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# Critical review ct maths3

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### Critical review ct maths3

1. 1. PS 4227 C & T: Mathematics III Critical Review1. Students’ 3D Geometry 2. Students’ Visualisations Thinking Profile of 3D Shapes Presented by: 08B0510 SUHANA BINTI HAMDAN
2. 2. “Students’ 3D Geometry Thinking Profiles”
3. 3. INTRODUCTION• Proceedings of the sixth Congress of the European Society for Research in Mathematics Education (CERME 6)• Focuses on the constructions, description and testing of a theoretical model for the structure of 3D geometry thinking• Tested the validity and applicability of the model in Cyprus
4. 4. PURPOSE OF THE RESEARCH1) Examine the structure of 3D geometry abilities by validating a theoretical model assuming that 3D geometry thinking consists of the 3D geometry abilities2) Describe students’ 3D geometry thinking profiles by tracing a developmental trend between categories of students
5. 5. SUMMARY 3D Geometry Abilities TheoreticalConsiderations 3D Geometry Levels of Thinking
6. 6. 3D Geometry Abilities1. The ability to represent 3D objects2. The ability to recognise and construct nets3. The ability to structure 3D arrays of cubes4. The ability to recognise 3D shapes’ properties and compare 3D shapes5. Calculate the volume and the area of solids
7. 7. 3D Geometry Levels of Thinking Compare solids on a global 1st level perception of the shapes of the solids without paying attention to properties Compare solids based on a global perception of the solids 2nd level leading to the examination of differences in isolatedVan Hiele’s mathematical properties Model Analyse mathematically 3rd level solids and their elements Anaylse the solids prior to any manipulation and their reasoning based 4th level on the mathematical structure of the solids including properties not seen but formally deduced from definitions or other properties
8. 8. METHODOLOGY• Sample – 269 students – From 2 primary schools and 2 secondary schools Grade No. of students 5th 55 6th 61 7th 58 8th 63 9th 42
9. 9. • Instruments – 3D geometry thinking test consisted of 27 tasks measuring the six 3D geometry abilities:
10. 10. RESULTS• The results are based on: 1) 3D Geometry Thinking 2) 3D Geometry Profile
11. 11. 3D Geometry Thinking
12. 12. 3D Geometry Thinking Profile
13. 13. EVALUATION• Format of the paper – Language – Organization of texts• Findings – Structure of 3D geometry thinking – Students’ 3D geometry thinking profile
14. 14. 3D Geometry Profiles Students were Students did not able to recognize have any difficulties and construct nets in the recognition and and represent 3D construction of nets shapes in a and representation sufficient way of 3D shapesStudents were Students wereable to respond able in all the only to the examined recognition of 4 tasks solid tasks Distinct Profiles
15. 15. CONCLUSION3D geometry thinking implies a large variety of 3D geometry tasksSix 3D geometry abilities are strongly interrelatedThe identification of students’ 3D geometry thinking profiles extended the literature in a way that those 4 categories of students may represent 4 developmental levels of thinking in 3D geometry
16. 16. “Students’ Visualisations of 3D Shapes”
17. 17. INTRODUCTION• Proceedings of the twenty-third Mathematics Education Research Group of Australasia (MERGA23)1• Report the investigation of students’ visualisations and representations of 3D shapes• Describes how students focused on critical and non-critical aspects of 3D for shapes and whether any differences exist between students’ visual images, verbal descriptions and drawn representations
18. 18. AIMS1. How well do students visualise 3D objects?2. In their visualisations, do students focus on critical or non-critical aspects of 3D objects? Are these aspects mathematical properties?3. What are the differences, if any, between students’ visual images, verbal descriptions and drawn representations of 3D objects?
