Geometry in NZC Shape - a teaching progression Sandra Cathcart LT Symposium 2009

Much research has been done about the way we think and reason in a geometric context In particular the Dutch couple van Hiele whose work began in Today it has had a major impact on the development of new Geometry curriculum world wide.

Shapes Classes of Shapes Properties of Shapes Relationships among properties Deductive systems of properties Analysis of deductive sytems 0 Visualisation 1.Analysis 2.Informal deduction 3.Deduction 4 Rigor The van Hiele Theory of Geometric Thought

What are the features of this model? The most prominent feature of the model is a 5 level hierarchy of ways of understanding spatial ideas. Each of the 5 levels describes the thinking processes used in geometric contexts. The levels describe how we think and what types of geometric ideas we think about, rather than how much knowledge we have. As we move from one level to the next there are increasingly complex objects of thought. They follow quite closely our own AO’s

Level 1 Sort objects by their appearance It is the appearance of the shape that defines it - a square is a square because it looks like a square. (van Hiele stage 0) Students can tell the shapes that are the same/different ie start to classify. May be able to use shape names May be able to do tangrams Recognise shapes in their world

Level 2 Sort objects by their spatial features with justification Classes of shapes rather than individual shapes become important eg students will start to talk about rectangles. They can focus in on 1 shape eg “what makes this shape a rectangle” Begin to realise that a collection of shapes go together because of particular properties. (triangles, squares,rectangles,circles, trapezium,cube, cylinder,prism,cone and pyramid) Regular and irregular figures Ideas about a particular shape can now be generalised to all shapes in that class.eg if a shape belongs to “triangles” then………………will be true. Should be able to draw simple shapes from instructions. Should be able to names shapes from simple descriptions. They cannot see subgroups yet.

Level 3 Classify plane shapes and prisms by their spatial features. Given a set of shapes, sort them and be able to describe them: “ these are all squares because they have 2 pairs of parallel sides,4 equal sides and 4 right angles” Definitions are developed - all triangles have three sides so any shape with 3 sides must be a triangle. 4 ways to describe this set

Level 4 - identify classes of 2 and 3D shapes by their geometric properties Development of sub groups ( classes within classes)

Each level has implications for teachers Teachers should be able to see some growth in geometric thought over a year. The use of hands on materials is essential. Emphasis on teaching where the student is. Most activities are adaptable for all levels.(encourage the poorer student and challenge the better student). Levels of geometric thought are not age determined.

So how do we move students through this progression? In early geometric thinking students need experience with a huge variety of 2D and 3D shapes. Along the way names and properties will be developed by the students themselves.

Activity one: Handout set of shapes - a set for each group of 4 There are several parts to this activity - do each in order a)Each student randomly selects a shape. In turn they tell the rest of their group 1 or 2 things about their shape. b)Each students selects 2 shapes. They now need to find 2 things that are the similar about their shape and 2 things that are different. c)The group selects one shape at random. Their task is now to find all the other shapes that are like the chosen shape, but all according to the same rule. d)Do a second sort with the target shape but different “rule”. Students share their sorting rules with the class. e)All students now draw a new shape that fits their rule explaining why it does. f)Do a secret sort (pick some pieces out that belong to a rule- leaving some that also do) - rest of class must find the rule and the other shapes. This can be used to help introduce a new property.

Activity 2 Building up(constructing) and Breaking down (dissection) of shapes Students need to explore how shapes fit together to form a larger shape and vice versa Mosaic puzzle from van Hiele Make this shape :with 2 pieces with 3 pieces four? Make a long parallelogram Make as many rectangles as you can. How many different sized angles can you find? Tangrams are popular but limited after early stages Can be made more difficult by reducing the outline, then proportional reasoning is involved.

Activity 3 - defining groups of triangles Sort the entire collection into three groups No triangle belongs to two groups Students write a description of the groups Repeat with a different reason for sorting. This activity results in the definition of 6 different types of triangles Now whole class copy and complete the diagram by sketching an example in each cell. 2 cells are impossible - why?

EquilateralIsoscelesScalene Right angled Acute angled Obtuse angles

Activity 4 Property lists for quadrilaterals On each sheet of paper there are 3 or 4 examples of a type of quadrilateral. Students work in groups of 3 or 4 and list as many properties as they can then report back to the class. Deal with one shape at a time and all class must agree before the final list is recorded. Each property listed must apply to all shapes. New set of terms (parallel,congruent,bisect,midpoint) New symbols  // etc

Acknowledgement Activities in this PP come from: 1. Teaching Student - Centered Mathematics Grades 5 to 8 John A Van de Walle & louAnn H Lovin 2. nzmaths website 3. NZC