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Angles and Triangles: Identifying and Overcoming Common Misconceptions
Feb 2015
Angles and Triangles:
Identifying and Overcoming
Common Problems and
Misconceptions
Name:
Subject: Mathematics
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
Feb 2015
Contents
Title Page
Contents
1. Introduction
2. Literature review
2.1 Multiple definitions of an angle
2.2 One sided angles
2.3 Vocabulary
2.4 Use of equipment
2.5 Identifying and classifying triangles
2.6 Base and height of a triangle
2.7 Memorising facts without
understanding their origin or being
convinced of their validity
2.8 Conclusion from the literature
3. Sequence plan
3.1 Lesson 1
3.2 Lesson 2
3.3 Lesson 3
3.4 Lesson 4
4. Data and ethics
5. Analysis of
observations
5.1 Vocabulary, and multiple definitions of
an angle
5.2 Vocabulary, and one-sided angles
5.3 Use of equipment
5.4 Memorising facts without
understanding their origin or being
convinced of their validity
5.5 Identifying and classifying triangles
5.6 Base and height of a triangle
6. Conclusion
7. References
8. Appendices
a. Pre-test and post-test
b. Transcript of focus group
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
Feb 2015
Introduction
‘Concepts of angle and rotation are central to the development of geometric
knowledge’ (Clements & Burns, 2000: 31) and it is well-documented that learners
experience ‘difficulty with angle, angle measure and rotation concepts (Clements and
Battiste, 1992; Krainer, 1991; Lindquist and Kouba, 1989; Mantaon et al, 1993;
Simmons and Cope, 1990)’ (Clements et al, 1996: 54). This then leads to problems
with triangles, and, since ‘triangles are the key building blocks for most geometrical
configurations’ (Atebe & Schäfer, 2008: 47-65), problems with shape in general.
Additionally, the focus of later parts of the geometry curriculum is on circle theorems,
Pythagoras’ theorem, and trigonometry, which are all heavily reliant on a solid
understanding of the properties of triangles. Addressing these early difficulties with
angles and triangles is therefore likely to lead to learners experiencing fewer
problems later in their mathematical education, so it is worth considering how they
can be overcome effectively.
Literature Review
Multiple definitions of an angle
The literature suggests that a major issue learners experience with angles is having
to move fluently between the multiple definitions of what an angle actually is, pointing
out that there are at least ‘three different perspectives from which we can define
angles: as a dynamic notion, as a measure and as a geometric shape’ (Fyhn, 2008:
21), or alternatively ‘as either an amount of turning between two lines meeting at a
point, a union of two rays with a common end point, or the intersection of two half
planes’ (Mitchelmore & White, 1998: 4), or, differently still, ‘as part of the plane
included between two rays meeting at their endpoints (the static definition) and as
the amount of rotation necessary to bring one of its rays to the other ray without
moving out of the plane (the dynamic definition)’ (Clements and Burns, 2000:31).
A lack of understanding of the connectedness of each of these definitions is likely to
have an effect on later learning; for example if a learner does not clearly understand
what an angle is, they will not be able to correctly compare angles by sight, leading
to the widespread misconception ‘that a small angle has short sides and a large
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angle has long sides (Clements 2003; Gjone and Nortvedt 2001)’ (Fyhn, 2008: 25).
This leads to difficulties with identifying and classifying angles, and, later, triangles.
Mitchelmore & White propose that one way of addressing this issue is by introducing
angles in a variety of physical situations that are known to the learners – for example
using ladders, scissors, and corners (1998: 4-27) These would provide a bank of
mental images of angles as both static and dynamic entities, encompassing the
multiple definitions, for pupils to recall later on.
Additionally, research including that carried out by Laroff and Nunez and quoted by
Fyhn supports the idea that ‘human mathematics is embodied, it is grounded in
bodily experience of the world’ (Fyhn, 2008: 19-35), suggesting that learners need to
be able to physically manipulate geometric objects, including both angles and
triangles, in order to fully understand them. Other authors have concluded ‘that
young children learn most effectively when they actively construct ideas and when
they are given the opportunity to engage in tasks which are both meaningful and
appropriate to their developmental level’ (Yelland & Masters, 1997: 84). The
common theme established by this research is that there is a need for a
constructivist approach to teaching and learning geometric ideas.
Building manipulation of real objects into a lesson sequence teaching angle concepts
would address each of these points and could allow pupils to construct their own,
deep understanding of what an angle is.
One sided angles
Another area in which learners tend to encounter problems is when applying the
theoretical idea of angles to real life situations, with it being widely agreed that it is
‘difficult for children to identify angles in slopes, turns and other contexts where one
or both sides of the angle are not visible’ (Fyhn, 2007: 22).
