From mathematics to numeracy and back: Reflections on aspects of teaching mathematics to adults in my career Gail E. FitzSimons

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www.monash.edu.au

From mathematics to numeracy and back:

Reflections on aspects of teaching mathematics to

adults in my careerGail E. FitzSimons

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Schools of Thought on Adult Learning and Development1. Liberal/conservative, intellectual,

paternalistic tradition [old humanist] 2. Behaviourist [technicist] approach to

teaching and learning3. Progressive education or student-

centred approaches 4. Individual self-actualisation, self-

direction, self-fulfilment 5. Social transformation 6. Organisational effectiveness

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1. Liberal/conservative, intellectual, paternalistic tradition [old humanist]

General: • Focus on individual as psychological being,

unquestioned [i.e. ‘politically neutral’] cultural transmission of disciplinary knowledge,

• the teacher as a figure of authority in the discipline and in full control of the teaching-learning process

• generally by lecture mode• This tradition was often apparent in workers’

educational associations, for example.

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1. Liberal/conservative, intellectual, paternalistic tradition [old humanist]Mathematics:

the discipline is treated as a fixed body of knowledge [i.e., underlying absolutist/Platonist philosophies],

• pedagogy is transmission-based; lectures on theory with worked examples and exercises set for the learners.

• The aim of the teacher is to explain clearly and motivate the learners.

• The role of the learner is to understand the material and apply it as appropriate.

• Between them, the text/s and the teacher are the sources of authority and assessors of correctness.

• Subjects are hierarchically organised, and mapped out in advance.

• Examinations are externally set and learners vary by their innate ability.

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Source/s of Motivation for the Adult Learner:

• Becoming an ‘educated person’• Learning the language of the discipline• Gaining a qualification• Inspiration from the teacher/lecturer or

even the popular media.• The goal is to learn mathematics; the

focus is on the individual’s abilities.

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Positive aspects:• very suitable for learners who already

had a love of learning and a thirst for knowledge.

• be already highly motivated, and even willing to seek out further information for themselves on the internet.

• proving to themselves that they really ‘can do it’ at this later stage of life

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2. Behaviourist [technicist] approach to teaching and learningGeneral: • Similar to #1 in approach to content, but the pedagogy is based on so-

called ‘scientific principles’ [Taylorism] • Belief among policy makers and senior bureaucrats that it is possible to

devise the ‘one best way’ of teaching, and even to make materials teacher-proof.

• The individual learner is still the focus, with assumptions of fixed ability realised by hard work.

• The curricular content is pre-determined elsewhere, following explicit hierarchies of knowledges and skills; the content is atomised into minute competencies.

• In the most extreme scenario of mastery learning, students sit at individual carrels and communication is only with the text/screen and the teacher.

• The learner is programmed to move through these individual competencies in a fixed order.

• The role of the teacher is to assist students who become stuck and to keep extensive records of achievement; these may be automated.

• Assessment is formative in the mastery learning system, but summative examinations may also be externally set.

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2. Behaviourist [technicist] approach to teaching and learning

Mathematics: • The role of the teacher [or worksheet] is to drill-

and-fill the learners with facts & algorithms. This low level transmission of skills with possible

‘applications’ is easily transposed into electronic forms of delivery and commonly available online & in CD-ROM versions.

• An extreme view would see no room for calculators in the classroom.

• Testing of the basic facts is decontextualised; if applications are given they are pseudo-contextualised.

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In both 1 & 2 (transmission-based) mathematics education:

• Mathematics education epitomises authoritarianism in the guise of simplistic right/wrong decisions — whether it is the final answer or even the correct workings being shown.

• The ‘back-to-basics movement’ is a prime example of the concept of using mathematics/numeracy as a means of social training in obedience.

• The role of the learner is to work hard, make an effort, practise continually, and even submit to rote learning where necessary. [cf first aid, etc.]

• Applications are tailored or constructed around the mathematical skills, with transfer assumed unproblematically

• Although there are certain justified social and economic needs for accuracy and efficiency, they are not universal.

• These perceived demands are a still a source of mathematics anxiety or mathematics avoidance for many adults today.

• The students are often assumed to be monocultural and even gender-neutral [i.e., male].

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• Gaining a qualification• Responding to government or other external

pressures• Experiencing repeated success from small

steps• Moving through the sheets as quickly as

possible (depending on the goal)• Passing entry level tests [often timed] such as

for the police, armed forces

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Positive aspects:• Mastery learning can give a wonderful feeling of

achievement — maybe for the first time for many adult learners of mathematics.

• It can really help to develop a feeling of self-confidence.

• Working at one’s own pace can reduce stress.• The certification as an achievement can be

cause for pride.

• The goal is to learn mathematics; the focus is on the individual’s behaviour.

