6th TO 7th GRADE PUPILS AND PRIMARY TEACHERS’ PERCEPTIONS REGARDING
THE NATURE OF ANGLES AND THEIR MEASUREMENT UNITS
Mitsoullis Christos
Math Teacher-PhD Candidate, Ltee Lab, University of the Aegean
Tzortzakakis Georgios
Primary Teacher, Ltee Lab, University of the Aegean
Sub-theme: Research on mathematical activity: Collaboration between teachers and
researchers
ABSTRACT
This paper focuses on the presentation and analysis part of the findings from two
questionnaires from 53 pupils (7th grade) and 35 primary teachers. These questionnaires
aimed at making explicit the perceptions of both pupils and teachers regarding the nature of
the angle and its measurement units (degrees) before our teaching experiment on the angles.
The interpretation of the results is based on a historical, epistemological and
phenomenological analysis of the angles and was also correlated with an analysis of textbooks
as well as teacher materials from the Greek Department of Education. After the discussion of
the analysis we realised that our basic hypothesis of confusion among a number of teachers
and pupils regarding the nature of the angles and their measurement units was confirmed.
This affirmation is strongly correlated to instructional materials, curricula and textbooks.
INTRODUCTION
Many researchers realize that the angle is a multifaceted concept that appears in diverse
contexts. The existing definitions seem to cover only a small part of its phenomenological
approaches and interpretations. Thus, one understands that pupils’ understanding of angles is
not easy and that pupils often meet various levels of difficulty in learning about angles. Many
of the difficulties that pupils meet seem to be connected to a static interpretation of the angles,
which is mainly rooted in Euclidean Geometry. Thus, a stereotype and confusion seems to be
created among pupils regarding the nature of the angles and their measurement units.
Teachers themselves seem also uneasy towards such multifaceted concepts. They tend to
simplify angle teaching despite their complexity. But this is an issue for the curricula and
school textbooks as well, which tend to treat various meanings of the angle isolating one from
another, in time difference among them, and in ways that do not make visible the multiple
contexts where angles and their inherent meaning appear. It seems that school curricula and
textbooks in Greece do not promote efficiently the integration of the different meanings of
angles.
This study is part of a broader action research-where teacher and researcher is the same
person-in the form of an instructional experiment of a middle duration. The teacher researcher, taught with the help of concrete materials-when it was possible and useful- the
nine chapters of the Maths textbook in regard to angle in a 7th grade class of a Greek high
school (12-13 years old) in Rhodes, Greece. The aim of this study is to identify the pupils’
conceptions in regard to the concept of the angles and their measurement, before the teaching
experiment, so that it would be useful in the design and the makeup of the hypothetical
learning trajectory. More specifically, we tried to identify the phenomenological way pupils
approach angles and to detect if there was any confusion regarding the nature of the angles
and their measurement units. Afterwards, we tried to interpret the pupils’ results. This process
was correlated with the evaluation of the pupils’ answers from various teachers of the primary
Education and also the instructions to the teachers on how they should teach the angles.
THEORETICAL FRAMEWORK
Angle is a multilateral concept which can be found in a lot and different contexts and in many
meanings. Freudenthal (1983, p.323), claims that: «I introduced angle concepts in the plural,
because there are indeed several ones; various phenomenological approaches lead to various
concepts though they may be closely connected. Even in practice one single version does not
suffice». Douek, (1998 p. 264) underlines: “the multiplicity of the angle concept: the
multiplicity of embodiments and reference situations, multiple relationships between dynamic
and static aspects … ».
As far as the definitions of angles are concerned, Euclid defines the plane angle as “ the
mutual bent of two lines in the same level”, without defining anywhere in the elements the
meaning of mutual bent or simply the bent which is used as a determinant of the angle
concept. Mitchelmore &White (2000a, p. 209) claim that «Three particular classes of angle
definitions occur repeatedly: an amount of turning about a point between two lines; a pair of
rays with a common end- point; and the region formed by the intersection of two halfplanes». Henderson & Taimina (2005, p.38) consider that: «… It seems likely that no formal
definition can capture all aspects of our experience of what an angle is. There are at least
three different perspectives from which we can define ‘angle’, as follows: a dynamic notion
of angle-angle as movement; angle as measure; and, angle as a geometric shape. A dynamic
notion of angle involves an action: a rotation, a turning point or a change in direction between
two lines. Angle as a measure may be thought of as the length of a circular arc or the ratio
between areas of circular sectors. If considered a geometric shape, an angle may be seen as
the delineation of space by two intersecting lines….Note that we sometimes talk about
directed angles or angles with direction».
