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Mathematical opportunties in the Early Years

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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.

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Mathematical opportunties in the Early Years

  1. 1. Creating Mathematical Opportunities in the Early Years Dr Tracey Muir University of Tasmania AAMT Connect with Maths 26th August 2014
  2. 2. Background Launceston
  3. 3. 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?
  4. 4. Connectionist, transmission and discovery beliefs Connectionist Transmission Discovery The use of methods of Primarily the ability to calculation which are perform standard both efficient and procedures or routines effective 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)
  5. 5. 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)
  6. 6. Principles of practice •Make connections •Challenge all pupils •Teach for conceptual understanding •Purposeful discussion •Focus on mathematics •Convey and instill positive attitudes towards mathematics
  7. 7. Teaching actions • Choice of examples • Choice of tasks • Questioning • Use of representations • Modelling • Teachable moments
  8. 8. How familiar is term PCK? (Vote)
  9. 9. • Teacher knowledge (Ball, Thames, & Phelps, 2008)
  10. 10. 5 Practices • Anticipating • Monitoring • Selecting • Sequencing • Connecting (Smith & Stein, 2011)
  11. 11. Anticipating • Planning for ‘teachable moments’ • Knowing your subject and the difficulties students might encounter • Experience, research and reflection • Highlighting possible misconceptions in planning
  12. 12. Pre-planning • What do I need to know and understand before teaching this topic? • What are the likely difficulties or misconceptions students may have?
  13. 13. Measurement opportunities
  14. 14. Devise task • What are some activities or tasks you could undertake to explore ‘Mr Splash’?
  15. 15. • 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?
  16. 16. • 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.
  17. 17. What teacher knowledge is required? • Content knowledge • Curriculum knowledge • PCK
  18. 18. 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)
  19. 19. In your groups, measure your heights, arm spans, hand spans and foot size (record these measures in the following table).
  20. 20. Name Arm span Height Wrist Neck Waist Zach 152 150 15 30 63
  21. 21. More informal measurement • Story • Book • Clip • Interactive website • Scenario • how-big-is-a-foot/ • 205
  22. 22. What do I need to know/consider about topic?
  23. 23. 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
  24. 24. 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
  25. 25. Estimating – a teachable moment
  26. 26. Counting and early number • Capitalise on ICT atch?v=aXV-yaFmQNk h?v=MGMsT4qNA-c
  27. 27. Songs and Rhymes
  28. 28. Sequencing
  29. 29. Consider: • What would be the advantages and disadvantages of doing this activity online as compared with using real materials in the classroom?
  30. 30. Capitalising on the ICT
  31. 31. Bridging 10
  32. 32. • 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]
  33. 33. • 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
  34. 34. TPACK Reproduced by permission of the publisher, © 2012 by
  35. 35. Subitising – making it purposeful
  36. 36. Press Here •
  37. 37. Thumbs up or down? Google Images
  38. 38.
  39. 39. Two of Everything Google Images
  40. 40. 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
  41. 41. 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
  42. 42. 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
  43. 43. More than ‘sharing’ Who did it a different way?
  44. 44. 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
  45. 45. 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) 13/10/2014
  46. 46. Place a number where you think it would fit…
  47. 47. 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)
  48. 48. 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!”
  49. 49. • 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.
  50. 50. Anticipating mathematical opportunities • Confusion between length, area and volume (and capacity) • Conversion of units • Multiplication • Dimensions of tins • Consistency for comparison
  51. 51. Self reflection/feedback Something that helped me learn … Something I wasn’t sure about … Something that stopped my learning
  52. 52. A couple of other favourites…. Google images
  53. 53. The wolf’s chicken stew
  54. 54. Useful resources
  55. 55. 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
  56. 56. Useful references 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, 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.