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Title: From mathematics to numeracy and back: Reflections on aspects of teaching mathematics to adults in m


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

3
Schools of Thought on Adult Learning and
Development
  • Liberal/conservative, intellectual, paternalistic
    tradition old humanist
  • Behaviourist technicist approach to teaching
    and learning
  • Progressive education or student-centred
    approaches
  • Individual self-actualisation, self-direction,
    self-fulfilment
  • Social transformation
  • 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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1. Liberal/conservative, intellectual,
paternalistic tradition old humanist
  • 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 individuals abilities.

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1. Liberal/conservative, intellectual,
paternalistic tradition old humanist
  • 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 learning
  • General
  • 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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2. Behaviourist technicist approach to teaching
and learning
  • Source/s of Motivation for the Adult Learner
  • 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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2. Behaviourist technicist approach to teaching
and learning
  • 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 ones 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 individuals 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
approaches
  • Mathematics
  • 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
  • Source/s of Motivation for the Adult Learner
  • 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
approaches
  • Positive 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 Engeströms
    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-fulfilment
  • General
  • Carl Rogers (self-actualisation) Freedom to
    Learn reflected in recent trends towards
    empowerment.
  • David Kolbs theory of learning style.
  • concrete experience, or being involved in a new
    experience
  • reflective observationobserving others in an
    experience, or developing observations about our
    own experience
  • abstract conceptualizationcreating concepts and
    theories to explain our observations and,
  • active experimentationusing the theories to
    solve problems and make decisions.
  • Criticised by
  • Loo, R. (2004). Kolbs 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-fulfilment
  • Andragogy
  • Malcolm Knowles established 4 basic underlying
    assumptions not rules
  • 1. Adults are self-directing
  • 2. Educators need to draw on learners experience
  • 3. The readiness to learn depends on need
  • 4. Learning should be problem-centred.
  • But, self-directedness is a goal of adult
    education, not a characteristic of adults
    (Stephen Brookfield).
  • Howard Gardiners 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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5. Social transformation
  • 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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5. Social transformation
  • 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 ones past history and
    experience

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5. Social transformation
  • 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 transformation
  • At 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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5. Social transformation
  • 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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5. Social transformation
  • 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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