As far as the difficulties with the angles, many researchers mention that are linked to the
integration of the different perspectives of definition or the different angle aspects or angle
concepts. Clements &Battista (1992, p.451) refer that there is a line of research which
indicates that students hold many different schema regarding not only angle concept but also
angle measure.Freudenthal (1983p.359-360), points out that “…Nevertheless, it is a fact in
teaching practice that the mastery of trigonometrical functions and the ability to use their
tables does not necessarily include any insight into what an angle is and how it is measured”.
In primary and secondary school the dominant mathematical framework is that of the
Euclidian Geometry and especially one that includes drawings of plane closed convex
polygons. What many researchers claim (Mitchelmore και White 2000, Vadcard 2002) and
our research also shows, is that this unilateral orientation is responsible for the creation of
stereotypes in regard to angle concepts and, as a result, many pupils recognize angles mainly
as angle-sectors (corners), that is, angles as parts of a polygon. Pupils recognize less frequent,
angles as pairs of half-lines and hardly angles as turns. Vadcard, (2002 p.79) claims that the
most common mistake made by pupils is that when measuring an angle they tend to
overconsider the length of its parts and more generally the spatial characteristics of the design
that represents the angle.Vadcard, cites Close (1982), who points out the various behavioral
responses of the pupils that are generated by the confusion between the mathematical concept
and its representation. She also refers to Balacheff (1988, p. 371) who claims that «…this
conception of the angle derives directly from identifying with the shape that represents it».
Clements &Battista (1992, p. 451) report that third graders frequently relate the size of an
angle to the length of line segments that form its sides, the tilt of the top line segment, the area
enclosed by the triangular region defined by the drawn sides, the length between the sides
(from points sometimes, but nor always equidistant from the vertex), the proximity of the two
sides, or the turn at the vertex (in logo experiences). They also refer that intermediate grade
students often possess one of two schemas (especially in logo experiences). In ‘45-90
schema’, slanted lines are associated with 45ο turns while horizontal and vertical lines with
90ο turns. In the ‘protractor schema’ inputs to turns are based on usage of a protractor in
‘standard’ position even if it is necessary to change position. Besides, an issue that creates
confusion among pupils (and even among teachers) is the determination of the nature of
angles and their measurement units. Α.Bouvier, M.George & F. Le Lionnais (1996) in
‘Dictionnaire des Mathematiques’, under vocabulary entry of ‘angle’ suggest that for decades
(within the mathematical community) this word has designated at the same time:
 an angular sector,
 a measure of an angular sector
 and a non oriented angle
This confusion has made vague the concept of adding angles (because two measures of
angular sectors do not give always as sum of their addition the measure of another angular
sector) not allowing the determination of the angle of a rotation in a simple way. Baruk
(1992) notes that this confusion between the angle as a shape and the angle as a
magnitude is not new, but rather has old roots. She considers that in regard to secondary
education we must distance ourselves from the old/ancient definition of the angle that is
persistent in everyday thinking. Baruk (1992) underlines that in maths the angle is no more
considered (and is not) a schema. She claims that in order for the curricula to simplify things
they signify a shape, thus making things even more difficult. She also reminds us that this
change of meaning which took place-in accordance with the change of meaning of a figure
and thus of the equality of figures -, at first, in France and elsewhere with the arrival of the so
called modern mathematics, during the seventies, was difficult to be accepted like other
changes were, since the curricula maintained the older tradition, which was ambiguous and
created confusion.
But what does an angle seem to be for the pupils when entering the high school level and for
their teachers in primary education? Is it a schema, like a segment or a magnitude, like the
size of the segment? Or is it both? What is the role of the context as this is expressed through
the teaching textbooks and the teaching-learning process, in the meaning making of what an
angle is in primary and the first classes of high school?
THE METHOD OF THE STUDY
At the initial stage what we did was to do an epistemological and historical analysis of the
angle concept. Afterwards, we analyzed teacher materials in relation with the curricula.
Finally, we studied and analyzed the answers from two questionnaires, given by pupils (12-13
years) and teachers of primary education.
The results refer to an analysis of the questionnaires and an attempt to interpret them in the
light of what instructional materials and curricula suggest for the teaching of the angle. The
first questionnaire was answered by 53 pupils of the 7th grade, where the teacher-researcher
taught during 2004-2005. The pupils’ answers were analyzed and categorized. The categories
of pupils’ answers of four out of ten questions were given for evaluation to 35 teachers of the
primary education during the school year 2005-2006. The teachers that taught in the schools,
where the vast majority of the pupils that answered the questionnaire attended, were between
the 35 teachers that evaluated the categorized answers of the pupils.