Research suggests that this gap in learners’ understanding can be addressed by first
introducing them to ‘physical angle contexts which most clearly involve two lines –
including crossings, corners and bent objects’ (Mitchelmore & White, 1998: 15), and
then extending and reinforcing the gained understanding by allowing them to
investigate contexts which they ‘do not naturally interpret using the standard angle
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model, either because the lines are not clear (e.g., rebounds) or because one or both
lines must be constructed (e.g., slopes, turns)’ (Mitchelmore & White, 1998: 26).
Introducing angle concepts in this order could give learners the necessary space to
properly identify contexts in which angles appear, and allow them to create a lasting
understanding of angles in real life situations.
Vocabulary
A third common issue concerns the use of specialist vocabulary: learners often find
the plethora of new vocabulary in an angles (or triangles) lesson overwhelming, and
even those who grasp the main ideas can struggle with articulating them – as
Mitchelmore & White identified, ‘children often seemed aware of physical relations
without being able to put them into standard (adult) language’ (Mitchelmore & White,
1998: 13). This can lead to disengagement with the topic as a whole.
One way in which the literature suggests that the issue of complex and extensive
specialist vocabulary could be addressed is by creating a word bank to collect
together all technical terms and labels, and allowing learners to readily refer to it
during lessons (Sears & Chávez, 2014: 767-780). Additionally, encouraging learners
to talk through the mathematics by encouraging group work and class discussions,
provides them with the opportunity to use this vocabulary in a less formal context,
and also has the added benefit of capitalising on ‘students’ informal learning style
and their natural propensity to teach one another’ (Bowers and Stephen, 2011: 286).
This approach can also help learners build confidence, empower them, and allow
them to consolidate their understanding (Crowley, 1987:1-16).
An alternative approach is summarised as follows:
‘She [Van Hiele-Geldof (1984, p.55)] recommends that teachers start with
everyday speech, referring first to situations that are familiar to the student,
and proceed from there to mathematical language’ (Towers, 2002: 129)
This approach differs from the others in that it initially places less emphasis on the
importance of the correct mathematical terminology, allowing to pupils to first
understand the main concepts. The use of correct language is seen as less
important here (at least to begin with).
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The two different approaches are not mutually exclusive however, and could easily
be combined by first introducing the concepts in everyday terms, and later
introducing the mathematical language, reinforcing its use through the use of word
banks, and allowing pupils to build proficiency and confidence in its use through
group work and discussion.
It is also important to recognise here that ‘just because children are using a word
does not mean they attach the same meaning to it as their listener’ (Crowley, 1987:
14), and so it is very important that teachers question learners effectively to ensure
they have correctly and fully understood new concepts.
Use of equipment
A related issue is with correct usage of angle-measuring equipment, with research
noting a ‘common blunder related to the use of a protractor… confused the readings
on the protractor and wrote the supplement of the desired angle’ (Sdrolias &
Triandafillidis, 2007: 163).
One way in which this could be addressed is by waiting to introduce measuring
angles until after the concept of angle classifiers (acute, obtuse etc.) have been
understood. If pupils are able to classify the angles by sight before measuring them,
they are much less likely to accidentally take the supplement of the required angle.
A common theme of the research on the issue of teaching the concept of angle
classifiers and angle estimation suggests that it can effectively be done through the
use of technology (namely, computer programs and Logo), with learners’ themselves
suggesting that ‘computers enable them to visualize and this helps them to learn
permanently’ (Ozerem, 2012: 723).
Identifying and classifying triangles
Many pupils come to secondary school with simplified and ‘incomplete notions of
basic geometric shapes and their properties’ (Burger & Shaughnessy, 1986: 46).
Since it is an important part of a teacher’s role to ‘challenge understanding and
broaden generalisations’ (Oberdorf & Taylor-Cox, 1999: 343), it is clear that these
ideas must be reformulated within the course of the lesson sequence, with some
research suggesting that they be tested for before teaching takes place (Özerem,
2012: 720-729).
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One such common misconception is that it is only possible to classify triangles in one
way: so a triangle is either scalene or isosceles or right-angled etc. Research
suggests that this stems from the fact that learners tend to define ‘shapes so as to
prohibit class inclusions’ (Atebe & Schäfer, 2008: 61), which in itself arises from the
common primary school practice of classifying in this way, which is ‘essentially
correct, but does not allow for growth in understanding’ (Oberdorf & Taylor-Cox,
1990: 340).
These authors suggest that this misconception can be addressed through promoting
cognitive conflict. One way of doing this, they suggest, would be to have pupils place
triangles within a Venn diagram with rings labelled scalene/right-angled or
isosceles/right-angled. This would force them to rethink and adjust their
preconceived ideas.
Base and height of a triangle
Another common misconception is that the orientation of a shape affects its
properties – for example, a right-angled triangle must have a vertical edge and a
horizontal edge, and the height of a triangle must be its height above the horizontal
in the orientation it is given in. If it is agreed that triangles are the key to
understanding geometrical figures, then it follows that these misconceptions are
serious and need to be addressed early on.
