maker. 13

There is a difference between the planned and the actual production, so manual and mental arithmetic are still necessary.

Functional understanding of numbers and mathematical skills is intercon- nected with and affected by other basic competences such as being literate and being able to use information technology. In order to understand and use written information in everyday life it is often important to work out the numbers in a text or to read graphs, tables or maps.

Functional mathematics and reading skills

At the beginning of the 1990s, surveys showed that many Danish adults had reading problems and among these were 20% of the semi-skilled workers. Since these findings have been published, the Danish education system has also directed its interest and initiatives towards adult literacy. The Danish Labour Market Authority carried out the project “Vocational Reading” and implemented a plan of action in order to meet with the needs of weak readers in the workplace Blanks from the supplier are received and have context. Where adults have insufficient been checked for quality control. Now a report reading and writing skills in comparison is to be made. with the demands of everyday life, they are offered reading courses by the Danish Ministry of Education.

We are also calculating while we are reading The following quotation is from a debate on changes in the unemployment benefit system printed in the Danish Federation of Semi-Skilled Workers magazine: ”Politicians rack their brains and adjust a little here and there. Employers have to pay the first two unemployment days after 72 hours of work within 4 weeks. That keeps a smaller number of employers from searching for more people.” (Fagbladet 31/95) The text contains some quantitative information. There are many questions to be posed, and one of them could be: “Does the employee have to be employed full time for four weeks before the employer is obliged to pay the first unemployment days?” To understand the objectives or use the information in the text to answer the questions the reader has to judge the amount of 72/4 hours compared to 37 hours (the typical working week in Denmark). The text example is of a type that we daily come across in newspapers, magazines and trade or professional journals. 16

The notice board at any large company has tables, charts and diagrams on sickness rates, bonus payment structures, service grades, etc.

Arithmetical skills are integrated in qualifications can be met through methodical, organisational and commu- traditional mathematics teaching. nicational qualifications/competences as In Vocational Mathematics it is well as in the specific vocational a fundamental assumption that competences. functional understandding of The concept of qualifications gives us numbers and arithmetical an opportunity to work professionally skills is a basic competence in with the relationship between education line with reading and writing and work. The advantage of separating skills in a number of common out a specific qualification and calling it and specific qualifications in “understanding of numbers and mathe- the semi-skilled labour market. matical skills” is that the specific mathe- A mathematical understanding or skill is matical competence becomes visible. functional when the mathematical ideas The disadvantage is that it is isolated or techniques can be used for solving a from the other competences which can task or a problem in everyday life. lead to the belief that the demands of

Functional mathematics and reading skills 17

Skills, understandings and attitudes You can see it." A long term of illness for towards mathematics are parts of one person affects the average. The workers’ general, specific and perso- group as a whole understands this but it nal qualifications is of no interest to Thomas how the figures are calculated or the graphs con- Thomas is a CNC operative in an auto- structed. Actually, his attitude is that all nomous production group at a metal these statistics are something the ma- company. There is no job rotation at the nagement sit doing in the office because lathe he operates and this suits him very they do not have anything else to do. well. When he is checking the objects that are turned, he reads a graph on the There are graphs showing the service screen where he evaluates whether the grades for each of the groups on a joint finished object fulfils tolerance require- notice board in the department. At the ments. He can also see whether produc- end of November Thomas’s group is 45 tion is stable, and this can have implica- hours behind. The service grade is down tions for the tools and the number of to 80. The production leader suggests objects to be checked. that they should organise the work in shifts so that they can come up to 100 There is a graph on the notice board of during December and not work between the production groups showing tables and Christmas and New Year. Thomas takes graphs of the sickness statistics for all no part in the group’s conversation about groups. The graphs also compare average organising the work so that the service absence due to illness for each depart- grade can be maintained, and he has no ment and for the company as a whole. intention of doing so. He just knows that This absence rate is up over 8% in it is a matter of working hard. Thomas’s group while the average is be- low 5% for the month of October. (The story has been constructed on the basis of authentic material and situations During the break Thomas speaks with the from different companies). other members of his group. He says: "That was me. That month in hospital.

Sickness statistics

9 8 7 6 5 Dep. D % 4 Group 2 3 Company 2 1 0

. h . c t Jan ar May July ep Nov. M S

The square is constructed at the bottom of the hole

There is an important difference in problems are not necessarily connected to examining the different requirements their skills in using data, formulas, and for functional skills in reading and graphs in the work context. calculations/ mathematics. The writ- Calculations at work can be more com- ten or “reading” texts are out there on plicated than in the classroom. Not be- view in workplaces while the “arith- cause of the degree of difficulty, but due metical problems” are not just lying to the complexity of the situation. In the there waiting to be solved. Though project, we asked a group of vocational tea- the texts are teeming with numbers, chers at AVT centres about their opinions plenty of numbers will only be vi- on students’ attitudes to arithmetic and sible once they are constructed (e.g. mathematics, and they identified one measurement of pipes and slabs or a problem: count of blanks). The mathematics is • The students consider theory as one often hidden in the technology and, thing, practice as another. Some students usually, formulas are used uncon- can solve problems in the school-based sciously of the applied mathematics material, but they are not able to use the embedded within them. theory and methods in practice. Their This means that the adults’ compe- skills are not functional. Other students tences of solving school mathematics experience just the opposite. Functional mathematics and reading skills 19

Meanwhile the teachers pointed out The score rates of semi-skilled another problem: students in eight mathematics- containing exercises • The students lack fundamental skills in arithmetic/mathematics. 45% Many teachers judge that the lack of 40% fundamental skills causes problems 35%

in vocational teaching and learning. 30% Forty-five vocational teachers drawn 25% from six AVT centres covering ten lines 20% of business completed the questionnaire 15% about their opinions on students’ atti- 10% tudes to arithmetic and mathematics. For 5% instance, they were asked to consider 0% 1-2 correct 3-4 correct 5-6 correct 7-8 correct and answer the following question: Do you experience students’ skills in This finding is supported by the student arithmetic and mathematics as a pro- survey in Vocational Mathematics where blem in your vocational teaching? students face problems in arithmetic and The vocational teachers answered: mathematics: 12% of the 108 semi- Yes, in most courses 15 skilled students only succeeded in 1-2 Yes, in some courses 20 exercises out of eight. 17% gave correct No 10 answers to 3-4 exercises; i.e. 29% – more than every fourth participant – only The same question was answered by 31 did well in half or less of the exercises. vocational teachers at a conference of the Business Committee of the Metal The eight exercises dealt with adding up Industry. prices (estimates), comparing prices, calculating areas and doing calculations The vocational teachers answered: with numbers. The type of the questions Yes, in most courses 14 was: “Can you buy the goods on the Yes, in some courses 16 receipt with 10 Euros?” – “Which is the No 1 cheapest juice?” – “How big is the area Thus, the vocational teachers are finding of the wall?” – “Find 30% of 150 that participants’ arithmetical and Euros.” The questions were of the type mathematical skills are causing that we may find everyday, at work and difficulties in their vocational teaching. in education.

There are systematic differences between mathematics in the workplace and in educational contexts.

