Title: From mathematics to numeracy and back: Reflections on aspects of teaching mathematics to adults in m
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2- From mathematics to numeracy and back
Reflections on aspects of teaching mathematics to
adults in my career - Gail E. FitzSimons
3Schools 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
41. 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.
51. 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.
61. 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.
71. 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
82. 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.
92. 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.
10In 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.
112. 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
122. 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.
133. 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.
143. 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.
153. 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.
163. 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.
174. 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.
184. 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.
194. 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.
205. 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.
215. 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.
225. 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
235. 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.
245. 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.
25Assessment 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.
26Assessment 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?
275. 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).
28Numeracy
- 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.
295. 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.
306. 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.
31Back 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.
32In 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?
33Issues of power
- Who decides adult mathematics/numeracy curriculum
and assessment in your country? - Learners?
- Teachers?
- Academic mathematicians?
- Employers?
- Government bureaucrats?
- Who should?
34Final 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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