Literature suggests that ‘varying the shape’s position during presentation will help
children understand that a shape remains constant regardless of its position in
space’ (Oberdorf & Taylor-Cox, 1990: 343), and that ‘the teacher should
continuously remind students that rotation of an object does not change its shape’
(Özerem, 2012: 728). It also emphasises the importance of clearly explaining that
the relationship between base and height is that they are perpendicular, not that they
are horizontal and vertical. These suggestions could be incorporated into a lesson
sequence fairly easily.
Memorising facts without understanding their origin, or being convinced of
their validity
Research has found that ‘accurate, fluent recall of facts frees up short-term memory,
allowing students to devote more attention to less obvious aspects of a problem’
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(Tait-McCutcheon et al, 2011: 321), so it is clearly important that learners are taught
relevant facts, and that they are emphasised regularly.
The problem with the teaching of these facts is that they are often ‘taught in a
mechanical way’ (Crowley, 1987: 13), with pupils being given facts to learn, without
having the opportunity to investigate their origins. A very common example of this
that the sum of the interior angles of a triangle is 180˚; as Crowley points out,
‘frequently, this fact is generalized after measuring the angles of a few triangles, or
worse, pupils are simply told the information’(Crowley, 1987: 13). Some researchers
argue that ‘learning and knowing basic facts requires the active construction and
meaningful memorisation of a body of knowledge’ (Tait-McCutcheon et al, 2011:
323), and so this prevalent method is inadequate for ensuring long-term recall and
correct application.
It is suggested that a better way of teaching these facts would be to have pupils
manipulate physical triangles by tessellation and then conjecture relationships
between the angles and a complete turn. Pupils could then test the correctness of
their conjectures by using computer software, or could be shown a formal proof either way would give them a more meaningful understanding. It is also pointed out
in the literature that this method is better, as it sets ‘the stage for further learning’
(Crowley, 1987: 13) since it can be extended to other geometric figures, such as
quadrilaterals. The issue with this approach, however, is that pupils with incomplete
understandings of the properties of shapes (in this case, triangles) might struggle,
initially at least, to ‘reason about shape in a formal way’ (Burger & Shaughnessy,
1996: 46).
Conclusion from the literature
It is clear that there are many potential issues which learners face in the topic of
angles and triangles, and which teachers must therefore be prepared for. It is also
clear that there has been a considerable amount of research into how these issues
can be overcome. The suggestions from this research will be tested within the
following sequence of lessons on the topic of Angles and Triangles, with a ‘middle
ability’ year 7 class in an outer London secondary school.
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Sequence Plan
An analysis of the literature suggested that a possible sequence of lessons to be
used to address the aforementioned common difficulties and misconceptions in the
‘angles and triangles’ topic could be as follows.
Lesson 1
Based on the reading, it seemed that a logical starting point would be to help pupils
construct an image bank for the often abstract idea of an angle, through introducing
angles in real life contexts. This would also embed the idea that angles do not
necessarily have two clearly identifiable straight arms, and are not always static
objects. In the course of this introduction, everyday terminology such as ‘steeper’
would be used to lead into more technical vocabulary relating to the size of the
angles, as suggested by Towers as a way of avoiding pupils feeling overwhelmed by
the amount of new vocabulary (2002:121-132), and also as a method for addressing
the misconception that ‘a small angle has short sides and a large angle has long
sides’ (Fyhn, 2008: 25).
In order to allow for an effective analysis of pupils’ learning through this sequence, a
short test (see appendix a) of pupils’ pre-existing knowledge would be carried out at
some point in the course of this first lesson.
Lesson 2
A logical progression from this point would be to move the focus onto angles in a
triangle, and specifically, the constant sum of angles in a triangle. In accordance to
research suggesting the importance of a constructivist approach to school geometry
(Yelland & Masters, 1997: 83-99), this could be introduced by having pupils
manipulate triangles, initially physically and later through the use of technology.
An adaptation of the tasks described by Crawley , suggest that one way of doing this
is to give each pupil a set of congruent triangles with each of the three angles
marked with a different symbol, and instructing them to tessellate the triangles and
observe the symbols of the angles appearing at each meeting point on a straight line
(1987: 1-15). The ‘sum to 180°’ property would be introduced from there, with pupils
being given the opportunity to check the generality of it through the Geogebra
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program, making use of technology and group work as suggested by Bowers &
Stephen (2011: 285-304).
Lesson 3
The preliminary focus of the third lesson would be to revisit the measuring of angles,
but with an initial exposure to estimating the size of an angle by making comparisons
with a ‘right angle’ and ‘straight angle’ (which most pupils are likely to be familiar
with). The initial estimation may help pupils avoid the common mistake of taking the
supplement of a required angle. If pupils are taught to first estimate an angle, then
taking the supplement would cause a cognitive conflict, thus forcing pupils to analyse
their work and correct their mistake.
