Geometry and Measurement
A Review of Literature
This report was produced under contract to the Ministry of Education, Contract No. 323-1642 by Andrew Tagg with the help of Derek Holton and Gill Thomas.
The Teaching and Learning of 1
Geometry and Measurement 1
A Review of Literature 1
The van Hiele levels 4
Development of geometric proof skills 6
Spatial representation 6
National Numeracy Strategy 7
NCTM Standards 7
asTTle Curriculum Map 8
Count Me Into Space 8
National Numeracy Strategy 10
NCTM Principles and Standards 10
asTTle Curriculum Map 11
Count Me Into Measurement 11
Learning approaches 12
Importance of play 12
Use of technology in geometry 12
Appendix A: Objectives from National Numeracy Strategy 18
Appendix B: Curriculum standards from NCTM Principles and Standards 20
Appendix C: Achievement objectives from asTTle project 23
Geometric Knowledge 23
Geometric Operations 24
Appendix D: Summary of ideas for consideration. 26
Appendix E: Annotated bibliography 27
This report aims to provide a theoretical background for the development of the Geometry and Measurement Strands in the New Zealand Mathematics Curriculum. As such it should be of assistance to the committee considering the Mathematics section of the current New Zealand Curriculum Review project.
We particularly concentrated on progressions in the two strands both within the area of higher level thinking (what we will refer to as ‘strategies’) and within in the area of content (‘knowledge’). While much has been written on the theoretical progressions, we were concerned that there is little to be found on progressions that could be of direct assistance to the classroom teacher. What we have found in this area comes from curricula from various countries.
We present our findings and make suggestions as to how the committee might move forward from here. The material is divided into the broad headings of Geometry, Measurement, Learning Approaches, and Suggestions, while the first two of these are broken down further into Theory and Curriculum/Implementation.
While not specifically stated as part of the scope of the review, we believe that it is also important to consider the position of geometry and measurement within the mathematics curriculum as a whole.
Within the Mathematics in the New Zealand Curriculum (MiNZC)(Ministry of Education, 1992) measurement and geometry are two of the five strands into which mathematics topics are divided, but this is not always the case internationally; The National Numeracy Strategy (DfEE, 1998) in the United Kingdom, which is also widely used internationally, and several other countries and regions including Hungary, Italy, Alberta (Canada), and British Columbia (Canada), group the two together in a strand called Space, Shape and Measures or similar. While we are used to considering measurement as a category of its own, grouping it with geometry in this way does resolve several issues with regard to certain topics. For example, when measuring area, clearly the geometric properties of shapes should be brought to bear. Angle, similarly, does not fit fully within either measurement or geometry; when referring to angle as a property of a shape we place it within geometry, but when measuring with a protractor, clearly measurement is more appropriate. Time, money and estimation are also topics that are often included within the scope of the measurement strand, but which could be seen as more logically positioned within the number strand (estimation is currently placed within number in the New Zealand curriculum) as their use is largely focussed around number rather than measurement.
For the purposes of this review however, geometry and measurement will be treated separately, with the recommendation that consideration be given to ending their status as separate strands.
The main emphasis of the theoretical writing on progressions in geometry tends to be on increasing sophistication of overall ‘understanding’ of geometry (how do students do geometry?), in contrast to the curriculum descriptions of geometry, which tend to be focused much more on the content of knowledge/ability (what do students do in geometry?).
Piaget and Inhelder’s (1956) theory describes the development of the ability to represent space. “Representations of space are constructed through the progressive organization of the child’s motor and internalized actions, resulting in operational systems” (Clements and Battista, 1992, p. 422). The order of development is seen to be: topological (connectedness, enclosure, and continuity); projective (rectiliniarity); and Euclidean (angularity, parallelism, and distance). They describe a sequence of stages in the development of children’s ability to distinguish between shapes when drawing them. These are:
Stage 0: scribbles (less than 2)
Stage I: Topological - irregular closed curves to represent circles, squares, etc. (2-4 years)
Stage II: Projective - progressive differentiation of Euclidean shapes (4-7 years)
Stage III: Euclidean - ability to draw Euclidean shapes (7-8 years)
(Piaget and Inhelder, 1956, pp. 55-57).
