1 The van Hiele levels of Geometrical thought in an in-service training setting in South Africa. Ronél Paulsen South Africa

2 Questioning  The importance of questioning  Use children’s innate curiosity  Probe to become more aware of thinking processes  Develop confidence to question, challenge and reflect

3 Mathematical thinking is a dynamic process,… enabling the complexity of ideas we can handle, expands our understanding (Mason)

4 Still drawing on John Mason What improves Mathematical thinking?  Practice with reflection

5 Still drawing on John Mason What supports Mathematical thinking?  Atmosphere of questioning, challenging, reflection.  This takes time to develop

6 Still drawing on John Mason What provokes Mathematical thinking?  Challenge, surprise, contradiction,

7 Japanese lesson plans emphasise what students will think, not what the teacher will say (Stigler 1998) Teachers anticipate students’ reaction Because teachers work collaboratively on lessons, each teacher contributes from his or her own experiences

8 Rohlen also emphasises that the goal for questioning in Japanese classrooms is to get students to think They pose questions and ask students to explain their thinking

9 Metacognition – knowledge of one’s own thought processes. (Romberg) Novice vs expert How do we teach students to become aware of their own metacognitive processes?

10 Tom Carpenter and Elizabeth Fennema found that although teachers have a great deal of intuitive knowledge about children’s Mathematical thinking, this knowledge is fragmented (CGI Programme at Wisconsin)

11 Teachers should take trouble to understand how children think Leads to fundamental changes in beliefs and practices This reflects on students’ learning (Fennema et al)

12 The van Hiele Levels In 1957, Dutch educators Dina van Hiele-Geldof and Pierre van Hiele proposed that the development of a student's understanding of reasoning and proof progresses through five distinct levels.

13 The van Hiele Levels  Students identify and reason about shapes and other geometric configurations based on shapes as visual wholes rather than on geometry properties Level 0: Visual  For instance, they might identify a rectangle as a "door shape"  They would identify two shapes as congruent because they look the same, not because of shared properties

14 The van Hiele Levels A learner on this level will name the following shapes as rectangle, parallelogram and square just on appearance without knowing any of their properties Level 0: Visual (continued)

15 The van Hiele Levels They might have problem to give the names of the shapes in the following orientations, “because they don’t look like a rectangle, a parallelogram or a square” Level 0: Visual (continued)

16 The van Hiele Levels The learners analyse figures in terms of their components and relationships between components, establishes properties of a class of figures empirically, and uses properties to solve problems Level 1: Descriptive/Analytic

17 The van Hiele Levels  Students recognise and characterise shapes by their properties Level 1: Descriptive/Analytic  For example, they can identify a rectangle as a shape with opposite sides parallel and four right angles

18 The van Hiele Levels Level 1: Descriptive/Analytic  Students at this level still do not see relationships between classes of shapes (e.g., all rectangles are parallelograms), and they tend to name all properties they know to describe a class, instead of a sufficient set

19 The van Hiele Levels Level 1: Descriptive/Analytic (continued) When learners investigate a certain shape they come to know the specific properties of that figure. For example, they will realise that the sides of a square are equal and that the diagonals are equal. Students discover the properties of a figure but see them in isolation and as having no connection with each other.

20 The van Hiele Levels Level 2: Abstract/Relational  Students are able to form abstract definitions and distinguish between necessary and sufficient sets of conditions for a class of shapes, recognizing that some properties imply others.  Students also first establish a network of logical properties and begin to engage in deductive reasoning, though more for organizing than for proving theorems.

21 The van Hiele Levels Level 2: Abstract/Relational (continued) When learners reason about and compare the properties of a figure they realise that there are relationships between them. The relationships being perceived:  Exist between the properties of a specific figure, and  Exist between the properties of different figures.

22 The van Hiele Levels Level 3: Formal Deduction and Proof  Students are able to prove theorems formally within a deductive system.  They are able to understand the roles of postulates, definitions, and proofs in geometry, and they can make conjectures and try to verify them deductively.

23 The van Hiele Levels Level 3: Formal Deduction and Proof (contd)  At this level the learner is able to make deductions. He/she is able to write proofs, understands the role of axioms and definitions, and knows the meaning of necessary and sufficient conditions. The learner reasons formally within the context of a mathematical system, complete with undefined terms, axioms and underlying logical system, definitions and theorems.

24 The van Hiele Levels Level 4: Rigour  The student at this level understands the formal aspects of deduction. Symbols without referents can be manipulated according to the laws of formal logic. The learner can compare systems based on different axioms and can study various geometries in the absence of concrete models.