This leads into the primary focus of the lesson, which is to introduce pupils to
classifying triangles. Research has suggested that primary school experience often
leaves pupils reluctant to classify triangles in more than one way, but that this can be
addressed through sorting various triangles into Venn diagrams, with rings
representing the various classifications. It is suggested by Oberdorf & Taylor-Cox,
that the cognitive conflict arising from a sorting activity such as this is beneficial to
pupils’ learning and understanding of geometry as whole, rather than as
disconnected parts (1999:340-345). The discussion with pupils regarding their
placements is also likely to aid understanding as it forces them to articulate ‘what
might otherwise be vague and undeveloped ideas’ (Crowley, 1987: 14).
Lesson 4
As the final lesson in the sequence, lesson four would be used to address the
remaining misconception arising from the research – namely, pupils’ difficulties in
identifying the base and height of a right angled triangle, particularly when given in a
non-standard orientation. In order to provide some context to this concept, it will be
introduced alongside area: pupils will first investigate the fact that every right angled
triangle can be formed by dividing a particular rectangle into two equal parts. It is
assumed that pupils will then be able to extend their understanding of how to
calculate the area of a rectangle to calculating the area of a right angled triangle.
This creates a need for identifying the base and height, and is therefore more likely
to engage pupils than a simple labelling exercise.
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This final lesson will end with a test, and pupils’ answers to it will be analysed in
order to assess the effectiveness of the tasks given to pupils in developing their
understanding of angles and triangles.
Data and Ethics
In order to evaluate the effectiveness of the lesson sequence in overcoming common
misconceptions, it was decided that it would be necessary to collect data on pupils’
prior knowledge (ie what they could recall from their primary school teaching) before
each idea identified in the literature review was introduced to them. Their ideas
presented at this stage would then be compared to their responses to a problem
posed at the end of the lesson, and, at the end of the sequence, their responses to a
post-test. The data collected at the end of each lesson will be used formatively – with
pupils’ responses being used to inform planning and teaching of the next lesson.
Since the same cannot be said of the data collected in the post-test, this will be kept
as short as possible, and be designed to help pupils collect their thoughts, so it acts
as a plenary to the sequence as well as an assessment of their learning.
Whilst it would be ideal to be able to carry out an in-depth analysis for every pupil in
the class, time constraints mean that realistically it will only be possible to do so for
six pupils (in order to maintain complete anonymity, they will be referred to only as
P1-6). These pupils will be selected so as to represent the class as a whole, and
include the higher achieving and lower achieving members of the class, pupils
seated in different areas of the classroom, and pupils of different genders.
These same six pupils will also be invited to attend a short focus group a fortnight
after the end of teaching of the sequence, to determine what they are able to recall
from the sequence of lessons. This will allow the effectiveness of the chosen tasks
on longer-term recall to be assessed. In order to meet ethical considerations, pupils
will be made aware that their participation in the focus group is completely voluntary,
that there are no repercussions for declining to attend, and that they are free to leave
at any point during its course.
In addition to these more formal methods of collecting data, data will also be
collected through listening to pupils’ discussions as they work in small groups,
looking at pupils’ work, and noting down pupils’ contributions to class discussions.
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These will be analysed to determine how pupils’ understandings are developing in
the course of the sequence of lessons.
In the course of the lessons, a reserved pedagogy will be employed, and pupils’
questions will be responded to by further questioning. The reasoning behind this that
a deliberate avoidance of giving pupils hints or answers will have the effect of forcing
them to re-examine their existing knowledge and use it to make connections and
infer relationships, therefore encouraging deeper, more meaningful, and longer-term
understanding.
Finally, it is necessary to note that the topic of ‘angles and triangles’ is one which is
on the Scheme of Learning for this particular class, and would be taught to them
whether or not research for this report was taking place. Hence pupils have not lost
learning time due to the writing of this report, and have not been disadvantaged by it.
Analysis of observations
The lesson sequence outlined earlier clearly states the order in which ideas which
were revealed by the literature to be difficult for pupils to grasp will be introduced to
pupils. What follows is an analysis of the success of the strategies suggested by the
literature in overcoming misconceptions associated with each of those ideas.
Vocabulary, and multiple definitions of an angle
Before teaching of the sequence began, pupils were asked to write down what they
recalled as the definition of an angle. There are a number of things worth noting
about pupils’ responses at this stage. Firstly, not all pupils responded to the task,
with many either writing or saying ‘I know what it is, but I can’t explain it’. This
appears to support Mitchelmore & White’s assertion that it is often the vocabulary
and articulation that pupils struggle with rather than the concepts themselves (1998:
4-27). Of those who did attempt to answer the questions, most drew an angle (often
a right angle) as an exemplifier, or a shape with the corners marked as ‘angles’. One
pupil wrote about using protractors to measure the size of an angle. Finally, although
some pupils mentioned angles as corners (ie as static objects), there was no
mention of them in a dynamic context. This is as expected, as much of the research
into this area mentions difficulties pupils have with the multiple definitions of angles
(Fyhn, 2008: 19-35).
