etl 421 teaching the curriculum / integrating numeracy...
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ETL 421 Teaching the Curriculum / Integrating Numeracy
Daniel_ Sameena_ s256688_ ETL421 Assignment 1
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Assignment 1: Curriculum through Numeracy
This document will attempt to illustrate the developments of numeracy and establish how
mathematical activities in a classroom are acquired and applied within an International
Baccalaureate context.
Documentation
What is numeracy?
The definition of numeracy has certainly evolved considerably over time; it has changed
from being similar, if not identical to literacy, to now being quite diverse and viewed as a
‘multifaceted.... cultural, social, emotional and personal aspects of each individual context
(Maguire & Donaghue 2002, cited in American Institutes for Research, 2006,p.6 in
COAG,2008). The Australian Association of Mathematics Teachers (AAMT) (1997,p.15 in
Perso) describes numeracy, ‘is to use mathematics effectively to meet the general demands
of life at home, in paid work, and for participations in community and civic life.’ This
definition mirrors many other professional literatures like Perso’s, who concurs with the
AAMT and suggests; that being numerate is having the ‘disposition and capacity to use
mathematics to function effectively and fully at home and in society’. Steen (2007,p.17
cited in Westwood)also suggests that ‘numeracy has no special content of its own but
inherits its content from its context’. Moreover, Mason and Johnston-Wilder claim that the
‘kinds of powers that are relevant for learning mathematics are innate’. (Mason et al,2006).
These claims broadly state that it is the ability of an individual to use numbers and
mathematics effectively in everyday life through positive attitude; mathematical skills and
thinking, concepts and numerical strategies that deem them as numerate. These views
later led to further diversification by looking at what type of numeracy would be required
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for those in certain communities or occupations and showed that numeracy would indeed
be different for everyone; which led to numeracy being applied to different contexts and
termed as ‘multiple numeracies’- ‘community numeracy’, ‘critical numeracy’ and ‘consumer
maths’ (Westwood, 2008) as just a few examples. With these aspects in mind and claims
from professional literature; it is evidently becoming apparent that we must acknowledge
the importance of ‘multiple numeracies’ in everyday life and therefore necessary for
schools to embrace and adopt learning skills required for participation in the wider
community or civic life (see fig 1).
Fig.1. Teaching Numeracy
Students Innate
Ability /
Positive Disposition
Problem Solving
Student Interest
Teachers Knowledge /
Strategies
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Moreover, throughout the readings, Perso, Westwood and Ginsburg et al talk about ‘the
affective component of numeracy includes beliefs, attitudes and emotions that contribute
to a person’s ability and willingness to engage in...mathematical thinking and
learning’(Ginsburg et al, cited in Westwood, 2008,p.10). Perso like other scholars also
agree that ‘numeracy is different from mathematics’ and this relationship is sometimes not
well understood by some teachers- ‘resulting in an imbalance in their approach’ (Perso,
2006a;2007 cited in Westwood) and consequentially focusing on computational methods
of teaching mathematics (Perso,2006). Steen (2007,p.18 cited in Westwood) also believes,
that ‘numeracy is often characterised as watered-down mathematics’. However,
DEETYA,(1997) states, ‘all numeracy is underpinned by some mathematics; hence school
mathematics has an important role in the development of young people’s numeracy’
(Westwood,2008,p.9) (fig.2). Gough (2007) proposes that: ‘the content of numeracy is
represented by most of what comprises the typical primary school mathematics course’.
Obviously, this basic introduction to simple arithmetic is essential to further
understanding. The acquisition and application of numeracy is now quite subjective to the
individual and their circumstances, but continues to refer to the effective use of
mathematics and numbers in a given situation.
ETL 421 Teaching the Curriculum / Integrating Numeracy
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Fig 2.
Inter-relationship between Mathematics and Numeracy
Learning
through
actions and
experiences
Basic
Mathematical
Skills
Specialised
mathematical
knowledge
Ability to
use
mathematics
in everyday
situations
Ability to
understand
further
mathematical
concepts
Further mathematical
learning
Knowledge gained
through necessary
activities
Broader mathematical
knowledge base adaptable
to many situations
Adapting knowledge
to real life situations
NUMERACY
MATHEMATICS
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Analysis
As a reference point this document will use a Grade 2 class from an International school in
Singapore. The Canadian International School (CIS) consists of expatriates from all over the
globe and is a large school that follows the International Baccalaureate Primary Years
Programme. The IBPYP program follows a constructivists approach to how children learn.
It is an inquiry based student-led learning process. The PYP curriculum follows a
transdisciplinary model and these themes cross the confines of traditional subject areas
(International Baccalaureate, 2009).
Grade 2 class comprises of 21 students which include:
4 ELL students of varying levels of English abilities
13 students are bi-lingual
10 countries are represented (9 from Eastern and Western Europe, 6 North
America, 6 Asia)
The students are at various stages of the IB continuum as they are not streamed
upon ability.
Teaching Numeracy within IB
Mathematics in the Primary Years Programme (PYP) at an IB school is very different to
mathematics in a traditional school. Following the constructivist approach, mathematics is
taught to children through a transdisciplinary programme (International Baccalaureate,
2013) (see fig.3). The work of Askew et al, suggests that teachers of numeracy tend to
reflect one of three possible orientations; connectionist-linking prior knowledge to new
knowledge; transmissionist-teach explicit information and discovery-learners discover
mathematical concepts for themselves (cited in Westwood,2008,p.36). It is very interesting
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to see how the IB also reflects the connectionist orientation with some elements of
discovery to their practices of teaching.
Fig.3 International Baccalaureate –How children learn mathematics
However, there is a realisation that not all students will be at the same part of the cycle at
the same time. The curriculum is designed on the recognition that the learning of
mathematics is a developmental process, which is not always age-related or linear. With
that in mind, the curriculum is based on phases or a continuum (International
Baccalaureate,2013). This does not mean there are not goals set for the grade levels. What
it does mean is teachers at CIS receive the grade level curriculum for their grade as well as
a grade below and a grade above. This allows the teacher to better tailor activities that are
inclusive to all abilities in the classroom. Vygotsky, suggests ‘optimum learning occurs
when tasks or problems are correctly tailored to be just a shade above a child’s current
Transferring meaning
Apply with
understanding
Constructing meaning
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level of ability but which a child can handle successfully with some support or guidance
from an adult’(cited in Westwood, 2008,p.29).
The Grade two mathematic curriculums are broken down into five sections:
Number benchmarks (50%)
Algebra –patterns and function benchmarks (10%)
Measurement (15%)
Statistics & Data handling (15%)
Shape and space (20%)
Although Number benchmarks and Algebra are separate sections they really go hand in
hand. Recognizing the patterns in maths, help the student create strategies to help them
with more complex problems. These segments coincide with Ginsburg et al (2006) views,
in such, that they suggest that ‘courses should be organised around four key strands:
number and operations sense; patterns, functions and algebra; measurement and shape
and data, statistics and probability (Westwood, 2008,p.7).
According to the IBPYP philosophy; it believes that ‘wherever possible, mathematics
should be taught through relevant, realistic context of the units of inquiry (UOI)...to help
students make connections between the different aspects of the curriculum’ (International
Baccalaureate,2012).
However, there are some points not clearly addressed in the IB Mathematics Scope and
Sequence curriculum; one being; teachers’ ability in numeracy and the other being; student
engagement. Perso emphasises the need for students to know how and when to apply
different types of mathematics or technology to solve problems and suggests that this is
best taught through a ‘task-centred’ approach. This is also highlighted through the
Annenberg Media entitled Mathematics: What's the Big Idea? (1997) where collaboration of
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student activity is encouraged in order to facilitate student engagement and hence aide
student learning. It is important for teachers to feel confident and competent about
teaching numeracy.
Inequalities
Being an independent international school in an affluent community, financial inequalities
are not prevalent, nor does the school lack funding or resources. However, inequalities are
apparent at the school and take shape in certain areas such as-language and culture
principally. The international school in Singapore uses English as its primary teaching
language and is made up of both local and international students, making it very transient.
As a result, there are several ELL learners; this can nevertheless cause complexities as the
classroom will inevitably have different abilities of spoken English which can create divide
and isolation. Also many students at CIS come from different cultures with little or no
knowledge of ‘western’ culture, this too can prove to be an issue for learners who are not
accustomed to ‘Canadian/Western’ culture as they may find it difficult to blend in; and
most teachers employed at the school are from a western background, who may have little
or no knowledge of different cultures of their learners which can lead to further
differences.
In addition, students experiencing this teaching approach for the first time may struggle
with group discussions, which can be a cause for concern for both students and parents
who come from more conventional traditional educational backgrounds, where education
culture is competitive and therefore difficult to appreciate and understand. Students from
traditional cultures or authoritative teacher led classes may have difficultly in the
cooperative learning and questioning approach. They may need time to learn and
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understand IB concepts. For many students the concept of thinking how to do something,
instead of being told is very foreign to them.
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Connecting theory to practice within IB
NUMERACY
EVERYDAY LIFE MATHEMATICAL
PROCESSES & SKILLS
WORK
CULTURAL COMMUNITY
EMOTIONAL RECREATIONAL
NUMBERS
SPATIAL ALGEBRA
DATA &
STATISTICS
SHAPE & SPACE
NUMERICAL
STRATEGIES
TEACHERS
UNDERSTANDING
COMPETENCE &
CONFIDENCE
DISPOSITION
TASK-CENTRED
APPROACHES
IB
TRANSDISCIPLINARY
INNATE ABILITY MULTIPLE NUMERACIES
CRITICAL NUMERACIES CONSUMER NUMERACIES
COMMUNITY NUMERACIES
MATHEMATICAL
KNOWLEDGE
CRITICAL THINKING
PROBLEM SOLVING
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Consolidation
The following is a broad outline of two rich inter-related activities that have been designed
for a Grade 2 class at CIS.
The activities will be outlined using the Learning by Design pedagogy.
Learning Focus__________________________________________________________________________________
To inquire into the following Transdisciplinary Theme:
UOI – Sharing the Planet
Central idea - Share resources with other people and with other living things.
Learning level audience – Professional colleagues and grade two students.
Learning outcomes-Grade Two Mathematics Continuum (International Baccalaureate
2013)
Collecting and displaying data in a range of graphs, and interpreting the results.
Using appropriate mathematical vocabulary and symbols.
Selecting and explaining appropriate methods for solving problems.
Finding a unit of a fraction by using halving or repeated halving.
Task 1: ‘Develop a community in the forest’. (Brief outline)
Students are given a forest to build a community and told that there will be 25 people, (all
different ages and gender) which they have to divide into appropriate families. Then they
must build a community (1 bedroom house =cut down 10 trees etc). Also, within the forest,
there are living things (birds, fish, rabbits, trees) and natural resources, every living thing
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that they eliminate to build will cost them etc. The idea is to build a community with the
smallest amount of loss.
Ex
pe
rie
nci
ng
Known Experiencing their own community (nature parks, forests etc). Looking at their individual families & how they are made up. Gathering learners’ prior knowledge about environment and purpose
New How does it relate to what they see in real life? Listening and responding to texts Illustrate dividing up of families as a graph.
Co
nce
ptu
ali
sin
g
Naming Through connection, sorting and
classifying. Thinking routine ‘think, pair, share’(TPS) to
discuss their representations and what it
shows
Theory What do living & non living things need to survive? Show data results in form of graph
An
aly
sin
g Functionally
Demonstrate the link between developing a community and
how many living things & natural resources are lost
Critically What purpose does it serve?
Group discussion about how much was lost building a community?
show data in form of graph
Ap
ply
ing
Appropriately Gather post data of how many living & non living things were impacted by building a community
Creatively How can we build a community and share (divide) resources? Devise problem solving strategies
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Task 2-(Brief outline)
‘If the Earth was an apple’- incorporate with International Year of Mathematics of Planet
Earth (IYMPE).
Start by asking ‘What is a fraction? Gather data and show results in the form of a graph.
Students are given an apple and told that if ¾ was water how much would be left? And
then it gets divided further and further, so the students have to work out fractions and see
what how much soil is left to grow the earth’s food consumption.
Ex
pe
rie
nci
ng
Known Gathering learners’ prior knowledge about fractions, environment and purpose?
New How does it relate to what they see in real life? Listening and responding to texts
Co
nce
ptu
ali
sin
g
Naming
Through connection, sorting and classifying. Thinking routine ‘think, pair, share’ to discuss
their representations and what it
shows
Theory How to slice apple in ½ ¾ etc. Show data results in form of fractions
An
aly
sin
g Functionally
Watching video- ‘If the earth was an apple’. What does it tell us?
Further T.P.S discussion
Critically What purpose does it serve?
Group discussion about how much we have left to grow, show data in
form of graph
Ap
ply
ing
Appropriately Gather post data and create pie chart
Creatively How can we connect to IYMPE? How does it help our understanding?
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Redirection Whilst working through the IB Mathematics teaching philosophy, I noticed how it mainly
centres its approach to task-centred activities; but should they also incorporate direct
instruction, for those who have little or no knowledge of English, e.g. ELL learners? And
how does the IB compare along traditional concepts of teaching mathematics? Is inquiry
based learning really the way forward to teach mathematics?
Furthermore, I have also noticed that there is very little mention of technology in the IB
curriculum. ICT is only first mentioned in the Grade 2 curriculum benchmark guide. Would
a calculator not give those struggling in maths a voice in mathematical discussions? Or are
calculators and other technologies a hindrance to a child’s learning, to their thinking
process? What impact does technology have on mathematics in elementary students?
Word count 2180
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References:
Annenberg Media (1997). Mathematics: What's the Big Idea? USA: Author.
Retrieved 10/08/13 from: http://www.learner.org/resources/series98.html#
Council of Australian Governments 2008, National Numeracy Review Report,
Commonwealth of Australia, Barton, ACT
International Baccalaureate. (2009). Making PYP Happen: A curriculum framework
for International Primary Education. International Baccalaureate (revised edition ed.).
Cardiff, Wales: Antony Rowe.
International Baccalaureate. (2013). Primary Years Programme: Mathematics Scope and
Sequence. Canadian International School, Singapore
Kalantzis, M., & Cope, B. (2012). Literacies. Melbourne: Cambridge University
Press.
Mason, J. & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks.
Milton Keynes: The Open University
Perso, T. (2006). Teachers of Mathematics or Numeracy?. amt 62(2), 36-40.
Retrieved 07/08/13 from
http://web.ebscohost.com.ezproxy.cdu.edu.au/ehost/pdfviewer/
pdfviewer?sid=d55ec1fa-53d3-45cb-b42df4b297f23633%
40sessionmgr10&vid=2&hid=28
Westwood, P. (2008). What teachers need to know about Numeracy. Victoria: ACER
Press