critical review ct maths3

PS 4227 C & T: Mathematics III

Presented by:08B0510 SUHANA BINTI HAMDAN

Critical Review

1. Students’ 3D Geometry Thinking Profile

2. Students’ Visualisations of 3D Shapes

“Students’ 3D Geometry

Thinking Profiles”

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

PURPOSE OF THE RESEARCH

1) Examine the structure of 3D geometry abilities by validating a theoretical model assuming that 3D geometry thinking consists of the 3D geometry abilities

2) Describe students’ 3D geometry thinking profiles by tracing a developmental trend between categories of students

SUMMARY

Theoretical Considerations

3D Geometry Abilities

3D Geometry Levels of Thinking

3D Geometry Abilities

1. The ability to represent 3D objects

2. The ability to recognise and construct nets

3. The ability to structure 3D arrays of cubes

4. The ability to recognise 3D shapes’ properties and compare 3D shapes

5. Calculate the volume and the area of solids

3D Geometry Levels of Thinking

Van Hiele’s Model

1st levelCompare solids on a global

perception of the shapes of the solids without paying attention to properties

2nd levelCompare solids based on a

global perception of the solids leading to the examination of

differences in isolated mathematical properties

3rd levelAnalyse

mathematically solids and their

elements

4th levelAnaylse the solids prior to any

manipulation and their reasoning based on the mathematical structure of the

solids including properties not seen but formally deduced from definitions

or other properties

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

• Instruments– 3D geometry thinking test consisted

of 27 tasks measuring the six 3D geometry abilities:

RESULTS

• The results are based on:

1) 3D Geometry Thinking

2) 3D Geometry Profile

3D Geometry Thinking

3D Geometry Thinking Profile

EVALUATION• Format of the paper

– Language

– Organization of texts

• Findings– Structure of 3D geometry thinking

– Students’ 3D geometry thinking profile

3D Geometry Profiles

4 Distinct Profiles

Students were able to respond

only to the recognition of

solid tasks

Students were able to recognize

and construct nets and represent 3D

shapes in a sufficient way

Students did not have any difficulties in the recognition and construction of nets and representation

of 3D shapes

Students were able in all the

examined tasks

CONCLUSION3D 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

“Students’ Visualisations of 3D Shapes”

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

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?

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, visualiserscould experience some disadvantages (Presmeg, 1986)

– 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)

• 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)

– 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)

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

• 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

Assessment TasksTask 1: Visualise a three-dimensional shapes

I want you to think about a cereal box, for example, a cornflakes or rice bubbles box. Tell me all you can

think about this box.

Task 2: Identify similar shapes

Can you think of any other things or shapes in the real world that are the same shape as this cereal box

shape? Why are they the same?

Task 3: Name the mathematical shape visualised

Do you know the mathematical name of this cereal box shape?

Task 4: Draw the visualised shape

Can you draw this cereal box shape for me? Can you explain your drawing to me?

Task 5: Describe a (held) shape

I’ve got a real cereal box here. You can pick it up and turn it around if you want to. Now can you describe

the shape of this box to me? (If the description was quite different from the original visualization, the

investigator said, “You said 4 sides before, and how you have told me there are 6 sides. Why did you say

4 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 shape

Here are some cardboard shapes. If you wanted to stick some of these shapes together to make one

cereal box, which shapes would you need? Hand them to me.

Task 7: Identify net of a shape

This 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 cornflakes

box shape?

Task 8: describe a blank (held) shape

Now look at this box (child was shown a muesli bar box which had been folded inside out so that all faces

appear blank to avoid distraction. The student was allowed to handle the box for a few seconds. Then it

was taken from view). Now describe the shape of the box.

– 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

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

Percentage of Students’ Responses by Category 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




Unable to name any math props correctly 75% 10% 9%


Name two prop correctly 0% 20% 27%

Name three prop correctly 0% 0% 9%


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




Unable to name any math props only 83% 30% 9%


Name two props correctly 0% 20% 45%

Name three props correctly 0% 0% 9%


EVALUATION

• Format of the paper

– Language

– Data analysis

– Organizations of texts

• Findings1. 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

3. Differences between drawn,

visualised and verbal descriptions There are considerable differences

between students’ abilities on these 3 aspects

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