Mathematics in work and in school

“No” is the most common answer to the solutions. (Accuracy in school and tole- question “Do you use mathematics in rance at the workplace are two different your everyday life?” This is despite the things.) Solving tasks has no practical fact that many of us use numbers and meaning: the results are not used for formulas on a daily basis. Most people anything except, maybe, solving more only associate mathematics with the tasks. In the so-called ‘problems’ the subject in school or the discipline. task-context is often practical problems, but the aim is to find the correct result That there are differences in mathema- by using the correct algorithm, not to tics at the workplace and in school has solve the practical problem. been a working hypothesis for the surveys in project Vocational Mathematics. In the workplace, the ‘tasks’ result The well-known activity “solving tasks” from solving a working task where the serves as an example: numbers are to be found or constructed with the relevant units of measurement In traditional mathematics in- (e.g., hours; kg; mm). It is the working struction, the task constitutes a central tasks and functions in a given technolo- element and structures the teaching. The gical context which control and struc- task is primarily used to practise skills ture the process, not the ‘task’. Some of (use of algorithms and concepts) and to these tasks look like school tasks (the test skills and understandings. Thus, the procedure is given in the work instruc- task is often solved by the individual tion) but the experienced worker has his/ student and it might be perceived as her own routines, and methods of mea- cheating to hand in a joint solution. The surement and calculation. Circum- task is formulated by the teacher, the stances in the production process might textbook or the program. The task has cause deviations from the instruction or, one correct solution and many wrong for example, that the number of random 22

samples in the quality control process is plan, distribution of products, a price, increased or reduced. It is characteristic etc. that tasks are solved in different ways In the figure, characteristic differences and that different procedures and are listed between mathematics in the solutions might be acceptable. workplace and in the traditional mathe- In the workplace solving tasks is a joint matics instruction the way most AVT- matter: you have to collaborate, not participants encountered it in primary compete. Solving tasks always has prac- and lower secondary school. tical consequences: a product, a working

Numeracy at work Mathematics in school

All numbers have units of measurement (mm; The numbers often appear as pure kg; Euros) or refer to something concrete. numerical quantities.

Numbers and tasks have to be constructed. Numbers and tasks are given.

A task often has different solutions. A task has only one correct solution.

Accuracy is defined by the situation. Accuracy is defined by the teacher. Right/wrong is negotiable. Right/wrong is not negotiable.

Solving tasks is a joint matter Solving tasks is an individual matter - i.e. collaboration. – i.e. competition.

Tasks are full of ‘noise’. Tasks are cleared of ‘noise’. The numbers are often ‘dirty’. The numbers are ‘clean’.

Reality requires the use of mathematical ideas Reality is a pretext to use mathematical and techniques. ideas and techniques.

Solving tasks has practical consequences. Solving tasks has no practical consequences.

Working tasks are defined and structured by the Mathematical tasks structure the teaching. technology.

Mathematics in work and in school 23

In AVT-teaching, teachers distinguish There is a difference between the between general and vocational calcu- ability to calculate in school and in lation activities that can be found “on everyday life. There is a difference in the the timetable” and practical calculation mathematical competence required to solve which takes place “at a corner of the formal mathematics exercises and the work table.” We have chosen to mathematics-containing competence compare mathematics at work with the required for solving practical tasks which traditional instruction. The difference involve similar types of mathematical ideas between work and school is distinct, yet and techniques. A person’s abilities are not the traditional mathematics instruction necessarily the same in all contexts — for can both be found in AVT and general example, in calculating percentages. Some adults’ education. Moreover, the adult people can work out percentages of pure participants’ experiences typically numbers, but not percentages of liquids or originate in mathematics instruction prices. Others experience the opposite. where solving tasks gave structure and It is usual to hear an AVT-student say: “In meaning to the course. This implies that everyday life I can easily solve this kind of many students expect a traditional problem but not in the maths class.” In the mathematics instruction with traditional student survey, we had a closer look at the task solving as a central activity. eventual differences between skills used in When mathematical instruction is solving a practical task which required the integrated within vocational use of mathematical techniques and skills, teaching with practical exercises and those used in solving a formal task or project tasks it is possible to using the same techniques. break this pattern. It can be 1. Calculate the obvious to the participants that area of a rect- the mathematical methods are angle with useful. The disadvantage is length = 3.5 simply that the mathematics can height = 2.3 2. Find 40% of be “integrated away”, so that 150 nobody (neither the teacher nor 3. Calculate the student) is conscious about 45(2,3+1,5) what is to be learned or what has been learned. This means among other things that the student doesn’t change his/her understanding of the The mathematical skills of 160 AVT-students mathematics or his/her own mathe- were tested. matics-containing competences.

Problem solving cartons in the crate. If they started out by calculating the area of he base of the milk “Milk” was the theme for the Lower Secon- crate [(37.5 x 23)cm2 = 862.5cm2] and then dary School graduation examination (year 9) divided by 49cm2 (the base area of the milk in 1990. One of the problems was calculating cartons) the result would have been 17.6. the area of the base of a milk carton. But, in fact, there is only room for 15 cartons (Calculation of areas is part of the curricu- in the crate. lum.) The base is a square and with a length of 7cm, so the area is equal to (7 x 7) cm2 = Why, then, are the pupils asked to calculate 49cm2. The next problem stated that the milk the area of the base of the milk carton? The crate is 37.5cm long and 23cm wide, and the reason is solely to show that they know how question was: “How many milk cartons are to calculate the area of a rectangle. In order there room for in the crate?” to have them demonstrate that they know and are able to use the area formula of a The pupils were not going to use the area of circle, some milk bottles are supposedly the carton base for anything at all, and found among granny’s old things. definitely not for calculating the number of 2. Milk carton – milk bottle The pupils in 9th grade are comparing how much room milk cartons and milk bottles take up in a crate. • Calculate the areas of the base of a milk carton and of a The milk cartons are put in crates that are 37. 5cm long and 23cm wide. • Make an accurate drawing of the base of the milk crate at size (on a scale of 1:2). • How many milk cartons are there room for in the base of • Show on your drawing how many upright milk bottles there the crate.

The survey showed, for instance, that theoretical higher education enrolled in 108 semi-skilled students from among the foundational AVT-education pro- the 160 students surveyed were, general- gram. The focus of the latter is on for- ly speaking, stronger in practical calcu- mal calculation rather than practical. lation than the corresponding formal Among the 160 students, 22 had fini- tasks. Thirty-two percent were able to shed a theoretical education. They were both do a practical calculation of a con- generally better at area calculation than crete wall area and find an answer for a the semi-skilled students, but it is remar- formal area task. Thirty-two percent kable that the number of students who were neither able to solve the practical were able to solve solely the formal task nor the formal task. Only 6% were able was larger than the number of students to solve the formal but not the practical only able to solve the practical task. task while 30% were only able to solve The AVT-students have varied the practical task. competences for solving practical and There are also skilled workers from theoretical tasks. These can be addressed other lines of industry and people with seriously in the classroom by using incomplete school education or else different approaches and methods. Mathematics in work and in school 25

Work task in the canteen time to place them in the refrigerated The canteen worker has no doubt how cabinet: “It lies in the hand.” She checks many milk cartons there are room for in that the delivery matches both the order the milk box. ½ litre cartons are put in and the invoice while filling up the the boxes in two layers, with 15 pieces in refrigerated cabinet. each layer. She takes them three at a

Lesser trained students Theoretically educated both wrong 32% both wrong 14%

practical right, formal practical right, formal wrong 30% wrong 14% formal right, practical formal right, practical wrong 6%, wrong 27%, both right 32% both right 40%

- 30% of the semi-skilled students were able to find the area of a wall by measuring and calculating, but not by solving the formal task about areas. 27% of the theoretically educated students were able to solve the formal task, but not the practical task. 26

AVT-participants come to mathematics with different attitudes and mixed feelings which are often related to their experiences of traditional teaching in primary and lower secondary schools. The teachers consider the negative attitudes as blocks and talk about lack of self-confidence. But they can also be signs of resistance to learning mathematics.