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3. Progressive education or student-centred approaches

General: • Stemming from the work done by Dewey in the early part of the 20th century,

reached a crescendo in the 1970s free-schooling movements. Widely adopted as policy by regular school systems prior to economic rationalist governments.

– a focus on the individual, but within a social context. – an emphasis on reflection and action, – the curriculum is ideally focused on the immediate problems and needs of

the learners. • Learning is seen as personal growth for the individual• The teaching methods could include

– problem solving, – scientific method or experimentation, – learning contracts, – the facilitator responsible for minimising the barriers to learning, e.g. by

organising caring and supportive work groups. • Learners are acknowledged to have varying abilities but these need to be

cherished. • The role of the teacher is to prevent failure and to facilitate personal exploration.

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3. Progressive education or student-centred approachesMathematics: • In the 1980s the use of problem solving and small group

work was highly fashionable as process superseded content as the main emphasis.

• There were efforts to have learners work ‘as mathematicians’ and this eventually led to the constructivist movement, where it was acknowledged that learners construct their own knowledge rather than receive it via a transmission process.

• Much research came out of the USA on how even young children could form communities of learners and negotiate the correctness of mathematics practices.

• The ideal was to solve, and perhaps even pose, problems in ways that reflected professional mathematics practice rather than the century-old school factory model.

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3. Progressive education or student-centred approaches


• Develop more fully as person in areas previously denied access

• Learn the ways of thinking of the discipline of mathematics, cf. mathematicians

• Learn that doing mathematics can be fun and a social, even (inter)cultural activity

• Inspiration from the self, possibly significant others, and perhaps the work-group.

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3. Progressive education or student-centred approachesPositive aspects:• This kind of teaching helped the women

develop a massive boost in self-confidence and a visible sense of agency with respect to mathematics and elsewhere.

• One woman overcame severe shyness and a lack of self-confidence to act as a role model and even tutor for her children and their friends.

• See FitzSimons, G. E. (2003). Using Engeström’s expansive learning framework to analyse a case study in adult mathematics education. Literacy & Numeracy Studies, 12(2), 47-63.

• The goal is to learn mathematics within a community of learners.

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4. Individual self-actualisation, self-direction, self-fulfilmentGeneral: • Carl Rogers (self-actualisation): Freedom to Learn;

reflected in recent trends towards empowerment.• David Kolb’s theory of learning style. 1. concrete experience, or being involved in a new

experience;2. reflective observation—observing others in an

experience, or developing observations about our own experience;

3. abstract conceptualization—creating concepts and theories to explain our observations; and,

4. active experimentation—using the theories to solve problems and make decisions.

Criticised by:Loo, R. (2004). Kolb’s learning styles and learning preferences: Is there a

linkage? Educational Psychology, 24(1), 99-108.

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4. Individual self-actualisation, self-direction, self-fulfilmentAndragogy: Malcolm Knowles established 4 basic underlying

assumptions [not rules]1. Adults are self-directing2. Educators need to draw on learners’ experience 3. The readiness to learn depends on need4. Learning should be problem-centred.• But, self-directedness is a goal of adult education, not a

characteristic of adults (Stephen Brookfield).Howard Gardiner’s multiple intelligences (linguistic, logical-

mathematical, spatial, bodily-kinesthetic, musical, naturalist, interpersonal, & intrapersonal). Criticised by:

Klein, P. D. (2003). Rethinking the multiplicity of cognitive resources and curricular representations: Alternatives to ‘learning styles’ and ‘multiple intelligences.’ Journal of Curriculum Studies, 35(1), 45-81.

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4. Individual self-actualisation, self-direction, self-fulfilment

Sources of motivation• Self-actualisation; content is secondary.

• In terms of teaching and learning mathematics, there are many andragogical approaches which adult educators may draw upon.

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5. Social transformation

General: • Education is used to achieve a new social

order. • The focus is on the collective, with the teacher

and learners as equal participants in a group, learning from each other.

• Problem posing and dialogue play an important role.

• Ability is seen as a cultural product and not fixed.

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Critical theorists:Paolo Freire (1972, 74): teachers as co-learners to

enable liberation from oppression; importance of dialogue; critical reflection.

Beyond Andragogy: Brookfield held that adult educators are morally responsible for contributing to direction of learning, more than just acting as facilitators

Jack Mezirow: transformative learning as the reassessment of perspectives or assumptions formed in childhood; emancipatory education.

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Sources of Motivation for the Adult Learner:

• To develop more fully as person• To make a difference on the local and/or

global scene• To gain respect for one’s past history

and experience

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Mathematics: • With a democratic socialist philosophy, social

constructivist view of mathematics, leading to emphasis on social justice and citizenship.

• Learning takes place by questioning, decision making and negotiation.

• Teaching encourages discussion, conflict, questioning of content and pedagogy.