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Following the suggestions that pupils need to be able to physically manipulate
geometric objects, such as angles, in order to fully understand them (Fyhn, 2008: 1935), a part of the introductory lesson involved pupils constructing models of Tower
Bridge. The suspended sections of these models were attached to the main body
using split pins, allowing these sections to rotate and thus demonstrating angles
appearing in dynamic contexts.
Initially, this task appeared to have helped pupils extend their ideas of angles to
dynamic contexts - as evidenced by pupils responding to the question, ‘where can
you see angles in this room?’, with responses such as ‘when you open a pair of
scissors’ and ‘if you open the door wide’, amongst the expected responses such as
‘corner of the table/board/book’. However, based on pupils’ responses to being
asked to ‘identify five places/objects where you might see angles’ in the post-test a
week later, this did not appear to have made a long-lasting impression: pupils
reverted to their prior static definitions, and bar a few ambiguous responses such as
hand, clock, laptop, and door, most mentioned only static objects such as corners of
tables, books, and whiteboards. It is important to note, however, that although the
dynamic context was explicitly discussed in this first lesson, the next three lessons
focussed on angles in triangles and other static situations. It is likely that a single
lesson on this theme was insufficient, and returning to it in subsequent lessons
would have helped pupils to fully grasp it and recall it later.
Vocabulary, and one-sided angles
A further aim of the sequence was to extend pupils’ understanding of angles from
situations in which both sides are clearly shown (as they would have encountered in
primary school) to situations in which only one side is clearly defined, a next step
suggested by Mitchelmore & White (1998: 4-27). These same authors also
suggested that this should be done by first introducing one-sided angles in contexts
with which the pupils are familiar (1998: 4-27). It was, therefore, decided that onesided angles would be introduced in the context of scooting up slopes forming
increasing angles with the horizontal, a situation which much of the class could relate
to. This was carried out as a whole class activity, and pupils were asked to first
decide whether the scooter would be able to ascend the ramp, and later articulate
how the ramps differed in each scenario.
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The activity proved to be an interesting one, with pupils engaging well and
challenging each other’s ideas. Pupils were inclined to use everyday vocabulary,
describing slopes as ‘higher’ or ‘steeper’, and angles as ‘bigger’ or ‘wider’. This was
encouraged, as much of the earlier research noted this as a useful and valid way into
the complicated mathematical language associated with geometry (Towers, 2002:
121-132; Crowley, 1987: 1-16).
Pupils first ventured into mathematical terminology when asked why it might be
possible for the scooter to ascend ramp 1 (inclined at 70° to the horizontal), but
definitely not possible to ascend ramp 2 (inclined at 110° to the horizontal): an initial
response to this was ‘to climb up ramp 2, the scooter would have to go upside down’,
with a pupil interjecting ‘because it’s obtuse, and the first one is acute’. The teacher
questioned this further, asking pupils for definitions of acute and obtuse, to which the
general response was that acute is smaller than a right angle, whilst obtuse is
greater than a right angle. Various responses were given by pupils when asked what
a right angle was: a number made L shapes with their hands; another said, ‘it’s a
right angle if a square can fit in the corner’; several referred to horizontal and vertical
lines meeting at a right angle, and a right angle being 90°.
Use of equipment
The gradual progression from everyday language in a context familiar to pupils, to
more technical mathematical language seemed to help pupils make sense of angle
sizes: a pre-test had revealed that the majority of the class (15 out of 22) had been
unable to determine the smallest and largest angles from a given set, with many
selecting the angle with the shortest sides as the smallest, and the one with the
longest sides as the largest – a common misconception identified by researchers
including Fyhn (2008: 19-35). At the end of the lesson in which slopes were
discussed, pupils were asked to work in pairs to sort a set of nine angles by size.
Most were able to do this without assistance, although some found it difficult as not
all the angles had a horizontal base line. A suggestion of turning the sheets so the
angles appeared to have a horizontal base line helped those with this difficulty.
Pupils then labelled each angle with its correct classification.
This exercise and its preceding discussion appeared to have had a positive impact
on pupils’ abilities to compare angles by sight, as indicated by the huge improvement
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in the pre-test exercise when presented in the post-test – all pupils with the
exception of four selected the correct angles in the post-test. Three of those who
selected different angles had chosen the right angle as the largest, which is only a
little smaller than the actual largest angle, and is likely to have been an error rather
than an indication of a misunderstanding or misconception.
