Vocational Mathematics

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Numbers and mathematics in AVT___________________________________

Vocational

Mathematics

Tine Wedege

Translated by Gail FitzSimons and Anna Folke Larsen

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Vocational Mathematics

Tine Wedege

Translation: Gail FitzSimons and Anna Folke Larsen

Photos:

2Maj/Mira cover

Dreamstime p. 14, 26, 44 Apart from these: Tine Wedege

Drawings: Anders Folke Larsen

First edition 1998:

Fagmat – tal og matematik i AMU.

The Danish Labour Market Authority, Copenhagen

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 educa-tion 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 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

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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 rea-ding, 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.

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Introduction

Why the project “Vocational Mathematics”?

At Grundfos, one of the world's lea-ding pump manufacturers, the semi-skilled workers receive a special offer to learn mathematics (at the AVT-centre or the Adult Educational Centre). One aim is to qualify the em-ployees for education and work; a second aim is that a number of em-ployees with blocks in the subject can move on. The Head of education at Grundfos has not been in doubt that the basic qualifications in mathema-tics are necessary in the labour

market today: “They are necessary in order to enter work. That is one hundred per cent sure, as calculation has not been automatised: the em-ployee has to decide upon the

numbers coming out of the machine.” In the 1990s, large national and

international surveys of adults’ literacy took place. Corresponding surveys did not exist in the area of calculation and mathematics, but it was striking that some of the reading tasks causing especial difficulties among both semi-skilled workers and others

presupposed that the reader had an understanding of numbers and the skills of reading maps and tables.

One could be led to ask: “But aren’t we well of without these competen-ces?” Yes, it is accepted that many adults are immensely competent in their work function after many years of experience, even though they do not have formal qualifications in mathe-matics. If you do not feel comfortable with numbers there are many strate-gies for avoiding them in the everyday life, but it will constrain your flexibi-lity. Furthermore, quality certification in the workplace implies rigid require-ments on the employees’ literacy and arithmetic skills.

The teaching in most AVT-education contains some calculation and voca-tional mathematics of some kind or other - either as general arithmetic in special modules or fitted into the voca-tional teaching, or as vocavoca-tional arith-metic integrated into the technical-vocational teaching.

Within a number of areas during the past 15-20 years, there have been new/stronger requirements for the participants to have mathematics-containing competences in the Danish AVT-education. For example, this applies to the courses in sewerage, CNC-turning, techniques of measure-ment, and logistics and co-operation.

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This can lead to some vocational and educational problems in teaching; but also in other places with no new requirements, problems can arise in the classroom. A vocational teacher expressed it this way:

“Calculations and mathematics are considered as a problem by incredib-ly many students. “I’m simpincredib-ly not able to do anything with numbers.” They give up beforehand. Maybe they have had bad experiences with mathematics... It is women my age who left school as thirteen-year-olds. They don’t have faith in themselves and they are left out. Here is a speci-fic example illustrating the problem: A pattern for a skirt is to be made with a waist of 100cm. The 100cm has to be divided by four. A typical student’s reaction is: “No, I can’t

figure it out. I’ve never been good in maths.”

Many adults have a frozen attitude towards mathematics. Some people call it mathematics anxiety, while others talk about blocks towards numbers.

What is project “Vocational Mathematics”?

Vocational Mathematics is an analysis and development project about num-bers and vocational mathematics in Adult Vocational training. We have posed three main questions in Vocational Mathematics:

• What skills and understandings in calculation and mathematics are needed in the semi-skilled jobs compared with the requirements in the AVT-teaching?

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• What are the difficulties in

numbers and mathematics that the AVT-students encounter, why do the problems arise, and what are the implications for those returning to vocational education.

• How can the mathematics-containing AVT-teaching be arranged such that it supports the students and provides them with opportunities to exploit their potentials?

In order to shed light on these ques-tions, project Vocational Mathema-tics consists of three surveys:

(1) The teaching survey on numbers and vocational mathematics in AVT-teaching in selected education pro-grams for semi-skilled workers. The survey consists of observations of the teaching in AVT-centres and exami-nation of educational documents. (2) The student survey on the AVT-students’ general understanding of numbers, arithmetic and mathemati-cal skills, their experienced needs for using these skills in the workplace, and their attitudes towards numbers and mathematics. The survey consists of qualitative interviews with 45 stu-dents at AVT-centres and structured interviews with 160 students at a commercial school and four AVT-centres.

(3) The workplace survey on the use of numbers, charts and formulas in semi-skilled job functions in a number of selected workplaces. The survey consists of observations and short interviews with nine core employees and an examination of seven existing qualification analyses.

In the surveys we were interested in numbers and vocational mathematics within the following four areas: Construction, commerce and office work, metal and transport.

Furthermore, our starting point was a questionnaire survey among vocatio-nal teachers at the AVT-centres. The survey was carried out in Spring 1995 in the framework of the cross-sectoral development project “Profile in Mathematics of Adults,” where the teachers were asked to evaluate the participants’ arithmetic and mathema-tical skills in relation to the vocational teaching.