The four questions which pupils addressed were:
 Can you name objects, situations or phenomena where the angle concept appears?
 How could you define (say) what an angle is to someone who does not know?
 What do you think an angle is? A schema? Like, for example, a segment? Or a
magnitude, like for example, the length of the segment?
 How could you define (say) to someone who does not know,
a) what a degree is? b) How much this degree is?
Teachers were asked to evaluate which of the categories of answers that the pupils gave in the
first question were closer to their teaching practice. In regard to the pupils’ answers in the rest
3 questions, teachers were asked to characterize them in a scale of: inadequate, adequate, and
correct (the wrong answers were also included in the category: inadequate).
Of the 35 questionnaires we excluded seven (7) which answered only a few questions. From
the rest, fourteen (14) out of twenty (28) teachers were women. Six (6) of the teachers were
less than 30 years old, seventeen (17) of them were between 30- 45 and five (5) were between
45- 65. Twelve (12) had a teaching experience of less than 10 years, another twelve (12) had
between 10-20 years and four (4) teachers had over 20 years teaching experience.
Our main hypothesis was that:
First, pupils’ static aspect about angles is dominant to their phenomenological approach of the
angles and there is confusion about the nature of the angles and their measurement units
(especially the degrees).
Second, pupils’ conceptions about angles, as estimated by the teachers’ evaluation, is
correlated to the teachers’ beliefs and instructional practices as they try, to use the textbooks
and to apply the instructions in order to implement the curriculum as suggested by the
Pedagogical Institute.
PRELIMINARY FINDINGS AND EXPECTED CONCLUSIONS
Our main hypothesis seems to be validated from preliminary findings. This is consistent with
other studies in the field of Mathematics Education (e.g. Baruk, 1992, Freudenthal, 1973)
which claim that among certain pupils and teachers there are stereotypes relative to the angle
concept and confusion relative to the nature of the angles and their measurement units. The
role of the teaching textbooks’ content in the primary education and its structure is crucial in
the meaning making of pupils and teachers of what an angle is.
A percentage of 80% of the pupils’ answers, in the first question, referred to reference
contexts where the angle considered as a plane angular sector, a 55% of the pupils referred to
reference contexts where the angle considered as two lines that are connected or branched, a
3% to contexts where the angle considered as an inclination from the horizontal or vertical
line, a 30% referred to contexts where angles considered as a spread of lines or surfaces
connected to a joint that allows movement, an about 20% referred to contexts where angles
considered as a point an about 10% to contexts where angles considered as a kind of
movement.
Two first categories were also considered by teachers as being closer to their instructional
practices. Furthermore, during the primary education and the first grades of the high school
the empirical basis of the pupils in relation to angles suggested by the textbooks is severely
limited so one should not expect more from them.
After categorizing the answers of the second question, we realized that the perception of the
angle as angular sector is prevalent to other perceptions, closely followed by the perception of
the angle as two lines that intersect each other. What followed was conception of the angle as
a point (a sharp). We also realized that in this question – in contrary to the first question-the
answers of the pupils referred almost completely to static meanings of the angles, whereas
dynamic approaches were almost absent (opening-closing, turns). At last it was obvious that
very few pupils described the angle as a magnitude.
The evaluation of the answers by the teachers revealed a similar tendency between the
teachers themselves. The large majority of them seemed to consider valid and adequate the
static notions of angles and only 1/3 of them considered valid the dynamic notions that were
expressed by few pupils.
Besides, the study of the teaching textbooks revealed that there is a one - sided orientation
throughout primary school towards, contextualized only through plane convex polygons
angles-sectors (corners) or de-contextualized angles-pairs of half lines. Other concepts of the
angle are missing thus contributing severely to the creation of stereotypes among pupils’ (we
believe and among some teachers’) conceptions relative to angles. This stereotypical way of
thinking increases the conception of angles as shapes.
The answers of the pupils in the third question reveal that some of the pupils conceptualize
the angle only as a shape because we can draw it, while others conceptualize it as a magnitude
because we can measure it. Some other pupils consider the angle both ways: as a magnitude
because we can measure it with degrees and as a shape because we can draw it. A small
number of pupils answered that they do consider it neither magnitude nor shape. We think
that the confusion is obvious.