It was only after this point in the sequence that it seemed appropriate to introduce
the idea of using a protractor to measure angles – the research suggested that until
pupils could confidently classify angles, they would be vulnerable to taking their
supplement when trying to measure (Sdrolias & Triandafillidis, 2007: 160-169). Using
a protractor to measure angles was introduced through demonstration, and pupils
were then given a short practice exercise. Pupils were advised to first classify their
triangle, and then use the protractor to measure it. During the lesson, the technique
of first classifying and then measuring appeared to help, and only a small number of
pupils needed help to read the protractors correctly. However, when this was tested
in a series of four questions in the post-test, only six pupils measured all four angles
correctly. The majority of the others measured the acute angles correctly, but took
the supplement of one or both of the obtuse angles. It is possible that either a
reminder to first classify the angles, or further practice of measuring angles, would
have avoided this error.
It is interesting to note how much better pupils performed at a measuring task when
they were reminded to first classify the angle, as they did in the lesson, compared to
when they missed out this first step. This clearly shows the effectiveness of the
‘estimate first’ technique.
Memorising facts without understanding their origin, or being convinced of
their validity
It was decided that the connection between angles and triangles would be made by
establishing the fact that the three angles inside any triangle sum to 180°. Since
there is strong evidence to suggest that a constructivist approach to establishing
ideas such as this helps develop understanding and long-term recall (Yelland &
Masters, 1997: 83-99), this idea was introduced through pupils actively constructing
the idea for themselves.
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The first element of this was a homework task set during the previous lesson: pupils
were to find out the sum of angles at any point on a straight line. The majority of the
class completed this task, and came to the following lesson with the knowledge that
the sum of angles at any point on a straight line is 180°. In addition, pupils paired up
and drew a triangle of their choice, marking each angle with a different symbol.
Adapting the strategy suggested by Crowley, the teacher created photocopies of
each pair’s triangle, and in the following lesson pupils were instructed to tessellate
them as if they were ‘tiling a floor’ (1987: 1-16).
An
example of one pair’s work is as follows:
Figure 1: Pupils' work on a task designed to introduce the idea that angles in a triangle sum to 180°
Individual pairs of pupils were then asked if they noticed a pattern in their
tessellation, and many pointed out a recurring ‘star, circle, hashtag’ pattern at every
point where the three triangles met. One pair made the connection between this
pattern and the fact that the sum of angles at a point on a straight line is always
180°, but didn’t extend this to the fact that the sum of angles in a triangles is always
180°, despite some prompting. In terms of engagement, pupils stayed focused
throughout the task – the idea of uncovering a pattern before their peers seemed to
be a good motivator.
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The second stage of the lesson involved pupils using Geogebra on computers to
draw various triangles and note down their angles. Pupils were very engaged in this
task and followed the instructions well. Bowers & Stephens suggested that paired
work is useful as it capitalises on pupils’ ‘natural propensity to teach one another’
(2011: 286) and this was evident when observing pupils work through this task –
they stepped in to correct each other’s work and explain why something had gone
wrong.
The lesson ended with pupils calculating the sum of the angles in each of their
triangles, and noting it on a list on the board. Pupils were interested to note that each
sum was approximately 180°, and were happy to accept that any slight divergence
from this was due to rounding errors. It is possible that pupils were happy to
generalise in this way as they had constructed the triangles themselves – and that
they would not have been so accepting had the triangles been drawn and handed to
them. This would suggest that computer programs are useful for these sorts of
generalisation exercises, which appears to hold true for this class, who all confidently
stated in the post-test that the sum of the three angles in any triangle all sum to 180°.
One particularly
succinct statement
was made which is
worth noting here:
Figure 2: A pupil’s response to being asked what he would expect the sum of angles in a triangle to be
This statement was read to pupils in the focus group a fortnight after the lesson
sequence had been taught (see appendix b for a transcript). Of the five pupils who
attended, three agreed with the statement. The other two seemed to be unsure of its
universal applicability, although they were happy to use it to calculate the missing
angle of a triangle presented to them. When questioned further, they spoke about
irregular triangles – which turned out to be three cornered shapes with curved
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‘edges’, or quadrilaterals in the shape of arrow heads. Other pupils were quick to
point out that neither of those shapes could be classified as triangles, and they
conceded to this.
Although time constraints meant that it wasn’t possible to check the whole class’
confidence in this fact, is likely that there were other pupils who weren’t entirely sure
of its universal application to triangles drawn on a flat surface. Spending a little more
time on establishing this fact would be advisable in future teachings of this topic.
Identifying and classifying triangles
Before teaching pupils the names given to triangles with various properties, the
teacher questioned pupils to check their prior knowledge, revealing that they recalled
the classifications from primary school teaching. Thus, the lesson quickly moved on
to a task in which pupils were each given a triangle and were asked to measure the
lengths of its sides and sizes of its angles, and then use this information to classify
the triangle.
A common misconception identified in the literature is that pupils tend to define
shape classifications in a way that excludes certain combinations of properties – for
example, given a triangle with angles 90°, 45°, and 45°, pupils will generally classify
it as a right angled triangle, ignoring the fact that it is also an isosceles triangle
(Oberdorf & Taylor-cox, 1999: 340-343). This was found to be the case within this
year seven class, with many pupils ignoring a property of a triangle in order to
classify it by another of its properties.