This publication is not a report on the three surveys but it is based upon the findings and on international research on adults, mathematics and work. By the translation in 2007, we have up-dated the text and generalised it to the broader European context whereever possible.

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

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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 technological development.

But, how is the teacher to judge what 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 to price a job – because “they don’t need that in the typical cleaning job”, as she says.

The tools are a setsquare, ruler, compass and pocket calculator in the small workshop.

The most important tools for controlling the storeroom of the machine shop of a big firm.

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10 Numbers and technology

The extent to which the education system should react exclusively to the demand for qualifications or whether it also should have an active function is a political question. One of the central education questions is about the need for qualifications. In Adult Vocational Training, the aim is to qualify the workforce to meet the needs of both the labour market and the individual, in line with technological development.

The participants might achieve the formulated objectives of the course, but what about their personal needs and further prospects for work and

education? What about the possibility

of them taking an active part in their job reorganisation, at some time in the future? 600 years ago the first mechanical clock was constructed. Up to that point all technical instruments had been an extension of the human arm - assisting or replacing human labour (e.g. the plough, the mill, the weaving loom). The clock was the first mechanical device. With this invention, time was divided into

random units (hours, minutes, seconds), and in Europe time gradually came to be perceived as the sum of these units. The mechanical clock extends the area for measurement and quantification: space, weight and time. Precise measurement of time became one of the central elements in the organisation of social and working life.

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Calculation in Midland Bank 1929.

Although it would be several hundred years before the clock became an everyday possession, it gradually

changed the relationship between human beings and reality. It introduced

objective measurements and made

possible objective mathematical rules for dealing with time. Time became an authority.

Twenty five years ago IBM released the first personal computers [PCs] onto the market. Today there is a PC in more than 50% of European homes and a simple pocket calculator is often given away when one buys other electronic products. In the space of less than a quarter of a century an infinite volume of calculation has been taken over by computers – at the same time, adults’ need to use

mathematical ideas and techniques has also changed.

While in its time the clock introduced and made visible figures for time, mathematics has now been made invisible by the new information technology. When a spread-sheet is first set up, the formula and calculations are hidden. It is only when reorganisations are necessary that formulas and mathematic-cal knowledge become visible. It is only when the daily routine is broken by a new

problem that the worker becomes conscious of his/her use of

mathematical ideas or techniques.

Changing technology and organisation of labour are important for the demands made on mathematics-containing qualifications. We will look further into this in chapters 5 and 6.

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12 Numbers and technology

Mathematics is hidden in the technology

In an airport, different functional groups involved in the handling of luggage co-operate by means of computer and telephone. When the aircraft is being loaded and

unloaded, the loading group and the load planner are in constant

computer contact. In the loading instructions, the planner has placed baggage, cargo, and mail in the four cargo compartments in front of and behind the wings. The ideal balance factor (38.0) and the limits

(5.9/51.6) also appear on these instructions. The loading group reads the balance factor of the aircraft on the screen during loading: In the loading report the actual balance factor for this specific aircraft is 28.2. Entering the distribution of weight between the four cargo compartments into the formula for the balance factor is not required. This figure is automatically

calculated when entering the cargo and the weight on the computer during loading.

When decisions are being made at the airport about loading an aircraft, the priorities are: 1) safety, 2) keeping to the timetable, and 3) service. Time is often scarce when loading and unloading an aircraft, and keeping to the timetable means that some cargo may have to be sent on a later plane. Meanwhile, giving safety first priority can lead to the flight being delayed, and the

level of service may not be so high if the balance factor is out. The fact is that safety considerations can mean that all the planned cargo has to be loaded even though it cannot be done on time. It is within the foreman’s competence to release the aircraft, but he has to consult the loading planner before taking any decision not to pursue the original

instructions.

Many years of experience give the foreman a background for judging weight and weight distribution, but in order to gain the formal competence – and maybe the real responsibility in a reorganisation of work – knowing the formula and being able to handle it is essential. This is one example where the necessary qualifications cannot solely be found in the

technique. On the computer one can find the balance factor of the actual load. This means that without

knowing the formula the foreman has a measure for what will happen if they leave out freight which is included in the instructions. In principle, he could take this type of decision, but if extra freight that is not mentioned appears, he cannot forecast precisely what will happen to the balance factor – if the freight is loaded in compartment 4, for

instance. The foreman would have to know and be able to use the formula for the balance factor, if – with the same technique – he was also the decision maker.

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

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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 infor-mation technology. In order to under-stand 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.

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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 context. Where adults have insufficient reading and writing skills in comparison with the demands of everyday life, they are offered reading courses by the Danish Ministry of Education.

Blanks from the supplier are received and have been checked for quality control. Now a report is to be made.

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

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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 methodical, organisational and commu-nicational qualifications/competences as well as in the specific vocational

competences.

The concept of qualifications gives us an opportunity to work professionally with the relationship between education and work. The advantage of separating out a specific qualification and calling it “understanding of numbers and matical skills” is that the specific mathe-matical competence becomes visible. The disadvantage is that it is isolated from the other competences which can lead to the belief that the demands of

qualifications can be met through traditional mathematics teaching.