19. 19. SUMMARY• Research on Students’ visualisation abilities – Students who differed in spatial visualisation skills did not differ in their ability to find correct problem solutions, but they concluded that an emphasis on spatial visualisation skills will improve mathematics learning (Fennema & Tartre, 1985) – While visual imagery did assist many students in solving problems, visualisers could experience some disadvantages (Presmeg, 1986)
20. 20. – Visual imagery, when properly developed, can make a substantial contribution at all levels of geometric thinking (Battista & Clements, 1991)– Visual imagery was important in young students’ noticing features of shapes and in deciding how shapes could be used (Owens, 1994)
21. 21. • Theories on the Development of Spatial Concepts – One of the most striking things about objects in images is how they mimic properties of real objects (Kosslyn, 1983)
22. 22. – The ability to draw correct diagrams stems from images that students possess and often these images do not reflect student understandings in terms of the properties of a given figure (Pegg & Davey, 1989)– Students with poor visual skills may focus on non-mathematical aspects of shapes and this may inhibit effective learning of geometric ideas (Gray, Pitta, and Tall, 1997)
23. 23. METHODOLOGY• Sample – 30 students – From a NSW Department of Education and Training school Year No. of students 1 10 students 3 9 students 5 11 students
24. 24. • Instrument – Interviewed based assessment of students’ understanding and visualisations of 3D shapes included 8 tasks • Adapted from instruments used in prior studies by Battista & Clements (1996), Shaughnessy (1999) in correspondence • The task also reflected sample activities from the NSW Mathematics K-6 syllabus (NSW Department of Education, 1989) • Were administrated on a one-one basis by chief investigator
25. 25. Assessment Tasks Task 1: Visualise a three-dimensional shapesI want you to think about a cereal box, for example, a cornflakes or rice bubbles box. Tell me all you canthink about this box. Task 2: Identify similar shapesCan you think of any other things or shapes in the real world that are the same shape as this cereal boxshape? Why are they the same? Task 3: Name the mathematical shape visualisedDo you know the mathematical name of this cereal box shape? Task 4: Draw the visualised shapeCan you draw this cereal box shape for me? Can you explain your drawing to me? Task 5: Describe a (held) shapeI’ve got a real cereal box here. You can pick it up and turn it around if you want to. Now can you describethe shape of this box to me? (If the description was quite different from the original visualization, theinvestigator said, “You said 4 sides before, and how you have told me there are 6 sides. Why did you say4 sides before? How was the picture in your mind different from the real box?”) Task 6: Identify shapes needed to make up into three-dimensional shapeHere are some cardboard shapes. If you wanted to stick some of these shapes together to make onecereal box, which shapes would you need? Hand them to me. Task 7: Identify net of a shapeThis is the shape of a cornflakes’ box flattened out. Now if you cut out these shapes (paper with nets A-E was shown) and folded them up along the dotted lines, which ones could you make into a small cornflakesbox shape? Task 8: describe a blank (held) shapeNow look at this box (child was shown a muesli bar box which had been folded inside out so that all facesappear blank to avoid distraction. The student was allowed to handle the box for a few seconds. Then itwas taken from view). Now describe the shape of the box.
26. 26. – Responses & Solution methods • Responses were recorded on an audiotape and students’ drawings and explanations were retained for later analysis • Solutions methods were coded for correct, incorrect, or non-response before being analysed for key mathematical aspects • Coding of responses was supervised and recoded by an independent coder
27. 27. RESULTS• Interview transcript were anaylsed and responses classified according to: 1. Student performance 2. The mathematical or non- mathematical aspects of the responses 3. Differences between drawn, visualised and verbal descriptions
28. 28. Percentage of Students’ Responses byCategory and Year Level for Tasks 1-8 Year 1 n = 12 Year 3 n = 11 Year 5 n = 11 TASK 1: visualize three-dimensional shape Described shape using non-math props only 17% 0% 0% Described shape using non-math and math props 83% 80% 73% Described shape using math props only 0% 20% 27% Unable to name any math props correctly 92% 60% 9% Name one prop correctly 8% 30% 27% Name two props correctly 0% 10% 55% Name three props correctly 0% 0% 9% Made incorrect estimate of either faces, corners or edges 50% 70% 36% TASK 2: identify other things with the same shape Unable to name anything with similar shape 33% 30% 18% Named one other thing with the same shape 58% 40% 45% Named more than one thing with the same shape 8% 30% 36% TASK 3: name the mathematical shape visualised Gave correct math name for shape 0% 20% 36% TASK 4: draw visualised shape Drew shape as 3D drawing 80% 30% 9% Drew shape as poor 3D drawing 20% 45% 36% Drew shape as a good 3D drawing 0% 25% 27% TASK 5: describe a held shape Described shape using non-math props only 17% 0% 18% Described shape using non-math and math props 83% 90% 82% Described shape using math props only 0% 10% 0% Unable to name any math props correctly 75% 10% 9% Name one prop correctly 25% 70% 55% Name two prop correctly 0% 20% 27% Name three prop correctly 0% 0% 9% Made incorrect estimate of either faces, corners or edges 8% 50% 45% TASK 6: identify shapes needed to make up into 3D shape Chose 6 correct shapes 0% 30% 36% Chose 6 shapes but incorrect ones 8% 20% 36% Chose 4 shapes only 33% 30% 18% Chose other incorrect combination of shapes 58% 20% 9% TASK 7: identify net of shape Identify correct nets 0% 10% 0% TASK 8: describe “blank” held shape Described shape using non-math props only 8% 0% 0% Described shape using non-math and math props 92% 100% 91% Described shape using math props only 0% 0% 9% Unable to name any math props only 83% 30% 9% Name one prop correctly 17% 50% 36% Name two props correctly 0% 20% 45% Name three props correctly 0% 0% 9% Made incorrect estimate of either faces, corners or edges 75% 70% 67%
29. 29. EVALUATION• Format of the paper – Language – Data analysis – Organizations of texts
30. 30. • Findings 1. Student performance  Students found difficulty in visualising 3D objects with an accurate awareness of their mathematical properties 2. The mathematical or non- mathematical aspects of the responses  Non-mathematical aspects featured strongly in students’ responses across grade levels
31. 31. 3. Differences between drawn, visualised and verbal descriptions  There are considerable differences between students’ abilities on these 3 aspects
32. 32. CONCLUSION The accuracy of drawing a 3D shape which a student has just visualised does not necessarily reflect the student’s visualisation ability The quality of some student’s visualisations may improve with grade level, but that students may remain focused on non-mathematical or non- critical aspects of shapes