Research Paper: Understanding Teachers' Conceptualisation of Angle by John Lawton Deakin University
The 15^{th} International Congress on Mathematical Education
Sydney, 714 July, 2024
UNDERSTANDING TEACHERS’ CONCEPTUALISATION OF ANGLE
John Lawton
Deakin University
Concept image, and its interaction with formal mathematics, is central to the learning of geometry. This paper discusses findings from a survey of teachers which maps their concept image for angle, and the formal angle definitions used by them in teaching. The findings are compared with a similar survey of university students in training to be teachers in Finland. In both surveys it was found that participants employ a diverse range of conceptualisations for angle. The concept of turn was included in the definition of angle in 45% of cases in the Australian group. Considerable diversity, and some difficulty, was experienced by the Australian group in responses to the angle concept image questions.
BACKGROUND
The importance of mental imagery in geometry learning was highlighted by Vinner (1991) when he argued that students learn primarily through the development of a concept image, that involves mental pictures and their associated properties and mental processes. Vinner described formally structured mathematics as concept definition and argued that although it was important for students to interact with these, they were not central to the learning of geometry. Concept image and concept definition became the basis for a survey into the conceptualisation of angle of 191 Finnish university students who were training to become teachers (Silfverberg & Joutsenlahti, 2014). This group were found to interpret the concept of angle in many ways.
The Finnish study categorised survey responses by using three conceptualisations for angle found by Mitchelmore and White (2000) to be used in schools. These are angle as a geometric structure formed by two lines meeting at a vertex, as a region formed by two intersecting half planes and as turn, about a point between two lines.
METHODOLOGY
The findings in this paper are from a survey of Australian teachers that explores their understanding of angle, to inform a PhD study. The survey assesses teachers’ conceptualisation and use of angle in their classrooms and their opinions of problems associated with angle as described in extant research. The survey was administered online on the Qualtrics platform. Respondents answered questions specific to the level that they currently taught most often. An average of 12 questions per level and 20 general questions were asked. The questions were mostly multi choice with respondents encouraged to provide written explanations for specific choices. A sketching tool in the survey was used to show how respondents typically drew an angle in class. The survey took about 30 minutes to complete.
The results reported in this paper are from three questions in the survey that were based on the Silfverberg and Joutsenlahti (2014) study, to determine how Australian teacher responses might compare. The questions asked about the angle representation in figure 1 were, Q 2.4 “Please indicate which of the points A to I are part of angle a, and which are not. Drag each point and drop it into one of the three possible boxes. Use the third box for any point that you are not sure about”. The following question, Q 2.5, repeated this instruction but asked respondents to drag all the points in figure 1 into “does” or “does not” boxes, reallocating any “unsure” points from Q 2.4 using a best guess.
Responses have been received from 40 teachers of mathematics. Fifteen are primary school teachers at levels three to six, and 25 are secondary teachers at levels seven and eight. They have on average been teaching for 18 years each. Nineteen respondents teach in government schools, 12 in Catholic schools and nine at independent schools. Twentyeight teachers in the survey work in the Australian state of Victoria and 12 work in the state of New South Wales (NSW). A different curriculum applies to teachers in each state.
FINDINGS  Teachers’ concept definition for angle
Question 2.1 in the survey asked teachers to give a yes or no answer to the question, “do you employ a formal definition of angle in your teaching?” Where a yes answer was given respondents were asked to provide the definition in writing. Analysis of the results are shown in table 1 .
Teachers’ concept definitions for angle 
Number 
% 