In Vocational Mathematics it is a fundamental assumption that functional understandding of numbers and arithmetical skills is a basic competence in line with reading and writing skills in a number of common and specific qualifications in the semi-skilled labour market. A mathematical understanding or skill is functional when the mathematical ideas or techniques can be used for solving a task or a problem in everyday life.

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Functional mathematics and reading skills 17

Skills, understandings and attitudes towards mathematics are parts of workers’ general, specific and perso-nal qualifications

Thomas is a CNC operative in an auto-nomous production group at a metal company. There is no job rotation at the lathe he operates and this suits him very well. When he is checking the objects that are turned, he reads a graph on the screen where he evaluates whether the finished object fulfils tolerance require-ments. He can also see whether produc-tion is stable, and this can have implica-tions for the tools and the number of objects to be checked.

There is a graph on the notice board of the production groups showing tables and graphs of the sickness statistics for all groups. The graphs also compare average absence due to illness for each depart-ment and for the company as a whole. This absence rate is up over 8% in Thomas’s group while the average is be-low 5% for the month of October.

During the break Thomas speaks with the other members of his group. He says: "That was me. That month in hospital.

You can see it." A long term of illness for one person affects the average. The group as a whole understands this but it is of no interest to Thomas how the fi-gures are calculated or the graphs con-structed. Actually, his attitude is that all these statistics are something the ma-nagement sit doing in the office because they do not have anything else to do. There are graphs showing the service grades for each of the groups on a joint notice board in the department. At the end of November Thomas’s group is 45 hours behind. The service grade is down to 80. The production leader suggests that they should organise the work in shifts so that they can come up to 100 during December and not work between Christmas and New Year. Thomas takes no part in the group’s conversation about organising the work so that the service grade can be maintained, and he has no intention of doing so. He just knows that it is a matter of working hard.

(The story has been constructed on the basis of authentic material and situations from different companies).

Sickness statistics 0 1 2 3 4 5 6 7 8 9 Jan . Mar ch

May July _Sept. _Nov.

%

Dep. D Group 2 Company

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The square is constructed at the bottom of the hole

There is an important difference in examining the different requirements for functional skills in reading and calculations/ mathematics. The writ-ten or “reading” texts are out there on view in workplaces while the “arith-metical problems” are not just lying there waiting to be solved. Though the texts are teeming with numbers, plenty of numbers will only be vi-sible once they are constructed (e.g. measurement of pipes and slabs or a count of blanks). The mathematics is often hidden in the technology and, usually, formulas are used uncon-sciously of the applied mathematics embedded within them.

This means that the adults’ compe-tences of solving school mathematics

problems are not necessarily connected to their skills in using data, formulas, and graphs in the work context.

Calculations at work can be more com-plicated than in the classroom. Not be-cause of the degree of difficulty, but due to the complexity of the situation. In the project, we asked a group of vocational tea-chers at AVT centres about their opinions on students’ attitudes to arithmetic and mathematics, and they identified one problem:

• The students consider theory as one

thing, practice as another. Some students

can solve problems in the school-based material, but they are not able to use the theory and methods in practice. Their skills are not functional. Other students experience just the opposite.

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Functional mathematics and reading skills 19

Meanwhile the teachers pointed out another problem:

• The students lack fundamental skills

in arithmetic/mathematics.

Many teachers judge that the lack of fundamental skills causes problems in vocational teaching and learning. Forty-five vocational teachers drawn from six AVT centres covering ten lines of business completed the questionnaire about their opinions on students’ atti-tudes to arithmetic and mathematics. For instance, they were asked to consider and answer the following question: Do you experience students’ skills in arithmetic and mathematics as a pro-blem in your vocational teaching? The vocational teachers answered: Yes, in most courses 15

Yes, in some courses 20 No 10

The same question was answered by 31 vocational teachers at a conference of the Business Committee of the Metal Industry.

The vocational teachers answered: Yes, in most courses 14

Yes, in some courses 16 No 1

Thus, the vocational teachers are finding that participants’ arithmetical and

mathematical skills are causing

difficulties in their vocational teaching.

The score rates of semi-skilled students in eight mathematics-containing exercises 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

1-2 correct 3-4 correct 5-6 correct 7-8 correct

This finding is supported by the student survey in Vocational Mathematics where students face problems in arithmetic and mathematics: 12% of the 108 semi-skilled students only succeeded in 1-2 exercises out of eight. 17% gave correct answers to 3-4 exercises; i.e. 29% – more than every fourth participant – only did well in half or less of the exercises. The eight exercises dealt with adding up prices (estimates), comparing prices, calculating areas and doing calculations with numbers. The type of the questions was: “Can you buy the goods on the receipt with 10 Euros?” – “Which is the cheapest juice?” – “How big is the area of the wall?” – “Find 30% of 150 Euros.” The questions were of the type that we may find everyday, at work and in education.

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There are systematic differences between mathematics in the work-place and in educational contexts.