25 The van Tall Hiele Levels Level zero?

26 Level Descriptors In order to place children on one if the levels of van Hiele, there are some descriptors which can be used to see what children can or cannot do. For the purpose of this talk, I only looked at a few of the descriptors, as set out by van Hiele.

27 Level 0 Descriptors The learners identify and operate on shapes (e.g., squares, triangles) and other geometric configurations (e.g., lines, angles, and grids) according to their appearance.

28 Level 0 Descriptors  The learner identifies instances of a shape by its appearance as a whole in a simple drawing diagram or set of cut-outs.  The learner identifies instances of a shape by its appearance as a whole in different positions

29 Three year olds can sort and classify (Level 0) Flat shapes and space shapes

30 Recognizing spheres and circles have the same shape

31 The visual effect!

32 Level 0 Descriptors A learner identifies parts of a figure but:  Does not analyse a figure in terms of its components.  Does not think of properties as characterising a class of figures.  Does not make generalisations about shapes or use related language.

33 Level 0 Descriptors Examples Does not analyse a figure in terms of its components. This is a rectangle This is not a rectangle

34 Level 0 Descriptors Examples Does not analyse a figure in terms of its components. This is a kite This is not a kite

35 Level 0 Descriptors Examples Does not analyse a figure in terms of its components This is a kite This is a rhombusThis is a square?  

36 Level 1 Descriptors The learners analyse figures in terms of their components and relationships between components, establishes properties of a class of figures empirically, and uses properties to solve problems

37 Level 1 Descriptors  The learner sorts shapes in different ways according to certain properties, including a sort of all instances of a class from non-instances.  Does not explain subclass relationships beyond specific instances against given list of properties.

38 Level 1 Descriptors  Does not explain how certain properties of a figure are interrelated  Does not formulate and use formal definitions

39 Level 1 Descriptors  Example A learner tells what shape a figure is, given certain properties. Quadrilaterals of which the diagonals bisect each other

40 Level 1 Descriptors  Example A learner tells what shape a figure is, given certain properties. Quadrilaterals of which the diagonals bisect each other

41 Level 2 Descriptors Learners formulate and use definition, give informal arguments that order previously discovered properties, and follows and gives deductive arguments.

42 Level 2 Descriptors The learner identifies sets of properties that characterise a class of figure and tests that these are sufficient.

43 Level 2 Descriptors Recognises the role of deductive arguments and approaches problems in a deductive manner, but  Does not grasp the meaning of deduction in an axiomatic sense (e.g. does not see the need for definitions and basic assumptions).

44 Level 2 Descriptors  Does not formally distinguish between a statement and its converse  Does not yet establish inter- relationships between networks of theorems.

45 Moving on… The learner must have completed levels 0, 1 and 2 in order to successfully cope with the proofs in Euclidean geometry. Levels are hierarchical

46 Level 3 Descriptors Learners establish, within a postulation system, theorems and inter- relationships between networks of theorems.

47 Level 3 Descriptors  Recognises characteristics of a formal definition (e.g. necessary and sufficient conditions) and equivalence of definitions.  Compares and contrasts different proofs of theorems

48 Level 3 Descriptors  Example Compares and contrasts different proofs of theorems Proving the theorem of Pythagoras

49 Level 4 Descriptors Learners rigorously establishes theorems in different postulational systems and analyses/compares these systems

50 Examples of teachers involving in in-service training programme Some video clips

51 The video Taken during in service training. All teachers are enrolled for an Advanced Certificate Programme at the University of South Africa. English second or third language speakers. All experienced primary school teachers Reading to do for preparation We discussed the van Hiele levels Flat shapes cut outs Simulate a classroom situation We concentrated basically on level zero van Hiele levels Teachers were required to do various activities: Sort the shapes with curves Sort shapes with three sides Find the squares Shapes with equal sides

52 Mathematical thinking? The person who took the video, also asked the questions. Look carefully and you will probably note many areas where the questioning techniques can be improved upon. Impromptu and unedited

53 Cut –out shapes

54 Thank you

55 Dropper Medicine spoon

56 Fenemma E, Carpenter TP, Franke, ML, Levi L, Jacobs VR, and Empsen SB A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education 27(4): Van de Walle, J.A Elementary and middle school Mathematics – teaching developmentally. Sixth Edition. New Jersey: Pearson Education. Rohlen, T and Le Trendre, (1998) Teaching and Learning in Japan. Cambridge University Press. Bibliography

57 Carpenter T, Fennema E, Franke ML, Levi L Empson SB (2000) Research Report, National Centre for improving student learning and achievement in Mathematics and Science. University of Wisconsin − Madison Mason, J Thinking Mathematically