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Creating Mathematical Opportunities in the Early Years
Presenter, Dr Tracey Muir, for Connect with Maths Early Years Learning in Mathematics community
As teachers, we are constantly looking for ways in which we can provide students with mathematical opportunities to engage in purposeful and authentic learning experiences. On a daily basis we need to select teaching content and approaches that will stimulate our children through creating contexts that are meaningful and appropriate. This requires a level of knowledge that extends beyond content, to pedagogy and learning styles. As early childhood educators, we can also benefit from an understanding of how the foundational ideas in mathematics form the basis for key mathematical concepts that are developed throughout a child’s school.
In this webinar, Tracey will be discussing the incorporation of mathematical opportunities into our early childhood practices and considering the influence of different forms of teacher knowledge on enacting these opportunities.
• What does effective mathematics teaching
• What types of knowledge does an effective
• How can we provide students with
authentic mathematical opportunities?
• What aspects of classroom practice should
we focus on in order to maximise
Connectionist, transmission and
Connectionist Transmission Discovery
The use of methods of
Primarily the ability to
calculation which are
both efficient and
procedures or routines
Finding the answer to a
calculation by any
Confidence and ability in
A heavy reliance on
paper and pencil
A heavy reliance on
need to be recognised,
made explicit and worked
the result of ‘grasp’ what
was being taught and
need to be remedied by
further reinforcement of
the ‘correct’ method
the results of pupils not
being ready to learn the
(Askew, et al., 1997)
Characteristics of effective
•Emphasise the importance of understanding mathematical
concepts and the connections between these
•Have high expectations that all children will engage seriously
with mathematical ideas
•Structure purposeful tasks that enable different possibilities,
strategies and products to emerge
•Probe and challenge children’s thinking and reasoning
•Build on children’s mathematical ideas and strategies
•Are confident in their own knowledge of mathematics at the
level they are teaching
(Groves et al., 2006)
Principles of practice
•Challenge all pupils
•Teach for conceptual understanding
•Focus on mathematics
•Convey and instill positive attitudes
• Choice of examples
• Choice of tasks
• Use of representations
• Teachable moments
• Planning for ‘teachable moments’
• Knowing your subject and the difficulties
students might encounter
• Experience, research and reflection
• Highlighting possible misconceptions in
• What do I need to know and understand
before teaching this topic?
• What are the likely difficulties or
misconceptions students may have?
• What are some activities or tasks you could
undertake to explore ‘Mr Splash’?
• This is a photograph of Mr. Splash. He loves to
have a bath in his pajamas. He seems to be a
bit big for the bath! I wonder how tall he is?
How could we find out?
• Does the tallest person have the longest feet?
• Does the tallest person have the greatest hand span?
• Are hand spans and foot lengths related?
• Are boys taller than girls?
• Draw a representation of Mr. Splash on the butcher paper.
• Estimate the following and explain your estimates:
– Mr. Splash’s height
– Mr. Splash’s height compared to the tallest person in class
– Mr. Splash’s height compared to the tallest person in your
– Mr. Splash’s height compared to the tallest person in the
• Explain whether you think Mr. Splash has the dimensions of a
What teacher knowledge is required?
• Content knowledge
• Curriculum knowledge
Anticipating student responses
• Selection of appropriate unit
• Measuring accurately with units (e.g., lining
up with no gaps, using the one unit)
• Confusion with formal units and conversions
(for older students)
In your groups, measure your
heights, arm spans, hand spans and
foot size (record these measures in
the following table).
Height Wrist Neck Waist
Zach 152 150 15 30 63
• Interactive website
What do I need to know/consider about
• Length refers to the measurement of something from end to end
• Sequence for teaching measurement
• Children need to experience the usefulness of non-standard units
• There are a number of principles to consider when asking
students to measure with non-standard units:
• The unit must not change – for example, we should select one
type of informal unit, such as straws, to measure the length of the
table, rather than a straw, pencil and rubber
• The units must be placed end to end (when measuring length),
with no gaps or overlapping units
• The unit needs to be used in a uniform manner – i.e., if dominoes
are being used to find the area of the top of a desk, then each
domino needs to be placed in the same orientation in order to
accurately represent the standard unit
• There is a direct relationship between the size of the unit and the
number required – i.e., the smaller the unit, the bigger the number
and vice versa
What do I need to know about students’
learning of topic?
• Possible misconceptions
• Physical difficulties with measuring (e.g., physically
lining up units, etc)
• Individual considerations – how to differentiate the
• Teacher: Great. OK. This is what we do when we
bridge ten. We make one of the ten frames up into
ten by moving the dots [shows two ten frames on the
board next to each other, one with eight and one
with seven counters or dots] Which would be the
sensible one to move the dots in up here?