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Mathematics

in

work

and in

school

“No” is the most common answer to the

question “Do you use mathematics in your everyday life?” This is despite the fact that many of us use numbers and formulas on a daily basis. Most people only associate mathematics with the subject in school or the discipline. That there are differences in mathema-tics at the workplace and in school has been a working hypothesis for the sur-veys in project Vocational Mathematics. The well-known activity “solving tasks” serves as an example:

In traditional mathematics in-struction, the task constitutes a central element and structures the teaching. The task is primarily used to practise skills (use of algorithms and concepts) and to test skills and understandings. Thus, the task is often solved by the individual student and it might be perceived as cheating to hand in a joint solution. The task is formulated by the teacher, the textbook or the program. The task has one correct solution and many wrong

solutions. (Accuracy in school and tole-rance at the workplace are two different things.) Solving tasks has no practical meaning: the results are not used for anything except, maybe, solving more tasks. In the so-called ‘problems’ the task-context is often practical problems, but the aim is to find the correct result by using the correct algorithm, not to solve the practical problem.

In the workplace, the ‘tasks’ result from solving a working task where the numbers are to be found or constructed with the relevant units of measurement (e.g., hours; kg; mm). It is the working tasks and functions in a given

technolo-gical context which control and

struc-ture the process, not the ‘task’. Some of these tasks look like school tasks (the procedure is given in the work instruc-tion) but the experienced worker has his/ her own routines, and methods of mea-surement and calculation. Circum-stances in the production process might cause deviations from the instruction or, for example, that the number of random

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samples in the quality control process is increased or reduced. It is characteristic that tasks are solved in different ways and that different procedures and solutions might be acceptable.

In the workplace solving tasks is a joint matter: you have to collaborate, not compete. Solving tasks always has prac-tical consequences: a product, a working

plan, distribution of products, a price, etc.

In the figure, characteristic differences are listed between mathematics in the workplace and in the traditional mathe-matics instruction the way most AVT-participants encountered it in primary and lower secondary school.

Numeracy at work Mathematics in school

All numbers have units of measurement (mm;

kg; Euros) or refer to something concrete. The numbers often appear as pure 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.

Right/wrong is negotiable.

Accuracy is defined by the teacher. Right/wrong is not negotiable. Solving tasks is a joint matter

- i.e. collaboration.

Solving tasks is an individual matter – i.e. competition.

Tasks are full of ‘noise’.

The numbers are often ‘dirty’. Tasks are cleared of ‘noise’. The numbers are ‘clean’. Reality requires the use of mathematical ideas

and techniques.

Reality is a pretext to use mathematical ideas and techniques.

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

Working tasks are defined and structured by the

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Mathematics in work and in school 23

In AVT-teaching, teachers distinguish between general and vocational calcu-lation activities that can be found “on the timetable” and practical calculation which takes place “at a corner of the work table.” We have chosen to

compare mathematics at work with the traditional instruction. The difference between work and school is distinct, yet the traditional mathematics instruction can both be found in AVT and general adults’ education. Moreover, the adult participants’ experiences typically originate in mathematics instruction where solving tasks gave structure and meaning to the course. This implies that many students expect a traditional mathematics instruction with traditional task solving as a central activity.

When mathematical instruction is integrated within vocational teaching with practical exercises or project tasks it is possible to break this pattern. It can be obvious to the participants that the mathematical methods are useful. The disadvantage is simply that the mathematics can be “integrated away”, so that nobody (neither the teacher nor the student) is conscious about 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 mathematics or his/her own mathe-matics-containing competences.

There is a difference between the ability to calculate in school and in everyday life. There is a difference in the mathematical competence required to solve formal mathematics exercises and the mathematics-containing competence required for solving practical tasks which involve similar types of mathematical ideas and techniques. A person’s abilities are not necessarily the same in all contexts — for example, in calculating percentages. Some people can work out percentages of pure numbers, but not percentages of liquids or prices. Others experience the opposite. It is usual to hear an AVT-student say: “In everyday life I can easily solve this kind of problem but not in the maths class.” In the student survey, we had a closer look at the eventual differences between skills used in solving a practical task which required the use of mathematical techniques and skills, and those used in solving a formal task using the same techniques.

1. Calculate the area of a rect-angle with length = 3.5 height = 2.3 2. Find 40% of 150 3. Calculate 45(2,3+1,5)

The mathematical skills of 160 AVT-students were tested.

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Problem solving

“Milk” was the theme for the Lower Secon-dary School graduation examination (year 9) in 1990. One of the problems was calculating the area of the base of a milk carton.

(Calculation of areas is part of the curricu-lum.) The base is a square and with a length of 7cm, so the area is equal to (7 x 7) cm2₌

49cm2_{. The next problem stated that the milk}

crate is 37.5cm long and 23cm wide, and the question was: “How many milk cartons are there room for in the crate?”

The pupils were not going to use the area of the carton base for anything at all, and definitely not for calculating the number of

cartons in the crate. If they started out by calculating the area of he base of the milk crate [(37.5 x 23)cm2_{= 862.5cm}2_{] and then}

divided by 49cm2_{(the base area of the milk}

cartons) the result would have been 17.6. But, in fact, there is only room for 15 cartons in the crate.

