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
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Appendix D: Summary of ideas for consideration.Geometry
Measurement
Appendix E: Annotated bibliographyThis appendix contains brief notes on some of the articles used in this review of literature. The notes are not intended to be a complete discussion of the articles. Ball, B., (2003). Teaching and learning mathematics with an interactive whiteboard. Micromath, Spring 2003, pp. 4-6. Summary This article discusses the use of the interactive whiteboard to teach mathematics. It describes strengths including its dynamic nature, the ability to carry out whole class activities on one computer, and the fact that a range of activities can be quickly switched between without the requirement for large amounts of writing. Bishop, A. J. (1983). Space and Geometry. In R. Lesh, and M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 175-203). London: Academic Press.
This article summarises previous research in the field of Space and Geometry, then discusses specific areas of the field with particular reference to a study carried out in Papua New Guinea. The article is more concerned with how to define ability in geometry than with describing a progression. Of particular interest are the two pairs of ‘ability types’ described and compared: Michael and McGee (1979):
Bishop (1980):
Spatial ability “As children develop they move from a position of total egocentrism in the perception of space, to a stage in which multiple viewpoints can be considered, to a position of being able to conceptualise and operate mentally on hypothetical space.” (p. 182) Clements, D. (2003). Teaching and Learning Geometry. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 151-177). Reston, VA: National Council of Teachers of Mathematics. Summary This chapter discusses the teaching and learning of geometry with particular references to the US education system. It has four main sections: theories of geometric thinking, learning, and teaching; instructional tools; selected geometry topics; and other issues in teaching and learning geometry. The chapter describes the poor performance of US students in geometry compared to other countries and discusses reasons and possible solutions to this problem. The van Hiele model of geometry concept development and computer based approaches to the teaching and learning of geometry receive particular emphasis. Clements describes the van Hiele levels as “increasingly sophisticated levels of description, analysis, abstraction, and proof” (p.152), and lists four levels: visual, descriptive/analytic, abstract/relational, and proof. He also mentions the possible existence of an earlier level, and discusses the proposed reconceptualisation of the visual level as syncretic, to better describe the way in which students at this level have been shown to perceive shapes. The van Hiele phases of learning are also described and discussed. Clements, D., and Battista, M. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York: Macmillan. Summary This chapter provides a good summary of many aspects of the teaching and learning of geometry, particularly in reference to spatial reasoning. Many of the key theorists including Piaget and Inhelder and the van Hieles are discussed. Clements, D., Swaminathan, S., Hannibal, M., and Sarama, J. (1999). Young childrens’ concepts of shape. Journal for Research in Mathematics Education, 30, 192-212
Discusses the criteria pre-school children use to distinguish between shapes, in particular using the van Hiele levels. The article concludes, among other findings, that an extra level should be added before the recognition level and that the visual level should be reconceptualised as syncretic. Department for Education and Employment (1998) The National Numeracy Strategy - framework for teaching. London: DfEE.
The National Numeracy Strategy was introduced in the UK in 1999 and provides a structure on which basis the daily teaching of mathematics can be built. Teachers are expected to teach mathematics daily using a structure which allows a short time at the start of the lesson for oral work and mental calculations, a main lesson, and a short plenary to review learning. It also provides objectives for each year level from reception to year 6 under 5 strands: Numbers and the number system; Calculations; Solving problems; Measures, shape and space; and Handling data. The bulk of the objectives are under the Numbers and the number system and Calculations strands, reflecting the emphasis that is put on operating with numbers in the strategy. Planning grids and supplements of examples are provided as an aid to teachers. Department of Education and Training (Undated). Count Me Into Measurement. New South Wales Department of Education and Training. Summary The Count Me Into Measurement project (NSWDET, Undated) aims to support teachers by giving a framework for measurement related mathematics. The framework consists of six generic levels which apply to measuring each of: length, area, volume/capacity, and mass.
Department of Education and Training (Undated). Count Me Into Space. New South Wales Department of Education and Training. Summary 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 the 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. Ell, F. (2001). Mathematics in the New Zealand Curriculum – a concept map of the curriculum document. Technical Report 11, Project asTTle, University of Auckland.
The development of the asTTle assessment tool included work carried out on the ‘mapping’ on Mathematics in the New Zealand Curriculum. The achievement objectives were grouped into categories and ordered by level, with several additional objectives added to fill perceived gaps in the curriculum. This work was carried out over the course of three years with initial work completed by Fiona Ell in 2001 and further development carried out by Thomas et al. in 2002 and 2003. Hoffer, A. (1983). Van Hiele-Based Research. In R. Lesh, and M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp.205-227). London: Academic Press.
Description of van-Hiele based research. Describes the thought levels, insight and phases of learning fundamental to this model of geometry learning. It also introduces several research projects related to the concepts, and discusses misconceptions and applications. Thought Levels Level 0: Students recognize figures by their global appearance. Level 1: Students analyse properties of figures. Level 2: Students relate figures and their properties. Level 3: Students develop sequences of statements. Level 4: Students rigorously apply rules to derive proofs. (p. 207)
Phase 1: Inquiry (discussion based) Phase 2: Directed orientation (as in directed teaching; teacher-sequenced exploration) Phase 3: Expliciting (refining use of vocabulary and establishing inherent structures) Phase 4: Free orientation (multi step tasks with no one route to solution) Phase 5: Integration (overview/summary) (p. 208) Holton, D., Ahmed, A., Williams, H., and Hill, C., (2001). On the importance of mathematical play. International Journal of Mathematics Education in Science and Technology, 32, 3, pp. 401-415. Summary This article discusses the characteristics of play and its importance in the development of mathematical ideas for both young children and older learners. Lehrer, R., Jaslow, L., and Curtis, C. (2003). Developing an understanding of measurement in the elementary grades. In D. H. Clements and G. Bright (Eds.), Learning and teaching measurement (pp. 100-121). Reston, VA: National Council of Teachers of Mathematics.
Lehrer describes 7 ‘important ideas’ in measurement, divided into two categories; conceptions of unit and conceptions of scale. Conceptions of unit Iteration: A unit is translated to obtain a measure. Identical unit: Subdivisions must be identical. Tiling: Units must fill the space. Partition: Units can be partitioned. Additivity: Measures are additive. Conceptions of scale Zero-point: Any point can serve as the zero point on the scale. Precision: All measurement is approximate. The choice of unit determines level of precision. Lehrer, R. (2003). Developing Understanding of Measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179-192). Reston, VA: National Council of Teachers of Mathematics. Summary Lehrer describes the conceptual development of measurement as “change in a network or web of ideas about measurement”. He lists eight components that provide a basis for this network: Unit-attribute relations: What units can/should be used? Iteration: Subdivision into congruent parts or repetition of a unit. Tiling: Gaps must not be left between the units. Identical units: If units are identical a count represents the measure. Standardisation: Using a standard unit makes communication of measures easier (possible). Proportionality: Different units can be used to measure and can be compared. Additivity: A line segment can be divided into several smaller line segments whose sum will equal the original length. Origin: When using a scale to measure it is important to identify the zero point. These eight components are seen as important in the development of understanding of all measurement attributes (length/area/volume/mass/angle/time). The development of a conceptual grasp of measures of the different attributes is neither simultaneous nor sequential in a linear way, but understanding of the eight components can be extended from one attribute to another. For example, a student who understands that when measuring a length using hand spans it is important not to leave gaps, is more likely to understand that measuring an area using tiles must also leave no gaps. The article discusses the development of the various attributes both theoretically and with reference to classroom studies. Mullis, I., Martin, M., Smith, T., Garden, R., Gregory, K., Gonzalez, E., Chrostowski, S., and O’Connor, K., (2003). TIMSS Trends in mathematics and science study: Assessment frameworks and specifications 2003. Boston: International Study Center, Lynch School of Education, Boston College. Summary A document describing the process for creating the Science and Mathematics frameworks used for TIMSS 2003. Sample activities are also included. National Council of Teachers of Mathematics, (2000). Principles and standards for school mathematics. Reston, VA: Author.
Principles and Standards for School Mathematics describes itself as “a resource and guide for all who make decisions that affect the mathematics education of students in pre-kindergarten through grade 12.” While the United States does not have a compulsory curriculum document, Principles and Standards provides national goals for the teaching of mathematics and guidelines for attaining them. Olson, A., Kieren, T., and Ludwig, S., (1987). Linking Logo, levels and language in mathematics. Educational Studies in Mathematics, 18, pp. 359-370.
This article discusses the links that can be made between the use of Logo and Turtle Geometry, everyday language, and the van Hiele levels in geometry. Outhred, L., Mitchelmore, M., McPhail, D., and Gould, P. (2003). Count me into measurement: A program for the early elementary school. In D. H. Clements and G. Bright (Eds.), Learning and teaching measurement (pp. 81-99). Reston, VA: National Council of Teachers of Mathematics.
This article describes Count Me Into Measurement, a research based measurement program developed along much the same lines as the successful Count Me In Too numeracy program. Count Me Into Measurement is implemented in classrooms of students in the first three years of schooling and aims to improve outcomes for students through the professional development of their teachers. The core of the program is the Learning Framework in Measurement, which aims to describe the stages students progress through in their developing understanding of measurement. The Learning Framework describes three key stages: Identification of the attribute (direct comparison/partitioning/conservation); Informal measurement (counting units/relating number of units to quantity/comparison of measurements); and Unit structure (replicating a single unit/relating size of units to number required). (p. 85) Students are perceived as passing through the same three stages in their understanding of each of length, area, and volume, though not at the same time, as increasing the number dimensions measured leads to increasing complexity of concept. The program also provides teaching activities to be taken with the aim of aiding students in progressing through the stages. While the program is in its early years, responses from teachers have been overwhelmingly positive, describing as particular strengths of the program: links between topics, clarification of measurement concepts, and explicit use of language (p. 91). Owens, K. (2003). Count me into space: Implementation over two years with consultancy support. Report for the New South Wales Department of Education and Training. NSW DET Professional Support and Curriculum Directorate. Summary This article describes results from the second year of implementation of Count Me Into Space, a research based measurement program developed along much the same lines as the successful Count Me In Too numeracy program. While little information is given on the actual structure of the program, the responses from teachers indicate that the material provided useful support in teaching geometry concepts. Owens, K. and Perry, B., (Undated). Mathematics K–10 Literature Review. University of Western Sydney. Available online at http://www.boardofstudies.nsw.edu.au/manuals/pdf_doc/maths_k10_lit_review.doc
This is a review of literature related to the teaching of mathematics in years K-12. The sections on measurement and geometry provide a good summary of much of the material found elsewhere. An interesting point raised is the suggestion that use of informal (nonstandard) units in measurement may reduce it to little more than a counting activity. It is suggested that this may not be an important step in developing an understanding of the need for standard units of measure. Perks, P., Prestage, S., and Hewitt, D., (2002). Does the software change the maths? Micromath, Spring 2003, pp. 4-6.
A brief discussion of how three programs (Omnigraph, Geometer’s Sketchpad, and Microsoft Excel) compare in terms of how they can be used to teach area and straight line graphs. While each was seen as useful, the extensions possible varied with the choice of program. Rosser, R., Lane, S., and Mazzeo, J., (1988). Order of acquisition of related geometric competencies in young children. Child Study Journal, 18, 75-90.
Performance of 4, 6, and 8-year-olds on four related geometric tasks were compared in a rigorously designed series of experiments. The four tasks were:
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. Success on the tasks was found to be related to age, and an order of difficulty was established for the 4 tasks. While tasks I and II were equivalently difficult for 6 and 8-year-olds, task II was more difficult that task I for 4-year-olds. Tasks IIIA and IIIB were equivalently difficult for all students, and more difficult than tasks I and II. The 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 and a hence an expected progression in the development of geometric ability should be expected. Ruddock, G. (1998). Mathematics in the School Curriculum: an International Perspective. (INCA Thematic study 3). London: Qualifications and Curriculum Authority. Summary This article outlines the structure, organization and implementation of the mathematics curriculum in 16 countries. Swafford, J., Jones, G., and Thornton, C. (1997). Increased knowledge in geometry and instructional practice. Journal for Research in Mathematics Education, 28, 467-483.
A report of an intervention program, where teachers were given training in geometry content and in research on students’ geometric cognition. The effect on their instructional practice was investigated in terms of van Hiele levels and geometry content knowledge. Describes the van Hiele levels as Levels 1-5 rather than 0-4: Level 1: recognition. Level 2: analysis. Level 3: informal deduction. Level 4: formal deduction. Level 5: rigor. Royal Society and Joint Mathematical Council (2001). Teaching and Learning Geometry 11-19. London: Royal Society.
A report produced by a working group on behalf of the Royal Society and the Joint Mathematical Council. The report aimed to consider all aspects of teaching and learning of geometry in schools and colleges. Several key principles of geometry education and a list of recommendations are included. From TKI | NZ Curriculum Marautanga Project | What’s happening | Mathematics | The teaching and learning of geometry and measurement page of http://www.tki.org.nzcurriculum/whats_happening/index_e.php |