From Another Perspective—training teachers to cope with problematic learning situations in geometry

Abstract

“From Another Perspective” is a year-long course for teachers of mathematics that is designed to enhance teachers’ awareness of the way that their students think when they are experiencing difficulties in geometry. It also aims at equipping teachers with tools needed to analyze and cope with Problematic Learning Situations in geometry (Gal & Linchevski, 2000). This paper reviews the rationale, content, and approach of the course, which is characterized as “the Back and Forth model”. It then reports on a study that tracked the changes that course participants (pre- and in-service mathematics teachers) passed through. The paper describes the results of the case study of Eti, one of the participants, who taught mathematics to junior high school students. The findings suggest that Eti was helped to achieving the goals of: (1) expanding and deepening her understanding of students’ ways of thinking; (2) increasing her awareness of her students’ processes of thinking in order to identify their difficulties; (3) equipping her with appropriate tools to analyze and cope with such difficulties; and (4) enhancing her ability to retrieve and utilize this knowledge while making instructional decisions. Conclusions and open questions for further study are drawn.

Appendix 1

Analysis of PLS1 (Gal & Vinner, 1997)

This interview suggests two difficulties: first, students find it hard to identify the right angle between two perpendicular lines; second, the term 90° angle may not have a visual representation and may not be linked to the concept of right angle.

This concept includes a number of elements, some of them are explicit and others are implicit:

• 1. Two lines (sometimes two segments or one line and one segment)

• 2. The lines intersect

• 3. A right angle is formed at the point of intersection

• 4. There are three more angles at the point of intersection

• 5. These three other angles are also right angles

• 6. A right angle is a 90° angle

A student facing this concept should already understand what lines and intersecting lines are (elements 1 and 2).

The student should also identify a right angle presented in its basic form (i.e., two rays emanating from a common point). An expected difficulty at this stage is a limited prototype of a right angle, with the student being able to identify a right angle only if one of its rays is horizontal.

A difficulty of this kind requires some training with the concept of right angles.

Let us therefore assume that the student can identify some “basic” forms of right angle (element 3). At this point, he or she still faces a number of difficulties:

A. Identification

In a configuration of perpendiculars, he or she must identify the right angle in its basic form. In other words, the student is required to identify a simple form of a right angle within a complex figure (intersecting lines forming four angles).

This difficulty can be explained by the Gestalt principles pertaining to the organization of perception: “The good continuity” principle explains this tendency of ours.

Thus, the student who observes two intersecting lines does not necessarily perceive an angle between them, and, in any case, does not know where to look for a right angle. This difficulty came up in the interview.

Let us examine how visual information is processed in the student’s mind, once it has been perceived and recorded in the cognitive system. After receiving visual information, it is organized unconsciously in units in such a way that each unit represents a part of the whole structure. Complex shapes are constituted of hierarchical units (Anderson, 1995, pp. 123–125). The figure , for example, may be decomposed into four distinguished sub-figures, where each sub-figure is a segment: . In this case, identifying an angle pattern within the complex pattern is not trivial (the student has to compose sub-figures producing an angle, then compare it with the right-angle pattern in his mind, e.g., he might identify an angle , while the pattern of the right angle in his mind is ). Of course, decomposing the above figure into or , will make the right angle identification easier. But, there is yet another barrier: what if the observed figure was:? It would most likely be decomposed into the following sub-figures: . Will the student be able to identify in this pattern the previous one of intersecting (perpendicular) lines, tilted by 45°? Devoting time to concrete examples of right angles and perpendicular lines, using paper cutouts and puzzles, and asking the students to transform them in the plane improves flexibility in dealing with patterns by making them more familiar.

B. Selection

The student has to decide on which of the four angles in front of him to focus (elements 3, 5). The understanding that if one of them is a right angle, then so are the rest, and that the selection is therefore arbitrary, cannot be taken for granted.

