Assessing students’ understanding of parallel lines and related angle properties in a dynamic geometry environment CHENG, Lo Carol True Light Middle School of Hong Kong

Assessment in mathematics education  should not just focus on rote memorization of number facts or ability for computational procedures but students’ thinking and learning potential in mathematics  mathematics education should help students to think mathematically and train 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 format  For example: use of calculator

Dynamic Geometry as Assessment Tools  DG software:  Sketchpad  GeoGebra  C.a.R.  Use of DG:  Exploration  Construction Can we use DG as assessment tool? DG Task example What can we tell from the assessment?

Learning and Teaching of Geometry  Perceptual Apprehension  It is about physical recognition (shape, representation, size, brightness, etc.) of a 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 Geometry  Sequential Apprehension  It is about construction of a figure or 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 Geometry  Discursive Apprehension  Mathematical properties represented in a drawing can only be clearly defined with 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 Geometry  Operative Apprehension  It is about making modification of a given figure in various ways:  the mereological way: dividing the whole given figure into parts of various shapes and combine these parts in another figure or sub-figures;  the optic way: varying the size of the figures;  the place way: varying the position or its orientation. 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 related to parallel lines

Tasks for orientation preference  Task 2a Task 2a Task 2a  Students’ answers Students’ answers Students’ answers

More than 70 students considered just the horizontal pair.

About 50 students considered both pairs

About 20 students considered just the vertical pair.

Tasks for orientation preference  Task 2b Task 2b Task 2b  Students’ answers Students’ answers Students’ answers

About 50 students considered just the horizontal (interior angles) pair.

About 40 students considered just the vertical (corr. angles) pair.

About 40 students considered both pairs.

Task for angle position  Task 3a Task 3a Task 3a  Students’ answers Students’ answers Students’ answers

More than 100 students got the correct answers 69  but still there are more than 30 made the angle 62 

Task for angle position  Task 3b Task 3b Task 3b  Students’ answers Students’ answers Students’ answers

More than 80 students the angle as 116  (i.e. 94   = 180  ). About 60 students got the values ranged form 111  to 113 

Tasks for making equal areas  Task 5a Task 5a Task 5a  Students’ answers Students’ answers Students’ answers

More than 50 students made all the angles equal 26 .

About 50 students tried to make a parallelogram.

Tasks for making equal areas  Task 5b Task 5b Task 5b  Students’ answers Students’ answers Students’ answers

More than 70 students gave correct answers but many of them tried to make a rhombus- like figure.

More than 100 students made a rhombus-like figure.