Adults’ blocks towards

learning mathematics

Many adults consider mathematics as a school or higher education. In both the student and subject that divides the population into the company surveys, present and former AVT- two groups: Those who can, and those students expressed the sentiment that mathematics who can’t. You often hear a statement was not their cup of tea, and that mathematics was like: “Numbers and maths has never only for engineers and other technicians. been me.” Formerly, the attitude among A story about the formation of attitudes This is a story from a mathematics class of quite many mathematics teachers was that a few years ago. It is about a senior teacher, Mr. being able to do mathematics or not Holm, and a bright pupil, Brian. The story could have been taken from the mathematics teaching being able to do mathematics was that many adults have experienced in reality: determined by the genes. Strangely The class is doing arithmetical problems. Question number 42 tells them to find the price enough, many more boys than girls were of 5 apples when 3 apples are 45 “øre” (or able to do mathematics. cents/pence etc.). After philosophising over the real prices of apples for some time, Brian is told off by Mr. Holm: Many of the older AVT-students in “Brian, my friend. You’re completely wrong. Denmark have experienced other sorting You’re not meant to think at all. This is an arithmetic lesson. You’re simply supposed to do mechanisms. They have never met what I’ve shown you so often. Use the unit! How mathematics as a subject during their much is one apple? … Proportionality is what you are expected to know by now, and with that you school lives. As a rough estimate, this can solve all eight exercises. was the situation for a fifth of the If you start asking stupid questions about trade in apples, you won’t get far in life. Anyhow, you participants in the courses in won’t get into one of the big shipping companies construction, metal, and transport. On or the Civil Service. They only want people with skills and who doesn’t pose questions at the the other hand, they might have come wrong moment.” (The story is made up by Tage Werner, former across the subject through their children senior lecturer in mathematics at the Royal who were taught mathematics from 1st Danish School of Educational Studies) grade onwards. This forms adults’ The story about Mr. Holm and Brian illustrates attitudes as well. what has been called “the hidden curriculum”. From their own experience or second At the same time as the pupils are taught arithmetic/ hand, students above 30 years of age in mathematics they learn something not written in particular find that mathematics is an the official curriculum: Mathematics does not have activity without any practical meaning. anything to do with reality, but if you want to get It is an activity that only belongs in anywhere you have to learn it. The story also shows 28

that the teacher’s understanding of estimate of the big amounts and afterwards mathematics is crucial. Is mathematics had a look at the small amounts. Some started presented as a collection of concepts and by adding up the tens in succession, then they methods (here, proportionality) that the added up the ones and finally the cents, while student has to acquire? Or is mathema- others summed the exact amounts in tics an activity that the student should succession. learn how to master? In the questionnaire circulated among voca- Another common experience of traditional teachers at the AVT-centres, only 2 out tional mathematical instruction is that of 76 teachers answered ‘no’ when asked there is only one correct method, namely whether they experienced that the students the teachers’ method. Surveys have had blocks to numbers and mathematics when shown that many adults feel guilty when starting the course. Between a third and half using a method in everyday life that is of the teachers answered that they had not to the same as what they have learnt experienced students with blocks in most in school. For example, you are taught courses. the formula for calculating percentage: 30% of 1200 = 1200/100 x 30 A common everyday method is to find From the interviews conducted at the AVT- 1/10 of 1200 and then multiply by 3. If centres we gained a background for distin- the percentage is 25, you can use the guishing blocks and resistance. Student blocks can originate from previous nega- same method and make use of 5 being tive experiences with mathematics, either half of 10. You can go far with this prac- in the teaching or because the student tice as long as the percentage ends in 0 was sifted out through testing. Resistance or 5. Some people might know that 25 is to learning mathematics can be related to exactly a fourth of 100 and simply the adult’s self-perception as a competent person coping with the challenges of his/ divide by 4. Others know how to use the her work life without the use of mathema- percentage button on their pocket tics. But resistance can also originate from calculator. experiencing that mathematics is of no use outside of school. The student survey in FAGMAT documents that the AVT-participants use Authentic problems with non-authentic various methods for calculating, all of solutions can provoke resistance to learning. which can lead to a correct result. The distinction between different kinds of Among other things, the 160 students authenticity should be made carefully. The were asked to make a rough calculation problem might be authentic while the solution of the total from a supermarket bill with- method is not. An example of an authentic out using a pocket calculator. Some problem is to choose where you want to rent people started by making a rough

Adults’ blocks towards learning mathematics 29

bicycles for your vacation, according to reaction is due to a suspicion that mathe- your own needs. (See example). matics only leads to irrelevant byways, When students attend AVT, their aim stealing time from the relevant vocational is to gain a vocational qualification qualification. A suspicion fed by their not to learn mathematics. On discove- experiences in school. It is understandable ring mathematics in the course, some that students are surprised when meeting students react with resistance or blocks. mathematics in this context because AVT “That wasn’t what we came for.” This does not advertise it – neither in educational documents nor in informational material. An authentic problem with a non-authentic solution In the material used at the AVT-centre, tended. The material does not suggest that the there is an example of three bicycle hire students are to practise the skills or understan- shops with different prices. In the practi- dings of arithmetical operations used in every- cal situation, people would solve the pro- day life, and it does not motivate them to ob- blem in many different ways. What you tain the skills and insights in systems of coordi- actually do, if you are on vacation at nates. On the contrary, it reinforces the adults’ Bornholm, depends on your calculation prejudices that, apart from the four arithmetic skills and the specific situation. If it is ob- operations, mathematics is only unnecessary vious from your vacation plans that you mystification. (Lindenskov, 1996) want to rent bicycles for four consecutive days, then some people would calculate the prices from advertisements from the three hire shops, while others would go around and ask the hire shops for the prices and eventually make a bargain with one. If the plans are not yet decided, one could calculate rental prices for different numbers of days and combine these with the bus prices. In this material there is only one method. But it is not an authentic method for finding a solution. When planning to rent bicycles for five days, nobody would draw two graphs and choose the lowest y- co-ordinate at x=5. People would simply calculate the three prices and choose the cheapest hire shop. The education material contains a non- communicated shift between everyday life and mathematics. Possibly, the educational idea is that the authentic problem about the bicycles will assist the adults’ learning of systems of co-ordinates. Yet, interviews and observations do not seem to show that the material worked as in-

In Vocational Mathematics it is a basic assumption that it is possible to identify a mathematics containing competence that everybody needs in principle in the labour market.

Numeracy in everyday life and

in education

In the previous chapters we have addressed workers’ skills and understandings in the “functional understanding of numbers and labour market. We also talk about needs mathematical skills”. We showed how that can be relevant to technological mathematics can be hidden in technology. changes (in technique and/or work organi- Also, how arithmetic skills are intervowen sation) or to workers’ perspectives of with literacy skills and form a part of the working life or supplementary training. general and technical-vocational qualifica- Among the semi-skilled workers in tions. We have pointed out systematic dif- workplaces and among the AVT- ferences between mathematics at work and participants, mathematics is often not in school. Now it is time to close in on this considered as something useful and the competence and name it. general attitude is that the AVT-courses In England at the end of the 1950s, contain too much mathematics. The reason “numeracy” was introduced as a parallel to might be that the tasks in problem solving “literacy” in order to catch the competence are not perceived as relevant, even though which enables people to handle practical they deal with things from the students’ mathematical demands in everyday life. In working lives. the English-speaking countries there are a For instance, with an authentic work large number of numeracy courses, instruction and the normal time given for a corresponding to literacy courses. We certain packing and controlling function, define numeracy as follows: the teacher can construct many different • Numeracy in the labour market tasks from this material. But the operator consists of functional mathematical will not actually encounter the need for skills and understandings that in prin- performing these calculations when carry- ciple all people in the work force need ing out the work function: “Packing and to have. controlling of lids”. The instruction states The expression “need to have” should not that every box should contain 75 pieces. be interpreted as expressing “necessity”, but When the operator is fetching containers for rather “relevance”. Thus, we are not only 600 pieces she has to calculate how many talking about the given requirements of boxes she needs. When there are new 32

Three problems constructed on the basis of a work instruction

1. Packing (The four basic arithmetical operations) 600 covers are to be checked and packed in boxes. There are to be 5 layers in each box. Each layer is to consist of 3 rows of 5. How many boxes are needed?

