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Assessing students’ Assessing students’ understanding ofunderstanding ofparallel lines and related angle parallel lines and related angle properties in a dynamic properties in a dynamic geometry environmentgeometry environment
CHENG, Lo CarolCHENG, Lo Carol
True Light Middle School of Hong KongTrue Light Middle School of Hong Kong
Assessment in mathematics education Assessment in mathematics education
should not just focus on rote memorization should not just focus on rote memorization of number facts or ability for computational of number facts or ability for computational procedures but students’ thinking and procedures but students’ thinking and learning potential in mathematics learning potential in mathematics
mathematics education should help mathematics education should help students to think mathematically and train students to think mathematically and train up their mathematical thinking. up their mathematical thinking.
Ginsburg, H. P., Jacobs S. F. & Lopez L. S. (1993). Assessing mathematical thinking and learning potential in primary grade children In M. Niss (Ed.), Investigations into assessment in mathematics education: An ICMI study (pp. 157–167). Dordrecht: Kluwer Academic Publishers.
Technologies shifts assessment formatTechnologies shifts assessment format
For example: use of calculatorFor example: use of calculator
Dynamic Geometry as Assessment ToolsDynamic Geometry as Assessment Tools
DG software:DG software: SketchpadSketchpad GeoGebraGeoGebra C.a.R.C.a.R.
Use of DG: Use of DG: ExplorationExploration ConstructionConstruction
Can we use DG as assessment tool?
DG Task example
What can we tell from the assessment?
Learning and Teaching of GeometryLearning and Teaching of Geometry
Perceptual ApprehensionPerceptual Apprehension It is about physical recognition (shape, It is about physical recognition (shape,
representation, size, brightness, etc.) of a representation, size, brightness, etc.) of a perceived figure. perceived figure.
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 143–157), Springer published in cooperation with NATO Scientific Affairs Division.
Learning and Teaching of GeometryLearning and Teaching of Geometry
Sequential ApprehensionSequential Apprehension It is about construction of a figure or It is about construction of a figure or
description of its construction. description of its construction.
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 143–157), Springer published in cooperation with NATO Scientific Affairs Division.
Learning and Teaching of GeometryLearning and Teaching of Geometry
Discursive ApprehensionDiscursive Apprehension Mathematical properties represented in a Mathematical properties represented in a
drawing can only be clearly defined with drawing can only be clearly defined with speech determination. speech determination.
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 143–157), Springer published in cooperation with NATO Scientific Affairs Division.
Learning and Teaching of GeometryLearning and Teaching of Geometry
Operative ApprehensionOperative Apprehension It is about making modification of a given figurIt is about making modification of a given figur
e in various ways:e in various ways: the mereological way: dividing the whole given figuthe mereological way: dividing the whole given figu
re into parts of various shapes and combine these re into parts of various shapes and combine these parts in another figure or sub-figures;parts in another figure or sub-figures;
the optic way: varying the size of the figures;the optic way: varying the size of the figures; the place way: varying the position or its orientatiothe place way: varying the position or its orientatio
n.n.Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 143–157), Springer published in cooperation with NATO Scientific Affairs Division.
Dynamic Geometry Tasks Dynamic Geometry Tasks related to parallel linesrelated to parallel lines
Tasks for orientation preferenceTasks for orientation preference
Task 2aTask 2a
Students’ answersStudents’ answers
More than 70 students considered just the horizontal pair.
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80 85 90 95 100 105 110 115 120
var3
Collection 1 Histogram
-2 0 2 4 6 8
Collection 1 Scatter Plot
About 50 students considered both pairs
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80 85 90 95 100 105 110 115 120
var3
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80 85 90 95 100 105 110 115 120
var4
About 20 students considered just the vertical pair.
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80 85 90 95 100 105 110 115 120
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0
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6
Tasks for orientation preferenceTasks for orientation preference
Task 2bTask 2b
Students’ answersStudents’ answers
About 50 students considered just the horizontal (interior angles) pair.
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82 84 86 88 90 92 94 96 98 100
var4
Collection 1 Histogram
Collection 1 Scatter Plot
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82 84 86 88 90 92 94 96 98 100
var3
Collection 1 Histogram
About 40 students considered just the vertical (corr. angles) pair.
About 40 students considered both pairs.
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82 84 86 88 90 92 94 96 98 100
var3
Collection 1 Histogram
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82 84 86 88 90 92 94 96 98 100
var4
Collection 1 Histogram
Task for angle positionTask for angle position
Task 3aTask 3a
Students’ answersStudents’ answers
More than 100 students got the correct answers 69 but still there are more than 30 made the angle 62
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56 58 60 62 64 66 68 70 72 74
Collection 1 Histogram
Task for angle positionTask for angle position
Task 3bTask 3b
Students’ answersStudents’ answers
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50
var3
102 104 106 108 110 112 114 116 118 120
Collection 1 Histogram
More than 80 students the angle as 116 (i.e. 94 + 116 = 180). About 60 students got the values ranged form 111 to 113
Tasks for making equal areasTasks for making equal areas
Task 5aTask 5a
Students’ answersStudents’ answers
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100
var4
22 24 26 28 30
Collection 1 Histogram
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var3
22 24 26 28 30
Collection 1 Histogram
More than 50 students made all the angles equal 26.
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var4
22 24 26 28 30
Collection 1 Histogram
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var3
22 24 26 28 30
Collection 1 Histogram
About 50 students tried to make a parallelogram.
Tasks for making equal areasTasks for making equal areas
Task 5bTask 5b
Students’ answersStudents’ answers
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80
var5
22 24 26 28 30 32 34 36 38 40
Collection 1 Histogram
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var6
22 24 26 28 30 32 34 36 38 40
Collection 1 Histogram
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var4
52 54 56 58 60 62 64 66 68 70
Collection 1 Histogram
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var7
52 54 56 58 60 62 64 66 68 70
Collection 1 Histogram
More than 70 students gave correct answers but many of them tried to make a rhombus-like figure.
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0 2 4 6 8
var1
Collection 1 Scatter Plot
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22 24 26 28 30 32 34 36 38 40
var5
Collection 1 Histogram
More than 100 students made a rhombus-like figure.
