Fractions Presentatie titel The Action model applied Swakopmund

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Fractions Presentatie titel The Action model applied Swakopmund, May 2018 Rotterdam, 00 januari 2007

Fractions Presentatie titel The Action model applied Swakopmund, May 2018 Rotterdam, 00 januari 2007

Fractions – The action model applied § § § § § The melon problem

Fractions – The action model applied § § § § § The melon problem – an example The action model – learning theory The action model – consequences for teaching fractions Fractions in the syllabi Fractions – understanding the 4 basic operations Workshop 1: Fractions part of a whole – the 4 basic operations Workshop 2: Fractions part of a number – the 4 basic operations Fractions – The action model applied: Fractions part of a whole

‘The melons problem’ in grade 6 § § The teacher miss Ank buys 6

‘The melons problem’ in grade 6 § § The teacher miss Ank buys 6 small melons on the market. She wants to give each learner in her class 2/5 part of a melon. How many learners can she give 2/5 part of a melon?

‘The melons problem’ in grade 6 § How many learners can she give 2/5

‘The melons problem’ in grade 6 § How many learners can she give 2/5 part of a melon? § 6 : 2/5 = 6 x 5/2 = 30/2 = 15 § Is this a solution of a grade 5 or a grade 6 learner? § Is this a solution of a grade 7 or a grade 8 learner?

‘The melons problem’ in grade 6 Martijn Noah

‘The melons problem’ in grade 6 Martijn Noah

The solution of Martijn and Noah Action level 1 How many learners can Miss

The solution of Martijn and Noah Action level 1 How many learners can Miss Ank give 2/5 part of a melon?

The solution of Martijn and Noah Action level 2 How many learners can Miss

The solution of Martijn and Noah Action level 2 How many learners can Miss Ank give 2/5 part of a melon?

The solution of Martijn and Noah Action level 3 A stripmodel of the question:

The solution of Martijn and Noah Action level 3 A stripmodel of the question: How many learners can Miss Ank give 2/5 part of a melon?

The solution of Martijn and Noah Action level 4 § § How many learners

The solution of Martijn and Noah Action level 4 § § How many learners can Miss Ank give 2/5 part of a melon? Focus on the number of parts: Martijn: 1 : 2/5 = 21/2 ; so 6 x 21/2 = 15 Noah: 2 : 2/5 = 5; so 3 x 5 = 15 The teacher: 6 x 5 = 30; so 30 : 2 = 15 The operation: 6 : 2/5 = 30/5 : 2/5 = 30 : 2 = 15

The action model of conceptual teaching of strategies 3 Representation – model/schematic 2 1

The action model of conceptual teaching of strategies 3 Representation – model/schematic 2 1 [performing formal operations] [representation of real life situations in thinking models] Representation – concrete [representation of objects and real life situations in concrete pictures] Informal acting in real life situations [to do, to experience, to see it happen] Van Groenesteijn, M (2011). Protocol ernstige reken en wiskunde problemen en dyscalculie, Assen: Van Gorcum. Act mentally Formal acting Communication Description 4

The action model of conceptual teaching of strategies – Not only for Fractions! §

The action model of conceptual teaching of strategies – Not only for Fractions! § § 5+3=? Level 1 ‘Living math’: Make two groups of learners and combine the groups. § § § 5+3=? Level 2

The action model of conceptual teaching of strategies – Not only for Fractions! §

The action model of conceptual teaching of strategies – Not only for Fractions! § § 5+3=? Level 3 § § 5+3=? Level 4 § 6, 7, 8; 5+3=8 § 5+3=8

The action model of conceptual teaching of strategies – Not only for Fractions! §

The action model of conceptual teaching of strategies – Not only for Fractions! § § 4 x 6=? Level 1 § § 4 x 6=? Level 2

The action model of conceptual teaching of strategies – Not only for Fractions! §

The action model of conceptual teaching of strategies – Not only for Fractions! § § 4 x 6=? Level 3 § § 4 x 6=? Level 4 § 4 x 6=6+6+6+6= 24 4 x 6 = 12 + 12 = 24 § 4 x 6 = 24 §