Question not answered 
5 
12.5 

No formal definition used 
5 
12.5 

Static definitions 
Geometric 
6 
15.0 
Geometric & region 
6 
15.0 

Dynamic definitions 
Turn only 
4 
10.0 
Turn & geometric 
12 
30.0 

Turn, geometric and region 
2 
5.0 

Total 
40 
100 
Table 1: Teacher definitions for angle
Seventy five percent of teachers gave a written definition for angle which included one or more of Mitchelmore and White’s (2000) three angle concepts. These are shown in bold in table 1. This contrasts with the Finnish study in which the students had difficulty defining angle. Their study did not report statistically on angle concept definitions. Ten teachers in the Australian study (25%) either did not answer the question or stated they do not use a formal definition for angle in teaching. The written definitions provided by the remainder were categorised according to which of the three concepts for angle defined by Mitchelmore and White (2000) were included in them. The most common definition, for instance, was a dynamic one that includes the notion of angle as turn, and which also refers to the geometric structure of angle as two arms and a vertex. An example, provided by one respondent is “angle is the amount of turn between two arms (lines) joined together at a vertex (a single point). Angle is measured in degrees”. Thirty percent of respondents reported using a static definition of angle in their teaching, which made no reference to turn. Of these about half referred to the geometric angle concept only (arms and vertex), and half also referred to the region between the arms of the angle.
FINDINGS  Teachers’ concept image for angle
The findings from survey question 2.5, described above, in relation to figure 1 are shown in table 2.
Figure 1. Angle illustration used in survey question.
This question came originally from a survey by Herschkowitz et al. (1987). The question was then adapted by Silfverberg and Joutsenlahti (2014) for their study.
Teachers’ concept image for angle 
Finnish study 
Australian study 

No. 
% 
No. 
% 

Angle not bounded by its lines 
1 
6 
0 
0 

Geometric image 
Finite 
C, E, F 
56 
32.0 
8 
32.0 
Infinite 
C, E, F, I 
66 
38.0 
11 
44.0 

Regional image 
Finite 
C, D, E, F 
4 
2.0 
1 
4.0 
Infinite 
C, D, E, F, H, I 
47 
27.0 
5 
20.0 

Total 
174 
100 
25 
100 
Table 2: Teacher concept image for angle
To answer this question, it is necessary to mentally construct a concept image for a plane angle and to decide which of the points in the illustration lie within the boundaries of that image. The collection of points chosen and reported by the respondent are then used to categorise the concept image that was engaged by them. Answers in this study are categorised into four concept images for angle shown in table 2. If points C, E, F only were chosen, it is argued that a mental image involving the two arms of the angle as line segments was activated and the answer was classified as geometric, finite. If points C, E, F, I only was chosen the answer was classified as geometric infinite, as the angle arms were argued to be seen as rays. A set of points C, D, E, F was classed as regional finite because point D is part of the region between the lines of the angle. C, D, E, F, I indicate an infinite region.
In both the Finnish and Australian studies, a similar range of concept images were chosen. Amongst the Australian group the question was not answered by six (15%) of the 40 respondents, a further nine (23%) gave answers that could not be classified. The Finnish study reported only on answers by 171 of their 191student total. For comparison purposes, the percentage calculation of the Finnish results has been adjusted to represent a population of 171.
The original survey by Herschkowitz et al. (1987) was of 518 grade five to eight students, 142 preservice, and 25 elementary teachers in Israel. Conceptualisations which viewed angle as an “infinite entity” (p. 223) including the region between its arms were taken in the Israeli survey as being correct. In terms of the points in the Australian survey question the answers C, D, E, F, H, I or D, H would have been taken as being correct in the Israeli survey. The answer D, H indicates a concept image in which the points on the arms of the angle are outside its boundary. Herschkowitz et al. found that less than half of the students, 60% of preservice teachers and 55% of teachers in their survey had what they argued to be a correct angle concept.
DISCUSSION
Angle is learned through concept image. The Australian teachers in this study have high levels of diversity and uncertainty in expressing what the concept image for angle should look like, and a diversity of different formal definitions for angle. High levels of diversity of angle conceptualisation were also observed by Silfverberg and Joutsenlahti (2014). In that study the authors were critical that in Finland, geometry was taught without ensuring that basic concepts such as angle were understood. It was recommended that differences in angle conceptualisations be discussed in their schools and critically debated, with the meaning of angle arrived at through classroom negotiation. The results from the Australian survey will inform experimental case study lessons which ask teachers to evaluate a more systematic approach to developing student concept image for angle. This will be done by asking students to search for angles in ways that build a holistic understanding of the various concepts for angle that need to be understood in school.
References
Herschkowitz, R., Bruckheimer, M., & Vinner, S. (1987). Activities with teachers based on cognitive research. In M. Lindquist & A. Schulte (Eds.), Learning and teaching geometry, K12: NCTM 1987 yearbook (pp. 222  235). Reston, VA: NCTM.
Mitchelmore, M., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics 41, 209238.
Silfverberg, H., & Joutsenlahti, J. (2014). Prospective teacher’s conceptions about a plane angle and the context dependency of the conceptions. In P. Liljedhal, C. Nicol, S. Oesterle, & D. Allen (Eds.), Proc.38th Conf. of the Int. Group for the Psychology of Mathematics Education Vol. 5 (pp. 185192). Vancouver, Canada: PME.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 6581). Dordrecht, The Netherlands: Kluwer Academic Publishers.