Why, then, are the pupils asked to calculate the area of the base of the milk carton? The reason is solely to show that they know how to calculate the area of a rectangle. In order to have them demonstrate that they know and are able to use the area formula of a circle, some milk bottles are supposedly found among granny’s old things.

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 108 semi-skilled students from among the 160 students surveyed were, general-ly speaking, stronger in practical calcu-lation than the corresponding formal tasks. Thirty-two percent were able to both do a practical calculation of a con-crete wall area and find an answer for a formal area task. Thirty-two percent were neither able to solve the practical nor the formal task. Only 6% were able to solve the formal but not the practical task while 30% were only able to solve the practical task.

There are also skilled workers from other lines of industry and people with incomplete school education or else

theoretical higher education enrolled in the foundational AVT-education pro-gram. The focus of the latter is on for-mal calculation rather than practical. Among the 160 students, 22 had fini-shed a theoretical education. They were generally better at area calculation than the semi-skilled students, but it is remar-kable that the number of students who were able to solve solely the formal task was larger than the number of students only able to solve the practical task. The AVT-students have varied

competences for solving practical and theoretical tasks. These can be addressed seriously in the classroom by using different approaches and methods.

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Mathematics in work and in school 25

Work task in the canteen

The canteen worker has no doubt how many milk cartons there are room for in the milk box. ½ litre cartons are put in the boxes in two layers, with 15 pieces in each layer. She takes them three at a

time to place them in the refrigerated cabinet: “It lies in the hand.” She checks that the delivery matches both the order and the invoice while filling up the refrigerated cabinet. L es s er trained s tudents both wrong 32% practical right, formal wrong 30% formal right, practical wrong 6% , both right 32%

T heoretic ally educ ated

both wrong 14% practical right, formal wrong 14% formal right, practical wrong 27% , both right 40%

- 30% of the semi-skilled students were able to find the area of a wall by measuring and calcu-lating, 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.

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AVT-participants come to mathematics with different attitudes and mixed feelings which are often related to their experiences of traditional tea-ching 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

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Adults’

blocks

towards

learning

mathematics

Many adults consider mathematics as a subject that divides the population into two groups: Those who can, and those who can’t. You often hear a statement like: “Numbers and maths has never been me.” Formerly, the attitude among many mathematics teachers was that being able to do mathematics or not being able to do mathematics was determined by the genes. Strangely enough, many more boys than girls were able to do mathematics.

Many of the older AVT-students in Denmark have experienced other sorting mechanisms. They have never met mathematics as a subject during their school lives. As a rough estimate, this was the situation for a fifth of the participants in the courses in

construction, metal, and transport. On the other hand, they might have come across the subject through their children who were taught mathematics from 1st grade onwards. This forms adults’ attitudes as well.

From their own experience or second hand, students above 30 years of age in particular find that mathematics is an activity without any practical meaning. It is an activity that only belongs in

school or higher education. In both the student and the company surveys, present and former AVT- students expressed the sentiment that mathematics was not their cup of tea, and that mathematics was only for engineers and other technicians.

A story about the formation of attitudes

This is a story from a mathematics class of quite a few years ago. It is about a senior teacher, Mr. Holm, and a bright pupil, Brian. The story could have been taken from the mathematics teaching that many adults have experienced in reality: The class is doing arithmetical problems. Question number 42 tells them to find the price of 5 apples when 3 apples are 45 “øre” (or cents/pence etc.). After philosophising over the real prices of apples for some time, Brian is told off by Mr. Holm:

“Brian, my friend. You’re completely wrong. You’re not meant to think at all. This is an arithmetic lesson. You’re simply supposed to do what I’ve shown you so often. Use the unit! How much is one apple? … Proportionality is what you are expected to know by now, and with that you can solve all eight exercises.

If you start asking stupid questions about trade in apples, you won’t get far in life. Anyhow, you won’t get into one of the big shipping companies or the Civil Service. They only want people with skills and who doesn’t pose questions at the wrong moment.”

(The story is made up by Tage Werner, former senior lecturer in mathematics at the Royal Danish School of Educational Studies)

The story about Mr. Holm and Brian illustrates what has been called “the hidden curriculum”. At the same time as the pupils are taught arithmetic/ mathematics they learn something not written in the official curriculum: Mathematics does not have anything to do with reality, but if you want to get anywhere you have to learn it. The story also shows

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that the teacher’s understanding of mathematics is crucial. Is mathematics presented as a collection of concepts and methods (here, proportionality) that the student has to acquire? Or is mathema-tics an activity that the student should learn how to master?

Another common experience of tradi-tional mathematical instruction is that there is only one correct method, namely the teachers’ method. Surveys have shown that many adults feel guilty when using a method in everyday life that is not to the same as what they have learnt in school. For example, you are taught the formula for calculating percentage: 30% of 1200 = 1200/100 x 30 A common everyday method is to find 1/10 of 1200 and then multiply by 3. If the percentage is 25, you can use the same method and make use of 5 being half of 10. You can go far with this prac-tice as long as the percentage ends in 0 or 5. Some people might know that 25 is exactly a fourth of 100 and simply divide by 4. Others know how to use the percentage button on their pocket

calculator.