The action model of conceptual teaching of strategies – 7 principles § 3 Representation

The action model of conceptual teaching of strategies – 7 principles § 3 Representation – model/schematic Always go from concrete 2 Representation – concrete 1 Informal acting in real life situations to abstract, From action in real life contexts and with real life materials : via pictures and photos, via schemes and models to ‘formal operations’, Apply the action model in the pre-primary and in grade 1 to grade 12. Act mentally § Formal acting Communication § 4

The action model of conceptual teaching of strategies – 7 principles § § 3

The action model of conceptual teaching of strategies – 7 principles § § 3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Start on the level that is mastered by the learners, Learners and teachers: always tell what you are doing, Go step by step to a higher mental action level, Important: the structures of the actions must be the same on the 4 levels. Act mentally § Formal acting Communication § 4

Inspiration of the The action model of conceptual teaching of strategies 3 Representation –

Inspiration of the The action model of conceptual teaching of strategies 3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Jean Piaget [1896 – 1980]; Switzerland, Cognitive development theory 0 -2 y Sensorimotor phase, § 2 -6 y Preoperational phase § 6 – 12 y Concrete operational phase § 12 -. . y Formal operational phase § § § Act mentally § Formal acting Communication § 4 Lev Vygotsky [1896 – 1934]; Russia Zone of proximal development theory Social constructivism, the basis of Realistic Mathematics education, § Language §

Formal acting 3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in

Formal acting 3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Act mentally 4 Communication The action model – consequences for teaching fractions § Use the 4 action levels always according the development of the learners, § In grade 1 – 5/6: ‘A fraction is never alone’,

§ § Formal acting 3 Representation – model/schematic 2 Representation – concrete 1 Informal

§ § Formal acting 3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations In grade 1 – 5/6: Use especially action level 1 – 2 – 3, From grade 6/7: Use especially action level 3 – 4, In all grades: communication and description will bridge the distance from the concrete action and the model action to the formal action. Act mentally § 4 Communication The action model – consequences for teaching fractions

Fractions in the syllabus – Grade 1 - 3 § JP Syllabus [2015], p

Fractions in the syllabus – Grade 1 - 3 § JP Syllabus [2015], p 52 § Fractions: a part of a whole, § a part of a group § § Action levels: 1 - 2 - 3

Fractions: a part of a whole or a part of a group § §

Fractions: a part of a whole or a part of a group § § Different situations, Same ‘math language’ ‘A fraction is never alone!’ A part of more than one whole A part of a group [or a part of a number]

The teacher selects the strategies, The teacher selects the action levels. Fractions in the

The teacher selects the strategies, The teacher selects the action levels. Fractions in the syllabus [2015] – Grade 4 – 7 in view of understanding & strategies Compare and order unit fractions and common fractions with the same denominators of up to 10. 4 Solve problems involving fractions 4 5 Compare and order common fractions with the same and different denominators up to tenths and mixed numbers. 5 Recognise equivalence of fractions within fraction families. 5 Perform simple calculations with fractions. 5 SP Syllabus [2015], p 6

The teacher selects the strategies, The teacher selects the action levels. Fractions in the

The teacher selects the strategies, The teacher selects the action levels. Fractions in the syllabus [2015] – Grade 4 – 7 in view of understanding & strategies Find equivalent fractions by extending or reducing fractions. 6 Do simple addition, subtraction and multiplication with common fractions. 6 Solve two-step problems involving common fractions. 6 7 Add, subtract and multiply common fractions. 7 Divide common fractions by a whole number and whole numbers by common fractions. 7

Fractions – understanding the 4 basic operations Fraction families – relations of fractions §

Fractions – understanding the 4 basic operations Fraction families – relations of fractions § ‘ 8–family’: 1 – 1/2 – 1/4 – 1/8 1 1/ 2 1/ 4 1/ § ‘ 9 -family’: 1 – 1/3 – 1/6 – 1/9 8 1 1/ 3 1/ 6 1/ Reducing and extending fractions with fraction families 9

Fractions – understanding the 4 basic operations Fraction families – relations of fractions §

Fractions – understanding the 4 basic operations Fraction families – relations of fractions § 1 ‘ 10 -family’: 1 – 1/5 – 1/10 1/ 5 1/ 1 1/ § ‘ 12 -family’: 1 – 1/2 – 1/3 – 1/4 – 1/6 – 1/12 1/ 1/ 1/ 2 3 4 6 1/ 12 Reducing and extending fractions with fraction families 10

Fractions – understanding the 4 basic operations Addition § § § 1/ + 1

Fractions – understanding the 4 basic operations Addition § § § 1/ + 1 /3 = Level 1: Learners cut a real pizza or…… and merge the pieces. Level 2: Reasoning with a picture of the pizza or……. . 2 § Level 3: Reasoning with the stripmodel. 1/2 1/6 § § 1/6 1/3 1/6 1/6 Level 4: 1/ + 1/ = 3/ + 2/ = 5/ 2 3 6 6 6 1/6

Fractions – understanding the 4 basic operations Subtraction § § § 3/ - 1

Fractions – understanding the 4 basic operations Subtraction § § § 3/ - 1 /2 = Level 1: Learners cut real chocolate bars or…… and look at the difference between pieces. Level 2: Reasoning with pictures of the chocolate bars or……. . 4 § Level 3: Reasoning with the stripmodel. 1/4 1/4 1/4 1/2 1/4 § Level 4: 3/ - 1/ = 3/ - 2/ = 1/ 4 2 4 4 4 1/4

Fractions – understanding the 4 basic operations Multiplication § § § 6 x 1

Fractions – understanding the 4 basic operations Multiplication § § § 6 x 1 /8 = Level 1: Learners cut a real baguette or…… and …. . § Level 2: Reasoning with pictures of the baguettes or……. . § Level 3: Reasoning with the stripmodel. 1/8 1/8 1/4 Level 4: 6 x 1 /8 = 6 /8 = 3 /4 1/8 1/4

Fractions – understanding the 4 basic operations Division § § § 2 : 1

Fractions – understanding the 4 basic operations Division § § § 2 : 1 /3 = Level 1: Learners cut 2 real baguettes or…… and …. . Level 2: Reasoning with pictures of the baguettes. § § Level 3: Reasoning with the stripmodel. 1/3 1/3 1/3 Level 4: 2 : 1 /3 = 6 /3 : 1 /3 = 6 : 1 = 6

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Act mentally § Formal acting Communication Summary Teaching with the action model [1] 4 Level 1: informal acting in real life situations – use the materials The teacher organises the real life situation with use of real life materials in the classroom. The students carry out the actions. 3/ 4 : 4=

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Act mentally § Formal acting Communication Summary Teaching with the action model [2] 4 Level 2: Representation – concrete – use the pictures The math book or the teacher provides photos and pictures of the real life situations.

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Act mentally § Formal acting Communication Summary Teaching with the action model [3] 4 Level 3: Representation – schematic – use the models The math book provides the models and the schemes in the book or the teacher provides the models and the schemes on an anchor chart in the classroom.

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life

3 Representation – model/schematic 2 Representation – concrete 1 Informal acting in real life situations Act mentally § Formal acting Communication Summary Teaching with the action model [4] 4 Level 4: Formal teaching – use the rules The math book describes the calculation strategies in the book or the teacher describes the calculation strategies on an anchor chart in the classroom.

§ Our common objectives: § Learners who understand their mathematics of the fractions!

§ Our common objectives: § Learners who understand their mathematics of the fractions!

§ Remember: ‘A fraction is never alone’ Jaap Griffioen – jaapgriffioen@jaap 2. com, Jaap

§ Remember: ‘A fraction is never alone’ Jaap Griffioen – [email protected] 2. com, Jaap de Waard – [email protected] 2. com