Similar seems to be the confusion among teachers. Indeed, 18 out of 28 consider pupils
responses about the angle as shape, adequate (7 teachers), or correct (11 teachers), 15 teachers
consider those answers that consider the angle as magnitude, as adequate (9 teachers) or
correct (6 teachers), 19 evaluate those answers that view the angle as both shape and
magnitude as adequate (4), or correct (15). Finally four (4) teachers consider the answers that
view angles neither as shape nor as magnitude as adequate (1) or correct (3).
The study of the teaching textbooks has also revealed an approach which define angles as
figures that we can measure it, a fact which seems to us that is strongly correlated with the
creation of misconceptions about the nature of the angles.
The fourth question included two sub questions: What a degree is and how much is a degree.
For the vast majority of pupils it seems that degrees are mere numbers on the protractor and
only 20% of the pupils realize the nature (as an angle) and the quantity (as 1/90 of the right
angle) of a degree and have also an intuitive perception about it.
The evaluation of the teachers was positive - 25 out of 28 for the nature (as an angle) of the
degree and 22 out of 28 for the quantity (as 1/90 of the right angle) of the degree-. At the
same time the teachers evaluated positively those answers that defined, just generally the
degree as either an angle measurement unit or more specifically as a number (the measure of
the measurement of a magnitude) as well as the quantity of one degree through its subdivision
(60΄, 3600΄΄).
School textbooks tend on the one hand, correctly to emphasize a procedural approach of the
degree through the use of the protractor in the measurement process. On the other hand they
unduly neglect for a long time the conceptual approach (the nature of the degree as an angle
and its quantity).
The above findings seem to put in place a broader speculation regarding the effectiveness of
the approaches of angles from curricula, school textbooks and teachers. More specifically it
appears reasonable that the design of the above materials and the design of teaching
trajectories should consider the following:
1. The multifaceted empirical approach (through objects, situations and phenomena) at
an early stage, of the different concepts of the angle (Freudenthal 1983, Mitchelmore
& White 2000α).
2. The explicit distinction between the concepts of the angles and their
representations(Close 1982, Balacheff 1988)
3. The explicit clarification of the nature of the angles and their measurement units
(Baruk, 1992).
4. The careful distinction in proper timing of the different meanings of the angles in the
various contexts, the appropriate bridging between them and their progressive
integration (Freudenthal 1983, Baruk, 1992).
REFERENCES
1. Balacheff, N. (1988). Une étude des processus de preuve en mathématiques chez les
2.
3.
4.
5.
6.
7.
8.
9.
élèves de Collège. Grenoble : Université Joseph Fourier et Institut National
Polytechnique.
Baruk, S. (1992). Dictionnaire de Mathématiques Elementaires. Editions du Seuil
Bouvier, A. Lionnais G M. F (1979). Dictionnaire des Mathématiques. Presses
Universitaires de France.
Clements, D. &Battista, M. ‘Geometry and Spatial reasoning’, Chapter 18 in Douglas
A. Grouws (Ed.) Handbook of Research on Mathematics Teaching and Learning, p.p.
420-65, Macmillan, New York,1992.
Close, G. S. (1982). Children’s understanding of angle at the primary/secondary
transfer stage. London: Polytechnic of the South Bank.
Douek, N. (1998). ‘Analysis of a long term construction of the angle concept in the
field of experience of sun shadows’. In A. Olivier and K. Newstead (Eds.),
Proceedings of the 22nd annual Conference of the International Group for the
Psychology of Mathematics Education (Vol. 2, pp.264-271) Stellenbosch, S. Africa:
Program Committee.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel
Publishing Company- Kluwer Academic Publishers Group
Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht:
D. Reidel Publishing Company-Kluwer Academic Publishers Group.
Henderson, D.W. Taimina D. (2005). Experiencing Geometry (Euclidean and NonEuclidean with History), Third Edition,, Pearson Prentice Hall, New Jersey.
10. Mitchelmore, M. & White, P. (2000a) ‘Development of angle concepts by
progressive abstraction and generalization’. Educational Studies in Mathematics, Vol.
41, pp.209-238.
11. Mitchelmore, M. C., & White, P. (2000b). ‘Teaching for abstraction: Reconstructing
constructivism’. In J. Bana & A. Chapman (Eds.), Mathematics education beyond
2000 (Proceedings of the 23rd annual conference of the Mathematics Education
Research Group of Australasia, pp. 432-439). Perth: MERGA.
12. Vadcard, L. (2002). ‘Conceptions de l’angle chez des élèves de seconde’. Recherches
en didactique des Mathématiques, vol. 22 no 1 pp. 77-120.