The student survey in FAGMAT docu-ments that the AVT-participants use various methods for calculating, all of which can lead to a correct result.

Among other things, the 160 students were asked to make a rough calculation of the total from a supermarket bill with-out using a pocket calculator. Some people started by making a rough

estimate of the big amounts and afterwards had a look at the small amounts. Some started by adding up the tens in succession, then they added up the ones and finally the cents, while others summed the exact amounts in

succession.

In the questionnaire circulated among voca-tional teachers at the AVT-centres, only 2 out of 76 teachers answered ‘no’ when asked whether they experienced that the students had blocks to numbers and mathematics when starting the course. Between a third and half of the teachers answered that they had experienced students with blocks in most courses.

From the interviews conducted at the AVT-centres we gained a background for distin-guishing blocks and resistance. Student blocks can originate from previous nega-tive experiences with mathematics, either in the teaching or because the student was sifted out through testing. Resistance to learning mathematics can be related to the adult’s self-perception as a competent person coping with the challenges of his/ her work life without the use of mathema-tics. But resistance can also originate from experiencing that mathematics is of no use outside of school.

Authentic problems with non-authentic solutions can provoke resistance to learning. The distinction between different kinds of authenticity should be made carefully. The problem might be authentic while the solution method is not. An example of an authentic problem is to choose where you want to rent

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Adults’ blocks towards learning mathematics 29

bicycles for your vacation, according to your own needs. (See example).

When students attend AVT, their aim is to gain a vocational qualification not to learn mathematics. On discove-ring mathematics in the course, some students react with resistance or blocks. “That wasn’t what we came for.” This

reaction is due to a suspicion that mathe-matics only leads to irrelevant byways, stealing time from the relevant vocational qualification. A suspicion fed by their experiences in school. It is understandable that students are surprised when meeting mathematics in this context because AVT 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, there is an example of three bicycle hire shops with different prices. In the practi-cal situation, people would solve the pro-blem in many different ways. What you actually do, if you are on vacation at Bornholm, depends on your calculation skills and the specific situation. If it is ob-vious from your vacation plans that you 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 educatio-nal 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-

tended. The material does not suggest that the students are to practise the skills or understan-dings of arithmetical operations used in every-day life, and it does not motivate them to ob-tain the skills and insights in systems of coordi-nates. On the contrary, it reinforces the adults’ prejudices that, apart from the four arithmetic operations, mathematics is only unnecessary mystification. (Lindenskov, 1996)

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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.

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31

Numeracy in

everyday life

and

in

education

In the previous chapters we have addressed “functional understanding of numbers and mathematical skills”. We showed how mathematics can be hidden in technology. Also, how arithmetic skills are intervowen with literacy skills and form a part of the general and technical-vocational qualifica-tions. We have pointed out systematic dif-ferences between mathematics at work and in school. Now it is time to close in on this competence and name it.

In England at the end of the 1950s,

“numeracy” was introduced as a parallel to “literacy” in order to catch the competence which enables people to handle practical mathematical demands in everyday life. In the English-speaking countries there are a large number of numeracy courses,

corresponding to literacy courses. We define numeracy as follows:

• Numeracy in the labour market

consists of functional mathematical skills and understandings that in prin-ciple all people in the work force need to have.

The expression “need to have” should not be interpreted as expressing “necessity”, but rather “relevance”. Thus, we are not only talking about the given requirements of

workers’ skills and understandings in the labour market. We also talk about needs that can be relevant to technological changes (in technique and/or work organi-sation) or to workers’ perspectives of working life or supplementary training. Among the semi-skilled workers in workplaces and among the AVT-participants, mathematics is often not considered as something useful and the general attitude is that the AVT-courses contain too much mathematics. The reason might be that the tasks in problem solving are not perceived as relevant, even though they deal with things from the students’ working lives.

For instance, with an authentic work instruction and the normal time given for a certain packing and controlling function, the teacher can construct many different tasks from this material. But the operator will not actually encounter the need for performing these calculations when carry-ing out the work function: “Packcarry-ing and controlling of lids”. The instruction states that every box should contain 75 pieces. When the operator is fetching containers for 600 pieces she has to calculate how many boxes she needs. When there are new

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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 a little and count, but soon it becomes routine. All calculations concerning work and normal time are performed by the computer.

But this specific factory is about to intro-duce autonomous production groups, which implies that the group members each take turns as the co-ordinator for 14 days. The working day starts out with printing the production plan. The factory meets orders

within a delivery time of five days (“just in time”) and, through the computer, the operator is in contact with the office recei-ving the order, the production department, and the sub-suppliers. There are many fac-tors to be taken into account when planning and co-ordinating the work. A little later, together with co-ordinators from the other departments, she participates in a meeting with the production leader and planner. In this meeting they walk through and discuss

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production plans and service grades. In such an autonomous group, there are higher requirements for the operators’ numeracy than in the ‘narrowly’ conceived job function. Co-operation with and co-ordination of colleagues and departments require understanding and conversion of information with numbers in complex contexts and further communication of this newly revised information.

The need for numeracy is changing 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 arith-metic 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 communi-cation 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 under-standings taken out of context. Numeracy is not just the four basic arithmetic operations or topics such as “dosage”. The dosage is always affected by its use and where it is carried out. For example, there is a big

difference in calculating quantities of medication or industrial cleaning agents. In order to describe numeracy in semi-skilled jobs, we have constructed an ana-lytical tool with four dimensions. One di-mension is context: What you are able to do and what you should be able to do is depen-dent upon whether it takes place in the supermarket, at work or in a test situation. A second dimension is the medium of appli-cation: The relevant numeracy depends on whether it is employed in oral communi-cation, or applied to reading a manual or measuring a heap of soil, even if the num-bers and arithmetic operations are the same.

Input and output on the computer The overview is important to the co-ordinator of the autonomous group, with a high number of factors involved in the decision-making process.

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The numbers on the wall are about flight safety.

A third dimension is personal intention: It is critical whether the intention is to gather information, to fill in a form, to plan pro-duction, to control the quality of a product, to kill time, etc. A fourth dimension is skills and understandings - for example, having a geometrical sense, a sense of scale, the ability to carry out rough estimations, and appropriate use of mathematical techniques.

The context of numeracy in Vocational Mathematics is the workplace. The actual context in the particular business/enterprise, with pieces of information about work or-ganisation and technology/machinery, should be considered in order to analyse the need for numeracy. Also, what is produced should be taken into account. As expressed by an operator in the quality control at a big electronics enterprise: “There is a difference between mistakes in an aircraft and a tele-vision.”

The medium covers the information and communication or the specific material and processes where numbers, formulas and figures are used, collected or constructed (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

texts Informative and instructive diagrams, charts, graphs, tables, maps, 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

of information “24m” – “Fourteen seventy-five” – “Three days” b. Longer or

shorter statements

“Fetch 12 strips at the store!” – “Jim is ill today so the three of you will have to handle the order without him.”

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

materials A heap of soil; 21 connectors; 45 sq cm or 45cm

2_{; 1mm aluminium}

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

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Personal intention. When solving

specific work tasks or carrying out a certain job function which calls for the use of or construction of numbers, formulas or fi-gures, the intention can be different accor-ding to whether the purpose is:

- to collect pieces of information - to collect data

- to control a process - 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

When the intention is, for example, to find a quantitative piece of information there is a difference in the necessary competence of collecting that information depending on

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 calcu-lating use of workplace materials. The formula for the circle circumference, 2π x

radius, is mathematical knowledge that

only can be used for calculating use of ma-terial when converted into π x diameter, and the person knows how to measure the dia-meter of the object.

Numeracy contains functional arithmetic skills

From the assembly department at a large factory, the following fault complaint is made: On the computer, a delivery is re-gistered as containing 1094 pieces, but it

is 160 pieces short.

In the delivery note the operator finds that they have sent 22 boxes of 42 blanks and a single box of 10 blanks. By means of the ten times table, the two times table, plus and minus she locates the error as a miscalculation.

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36

Conversely, the multiplication tables of two, five and ten can take you far in prac-tice as long as know how to set up the cal-culation for solving the task.

In the workplace it is not always enough to be able to find “the correct answer” to the arithmetic problem. You also have to be able to judge the validity of result and the method in relation to the

specific application.

In the investigation at selected workplaces, the analytical tool containing the four di-mensions (context, medium, personal inten-tion, understanding and skills) has been shown to be useful for describing the need for numeracy in specific job and work func-tions. The following description of the job function goods reception and quality control will serve as an example.

Context: Annette is an industry operator in a big electronics enterprise. She works in the department of inward goods and quality control (at present, six females and one

male technician). The department is con-nected to other departments via an internal network. The work is independent and is organised based on rules of prioritising particular tasks. In relation to Annette and the others, the technician’s function is mostly to clarify specific technical ques-tions. Formerly, he acted as a work team-leader and was to be involved in all re-ported fault complaints. For every type of blank Annette is controlling, there is a detailed procedure with a specific and a general instruction/specification. Reception, control and delivery are all documented both on paper and on screen.

Media: Numbers, formulas and figures are found in, or constructed from:

(1) Written information and communication

1a) Prose texts: General instructions for quality control; specific control instructions (for every blank)

1b) Reference texts: Tables and work

Quality also involves accurate measuring. Here the tolerance is 1/10mm

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37

drawings (in instructions and specifica-tions), archives, delivery notes, reception reports (on both paper and screen), pro-duction 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

2a) Short piece of information: “Who has number 243 343?”

2b) Longer or shorter statement: “Ben has set up the formula we need in 243 453.” 2c) Dialogue: “Are you driving with 83-23 now?” – “I’m not driving with anything now but I’ve put something up.”