2. Basis time (The four basic arithmetical operations, percentages) The basis time for this work operation is 4.63. How many pieces per hour does this corre- spond to? The working time needed for checking and packing 248 covers is 1 hour. What percentage of basis time is the working time?

3. Quality control (The four basic arithmetical operations, percentages) The surface is to be checked on all covers (100%). In every box, the leading varnish is measured on 3 covers. What percentage of the covers is to be checked?

blanks coming in, she has to read, calculate within a delivery time of five days (“just in a little and count, but soon it becomes time”) and, through the computer, the routine. All calculations concerning work operator is in contact with the office recei- and normal time are performed by the ving the order, the production department, computer. and the sub-suppliers. There are many fac- But this specific factory is about to intro- tors to be taken into account when planning duce autonomous production groups, which and co-ordinating the work. A little later, implies that the group members each take together with co-ordinators from the other turns as the co-ordinator for 14 days. The departments, she participates in a meeting working day starts out with printing the with the production leader and planner. In production plan. The factory meets orders this meeting they walk through and discuss 33

production plans and service grades. difference in calculating quantities of In such an autonomous group, there are medication or industrial cleaning agents. higher requirements for the operators’ In order to describe numeracy in semi- numeracy than in the ‘narrowly’ conceived skilled jobs, we have constructed an ana- job function. Co-operation with and co- lytical tool with four dimensions. One dimension is context: What you are able to do ordination of colleagues and departments and what you should be able to do is depen- require understanding and conversion of dent upon whether it takes place in the information with numbers in complex supermarket, at work or in a test situation. A contexts and further communication of this second dimension is the medium of appli- newly revised information. cation: The relevant numeracy depends on whether it is employed in oral communication, or applied to reading a manual or measuring a heap of soil, even if the num- The need for numeracy is changing bers and arithmetic operations are the same. in the labour market along with the technological development. Since the 1980s, there have been two contradictory tendencies: On the one hand, pocket calculators and information technology have changed the need for manual calculation, mental arithmetic and manual design work. This makes different demands on workers’ numeracy. Some would say fewer demands. On the other hand, the same technology opens up opportunities to organise the work differently. The need to have an overview of the process, and the need for communication and co-ordination grows. This makes new demands on workers’ numeracy. Some would say more demands. People’s numeracy cannot be determined solely as a collection of skills and understandings taken out of context. Numeracy is not just the four basic arithmetic operations or topics such as “dosage”. The dosage is Input and output on the computer The overview is important to the co-ordinator always affected by its use and where it is of the autonomous group, with a high carried out. For example, there is a big number of factors involved in the decision-making process. 34

The context of numeracy in Vocational Mathematics is the workplace. The actual context in the particular business/enterprise, with pieces of information about work organisation and technology/machinery, should be considered in order to analyse the need for numeracy. Also, what is produced The numbers on the wall are about flight safety. should be taken into account. As expressed by an operator in the quality control at a big A third dimension is personal intention: It is electronics enterprise: “There is a difference critical whether the intention is to gather between mistakes in an aircraft and a tele- information, to fill in a form, to plan pro- vision.” duction, to control the quality of a product, to kill time, etc. A fourth dimension is skills The medium covers the information and and understandings - for example, having a communication or the specific material and geometrical sense, a sense of scale, the processes where numbers, formulas and ability to carry out rough estimations, and figures are used, collected or constructed appropriate use of mathematical techniques. (See chart and illustration).

(1) WRITTEN INFORMATION AND COMMUNICATION* a. Prose texts Informative and instructive texts can be found in: Manuals, directions for use, work instructions, safety rules and instructions, quality control materials, booklets, catalogues, brochures, handbooks, technical books and periodicals. b. Reference Informative and instructive diagrams, charts, graphs, tables, maps, texts drawings, signs, scales etc. are found in: Manuals, work instructions etc., and also on wrappings, labels, signs, delivery notes, invoices, in duty rosters, work and production plans, timetables, measuring instruments, displays, price lists etc. c. Fill-in texts Tables, schedules, charts, diagrams to be filled in are found in: Daily and weekly reports, production plans, labels, delivery notes, reception reports, route diagrams, invoices, control forms, accident reports etc. (2) ORAL INFORMATION AND COMMUNICATION a. Short pieces “24m” – “Fourteen seventy-five” – “Three days” of information b. Longer or “Fetch 12 strips at the store!” – “Jim is ill today so the three of you will shorter have to handle the order without him.” statements c. Dialogue “They cheat with the concrete. Some of it is always missing.” – “Yes, eight to ten per cent every time.” – “Next time I will order more than we need.” (3) SPECIFIC MATERIALS, TIME AND PROCESSES a. Concrete A heap of soil; 21 connectors; 45 sq cm or 45cm2; 1mm aluminium materials b. Time Bonus time; 5 hours; 12 working days until Christmas; delivery time c. Processes Order production with a delivery time of five days; “Just in Time”; warehouse production * All three kinds of texts are found both on paper and screen 35

whether it is intended for specific use in the workplace or simply as part of background information.

Skills and understandings can be broken down in the following way: - to make a rough estimation of numbers and sizes - to have a sense of number - to be able to manipulate numbers - to have a sense of geometry and dimensions - to set up a formula and use it for calculations In numeracy, skills and understandings are functional. It is not sufficient to know the multiplication table up to 12 x 12 if the calculation skill cannot be converted into a competence when measuring and calculating use of workplace materials. The formula for the circle circumference, 2π x Personal intention. When solving radius, is mathematical knowledge that specific work tasks or carrying out a certain only can be used for calculating use of ma- job function which calls for the use of or terial when converted into π x diameter, and construction of numbers, formulas or fi- the person knows how to measure the dia- gures, the intention can be different accor- meter of the object. ding to whether the purpose is: Numeracy contains functional - to collect pieces of information arithmetic skills From the assembly department at a large - to collect data factory, the following fault complaint is - to control a process made: On the computer, a delivery is re- gistered as containing 1094 pieces, but it - to make a judgement - to pass on information - to make a report, verbal or written - to construct a model, etc.

