1. The Singapore Approach

2.  The TIMSS achievement scale has a centre point of 500 and a standard deviation of 100. It is scaled to remain constant from assessment to assessment, allowing comparison over time.  The highest performing countries were those in the Asian Pacific Rim (four such countries for TIMSS 2007 and five in 2011). In Europe, only Northern Ireland significantly outscored England in mathematics at this age in 2011.

3.  Inspire Maths provides a highly scaffolded learning framework with problem solving at its heart. It is built on a focused, coherent and cumulative spiral curriculum that continuously builds and consolidates knowledge to reach deep understanding. The programme encourages extensive practice to develop fluency and mastery, so that every child-across all abilities – can succeed at mathematics.

4.  Problem solving curriculum  Emphasis on the development of intellectual competence such as the ability to visualise  Emphasis on conceptual understanding (representing concepts through objects and pictures, making connections between different representations)  Systematic development of skills and concepts - the spiral approach.  Emphasis on the C-P-A approach (concrete-pictorial-abstract)

5. 3 is abstract – it is a concept. A representation of 3 can be shown i.e. counters, fingers, numerals

6.  Students learn a new skill by using concrete materials, reinforcement is achieved by going back and forth between the representations. Concrete Enactive – internalised action. Pictorial Iconic – this can also be sensory imagery. Abstract Symbolic – arbitrary symbols (words, numerals)

7. Children are expected to use and understand numbers, which are abstract concepts. Many children struggle with this and so their first experience of mathematics is confusing, leaving them with no solid foundation to build on for later learning. The CPA approach helps children achieve secure number sense – what numbers really represent and how to use them mathematically. This is done through a series of carefully structured representations – first using physical objects (concrete), then diagrams or pictures (pictorial), and ultimately using representations such as numerals (abstract). [teachers guide 1A pg 7]

10. The National Curriculum for mathematics aims to ensure all pupils:  ‘become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and are able to recall and apply their knowledge rapidly and accurately.’  reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, or proof using mathematical language.  can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

11. When children have achieved mastery in a particular area of mathematics they:  Are fluent in the facts and procedures  Can transform these facts, make connections and transfer their understanding to unfamiliar situations applying a higher level of mathematics and thinking.

12. Conceptual Understanding Blobs= items of knowledge More links = conceptual understanding. Networks, discussion and practical work help form these links. Valsa Koshy

13. Heuristics refers to the different strategies that children can adopt to solve unfamiliar or non-routine problems.  Bar modelling  Pattern spotting  Using diagrams  Making conjectures, estimating (guess and check)

14. The bar model is a step-by-step method that helps children to understand and extract the information within a calculation or word problem. By drawing a bar model, children translate a calculation or word problem into a picture. The approach helps children process information given in the problem, visualise the structure, make connections and solve the problem.

15.  To help pupils to process the information in the question. (Read)  Make sense of the question or problem. (Think and Plan)  Just one of the ways to comprehend the problem & from the models, pupils will be able to work out the solution.

16. Different types of models  Discrete Model  Part-Whole Model  Comparison Model  Ratio Model  Change/Transforming Model or before and after model.

17. Representations not pictures. Each box represents 1 discrete item. Theo has 4 cars and 3 lorries. How many toys has he got?  Theo has got _____ toys. c c c c l l l

18. Sofia has 7 sweets and Finley has 5 sweets. How many sweets do they have altogether? ? 7 5 Always a question mark where the answer goes – this is what we are trying to find.

19. The model represents a quantitative relationship among three variables: whole, part 1 and part 2. Given the values of any two variables, we can find the value of the third one by addition or subtraction. Whole Part 1 Part 2

20. Michelle baked 12 cakes. She gave away 7 cakes. How many cakes did she have left? Michelle has _____ cakes left. 12 7?

22. William has 95p. He has 20p less than his brother. How much money do William and his brother have altogether? They have ____ altogether. 95p 20p W B ? 0.95 1.15 £2.10

23. The model represents a quantitative relationship among three variables: larger quantity, smaller quantity and difference. Given the values of any two variables, we can find the value of the third one by addition or subtraction. Smaller quantity Larger quantity Difference

24.  Bryan has 16 marbles.  He has 4 more marbles than Christopher.  How many marbles does Christopher have?  Christopher has ____ marbles. 16 ? 4

26. The following part-whole model represents a whole divided into 3 equal parts: Whole part The model illustrates the concepts of multiplication as: One part x Number of parts = Whole

27. 222

28. Jerry has 30 marbles. Tom has 4 times as many marbles as Jerry. How many marbles do they have altogether? Jerry and Tom have ____ marbles altogether. ? 30 Jerry Tom

29.  One quantity is a multiple of the other, e.g. Larger quantity Smaller quantity The larger quantity is 3 times as much as the smaller quantity, and the smaller quantity is equal to 1/3 of the larger quantity.

30. The difference between two numbers is 516. If the larger number is 4 times the smaller number, find the sum of the two numbers. 516 ? / 3= 172 172 172 x 5 = 860 ? = 860

31. The following part-whole model shows a whole divided into 3 parts A, B and C in the ratio 2:3:4. Whole Part A Part BPart C The following comparison model shows three quantities A, B and C which are in ratio 2:3:4. A B C The ratio 2:3:4 means “2 units to 3 units to 4 units ”

32.  The ratio of the number of Katie’s marbles to Emma’s marbles is 4:5. Katie is given 12 more marbles.  Katie now has twice as many marbles as Emma. How many marbles did Katie have to begin with?  Katie had ____ marbles to begin with.

33.  The ratio of Katie’s marbles to Emma’s marbles is 4:5.  Katie is given 12 more marbles and now has twice as many marbles as Emma. If Emma has 5 parts then Katie now has 10 parts meaning 6 are added to her original 4 parts.  12 is the difference so 12/6 =2. Each part has the value of 2.  Therefore 2+2+2+2=8.  Katie had 8 marbles to begin with. K E ? Double E =10 parts 12/6=2 2222 2 2 22 22 ? = 8 THIS IS A YEAR 6 LEVEL QUESTION!

34.  This type of model can be used to solve complex problems.  The parts can be transformed into smaller units.  This type of model is useful for tackling problems which involve before-and-after situations.

35.  A shop keeper sold 1/3 of his balloons in the afternoon and 2/5 of the remainder in the evening.  If he had 150 balloons left, find the number of balloons he had at first.  He had ____ balloons at first.

36. 150 /3=50 50 50+50+50+50+50= 250 250/2=125 250+125=375 He had 375 balloons at first. 150 50+50+50+50+50= 250 250/2=125

37.  Read the problem.  Rewrite the question in sentence form leaving a gap for the answer.  Think about what is being asked and which model supports the question.  Draw the bars.  Partition or chunk the bars and note which section represents the answer.  Discuss the question and think about what is being asked.  Write the answer in the sentence and check that the answer makes sense.  Is there another way to draw the model and represent the question?  What further questions could you ask using the model as a prompt?

38.  Let’s Learn! To build firm foundations  Guided practice to develop deep understanding  Let’s explore! And games to investigate and apply learning  Math’s Journal to reflect  Put on your thinking caps! To challenge  Home maths to encourage mathematical conversations  Practice Books to develop fluency and consolidate  Assessment Books to create a record of progress

39. Any questions?