This has not been widely accepted – even young children may be able to operate with some Euclidean concepts. It seems more likely that topological, projective and Euclidean notions all develop over time and their usage becomes increasingly integrated.
In the 1950s Pierre van Hiele and Dina van Hiele-Geldof developed a series of thought levels that they perceived as describing a progression of increasing sophistication of understanding of geometry. Initially, five discrete hierarchical levels were described, numbered 0-4; variations on these levels continue to provide the basis for many models used to understand learning in geometry. In recent years the original five levels have more commonly been renumbered as levels 1-5 (Swafford et al., 1997), and many researchers have described the existence of an earlier, pre-recognitive level (Clements and Battista, 1992, p. 429; Clements et al., 1999). It is this more recent numbering that will be used in the following discussion.
Level 0: Pre-recognitive
At the pre-recognitive level students cannot reliably distinguish between different classes of figures. For example, while they may be able to distinguish between squares and circles, they may not be able to distinguish between squares and triangles.
Level 1: Visual
At the visual level students recognise figures by their global appearance, rather than by identifying significant features, for example a rectangle would be recognized as a rectangle “because it looks like a door”. Some researchers (Clements et al, 1999) believe that this level can be better described as syncretic, as students at this level often use a combination of verbal declarative and visual knowledge to differentiate between shapes. That is, at Level 1 some children may apply a combination (synthesis) of overall visual matching with limited feature analysis to identify shapes.
Level 2: Descriptive/Analytic
At the descriptive/analytic level students differentiate between shapes by their properties. For example a student might think of a rectangle as a shape with four sides, and label all shapes with four sides as rectangles. However they might refuse to accept a square as a rectangle “because it is a square”.
Level 3: Abstract/Relational
At the abstract/relational level students relate figures and their properties. They can provide definitions, and differentiate between necessary and sufficient conditions for a concept. They can classify figures hierarchically, and produce some geometric arguments.
Level 4: Formal deduction
At the formal deduction level students develop sequences of statements that logically justify a conclusion; constructing simple, original proofs.
Level 5: Rigour
At the final level, students rigorously apply rules to derive proofs within a mathematical system.
Phases of Learning
As well as the levels of understanding the van Hieles also described 5 phases of learning through which students can be taken in advancing to the next level (Hoffer, 1983, p. 208).
Phase 1: Inquiry
In this phase the teacher engages the student in two-way conversation about the topic. Vocabulary is established and the teacher sets the ground for further study.
Phase 2: Directed orientation
Here the teacher directs the path of exploration in such a way as to ensure that the student becomes familiar with specific key ideas related to the topic.
Phase 3: Expliciting
Now the students work much more independently, refining their understanding and use of vocabulary.
Phase 4: Free orientation
In this phase the students encounter multi-step tasks with no one route to solution, and explore their own methods to obtain solutions.
Phase 5: Integration
Finally the students review their learning and produce an overview of their understanding. The teacher aids them in summarising their key ideas.
Development of geometric proof skills
Clements and Battista (1992, p. 439) describe three levels of the development of proof skills:
Level 1 (Up to age 7-8): At this level there is no integration of ideas.
Level 2 (7-8 through to 11-12): At this level students begin to make predictions on the basis of results they have seen in previous experiments. For example, they may, after experimenting with triangles, state that the angles add to make a straight line for each triangle.
Level 3 (Ages 11-12 and beyond): At this level students are able to apply deductive reasoning to any assumptions.
Rosser et al. (1988) describe a sequence of mastery of conceptualization of geometry operations related to reproduction of a simple pattern. The sequence is:
Reproducing a geometric pattern, constructed from blocks.
Reproducing a similar pattern, which was covered after an initial 6 second observation period.