One specific case occurred with P6, who quickly identified her triangle as being right
angled. When questioned about this classification, she correctly identified that the
triangle was right-angled as it had an angle of 90°. A prompt by way of asking her to
check the definition of a scalene triangle led her to state that the triangle was
scalene. She seemed somewhat confused by this, and asked whether it was
possible for a triangle to be both right angled and scalene. A response of ‘is there a
reason why it couldn’t be both?’ led her to conclude that actually it was possible for a
triangle to be classified in more than one way. It is worth noting that a similar
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conversation was had with a number of pupils in the class, and all insisted that they
had never before classified a triangle in more than one way.
In order to extend this idea to the whole class, each pupil was given some BlueTack
and asked to come to the board at the front of the room and stick their triangle in the
correct section of a Venn diagram labelled with the triangle classifiers. As they did
so, they were required to explain their reasoning to the class, and the class had the
opportunity to agree or disagree with their classification. The use of Venn diagram in
this way was suggested by Oberdorf & Taylor-cox, as a useful way of encouraging
cognitive conflict which is often necessary to modify and correct previous knowledge
(1999: 340-343). Separately, by having pupils explain their reasoning, they were
required to use mathematical language in order to articulate mathematical ideas,
which is beneficial to their own learning and development (Crowley, 1987:1-16), and
the learning of the class as a whole. Pupils were very supportive of each other
throughout the course of this task, and remained engaged for the duration. It is
possible that this was due to the fact that it was different to their usual (more
repetitive) tasks, and involved them talking to each other.
Pupils’ recall of their work on classifying triangles was tested through a starter
activity at the beginning of the next lesson, in which they were given the lengths of
the sides of an isosceles right angled triangle, and asked to classify it. Pupils
managed this fairly easily, although a few did require prompting with the question, ‘is
there any other way you could classify that triangle?’
Many pupils moved on to the extension task, which required them to work out the
sizes of the two missing angles. Those who were able to work it out easily were
asked to explain their reasoning to others at their table, and listening to their
explanations provided a valuable insight into their understanding. It was clear that
many pupils had grasped the concepts being taught in the previous lesson, and the
lesson on the sum of angles in a triangle, and were now able to apply those
concepts to problem solving.
Base and height of a triangle
The literature suggested that identifying the base and height of a triangle is
something pupils struggle with, particularly in situations where the base of the
triangle is not parallel to the bottom of the page. Suggested methods for overcoming
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this difficulty were first, ensuring that shapes are not always shown in standard
position (ie base parallel to the bottom of the page) (Oberdorf & Taylor-Cox, 1999:
340-343) and second, emphasising the fact that shapes can be rotated without
altering their properties (Ozerem, 2012: 720-729). Both of these suggestions were
accounted for in the preceding lessons.
In order to assess the effectiveness of these suggestions, it was decided that pupils
would be asked to find the area of various right angled triangles. This would allow an
assessment of their ability to identify the base and height of the triangle to take
place, without making it the sole focus of the lesson.
Pupils were introduced to this task through finding the area of a rectangle, and then
halving it to find the area of the right angled triangle formed by drawing a diagonal
line connecting two of its vertices. Initially, pupils were asked to do this for triangle
drawn on plain paper, but the constructing of a rectangle without grid lines to use as
a guide turned out to be difficult for a large proportion of the class. This then led to
some disengagement and off-task behaviour.
However, when an adapted (to include grid lines) version of the exercise was
presented as a starter to the following lesson, pupils were able to find the area of the
triangles, and were also able to explain how they knew their answers were correct. In
addition, a small number of pupils, including P3 and P5, were able to calculate the
area without drawing the associated rectangle. Pupils were engaged by the activity,
and, marking their work after the lesson revealed that it had helped them identify the
base and height, which had been the intended outcome.
Conclusion
Through first teaching a sequence of lessons on angles and triangles, and later
assessing what pupils had learnt from the sequence, it has become clear that some
of the strategies suggested by the research have been more effective than others.
For example, pupils’ exploration of angles as a dynamic notion appears to have been
initially successful, but failed to make an impression in the longer-term. This is quite
possibly because it was quickly overshadowed by the static notion, since this is what
is required in the study of shape. An improvement to the sequence may be to refer
back to the dynamic notion in later lessons, and introduce problems involving it. This
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Feb 2015
would make it more relevant to pupils’ work and thus make a greater impression on
them.
In contrast, the suggestions made in the literature for allowing pupils to construct
their own understanding of the idea that the three angles in a triangle sum to 180°
were very successful and the vast majority of pupils were comfortable to generalise
from the work carried out in that lesson. The use of group work is widely suggested
to help pupils develop their thoughts, and in this case it appeared to do so.