Such an understanding can be considered as “visual understanding” (level 1, visualization according to van Hiele’s theory, e.g., Hoffer, 1983). As preparation, two perpendicular lines could be presented, where each of the four angles is emphasized in its turn. Changing the focus from one angle to another and then to the lines and vice versa may help perceiving the relations between the angles (and between them to the lines). Such an understanding can also be considered as a level 2, analysis (e.g., by taking apart and assembling the four right angles to form the perpendiculars with their four right angles; or by folding paper into four). This analysis involves mental activity, and it is advisable to follow it up with a verbal description.

It may be assumed that, when “quickly overviewing” the four angles before him/her, a student will check the one “closest” to the image he or she has in mind of what a right angle is. “Closest” in what sense? Is the angle’s size the criterion, or its orientation in the plane? How flexible are the mental transformations, and do they enable a comparison between different positions in the plane?

Now, if we return to the problem of “selection” and the need to identify a certain angle as a right angle, one might assume that the angle selected will be the angle which bears the greatest resemblance (“resemblance” in one of the meanings mentioned) to the pattern in the student’s mind.

Inference

The inference concerning the other three angles (in the configuration of the perpendicular lines) is not a trivial matter. There are two possibilities here. If the student knows that all four angles are right angles, in case one of them is a right angle, then the other three will be conceived as right angles. Otherwise, the student might fail to realize that all the other three angles are also right angles. (In such a case, if the teacher talks about an angle other than the one the student has chosen, it should come as no surprise if the student does not realize that it is a right angle). Moreover, concerning one angle, failing to recognize a right angle would cause a failure in recognizing perpendiculars though concerning another one could make the student succeed!

So far, we have dealt with right angles, without relating to its measure. The right angle is often defined in class as a 90° angle.

This time we have one more difficulty to boot: we do not know what concept image the student has of the right angle and whether it coincides with a 90° angle. The impression from the interview is that the students have heard the notions of 90° and right angle and are aware that they are synonymous. This does not mean that these terms necessarily have a meaning, and even if they do, it is not necessarily the same for both. We will use the term “conceptual behavior” (denoting the result of conceptual thinking processes, dealing with concepts, relations between them, ideas to which these concepts are related, logical relations, etc.) as opposed to “pseudo-conceptual behavior”, which might look like conceptual behavior, but which is brought about by mental processes which do not characterize conceptual behavior (Vinner, 1997). According to Vinner, in mental processes which lead to conceptual behavior, words are connected to ideas, whereas in mental processes which lead to pseudo-conceptual behavior, words are connected to words, without any ideas behind them.

The interview above demonstrates a pseudo-conceptual behavior. The students deal with the notions of angles, perpendiculars, 90°, but seem to be unclear about the relations between them (if any such relations exist in their minds), and ideas linking the concepts are not known (at least to the teacher). The words “90 degrees” are associated with “perpendicular lines”, but there are probably no ideas behind them, and therefore the students do not know where to look for the “90 degrees”.

Teachers frequently approach right angles “numerically”: calculating angles, ascertaining perpendicularity according to the numerical size of the angle, etc. Weak students are also capable of solving such assignments. They rely on arithmetical knowledge, constructing their answers on verbal cues. The real situation will be exposed when moving on to non-computational problems.

Let us go back now to Ora and Keren. Ora has noticed that the students fail to “find” the angle (Ora: “They don’t understand where the angle is drawn, … they don’t know where they need to check this angle which is indeed 90°”). Nothing of what she says indicates that she understands the source of the problem. She does not refer to the difficulty of identifying a simple figure inside a complex one nor to the tendency to see lines which form “good continuity”. On the other hand, she does seem to define better the problem underlying the use of the notions 90° and right angle (Ora: “...I am not sure that they know this for real, but they say 90° straight out…When I ask them what kind of angle is formed between two perpendicular lines, they tell me: aha 90°, but then they don’t know what they have to check”). Keren sensed this too (Keren: “They know that perpendicular lines represent 90° but do not understand what this means”). Despite this, they fail to make a further analysis and to characterize the problem accurately. In any event, they have no idea how to improve understanding in students! (Keren: “I really don’t know”).