Creating Mathematical Opportunities in the Early Years

Dr Tracey Muir

University of Tasmania

AAMT Connect with Maths

26th August 2014

Background

Launceston

Overview

• What does effective mathematics teaching look like?

• What types of knowledge does an effective teacher require?

• How can we provide students with authentic mathematical opportunities?

• What aspects of classroom practice should we focus on in order to maximise mathematical opportunities?

Connectionist, transmission and discovery beliefs

Connectionist Transmission Discovery

The use of methods of calculation which are both efficient and effective

Primarily the ability to perform standard procedures or routines

Finding the answer to a calculation by any method

Confidence and ability in mental methods

A heavy reliance on paper and pencil methods

A heavy reliance on practical methods

Pupil misunderstandings need to be recognised, made explicit and worked on

Pupils’ misunderstandings are the result of ‘grasp’ what was being taught and need to be remedied by further reinforcement of the ‘correct’ method

Pupils’ misunderstandings are the results of pupils not being ready to learn the ideas

(Askew, et al., 1997)

Characteristics of effective numeracy teachers

•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

•Make connections•Challenge all pupils•Teach for conceptual understanding•Purposeful discussion•Focus on mathematics•Convey and instill positive attitudes towards mathematics

Teaching actions

• Choice of examples• Choice of tasks• Questioning• Use of representations• Modelling• Teachable moments

How familiar is term PCK?(Vote)

• Teacher knowledge

(Ball, Thames, & Phelps, 2008)

5 Practices

• Anticipating• Monitoring• Selecting• Sequencing• Connecting

(Smith & Stein, 2011)

Anticipating

• Planning for ‘teachable moments’• Knowing your subject and the difficulties

students might encounter• Experience, research and reflection• Highlighting possible misconceptions in

planning

Pre-planning

• What do I need to know and understand before teaching this topic?

• What are the likely difficulties or misconceptions students may have?

Measurement opportunities

Devise task

• 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

family– Mr. Splash’s height compared to the tallest person in the

world• Explain whether you think Mr. Splash has the dimensions of a

real person.

What teacher knowledge is required?

• Content knowledge• Curriculum knowledge• PCK

Anticipating student responses

Measurement considerations:• 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).

Name Arm span

Height Wrist Neck Waist

Zach 152 150 15 30 63

More informal measurement

• Story• Book• Clip• Interactive website• Scenario

• http://www.mwpenn.com/lesson-plans/lesson-plan-for-how-big-is-a-foot/

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L205

What do I need to know/consider

about topic?

What do I need to know/consider

about topic?• 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 task

Estimating – a teachable moment

Counting and early number

• Capitalise on ICT

https://www.youtube.com/watch?v=aXV-yaFmQNk

https://www.youtube.com/watch?v=MGMsT4qNA-c

Songs and Rhymes

Sequencing

Consider:

• What would be the advantages and disadvantages of doing this activity online as compared with using real materials in the classroom?

Capitalising on the ICT

Bridging 10

• 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 [ten frame]

• 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 yellow frame]

• 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

TPACK

Reproduced by permission of the publisher, © 2012 by tpack.org

Subitising – making it purposeful

Press Here

• https://www.youtube.com/watch?v=Kj81KC-Gm64

Thumbs up or down?

Google Images

http://illuminations.nctm.org/Activity.aspx?id=3528

Two of Everything

Google Images

Monitoring

• 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

In Out

3 7

5 11

4 9

10 21

In Out

2 4

3 6

5 10

1 2

In Out

8 11

10 15

6 7

20 35

Connecting

• Drawing connections between students’ solutions and the key mathematical ideas

• Goal is to have student presentations build on one another to develop powerful mathematical ideas

More than ‘sharing’

Who did it a different way?

Factors influencing the planning and uptake of mathematical opportunities

• Teacher knowledge Content knowledge, Pedagogical Content Knowledge (PCK)

(Shulman, 1987)

• 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 mathematical opportunities

• 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)

09/04/2023

Place a number where you think it would fit…

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 the door!”

(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.

Anticipating mathematical opportunities

• Confusion between length, area and volume (and capacity)

• Conversion of units• Multiplication• Dimensions of tins• Consistency for comparison

Self reflection/feedback

Something that helped me learn …

Something I wasn’t sure about …

Something that stopped my learning

A couple of other favourites….

Google images

The wolf’s chicken stew

Useful resources

Conclusions

• Useful to consider the types of knowledge required by teachers

• Mathematical opportunities can be anticipated

• 5 practices are useful for orchestrating productive mathematical discussions

• Mathematical opportunities are everywhere – be creative but make them purposeful

Useful referencesArafeh, 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, Mexico.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, VA: NCTM.