• Student: Move from the yellow one to the purple one
• Teacher: Would you do that Jim? Would you fill up
the ten frame, the purple ten frame, with eight in it?
Would you be able to put the dots on the other side?
• [Jim moves the dots to the ten frame, and leaves a
column of three dots and a column of two dots in the
• Teacher: Now can you arrange the other frame
so that all the dots are in a straight line?
• [Jim moves the dots so that they form a
column of 5]
• Teacher: Great, so what have we got?
• Jim: Five and ten
• Teacher: Which make?
• Jim: Fifteen
• Paying close attention to students’
mathematical thinking and solution strategies
as they work on task
• Can assist through creating a list of possible
solutions before lesson (anticipating)
• Questioning – more than observing
Selection and Sequencing
• Drawing connections between students’
solutions and the key mathematical ideas
• Goal is to have student presentations build on
one another to develop powerful
More than ‘sharing’
Who did it a
Factors influencing the planning and
uptake of mathematical
• Teacher knowledge Content
knowledge, Pedagogical Content Knowledge (PCK)
• Teacher beliefs What it is to be numerate
pupil, how pupils learn to become numerate, and
how best to teach pupils to become numerate
Choosing tasks to elicit
• Connect naturally with what has been taught
• Addresses a range of outcomes in the one task
• Are time efficient and manageable
• Allow all students to make a ‘start’
• Engage the learner
• Provide a measure of choice or openness
• Encourage students to disclose their own
understanding of what they have learned
• Are themselves worthwhile activities for students’
learning (Downton, et al., 2006, p. 9)
Counting on Frank
“I don’t mind having a bath – it
gives me time to think. For
example, I calculate it would take
eleven hours and forty-five minutes
to fill the entire bathroom with
water. That’s with both taps
running. It would take less time to
empty, as long as no one opened
(Clements, 1990, unpaged)
Is it accurate?
“Going shopping with mum is a big event. She is
lucky to have such an intelligent trolley-pusher.
It takes forty-seven cans of dog food to
fill one trolley, but only one to knock over one
hundred and ten!”
• Does it really take 47 cans of dog food to fill
one trolley? Even allowing for different sized
trolleys and cans of dog food, this seems a
gross under-estimation. In order to test this
claim, a 12 year-old student, using real cans of
dog food (680 gram; 23.94 ounce tins) found
that it took approximately 47 cans of dog food
just to fill the base of a shopping trolley, and
that it would take closer to 216 cans to fill your
average shopping trolley.
• Confusion between length, area and volume
• Conversion of units
• Dimensions of tins
• Consistency for comparison
Something that helped me learn …
Something I wasn’t sure about …
Something that stopped my learning
• Useful to consider the types of knowledge
required by teachers
• Mathematical opportunities can be
• 5 practices are useful for orchestrating
productive mathematical discussions
• Mathematical opportunities are
everywhere – be creative but make them
Arafeh, S., Smerdon, B., & Snow, S. (2001, April 10-15). Learning from teachable moments: Methodological
lessons from the secondary analysis of the TIMSS video study. Paper presented at the Annual Meeting of the
American Educational Research Association, Seattle, WA.
Askew, M. (2005). It ain't (just) what you do: effective teachers of numeracy. In I. Thompson (Ed.), Issues in
teaching numeracy in primary schools (pp. 91-102). Berkshire, UK: Open University Press.
Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective teachers of numeracy. London:
School of Education, King's College.
Clarke, D., & Clarke, B. (2002). Challenging and effective teaching in junior primary mathematics: What does it
look like? In M. Goos & T. Spencer (Eds.), Mathematics making waves (Proceedings of the 19th Biennial
Conference of the Australian Association of Mathematics Teachers, pp. 309-318). Adelaide, SA: AAMT.
Groves, S., Mousley, J., & Forgasz, H. (2006). Primary Numeracy: A mapping, review and analysis of Australian
research in numeracy learning at the primary school level. Canberra, ACT: Commonwealth of Australia.
Muir, T. (2008). “Zero is not a number”: Teachable moments and their role in effective teaching of numeray. In
M. Goos, R. Brown & K. Makar (Eds.), Navigating currents and charting directions (Proceedings of the 31st
annual conference of the Mathematics Education Research Group of Australasia, Brisbane, pp. 361-367).
Adelaide, SA: MERGA.
Muir, T. (2008b, July 6-13). Describing effective teaching of numeracy: Links between principles of practice and
teacher actions. Paper presented at the 11th International Conference on Mathematics Education (ICME-11) for
Topic Study Group 2: New developments and trends in mathematics education at primary level, Monterrey,
Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics teaching: Reflecting
on practice with the knowledge quartet. London: SAGE Publications Ltd.
Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston,