1. An NCETM research study module Generalisation, abstraction and angles

2. This module is based on Mitchelmore, M. Mitchelmore, M. (2002) The role of abstraction and generalisation in the development of mathematical knowledge. Mike Mitchelmore teaches at Macquarie University, Sydney. He has a special interest in students’ learning of mathematical concepts through real-life contexts.

3. Background There are many sources of difficulty that students have with understanding what an angle is, and what is being measured, and how it is measured. This paper is written for a research audience and this module provides guided reading for busy teachers. Add one to the year numbers to match them to UK school years, for example Year 2 in the paper is the same as year 3 in UK.

4. Why this paper is interesting and useful It gives insight into learners’ understanding of angle. It provides a context for thinking about abstraction and generalisation more generally. There is a lesson sequence about angles which can be discussed and critiqued. Only half the students came to understand an abstract concept of angle. These are for year 3, but teachers at all school levels can use them to think about how generalisation and abstraction can be fostered, or not, through lesson design.

5. How this module could be used This offers PowerPoint : Access to a full research paper Key ideas Discussion points Summaries Reading excerpts Reflections on research Suggestions for development work

6. Key idea: students’ difficulties with angle The main problems for young students the author had found were: connecting the idea of turn to static angles, such as when measuring with a protractor students could identify angles at junctions, between walls, made by scissors, fans and signposts quite well, but not angles made by an opening door, the slope of a hill, or a turning wheel.

7. Discussion point 1: understanding angle How do your students come to understand angle?

8. Discussion point 2: understanding angle Do they develop both dynamic and static understandings? dynamicstatic

9. Discussion point 3: understanding angle When they measure angles, what difficulties are due to the equipment and what difficulties are due to their understanding?

10. Discussion points: ‘generalisation’ and ‘abstraction’ What do you mean, for your students, by ‘generalisation’ and ‘abstraction’? There are multiple meanings throughout the mathematics education literature. Sometimes these relate to the age of the students and/or the complexity of the mathematical ideas being discussed.

11. Abstraction This excerptThis excerpt explains what this author means by abstraction in this paper.

12. Key idea about abstraction Mitchelmore quotes Skemp, seeing abstraction as a process: ‘the activity by which we become aware of similarities’ which ‘enables us to recognise new experiences as having the similarities of an already formed class’ Empirical abstraction is based on seeing superficial similarities in everyday experiences. Reflective abstraction involves reflecting on actions, not just on what we see. What is the role of the teacher for this to happen?

13. Generalisation This excerptThis excerpt explains what the author means by generalisation in this paper

14. Learning about angle is not just a case of generalising all angles met so far, because learners may not even recognise the presence of an angle in many situations. Mitchelmore uses ‘generalisation’ to indicate the widening of a class of objects that all have the same relationships and properties, beyond what the learner already knows. Key ideas about generalisation

15. Discussion points: the role of the teacher What is the role of the teacher if students are going to broaden and connect their ideas of angle? Can the processes of generalisation and abstraction be taught or do they occur naturally?

16. Summary: three possible teacher roles Mitchelmore suggests that there are three methods teachers use extend and connect students’ understanding of angle: the generalisation (a definition) is taught before learners have much experience with angles, and they may not think to apply it and may forget it exploratory methods: learners generalise from a range of experiences but this may lead to false reasoning based on limited experience problem solving that includes generalisation towards the abstract idea

17. Horizontal and vertical mathematisation: first step Excerpt from paper Treffers (1991, p. 32) calls the first step horizontal mathematising: “The modelling of problem situations [so] that these can be approached with mathematical means.” Mitchelmore’s interpretation Develop rules of operation in several specific, familiar, everyday contexts

18. ...second step The second step consists of the recognition of structural similarities. Demonstrate that the same structure is present in several such contexts.

19. and the third step the construction of a new mental object. Treffers calls these steps vertical mathematising, which is “directed at the perceived building and expansion of knowledge within the subject system, the world of symbols.” We can recognise the three steps as together constituting abstraction and generalisation. Formulate, symbolise and study the common structure...third step

20. The author designed ten lessons for teachers to teach 7 year-olds about angle using familiarity, similarity and reification in practice. 192 students were involved. Half the students came to understand angles without two lines and a vertex. They give an outline of the lesson content.lesson content Research method

21. Key idea: task design principles In Realistic Mathematics Education tasks are designed to include: familiarity (experience) similarity (identify superficial and structural similarities) reification (how to use the idea, how it relates to other ideas, what its special cases are). An elaboration of these principles can be found here.here

22. Summary of students’ understanding of angle Some classroom research suggests that the final stage does not always take place. In this study all students came to understand angle when two lines and a vertex were present, but only half came to understand it in a more abstract way.

23. Discussion point: matching to your experience How does this finding match with your experience? Should we only expect half the students to learn? Do the lessons he used need tweaking? Is the theory at fault? What needs to be added to/altered in the theory or the lessons to enable more to learn?

24. Discussion point: abstraction and generalisation How do the basic ideas of generalisation and abstraction relate to what you are currently teaching?

25. Suggestions for development: researching in your school How can you find out your students’ understandings of angle? Do you and your colleagues provide tasks so that learners can develop abstract ideas through familiarity, similarity and reification? The module on randomness deals with similar issues in a different way, in a different mathematical context.

26. Reading the paper This might be a good time to read the whole paper. whole paper

27. Reflecting on the project Revisit the research methods: the original assumptions, the sample used, the task design, the analysis methods, the presentation of findings. Are these appropriate for the research questions? Are the links between them clear? Do the conclusion and recommendations follow from the study? Is it helpful for your thinking?

28. Further reading Harel, G., & Tall, D. O. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42. Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32, 195-222. Mitchelmore, M. C. (1998). Young students’ concepts of turning and angle. Cognition and Instruction, 16, 265-284. Mitchelmore, M. C., & White, P. (2000a). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41, 209-238. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.

29. Reflective abstraction To recognise their angular similarity, a student must firstly focus on one spoke of the wheel and then relate its starting and finishing positions. This process is particularly difficult if the wheel does not have spokes! But even if it does, in order to form the angle either the starting or the finishing position has to be imagined or remembered - whereas the two arms of the angle on a tile are both present at all times. The formation of a general angle concept (i.e., one that includes all possible angle contexts) is thus a particularly analytical, constructive process. One may ask what action … drives such abstraction. Perhaps it is angle combination and measurement. For example, students may recognise the similarity between four square tiles fitted around a point and a toy ballerina making four quarter-turns, and then begin to compare corners and turns in general. Whatever the action, it is probably more the result of explicit instruction than spontaneous investigation. Reflective abstraction occurs late in the development of children’s angle concepts. There is no superficial similarity between tiles and wheels, for example. Back to main presentation

30. Generalisation as extension There are at least three senses in which generalisation means extension (G2). Empirical extension (called primitive generalisation by Dienes, 1963, and expansive generalisation by Harel & Tall, 1991) occurs when a person finds other contexts to which an existing concept applies. For example, an infant recognises a large, white, shaggy animal as a dog even when she has only had experience of small, brown, sleek dogs. Or an older child accepts a scalene triangle as a triangle even though his textbook only has diagrams of isosceles and equilateral triangles. Mathematical extension (called mathematical generalisation by Dienes, 1963, and reconstructive generalisation by Harel & Tall, 1991) occurs when one class of mathematical objects (e.g., the whole numbers) is embedded in a larger class based on a different similarity (e.g., the integers). In this case, the similarity relating members of the first class has to be viewed in a new light (i.e., reconstructed) in order to also relate members of the larger class. Mathematical invention (called Cartesian abstraction by Damerow, 1996, and creative generalisation by Fischbein, 1996) occurs when a mathematician deliberately changes or omits one or more defining properties of a familiar concept to form a more general concept. The most familiar example of this kind of generalisation is probably the invention of non-euclidean geometry by varying the parallel axiom. Such generalisation can only create abstract-apart concepts. Some teachers who read this text found it helped to draw diagrams about its meaning. Back to main presentation

31. Lesson content The first three lessons focussed on corners, including the corners of pattern blocks, corners in the room, and measurement of the size of a corner using a primitive “angle tester” (a paper protractor consisting of six lines through a point intersecting at 30  ). Students investigated how scissors moved in Lesson 4, and in Lesson 5 they investigated other scissors-like objects. In all lessons, students matched angles in different contexts by superimposing one angle on the other. It was hoped they would be able to recognise that all the examples of angles involved (a) two lines, (b) a point where the lines meet, and (c) an amount of opening between the lines. Lessons 6-8 each introduced one 1-line angle context: the hour hand of a clock, a door, and a sloping ruler. In each lesson, students firstly studied how the object moved and the significance of this movement. (For example, the hour hand of a clock moves from 2 o’clock to 5 o’clock in 3 hours.) They then investigated how to describe the size of such movements (e.g., by matching the 3-hour movement to a corner of a square pattern block), thereby linking back to the angles developed in earlier lessons. The learning activities were designed to help children identify the second, missing line of the angle in each context. Lesson 9 was an attempt to highlight the similarities between all the angle contexts studied in the unit, and Lesson 10 was an open-ended, creative activity designed to generalise the angle concept to other curriculum areas besides mathematics. Back to main presentation

32. Principles “The familiarity principle. Students should become familiar with several examples of the concept (i.e., several contexts from which the concept will be abstracted) before making any attempt to abstract the concept itself. These examples - which may be objects, operations, or ideas - should be discussed using the natural language peculiar to each context, not the mathematical language related to the concept to be abstracted. All the examples should be familiar to children’s experience, and not include abstract models “embodying” the concept. However, the teacher should anticipate the abstraction to be made later (e.g., by drawing children’s attention to crucial characteristics that may not be obvious). The similarity principle. The teacher teaches the concept by leading students to identify the similarities underlying familiar examples of that concept. The similarities may be superficial or structural. Whichever it is, the teacher directs the students’ attention to the critical attributes which define these similarities and which are encapsulated in the concept to be abstracted. The teacher then introduces the mathematical language associated with the concept and uses this vocabulary to show how the concept relates to the similarities on which it is based. Abstract physical models that embody the concept may be introduced at this stage if they help children recognise the similarity between different contexts. The reification principle. As students explore the concept in more detail, it becomes increasingly a mental object in its own right, detached from any specific context. Almost any use of the concept is likely to assist its reification, providing the relation between the abstract concept and familiar examples of the concept is maintained. Some possibilities include applying the concept in practice, investigating how to operate on the abstract concept, working with special cases, and looking for generalisations (…) relating the concept to other concepts already learnt.” Back to main presentation