- to co-ordinate or lead is 160 pieces short. When the intention is, for example, to find a In the delivery note the operator finds that they have sent 22 boxes of 42 blanks quantitative piece of information there is a and a single box of 10 blanks. By means of the ten times table, the two times difference in the necessary competence of table, plus and minus she locates the error as a miscalculation. collecting that information depending on 36

Conversely, the multiplication tables of male technician). The department is con- two, five and ten can take you far in prac- nected to other departments via an internal tice as long as know how to set up the cal- network. The work is independent and is culation for solving the task. organised based on rules of prioritising

In the workplace it is not always enough particular tasks. In relation to Annette and to be able to find “the correct answer” the others, the technician’s function is to the arithmetic problem. You also have mostly to clarify specific technical ques- to be able to judge the validity of result tions. Formerly, he acted as a work team- and the method in relation to the specific application. leader and was to be involved in all re- ported fault complaints. For every type of In the investigation at selected workplaces, blank Annette is controlling, there is a the analytical tool containing the four di- detailed procedure with a specific and a mensions (context, medium, personal inten- general instruction/specification. Reception, tion, understanding and skills) has been control and delivery are all documented shown to be useful for describing the need both on paper and on screen. for numeracy in specific job and work functions. The following description of the job Media: Numbers, formulas and figures are function goods reception and quality found in, or constructed from: (1) Written information and control will serve as an example. communication Context: Annette is an industry operator in 1a) Prose texts: General instructions for a big electronics enterprise. She works in quality control; specific control instructions the department of inward goods and quality (for every blank) control (at present, six females and one 1b) Reference texts: Tables and work Quality also involves accurate measuring. Here the tolerance is 1/10mm 37

drawings (in instructions and specifications), archives, delivery notes, reception reports (on both paper and screen), production plans, labels, specifications on blanks (e.g. date), Vernier callipers, and digital instruments for weight, and other measurements. 1c) Fill-in texts: reception reports (paper and screen), labels (printed on a certain machine) (2) Oral information and communication 1000 small blanks are to be counted out in bags of 2a) Short piece of information: “Who has 250 each. Weighing them helps, but it does not take number 243 343?” into account methodical considerations. 2b) Longer or shorter statement: “Ben has Skills and understandings: Annette needs set up the formula we need in 243 453.” her sense of numbers (every object has a 2c) Dialogue: “Are you driving with 83-23 nine-digit code, so has every task, customer, now?” – “I’m not driving with anything product, specification); measuring; counting now but I’ve put something up.” (manually and by weighing); geometrical (3) Specific materials, time and processes sense (symmetry of the blank, use of work 3a) Specific materials: the blanks drawings); manipulation of numbers (set up (connectors, pins, etc.) and use formulas). Annette has to follow a 3b) Time: normal time, time consumption, certain procedure but she also has to be able production plan (when they need this blank to adapt to any given situation. in the production process) The specific vocational competence gives an 3c) Processes: Fault complaints and rejec- opportunity for flexibility and independence Annette takes a bag containing 9 small connectors tion of a delivery will occur after a judge- for flat cables, which must be checked in relation to ment of the economics and the consequen- various standards. It is a new brand and one of the measurements, taken with a digital slide gauge, does ces for other places in the production not quite fit with the drawing. Thus she draws atten- process. tion to a changing of the documentation. The measurement of the connector is 15.58mm and Personal intention: Annette is collecting it should be 16.00mm according to the drawing. The tolerance is 0.01, but the worker’s experience from pieces of information to be used directly in the production department now benefits her - and the work process, giving information, the factory. She knows that the discrepancy in this measurement and this connector has no practical collecting data on the blanks, controlling significance. Had the measurement been over 16.01 the blanks and their delivery, judging, she would have rejected the items. She can also see on her computer screen that during the day the pro- deciding, reporting, co-ordinating with the duction department will be short of this type of con- needs of the production department. nector and she takes this into account when making her decision.

38______

Many numbers, formulas and charts are used in semi-skilled work and job functions. Advanced mathematics is not needed for them, but mathematical ideas and techniques are employed in complex situations. Internal and external communication through the computer increases the number of factors in the decision-making process.

Numbers and Vocational mathematics in semi-skilled jobs

Development tendencies in the • to use the four dimensions in order to present European labour market towards a detailed description of the work tasks and broader jobs, rotation and autonomous functions containing manipulation of numbers/ groups make new demands on nume- charts, quantification and judgement of quanti- racy. For example, it is not sufficient ties or spatial relations. just to be able to read a graph. An understanding of how it is drawn is also important in order to extract the consequences of the embedded information. This is confirmed by the workplace and student survey in Vocational Mathe- matics. We chose a number of workplaces and followed core employees in semi-skilled job functions. The aim was to describe the workers’ numeracy in order to make a comparison with the demands in AVT-teaching. We wanted to extend our knowledge of the mathematical ideas and techniques employed (or needed) in semi-skilled jobs and how these competences form a part of the general vocational and technical vocational qualifications. In the workplaces the task was • to collect authentic job material which directly or indirectly makes demands on numeracy, Cuttings from written materials used in semi-skilled job functions 40

Experienced needs at work Regularly % Occasionally % Never % Counting (e.g. blanks or money) 75 17 8 Solving arithmetic problems (Adding, subtracting, dividing) 73 16 11 Fill in week notes 62 8 30 Measure lenghts or thickness 61 23 16 Fill in other forms, e.g. requisitions 46 21 33 Read numbers on labels 46 24 29 Mix something in a certain mixing proportion, e.g. liquids or gravel and concrete 44 26 30 Write messages with numbers or drawings 42 28 30 Calculate an area 39 18 43 Use working drawings 38 28 34 Calculate a percentage 38 26 36 Calculate a weight 34 21 45 Calculate a price 31 21 48 Read numbers or diagrams on the notice board 26 24 50 Use formulas 25 23 52 Check of pay slip 70 18 12

In the student survey, 160 parti- out arithmetic problems using the cipants in AVT-courses in the four basic arithmetic operations four lines of business were (73% needed the skill regularly). To

asked about their need for six- read numbers or diagrams on the teen different functional mathe- notice board was only needed by matical skills in their daily 50% of the participants. work. One example of a ques- Underlying the work place survey tion was: “Do you need to were six working hypotheses: calculate percentages?” – Between 50% and 90% of the Working hypothesis 1: students needed the sixteen In every semi-skilled job, problems arise that can only be solved by skills regularly or occasionally. The biggest need was to count quantification and use/evaluation of

blanks or money (75% needed quantitative units. The cash the skill regularly) and to work In the workplace survey all the register is observed job functions required the too slow when many use of and judgement of numbers. people are From the counting of blanks and in line and reporting of work notes, through the the sales ti- measuring and judgement of quality cket func- checks, to the setting up models for tion is fulfilling the required service turned off. grades. Numbers and vocational mathematics in semi-skilled jobs 41

Zone 43 – Zone 1 = 1 x 3 zones punch + 1 x 2 zones punch At the bus stop, the bus driver opens the door and an older woman gets into the bus, together with a lot of other people. She has a yellow ticket coupon in the one hand and a blue in the other hand. She shows the yellow to the bus driver, asking him: “I am going to the Town Hall Square. Is it necessary to add a blue punch to the yellow one?” The bus driver takes the yellow ticket. He studies it for a moment and answers: “Yes, a blue punch.” Meanwhile, 10-15 passengers have pushed their way into

the bus all showing their monthly season ticket to the bus driver. Working hypothesis 2: writing is required. Tasks and functions of semi-skilled Interviews with production leaders and workers require relatively simple formal employees confirmed that the attention skills and understandings in mathematics given to mathematics is connected with but, informally, numeracy is developed in writing - for example, when working for a complex working situations and the use of quality certification where the work pro- mathematical ideas and techniques is cess and quality checks are to be advanced. documented. The mathematical techniques used by the Working hypothesis 4: observed employees were relatively basic. There are systematic differences between Many of the arithmetic problems are mathematics in the workplace and questions of learning by heart in the sense mathematics in traditional teaching. that the same numbers are multiplied or added up every day in the given work The survey supported this working hypo- function. These employees did not use thesis. In work, numbers and arithmetic advanced mathematical knowledge in problems are to be constructed, while they order to handle the tasks and work are given in traditional mathematics in- functions but, on the other hand, their use struction. Solving the problem has a prac- of mathematical techniques was advanced. tical consequence in work, but not in teaching. In work the problems are deter- Working hypothesis 3: mined and structured by technology while, The need for arithmetic and mathematics on the other hand, the mathematical pro- is not discovered at the factories until blems determine the course of teaching. 42