(3) Specific materials, time and processes

3a) Specific materials: the blanks (connectors, pins, etc.)

3b) Time: normal time, time consumption, production plan (when they need this blank in the production process)

3c) Processes: Fault complaints and rejec-tion of a delivery will occur after a judge-ment of the economics and the consequen-ces for other placonsequen-ces in the production process.

Personal intention: Annette is collecting pieces of information to be used directly in the work process, giving information, collecting data on the blanks, controlling the blanks and their delivery, judging, deciding, reporting, co-ordinating with the needs of the production department.

1000 small blanks are to be counted out in bags of 250 each. Weighing them helps, but it does not take into account methodical considerations.

Skills and understandings: Annette needs her sense of numbers (every object has a nine-digit code, so has every task, customer, product, specification); measuring; counting (manually and by weighing); geometrical sense (symmetry of the blank, use of work drawings); manipulation of numbers (set up and use formulas). Annette has to follow a certain procedure but she also has to be able to adapt to any given situation.

The specific vocational competence gives an opportunity for flexibility and independence

Annette takes a bag containing 9 small connectors for flat cables, which must be checked in relation to various standards. It is a new brand and one of the measurements, taken with a digital slide gauge, does not quite fit with the drawing. Thus she draws atten-tion to a changing of the documentaatten-tion.

The measurement of the connector is 15.58mm and it should be 16.00mm according to the drawing. The tolerance is 0.01, but the worker’s experience from the production department now benefits her - and the factory. She knows that the discrepancy in this measurement and this connector has no practical significance. Had the measurement been over 16.01 she would have rejected the items. She can also see on her computer screen that during the day the pro-duction department will be short of this type of con-nector and she takes this into account when making her decision.

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38____________________________________________________________

Many numbers, formulas and charts are used in semi-skilled work and job functions. Advanced mathematics is not needed for them, but mathemati-cal 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.

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39

Numbers

and

Vocational mathematics

in semi-skilled jobs

Development tendencies in the European labour market towards broader jobs, rotation and autonomous groups make new demands on nume-racy. For example, it is not sufficient just to be able to read a graph. An understanding of how it is drawn is also important in order to extract the conse-quences of the embedded information. This is confirmed by the workplace and student survey in Vocational Mathe-matics. We chose a number of work-places 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 mathe-matical 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,

• to use the four dimensions in order to present a detailed description of the work tasks and functions containing manipulation of numbers/ charts, quantification and judgement of quanti-

Cuttings from written materials used in

ties or spatial relations.

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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

The cash register is too slow when many people are in line and the sales ti-cket func-tion is turned off.

In the student survey, 160 parti-cipants in AVT-courses in the four lines of business were asked about their need for six-teen different functional mathe-matical skills in their daily work. One example of a ques-tion was: “Do you need to calculate percentages?” – Between 50% and 90% of the students needed the sixteen skills regularly or occasionally. The biggest need was to count blanks or money (75% needed the skill regularly) and to work

out arithmetic problems using the four basic arithmetic operations (73% needed the skill regularly). To read numbers or diagrams on the notice board was only needed by 50% of the participants.

Underlying the work place survey were six working hypotheses:

Working hypothesis 1:

In every semi-skilled job, problems arise that can only be solved by quantification and use/evaluation of quantitative units.

In the workplace survey all the observed job functions required the use of and judgement of numbers. From the counting of blanks and reporting of work notes, through the measuring and judgement of quality checks, to the setting up models for fulfilling the required service grades.

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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, toge-ther with a lot of otoge-ther people. She has a yel-low 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.

Tasks and functions of semi-skilled workers require relatively simple formal skills and understandings in mathematics but, informally, numeracy is developed in complex working situations and the use of mathematical ideas and techniques is advanced.

The mathematical techniques used by the observed employees were relatively basic. Many of the arithmetic problems are questions of learning by heart in the sense that the same numbers are multiplied or added up every day in the given work function. These employees did not use advanced mathematical knowledge in order to handle the tasks and work

functions but, on the other hand, their use of mathematical techniques was advanced.

The need for arithmetic and mathematics is not discovered at the factories until

writing is required.

Interviews with production leaders and employees confirmed that the attention given to mathematics is connected with writing - for example, when working for a quality certification where the work pro-cess and quality checks are to be

documented.

There are systematic differences between mathematics in the workplace and

mathematics in traditional teaching.

The survey supported this working hypo-thesis. In work, numbers and arithmetic problems are to be constructed, while they are given in traditional mathematics in-struction. Solving the problem has a prac-tical consequence in work, but not in tea-ching. In work the problems are deter-mined and structured by technology while, on the other hand, the mathematical pro-blems determine the course of teaching.

Vocational Mathematics

Vocational

Mathematics

Tine Wedege

Contents

Preface

Introduction

Numbers

and

technology

Functional

mathematics

and reading skills

Mathematics

in

work

and in

school

T heoretic ally educ ated

Adults’

blocks

towards

learning

mathematics

Numeracy in

everyday life

and

in

education

Numbers

and

Vocational mathematics

in semi-skilled jobs