• Resources are socially relevant and authentic, assessment takes a variety of modes and incorporates social issues and content.

• The accommodation of social and cultural diversity is a necessity.

• Critique of mathematics and the role it plays in society are important.

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5. Social transformationAt this workplace, the motivation of the workers was:• to gain a credential in recognition of the work they were

already competently performing• to keep their current job [under subtle but external pressure

from management]• possibly to gain a promotion or the opportunity to switch

jobs• to prove they could do it.The motivation of management was:• to increase productivity• to comply with the German headquarters ethic of educating

all workers• to assist the workers to become familiar with the vocational

and higher education system in Australia • to possibly find workers worthy of higher duties, e.g.

management or training roles.

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Assessment example 1

Your work site:• Sometimes things go wrong in the

workplace. Think about one thing that can go wrong in your area such as counting, measuring, or locating (finding something). Use your knowledge of mathematics to explain what went wrong and how it could be fixed.

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Assessment example 2

Explain the purpose of computers and their impact on society.

• Make a list of at least five places where you see computers being used outside your workplace. Briefly describe what they are used for.

• Collect at least five articles from newspapers and magazines that discuss the uses and/or abuses of computers in society. Give your opinion about them.

• …

Describe the different computer systems used in the pharmaceutical industry.

• Make a list of all places where you see computers being used at Bayer. Briefly describe what they are used for. How are they linked up?

• Describe how computers might be used to control machine operations, such as the Marchesini tablet machine.

• What occupational health and safety issues do you need to think about?

• How might computers improve GMP?

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Positive aspects:• increased worker confidence and participation

in workplace discourse• workers gained a credential which recognised

what they were actually doing on the job• improved productivity.

[See also Teaching and Learning Research Project, Institute of Education, University of London, e.g.

Bakker, A., Hoyles, C., Kent, P., & Noss, R. (2006). Improving work processes by making the invisible visible. Journal of Education and Work, 19(4), 343-361).]

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Numeracy

This kind of education comes closest to my interpretation of numeracy, where mathematics is but part of the discourse.

The workers are operating in a socially, culturally, and historically rich environment.

Learning and using mathematics is an important but secondary goal.

The motivation is to get the job done.

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• FitzSimons, G. E. (2000). Lifelong learning: Practice and possibility in the pharmaceutical manufacturing industry. Education & Training, 42(3), 170-181.

• FitzSimons, G. E. (2001). Integrating mathematics, statistics, and technology in vocational and workplace education. International Journal of Mathematical Education in Science and Technology, 32(3), 375-383.

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6. Organisational effectiveness

General: • The development of desired skills and attitudes

in workers to conform with perceived management needs e.g. Argyris & Schön.

• The curriculum is determined by the organisation to help it run more effectively and to achieve its goal, with a variety of teaching techniques and assessment of the objectives achieved.

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Back to Mathematics: Techno-mathematical literacies

• Phillip Kent & colleagues describe Complex modelling as where “employees are required to manipulate qualitative and quantitative data to diagnose problems, search for solutions and carry out process improvement.”

• [It] requires some understanding of the sophisticated concepts of variable and functions, however not in an abstract (mathematical) sense but situated in the workplace context, and supported by intuitions for the meaning of these concepts.”

• Kent, P., Hoyles, C., Noss, R., & Guile, D. (2004). Techno-mathematical literacies in workplace activity. Paper presented at International Seminar on Learning and Technology at Work, Institute of Education, London, March, 2004.

• I have found similar requirements at a range of Australian workplaces, at all different occupational levels.

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In Summary

• Why are we teaching adults mathematics?• Are we teaching for:

– Conformity & obedience? – Qualifications? – Learner empowerment? – Social critique and democratic change?

• How do qualifications reconcile with workplace demands? Is there more at stake than just a certificate or statement of mathematical skills?

– How are learners, once in the workplace, prepared for:Communication upstream and downstream?Multi-skilling?

Up-skilling?• How does adult mathematics education support cognitive,

technical, & behavioural skills in a technological world?

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Issues of power:

• Who decides adult mathematics/numeracy curriculum and assessment in your country?

• Learners? • Teachers? • Academic mathematicians?• Employers? • Government bureaucrats?• Who should?

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Final Questions

• How is it that in many countries adult mathematics education is tied to school mathematics curriculum and assessment?

Clearly, this activity has failed to prepare many school leavers, young and old, at all levels, for full participation in work and civil society.

• Is it because, unlike the actual, often unruly practices of adults, school-type activities are easy to measure and to tie down [restrict]?[meeting accountability needs of politicians]

• Lastly, what are adults’ actual motivations for learning mathematics, and how are they taken into consideration?

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From mathematics to numeracy and back: Reflections on aspects of teaching mathematics to adults in my career Gail E. FitzSimons