Working hypothesis 5: architects etc.). The same attitude While semi-skilled workers was discovered in the interviews with students at the AVT-centres. think that mathematics is very important in the labour market, Working hypothesis 6:

they do not regard mathematics Semi-skilled workers are not con-

as something of personal scious of their mathematics acti- relevance to them. vities in their daily work and, thus, Several observed employees of their ‘mathematical’ competen- were surprised that we could be ces. This consciousness only interested in their job since it appears in a situation where there

did not contain any mathema- is a job they cannot manage due to tics. As they said: “Too bad that their lack of mathematics skills. you are here now, because Working hypothesis (6) is connected nothing interesting is going on.” with (5) and similar comments can Beforehand, the production be made. The workers often came up leader in a metal factory had with answers like: “That is simple

decided that we were to follow common sense” – “that is logical” or employees by turn because their – “it is as plain as your face”, in work was too routine. In the response to specific questions on breaks the groups automatically their use of mathematical ideas and

started to talk about the project techniques when solving tasks. and numbers, formulas and Another answer could be: “I’ve diagrams in the work. Many never considered that.” In the check of quality at an electronics were of the opinion that the AVT-courses contained too enterprise, the employees considered

much mathematics. It was a the technical support person as the widespread attitude that worker with the mathematical skills: mathematics is for “the others” He set up formulas used for the (production leaders, engineers, measurements when the other employees could not manage to “A bucket of figure it out. concrete” of Lacking consciousness of their own ¾ cubic numeracy can lead workers to lack metres is the self-confidence in relation to unit of mea- arithmetic and mathematics. They surement at know what they cannot do, but they the construction site. do not know what they actually are able to do. Numbers and vocational mathematics in semi-skilled jobs 43

Crane work can be dangerous. For safety’s sake, arithmetic skills are crucial.

In all construction work, there is crane work which is risky for people and materials. Thus certification is required. Decisions built on an understanding of numbers, geometric sense and calculations are to be made every time something is to be allocated. A right or wrong decision can be a question of life or death.

The correct gear has to be chosen for the task. The crane driver needs the ability to read and understand lengths, volume and weight of the blanks. But, most commonly, these pieces of information are not given. He has to make a rough estimate of the weight if it is not given. Finding the volume is a complicated matter. Maybe the crane driver knows the specific gravity by heart or else he has to look it up in a table. Then the volume is to be multiplied by the specific gravity and the units have to fit.

All parts of the crane have to have a With two straps the crane driver can lift distance to fixed objects of more than heavier goods than with one strap 0.5 metres while the crane is working. alone. How much depends on the So the crane driver must have a certain spreading angle of the straps. The two sense of a distance of 0.5 metres. straps can carry four thirds of the so- When hitching long goods the straps called WLL (=Working Load Limit) when have to be fixed according to specific the spreading angle is between 30 and ratios: 90 degrees. They can carry double WLL if the angle is smaller than 30 degrees.

Also the gear has to be examined. Calculations of five per cent, ten per cent and a third are part of the safety regulations.

Correct hitching of long goods

The Adult Vocational Training [AVT] course is filled with numbers, formulas and charts. They are all through the general and specific vocational instruction and in the theoretical and practical tasks. But the teachers are not really guided by educational documents or teacher training in their organisation of the arithmetic and mathematics teaching.

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Numbers and Vocational mathematics in AVT-teaching

In many places, the mathematics component of Column up, column down AVT-teaching is driven by tasks like those In many textbooks, you find droves of found in traditional school teaching. exercises. According to the vocational It seems natural to include calculation teachers, many of the students feel comfortable with the textbooks because exercises in educational documents and they resemble what they know from courses. But workshop teaching, which school. includes the solution of practical tasks in workplace contexts, is different from solving mathematical tasks in isolation from any context.

Tasks are found in teaching and at work. In teaching it is important to distinguish between exercises and problems in order to determine the aim of the task as an educational tool. A task is a routine task at work and an exercise in teaching if you can handle it immediately and without complications. The students can use it to practise skills in mathematical ideas and techniques. When you cannot handle the task immediately it is a non-routine task at work and a challenge and a problem in teaching. Problem solving can be used as an occasion for the participants to acquire new mathematical ideas or techniques or to simply learn how to use what they already know.

Exercises for acquiring and practising skills A routine Tasks in the form of exercises are widespread in AVT- task in the packing of teaching. In the course “Understanding of drawing”, the blanks allows main educational idea is for the students to learn to under- for practice stand others’ drawings by mastering the techniques. The of the twenty individual student learns through a long introduction. three times The teacher is the master; the student is the novice or table.

trainee. This is reflected in both teaching and teaching

materials. The master personifies the objective of the course because he masters what the trainee wants to learn a little of. The master sets the trainee minor tasks and gives instructions and thus lets in the trainee in his world little by little. The trainee undertakes the small tasks one by one, and s/he never knows what will be the next task. S/he completes the tasks by the master’s directions and

receives critique instantly. If s/he did well enough s/he solved and the solution is used to get will be given the next exercise. If not, s/he has to do it on with the work task. When the task once more. (Lindenskov & Wedege,1998) is routine the students will barely The mathematics-containing instruction in AVT can notice the mathematics in the task be described as an activity where, in order to solve a unless it is pointed out by the work task, the participants are constructing calcula- teacher. But when there are problems tions or charts by means of mathematical ideas and with solving a task and the reason is techniques. Subsequently, the mathematical task is a lack of mathematical skills, then

The mathematical formulas always reduce real life students acknowledge that “I am not problems to those parts which can be quantified. able to do mathematics”. The result of a mathematical task also has to be evaluated by taking the solution of the work task into account. We only The functional skills and under- encountered once any educational material which considered standings in mathematics, cal- the final use of the results. That was in the document: led “numeracy”, enter into “Ordering of goods in the shop”. The following formula is both general vocational qualifi- used for deciding how many units to order per consignment: cations (methodical, organisa- 2 x sale x purchasing costs per consignment tional and communicative) and price per unit x rate of interest (decimal) specific qualifications. Further- It says in the educational instruction that the teacher should more, specific qualifications discuss the limitations of the formula in the classroom. For can require a specific mathe- example, the formula does not include a discount for quanti- matical-vocational competen- ty and, furthermore, it can be difficult to decide the accurate ce, e.g. knowledge of certain purchasing costs for every order in advance. formulas or skills in certain geometrical constructions. Numbers and vocational mathematics 47

