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Mastery Unlocked

This session explores the importance of representations in mathematics and how they facilitate understanding and access to abstract concepts. It covers the stages of representation, constructing meaning with mathematical tools, and progression through the stages. It also introduces various representations, such as ten-frames and bar models, and demonstrates how they help visualize the structure of mathematics. The session concludes with an exploration of doubles and missing numbers using the bar model.

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Mastery Unlocked

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  1. Mastery Unlocked Session 2 – Representation and Structure

  2. The 5 big ideas

  3. Representation and structure • Mathematics is an abstract subject, representations have the potential to provide access and develop understanding. • A representation needs to pull out the concept being taught. It exposes the underlying structure of the mathematics.

  4. Creating a conjecturing community….. • Jot down a 2 digit number. • Reverse the digits. • Add them together. • What do you notice? Can you make a conjecture? • How many examples do you need to do to spot a pattern?

  5. The stages of representation... Famously, the educational psychologist Jerome Bruner recognised that children have difficulty accessing abstract concepts – like the value of a number compared to the written numeral. Success can be achieved by experiencing three stages of representation: Concrete / enacting Pictorial / iconic Abstract

  6. Concrete, Pictorial(iconic), Abstract

  7. Constructing meaning • Mathematical tools should be seen as supports for learning. But using tools as supports does not happen automatically. Students must construct meaning for them. This requires more than watching demonstrations; it requires working with tools over extended periods of time, trying them out, and watching what happens. Meaning does not reside in tools; it is constructed by students as they use tools” (Hiebert 1997 p 10) Cited in Russell (May, 2000). Developing Computational Fluency with Whole Numbers in the Elementary Grades)

  8. Progression through the stages • Struggling learners often get “stuck” in the concrete stage. • It is important that we move them on and show them how to move through each stage and onto the abstract. • Fast graspers often view the concrete as unnecessary or even ‘beneath them’.

  9. Resources and Representations of Mathematics • Resources to help build concepts Ofsted 2013

  10. Here are two representations of numbers to 10 These are both very helpful representations of number but, crucially, they are representing different structures. When would you choose each one?

  11. + 17 = 15 + 24 99 - = 90 - 59

  12. Which is the most useful representation of 7 X 6? Why? 7 x 6 = 42 6 x 7 = 42 7 x (5+1) = 42 (5+2) x (5+1) = 42 7x6 = 42 6x7 = 42

  13. Arrays extend into upper KS2 and beyond 4/7 x 7/8 = The overall array has been divided into 8 x 7 smaller parts, hence the denominator is 56. The shaded part is 7 x 4 = 28. So the answer is 28/56 or 1/2. “Models in Mind,” Mike Askew nrich.maths.org

  14. Part 2 • Familiarisation with 2 different representations… • The tens frame • The bar model There are many other representations which help children visualise the structure of mathematics, these are just 2 of them.

  15. The ten-frame • Ten-frames are two-by-five rectangular frames into which counters are placed to illustrate numbers less than or equal to ten. Ref: Nrich – “Number sense series: A sense of “ten” and place value.” Jenni Wray

  16. How many dots?How do you see them?

  17. “There are 8 because 2 are missing.” This child has a strong sense of ten and its subgroups, an essential stage from counting to calculating.

  18. 2 digit numbers and place value 3 1 (ten) • Ten frames can be a useful tool to build an understanding of place value by the introduction of a second frame. • Notice how the labels make a connection with the abstract.

  19. Using the structure of the tens frame Using your tens frames illustrate this calculation: There are 7 daffodils and 5 roses How many flowers are there altogether?

  20. Bridging 10 How can we use 10 to solve the addition problem?

  21. Number facts: Doubles

  22. Representation and Structure: Doubles

  23. 1 + 1 = 2 1 2 2 1 1 1

  24. 2 + 2 = 4 2 4 4 2 2 2

  25. 3 + 3 = 6 3 6 6 3 3 3

  26. 4 + 4 = 8 4 8 8 4 4 4

  27. 5 + 5 = 10 5 10 10 5 5 5

  28. 6 + 6 = 12 6 12 6 12 6 6

  29. 7 + 7 = 14 7 14 7 14 7 7

  30. 8 + 8 = 16 8 16 8 16 8 8

  31. 9 + 9 = 18 9 18 9 18 9 9

  32. 20 10 + 10 = 10 20 10 20 10 10

  33. Can you imagine…? Doubles

  34. Can you imagine double the number of counters? double 2 = 2 + 2 = 4 4

  35. 14 double 7 = 14 7 + 7 =

  36. 18 double 9 = 18 9 + 9 =

  37. 16 double 8 = 16 8 + 8 =

  38. What can you see…? Doubles

  39. 3 double = 6 half of 6 = 3 3 3 + = 6

  40. 5 double = 10 half of 10 = 5 5 5 + = 10

  41. 6 double = 12 half of 12 = 6 66 + = 12

  42. 9 double = 18 half of 18 = 9 99 + = 18

  43. Missing Numbers: Doubles

  44. 3 double = 6 half of 6 = 3 3 3 + = 6

  45. 6 double =12 half of 12 = 6 66 + = 12

  46. 9 double = 18 half of 18 = 9 99 + = 18

  47. The bar model • “The bar model is used in Singapore and other countries, such as Japan and the USA, to support children in problem solving. It is not a method for solving problems, but a way of revealing the mathematical structure within a problem and gaining insight and clarity as to how to solve it. It supports the transformation of real life problems into a mathematical form and can bridge the gap between concrete mathematical experiences and abstract representations.” NCETM

  48. Addition and subtraction model 8 3 ? Identification of relationships and making Connections supports depth and sustainable learning and paves the way for later learning + = 3 3 + = - = - =

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