Son, Ji-Won (2006). “Investigating preservice teachers’ understanding and strategies on a student’s errors of reflective symmetry.”

From MathWiki

Son, Ji-Won (2006). “Investigating preservice teachers’ understanding and strategies on a student’s errors of reflective symmetry,” In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 145-152. Prague: PME.

Table of contents

Study Outline & Method

Son (2006) sets out to explore teachers’ content knowledge of reflective symmetry and their related pedagogical strategies for teaching reflective symmetry to their students. Thus, the study explores how teachers understanding reflective symmetry, and how they would react to students’ errors in learning and applying reflective symmetry.

54 pre-service teachers participated in the study – 32 prospective elementary (P/J) teachers, and 22 prospective middle/secondary (I/S) teachers with a major in mathematics.

Each participant engaged in two written tasks: a content knowledge task (addressing the question, “How do preservice teachers understand reflective symmetry?”) and a pedagogical strategies task (addressing the question, “What types of pedagogical strategies do preservice teachers refer to in order to teach students reflective symmetry?”).


Results

P/J vs. specialized I/S teacher abilities

I was hoping that the author would explore any potential differences between the two groups of participants – P/J VS. I/S – but as the author states, “there was no difference between two groups. Therefore, the study did not compare with and contrast to difference of understanding between two groups” (Son, 2006, p.147). The fact that there was no difference between the groups should perhaps raise other questions as well regarding the content knowledge and pedagogical strategies of specialized (math major) I/S teachers.

How do preservice teachers understand reflective symmetry?

In the area of content knowledge, many of the teachers showed a lack of understanding of reflective symmetry. The teachers were confused about lines of symmetry, and relied heavily on procedural knowledge (such as folding paper) when teaching the subject.

Item 1: Lines of symmetry

Teachers were asked, “How many lines of symmetry are there in a parallelogram?”

76% (41 out of 54) answered correctly. Of the 13 teachers that answered incorrectly, 9 thought that a parallelogram has two lines of symmetry.

Item 2: Symmetry, Reflection & Rotation

In a multiple-choice format, teachers were asked to choose the best option for describing a line of symmetry.

64% of the participants chose the intended correct answer, “Every vertex is the same distance away from the mirror.”

36% of the participants incorrectly selected the option that “when the original drawing is rotated, the two drawings match.”

The author asserts that, “many of the prospective teachers confused the property of reflection and those of rotation” (p.150).

Item 3: Performing Reflection

The participants were asked to reflect a “flag” shape. 83% of the teachers (46 out of 54) completed this task correctly. 17% (8 teachers) had difficulty with this task and answered incorrectly.

Item 4: Explaining the task of reflection

Referring back to the task of reflecting the “flag” shape, teachers were asked to explain how they performed the reflection.

43% (23 teachers) explained reflection through the properties of reflective symmetry (the author describes these through the use of a perpendicular line, and equal distances, p.148).

54% (29 teachers) explained the reflection “in terms of creating the reflected image” (p.150). This could include “using a mirror, flipping, turning the paper, folding, tracing, and coordinating” (p.148).


What types of pedagogical strategies do preservice teachers refer to in order to teach students reflective symmetry?

The participants were given a student’s work that showed errors in the student’s understanding of reflective symmetry. They were then asked to identify the errors and respond to them.

Most of the teachers (56%) “identified Emily’s errors in terms of knowing the [conceptual] properties of reflection,” and 19% of the teachers focused their analysis on the student’s procedural act of “creating the reflected image” (Son, 2006, p.151). (The rest of the participants’ responses were classified as “informal expression” or “misinterpretation”.)

However, while most of the teachers identified the error to be conceptual, this was not apparent (in the same percentage, at least) through their responses to student’s work.

46% (25 out of 54) of the teachers said they would respond do the student through referring to the properties of reflective symmetry (conceptual), while 41% (22 out of 54) said that they would address the process of creating the reflected image (e.g. through folding, flipping etc). (Again, the rest of the participants’ responses were classified as “informal expression” or “misinterpretation”.)


Some "reflections" on 'reflections' by Walter Whiteley

This paper raised a number of issues which also arise in other Mathematics Education studies of the learning of geometry. In general, I have concern that the way I, as a geometer, would view 'answers' to geometry questions, and even how the questions would be posed, is different than how the researchers pose, analyze and describe the geometric hierarchies, vocabulary, concepts, connections.

Geometry as currently understood and practiced is significantly different than geometry as studied by most geometers. There is a key question about what geometry 'is' to students learning it, and what it will 'be' to people later using this geometry, in mathematics, or outside of mathematics. One of the key differences, relevant to this study, is the new emphasis on transformations as 'key properties (invariants)' of figures, rather than lists of measurements and related 'features', which are secondary features. So the very hierarchy of properties and processes which this paper uses can be challenged as inappropriate to the underlying objectives of 'learning geometry'.

There is a problem with the vocabulary of 'lines of symmetry'.

  • there is a meaning for 'line of symmetry' for objects in 3-space, which is an axis for a half turn of the pattern. This is not really the intended meaning when used in the plane.
  • there is a naive association of these two words 'line' and 'symmetry' which would be, in the plane, to cut the object with a line and see if there is a symmetry of the pieces. This, of course, includes both mirrors and rotations.
  • a significant number of the 'errors' found in the study showed this interpretation.
  • this interpretation is 'interesting' and could be used as a source of learning, even of superior insight.
  • the person who has a larger sense of 'symmetry' has a good starting point which should not be side-tracked.

My conclusion is that when a mirror in the plane, or a reflection, is intended, than that word 'reflection' should be used, rather than the overarching word 'symmetry'. We do not speak about a 'point of symmetry' as a weak short hand for a rotation.

The confusion is exaggerated when 'plane reflection' is conflated with 'flip' which really is a rotation in 3-space.

This use of 'line of symmetry' seems to be fairly common - but not universal in the English Language mathematics curricula. It would be interesting to hear any arguments for why this is superior to, or even as good as, mirror line or line of reflection.

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