Problem solving in order to test understandings and skills in a new context

Tasks which work as problems for the have to be employed with care when participants are found in the Danish AVT calculating store and shortage costs. - e.g. project tasks and management The game gives the mathematics games. The course “Logistics and Co- elements a motivating authenticity. operation” contains a management game Furthermore, it provides an occasion for with many numbers, calculations and co-operation among the students, and geometry. Throughout the game the stu- between the students and the teacher dents experience the connection between about quantity, signs, calculations, the different departments within a firm graphs, diagrams and spreadsheets. The and the possibilities of optimising students who know the mathematics production under different forms of elements beforehand can practise and organisation. In the game, one or two strengthen them in a work relevant students have the responsibility for the context. For the students who do not work done in every department and for know the mathematics elements documenting that work with numbers beforehand, the game works as an and graphs. The communication between introduction to the elements and a the departments follows fixed rules. All motivation to learn them later on. students participate in balancing the ac- The game makes the mathematics visible counts of the department, plot them in a and contributes to the consciousness of spreadsheet on the computer and eva- mathematics as an important tool for luate the economic result. Production, decision making in production and work management and accounts are organisation. But there is no room or mathematics-containing. There are time allowed in the game for the several difficult mathematics elements, acquisition or practice of new skills. such as graphs. Some of the simpler (Lindenskov & Wedege, 1998) elements, e.g. addition and subtraction, Mathematics instruction in function they are aiming at. While a subject Adult Vocational Training can like “area/volume” is directly relevant to a be either general vocational number of job functions within the cons- or specific vocational. General truction, clothing, metal, cleaning and trans- vocational instruction can ap- port industries, not every office worker needs pear as a general subject that is relevant for “everybody” in to know how to calculate an area. But a the labour market or across functional skill like calculating areas and one or more lines of business. volumes can give flexibility in the job For example, the subjects known as construction for an office assistant in a “the four arithmetic operations/ production firm or in a transport enterprise. pocket calculator” and “diagrams/ The specific vocational instruction is tables” are relevant to all AVT- either vocational mathematics for participants no matter what job application in a specific work 48

function or practical arithmetic bute to the vocational courses of which enters into the solution of a education for qualification of semi- vocational task. The vocationnal skilled adults. Each contributes in the mathematics is visible: It is on the time- areas where they are better. As a subject table in vocational education, and it is in Formal Adult Education (FAE), explicitly formulated (e.g. “calculation mathematics is founded in the school of use of materials”, “dosing”, “calcula- subject and in the discipline of mathe- tion of speed”, “right-angled project- matics, but the teaching is based on adult tion”). The practical calculations are teaching and learning principles. In FAE done “at a corner of the work table”. the starting point for the curriculum is They are invisible except for situations mathematics as a system and method, where something goes wrong and then while in AVT the starting point is mathe- the blackboard/whiteboard is used. matics as a tool. This fact can lead to Mathematics is a central subject in the some confrontations when combined cross-sectoral co-operation between the with the fact that the students do not Danish AVT-centres and the Adult choose AVT courses in order to learn Educational Centres. The basic idea is mathematics. that the two education systems contri

Two different conceptions of mathematical knowledge In a cross-sectoral course a FAE mathe- We join the class at the point where the tea- matics teacher and an AVT specialist teacher cher, having reached agreement with the are teaching the same group. The objective participants on the "diameter", is now pre- is to ensure that the participants in one and paring to calculate the size of each edge that the same course achieve a formal general is, the base of the equilateral triangles. competence in mathematics and a profes- One of the participants spontaneously leaves sional competence in the industry: During the room with a remark to the effect that she the preceding week, the participants had is going down to measure how long they are. been working in the cabinet making work- The teacher continues to explain how the shop. They had marked and sawed out a baseline can be measured. In the middle of table top shaped like an equilateral octagon. all of this, the participant returns. The teaching of mathematics was therefore The teacher breaks off, asking: “Well, how concerned with octagons, including the equi- long was it then?” “It was 43.3, that is one of lateral triangles that they are composed of. them ... the other was 44.2!” The teacher: “But it can’t be right! They ought to be the same size.” Participant, quickly: “But it’s true.” “Alright, if you have measured it we’ll write it”, answers the teacher obligingly. Whereupon the participant spontaneously retorts: “Here we go again! You’ve got your theories; it’s different in practice.” And there is scattered applause and remarks by the

audience. (Scavenius & Wahlgren, 1994) Numbers and vocational mathematics 49

The students are the cause of the pro- tary teaching in the workshops. blems with numbers and mathematics “Theory-practice” in the classroom, according to the 1. A key point in the learning process vocational teachers responding to the is the students’ conception of the questionnaire in the AVT-centres. It consequences of the theory for is the participants who need the basic practice. – Some students think skills. The participants have problems that theory does not tell them handling the relation between theory anything about real practice; others and practice. The participants lack think that the theory and the self-confidence and have mental practical experiences are identical, blocks when it comes to numbers and practice just being more mathematics. complicated. 2. An important point for motivation Other reasons are pointed out in Lena is the students’ conception of the Lindenskov’s student survey, which authenticity of materials and includes observations and interviews activities. Are the medium, the concerning AVT-participants’ experien- method and the personal intention ces and potentials in vocational mathe- all authentic? matics. This survey suggests that the 3. Crucial to motivation are the teachers lack vocational and educational students’ longer-term perspectives. skills. Utilising the numeracy concept as Will there be an opportunity to defined previously, and the four dimen- gain individually or collectively sions (context, media, personal intention, from the new theoretical dimension skills and understandings), teachers of their numeracy in future work? might attend to the following points: “Self-confidence and blocks” “Basic skills” 1. The students’ self-confidence is 1. Some students have basic skills and decisive and it is affected potential based on numeracy throughout their whole experience developed in other contexts which is of education, including the current not exploited in the teaching. course. 2. Some students have a need for spatial 2. Some students have blocks caused conceptions and images. Characters by their educational backgrounds and letters on paper and screen alone in particular. – Some students are insufficient and other media must show resistance provoked by the be involved in the teaching. actual situation in the classroom 3. There is a need for differentiated and also by perceptions of their teaching and better exploitation of the future prospects. possibilities of individual supplemen- 50

When accommodating mathematics teaching within AVT, the aim is to give the participants the possibility of qualifying for their contemporary or future jobs. On the one hand, the teaching has to be relevant to the needs of the qualifications in the respective lines of business and the expected technological developments. On the other hand, the teaching has to be relevant to the individual participants’ need for qualification.

______51

Relevance and

visibility of mathematics

In Vocational Mathematics, the connection When attending AVT, the students’ aim between teaching, students’ needs, and the is to gain a vocational qualification and requirements of the workplaces has been not to learn mathematics. On meeting illustrated in the respective surveys. The mathematics in their vocational educa- need for numeracy in semi-skilled jobs has tion some react with resistance or been compared with the requirements in blocks. In AVT there is the possibility AVT-teaching, and AVT-students’ difficul- of breaking with tradition in the mathe- ties with numbers and mathematics in the matics teaching, but it requires the vocational teaching have been considered teachers to have both vocational and from different perspectives. educational qualifications. The surveys showed that the numeracy With technology, formulas and needed in the labour market is different mathematical competences are usually from a pure understanding of numbers only visible when the technology and mathematical skills. An important changes. The worker is only conscious part of the competence is to be able to of his/her use of mathematical tech- apply numbers to real life, as there are no niques or numeracy when the daily mathematical “exercises” laying around routine is disrupted by a new problem. If in the workplaces. Here, the semi-skilled the teacher does not draw attention to workers use mathematical ideas and tech- the mathematics in practical exercises, niques in a specific work context with a the students will not change their starting point in different media and with attitude towards mathematics as some- a specific intention. That is important for thing they cannot do. what competences they need in a particular sense. The solution of a work task The three surveys provide pieces of always has practical consequences. information for an action-orientated Another difference between the use of answer to the question of how the mathematics on the job and mathematics mathematics-containing AVT-teaching in traditional teaching is “the different can be arranged so that it supports the procedures” compared with “the unique students and gives them the opportunity correct way”. to exploit their potentials. These pieces, 52

which can be subject to further development AVT-centres, vocational schools etc. who through experiments and developmental will evaluate the specific needs of edu- activity, can be characterised under the two cation, describe the aims and scope for main titles of “relevance” and “visibility” of the educational courses, prepare educa- mathematics. tional materials and are responsible for

the teacher training, Relevance • teachers at the AVT-centres, vocational The mathematics teaching in AVT has to be schools, adult educational centres etc. who relevant to the participants’ vocationally plan and carry out the actual teaching, oriented needs for learning. Thus, the content • participants in the AVT-educational has to be prioritised and arranged carefully in courses who bring personal motivations order to achieve the action-orientated aim of and perspectives to learn (or not to learn) the education. The relevance of the mathematical ideas and techniques has to be visible • workplaces whose employees participate to the participants both in the teaching acti- in the AVT-educational courses and who vity and in their student materials. hence provide (or not provide) them with the possibility of training and job development in connection with their education. Visibility This publication about numbers and voca- Use of mathematical ideas and techniques tional mathematics in AVT can be concluded has to be visible in AVT-teaching. The with some good advice for teachers. teaching must be arranged such that the students’ everyday competences for em- ploying these ideas and techniques in the The teachers’ job is to turn the educational work (their numeracy) are made visible. In documents into mathematics-containing this way it can overcome the problem that teaching, giving the participants the oppor- students solely consider mathematics as that tunity for engaging in relevant learning which they cannot do. processes. The teachers have direct contact with the participants and maybe with the workplaces as well. Founded in the surveys When implementing the two principles of and other international research we have the relevance and visibility in the mathematics following advice for teachers: teaching it is worth remembering that the aims and scope in AVT are set up by dif- (1) Remember to clarify the relevance and ferent types of players: connection between the mathematical subjects and the vocational qualifications • educational planners in the labour market which are the aim of the course. Allow and education authorities, and in the

Relevance and visibility of mathematics 53

the students to exchange experiences about (6) Remember that the students learn in dif- the contexts in which they are going to use ferent ways, and this requires the use of their qualifications. different types of media and materials in the (2) Remember that the personal intention in instruction. When a new mathematical con- the workplace is never just to do arithmetic. cept is introduced, more than one sense There is always an intention beyond just should be stimulated. finding the right answer; namely to make a (7) Remember that the students’ blocks report, check the quality, evaluate, co-ordi- towards numbers and mathematics do not nate, etc. Thus, it is an important activity for always originate from negative school the students to formulate tasks in different experiences. They can be an expression of contexts. adults’ resistance towards learning some- (3) Remember that an authentic problem is thing new. A resistance is specifically acti- not sufficient. The method has to be authentic vated if the relevance of the mathematical too. Study the workplaces for typical proce- ideas and techniques are not made visible in dures and techniques and ask the participants the instruction. about their own and their colleagues’ usual (8) Remember to exploit the opportunity methods. for dialogue with the students; also about (4) Remember to clarify the aim of the their attitudes and barriers towards specific task for the students (and with them); mathematics. In certain classes a question whether it is to revise skills and under- about their attitudes towards mathematics standings, to reinforce existing skills or to can lead to fruitful conversations which learn something new. Whether the individual may help relieving the blocks and participants consider a task as an exercise or increasing the inclination to learn. a problem obviously depends on their own personal skills and understandings. (5) Remember that the students have many different methods for calculating and solving specific tasks and make these methods visible in the teaching. From former mathematics teaching, many participants have had the experience that there is only one correct method, namely the teacher’s. The students can enjoy a new, positive experience if the teacher shows an interest in their methods as well.

Literature

(…) refers to the chapters in “Vocational Mathematics”. AMU-direktoratet (1988). Almene kvalifikationer og uddannelsesmæssige konsekvenser inden for specialarbejderuddannelsen. Rapport fra arbejdsgruppen vedr. almene emner i specialarbejderuddannelsen. [General qualifications and consequences for vocational training of semi-skilled workers.] Copenhagen: AMU-direktoratet. (Chap 2 & 7) Arbejdsmarkedsstyrelsen (1996). Projekt Faglig Profil i Matematik. Delrapport 1. [Project “Mathematical profile”] Copenhagen: The Labour Market Authority. (Chap 2, 4 & 7) FitzSimons, G. E. (2002). What counts as mathematics? Technologies of power in adult and vocational education. Dordrecht: Kluwer Academic Publishers. (Chap 3, 6 & 7) FitzSimons, G. E. (2006b). Divergence and convergence in education and work: The case of mathematics and numeracy. VET & Culture conference 2006: Divergence and convergence in education and work. Retrieved September, 2007, at www.peda.net/veraja/uta/vetculture. (Chap 3, 4, 6 & 7) FitzSimons, G. E., & Wedege, T. (2007). Developing numeracy in the workplace. Nordic Studies in Mathematics Education 12(1), 49-66. (Chap 6) Lindenskov, L. (1996). "Det er fordi jeg mangler billeder ..." - AMU-kursisters oplevelser og potentialer i faglig regning og matematik. [“I simply miss pictures…” – AVT students’ experiences and potentials in vocational arithmetic and mathematics] (Chap 4, 6 & 7) Lindenskov, L. & Wedege, T. (1998). FAGMAT - Rapport fra et analyse- og udvik- lingsprojekt om tal, formler og figurer i arbejdsmarkedsuddannelserne. [This is the research report from project “Vocational Mathematics”] Roskilde: Roskilde Univer- sity, IMFUFA Text no. 349. (Chap3-7) Lindenskov, L. & Wedege, T. (2001). Numeracy as an Analytical Tool in Adult Education and Research. Centre for Research in Learning Mathematics, Publication no.31, Roskilde University. Retrieved September, 2007, at www.statvoks.no/emma/. (Chap 5) OECD (2000). Literacy in the information age: final report of the International Adult Literacy Survey (IALS). Paris: OECD. Retrieved September, 2007, at www.oecd.org (Introduction) 55

OECD (2005). Learning a living: first results of the Adult Literacy and Life Skills Survey (ALL). Paris: OECD. Retrieved September, 2007, at www.oecd.org (Introduction) Scavenius, C. & Wahlgren, B. (1994). Tværsektoriel undervisning. Femte og afsluttende rapport om samarbejdet mellem AMU og VUC. [Cross-sectoral education. Co- operation between AVT and FAE.] Copenhagen: Danmarks Lærerhøjskole. (Chap 7) Wedege, Tine (1995). Teknologi, kvalifikationer og matematik. [Technology, qualifications and mathematics]. Nordic Studies in Mathematics Education, 3(2), 29-51. (Chap 1-4 & 6-7) Wedege, Tine (2002a). Numeracy as a basic qualification in semi-skilled jobs. For the Learning of Mathematics – an International Journal of Mathematics Education, 22(3), 23-28. (Chap 3, 5 & 6) Wedege, Tine (2002b). ”Mathematics – that’s what I can’t do” – Peoples affective and social relationship with mathematics. Literacy and Numeracy Studies: An International Journal of Education and Training of Adults, 11(2), 63-78. (Chap 4) Wedege, Tine (2004). Mathematics at work: researching adults’ mathematics-containing competences. Nordic Studies in Mathematics Education, 9 (2), 101-122. (Chap 5 & 6) Wedege, T. & Evans, J. (2006). Adults’ resistance to learn in school versus adults’ competences in work: the case of mathematics. Adults Learning Mathematics: an International Journal 1(2), 28-43. Retrieved September, 2007, at www.alm- online.net (Chap 4)