IIIA. Reproducing the result after rotation of a similar pattern, which was covered after an initial 6 second observation period, and then rotated.
IIIB. Reproducing a perspective view of a similar pattern, with the original pattern remaining available.
The order of difficulty of the tasks was shown by experiment to be IThe conclusion reached was that the operations associated with the lower level tasks are required for solving the higher level tasks, and that a fixed order of task mastery should be expected.
National Numeracy Strategy
The National Numeracy Strategy (DFEE, 1999) describes objectives for students from Reception to Year 6. Within the measures, shape and space strand, the objectives related to shape and space can be broadly grouped into five categories: describing and classifying shapes; making patterns, shapes and objects; symmetry and transformations; position and movement; and angles. The objectives within each of these categories describe a clear progression of complexity from simplest to most complex. For example, the progression of objectives related to direction and movement is:
Reception: Use everyday words to describe position, direction and movement.
Year 1: Use everyday language to describe position, direction and movement.
Year 2: Use mathematical vocabulary to describe position, direction and movement.
Year 3: Read and begin to write the vocabulary related to position, direction and movement.
Year 4: Recognise positions and directions. Use the eight compass directions.
Year 5: Recognise positions and directions. Read and plot coordinates in the first quadrant.
Year 6: Read and plot coordinates in all four quadrants.
Principles and Standards for School Mathematics (NCTM, 2000) describes objectives for students from pre-kindergarten to grade 12. Within the geometry strand, the objectives are grouped into three categories:
Specify locations and describe spatial relationships using coordinate geometry and other representational systems;
Apply transformations and use symmetry to analyze mathematical situations; and
Use visualization, spatial reasoning, and geometric modeling to solve problems.
The objectives within each of these categories describe a clear progression of complexity from simplest to most complex.
The New Zealand Curriculum Exemplars (Ministry of Education, 2003) describe a progression in development of spatial understanding related to tessellations.
Level 1: Fit shapes together to form a tessellation.
Level 2: Identify common shapes that tessellate.
Level 3: Use right angles to explain the tessellation of objects.
Level 4: Know that tessellating objects fit together round a point.
Level 5: Use angles to show that shapes will or will not tessellate.
asTTle Curriculum Map
The development of the asTTle assessment tool included work carried out on the ‘mapping’ on Mathematics in the New Zealand Curriculum (Thomas et al., 2003). The achievement objectives were grouped into categories and ordered by level, with several additional objectives added to fill perceived gaps in the curriculum. An attempt was made to identify progressions through the levels of MiNZC. The subcategories within geometry identified in the asTTle map were:
Two dimensional and three dimensional shapes
Symmetry and Transformations
Construction and Drawing
The Count Me Into Shape project (NSWDET, Undated) aims to support teachers by giving a framework for space related mathematics. The framework consists of two key aspects: part-whole relationships, referring to how a shape can be part of a larger shape, and features of shapes; and orientation and motion, referring to perspective and manipulation of shapes. Each of these aspects is seen to have the same sequence of development of types of strategies.
Emerging strategies: Students are beginning to attend to spatial experiences, exploring grouping like objects, and assigning words to groups of shapes and concepts related to position and movement.
Perceptual strategies: Students are attending to spatial features and beginning to make comparisons, relying particularly on observations and manipulations of physical materials.
Pictorial imagery strategies: Students are developing mental images associated with concepts and increasingly using standard language to describe their understanding.
Pattern and dynamic imagery strategies: Students are developing conceptual relationships and using pattern and movement in their mental imagery.
Efficient strategies: Students select from a range of spatial strategies for a given context. They use imagery, classification, part-whole relationships and orientation.
The TIMSS (Trends in Mathematics and Science Study) 2003 framework is intended to describe “important content for students to have learned in mathematics and science” (Mullis et al., 2003, p. i). Within the Geometry strand the objectives are grouped into five categories:
Lines and angles
Two- and three-dimensional shapes
Congruence and similarity
Locations and spatial relationships
Symmetry and transformations
There appears to be no attempt to discuss progressions here.