Similarly, introducing angle classifiers before introducing the measuring of angles
using a protractor appeared to help many pupils avoid the common error of taking
the supplement of a given angle. However, some pupils still appeared to have
difficulty with measuring obtuse angles, especially when they appear in the third
quadrant (taking the point of intersection of the line segments as the origin). This is
likely to be an issue which can be overcome with practice, and time for this will be
built into future teaching of this sequence.
Likewise, pupils were quick to grasp the idea of classifying triangles in different
ways: the use of the Venn diagram as suggested by the research introduced
cognitive conflict, as the authors had indicated, which proved to be useful for
developing pupils’ understanding of classifying triangles. The opportunity to use
mathematical terminology in justifying decisions to others was also useful, and
helped pupils develop their ability to use such terminology correctly.
In general, pupils found it surprisingly difficult to find the area of a right angled
triangle, unless it was drawn on grid paper. Their working suggests that the difficulty
they had was with constructing the corresponding rectangle, as they struggled to
draw the lines which would fulfil the conditions for a rectangle. Once they had done
this, they were able to identify the base and height of the triangle, and use it to
calculate the area of the triangle fairly easily. In future teachings of this subtopic, it
would be useful to begin with triangles drawn on grid paper, and only introduce
triangles drawn on plain paper as an extension activity.
Reflecting on the sequence as a whole, it could be said that an improvement would
be to dedicate a slightly longer period to this topic. This would allow pupils to
consolidate their learning and would likely lead to better outcomes, particularly in
areas which require some practice (such as measuring angles). In its current form,
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Feb 2015
some parts of the lessons felt a little rushed and did not allow pupils time to reflect on
what they had learnt, meaning that they could not recall it later. Generally, however,
the objectives were met and the suggestions made in the literature did help pupils
overcome common difficulties and misconceptions.
Throughout the lesson sequence, a number of different types of activities (adapted
from researchers’ suggestions) were presented to pupils: they had access to
technology; they manipulated physical objects, they worked individually, and in pairs;
they took part in whole class discussions; and they justified their reasoning to their
peers. Of the five hours of teaching on this topic, pupils worked from a worksheet for
less than an hour. Generally, this class has a reputation for being fairly loud and
difficult, and their class teacher generally manages this through setting textbook or
worksheet based work for them to do individually and quietly. Due to this, there was
initially some concern about how they would react to the less structured, more active
lessons presented in this sequence. However, it became clear early in the sequence
that this concern had been unnecessary: the variety of tasks motivated the class to
stay focussed, and engaged with their learning, as evidenced by their eagerness to
participate in discussion, and the generally high quality of both classwork and
homework they produced. This is something that will be taken forward and applied in
the planning of future lesson sequences.
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References
Atebe, H.U. & Schafer, M. (2008) '“As soon as the four sides are all equal, then the angles must be
90° each”. Children's misconceptions in geometry', African Journal of Research in Mathematics,
Science and Technology Education, 12(2), pp. 47-65.
Bowers, J.S. & Stephens, B. (2011) 'Using technology to explore mathematical relationships: a
framework for orienting mathematics courses for prospective teachers',Journal of Mathematics
Teacher Education, 14(4), pp. 285-304.
Burger, W.F. & Shaughnessy, J.M. (1986) 'Characterizing the van Hiele levels of development in
geometry', Journal for Research in Mathematics Education, 17(1), pp. 31-48.
Clements, D.H., Battista, M.T., Sarama, J., Swaminathan, S. (1996) 'Development of turn and turn
measurement concepts in a computer-based instructional unit',Educational Studies in
Mathematics, 30(4), pp. 313-337.
Clements D.H. & Burns B.A. (2000) 'Students' development of strategies for turn and angle
measure', Educational Studies in Mathematics, 41(1), pp. 31-45.
Crowley, M.L. (1987) 'The van Hiele Model of the Development of Geometric Thought ', in Lindquist,
M.M. (ed.) Learning and Teaching Geometry, K-12: 1987 Yearbook.Reston, VA: National Council of
Teachers of Mathematics, pp. 1-15.
Fyhn, A.B (2008) 'A climbing class’ reinvention of angles', Educational Studies in Mathematics, 67(1),
pp. 19-35.
Mitchelmore, M. & White, P. (1998) 'Development of angle concepts: a framework for
research', Mathematics Education Research Journal, 10(3), pp. 4-27.
Oberdorf, C.D. & Taylor-Cox, J. (1999) 'Shape up! Uncovering the roots of misconceptions', Early
Childhood Corner, 5(6), pp. 340-345.
Ozerem, A. (2012) 'Misconceptions in geometry and suggested solutions for seventh grade
Students', Procedia - Social and Behavioral Sciences, 55(), pp. 720-729.
Sdrolias, K.A. & Trandafillidis, T.A. (2008) 'The transition to secondary school geometry: can there be
a “chain of school mathematics”?', Educational Studies in Mathematics, 67(2), pp. 159-169.
Sears R. & Chavez O. (2014) 'Opportunities to engage with proof: the nature of proof tasks in two
geometry textbooks and its influence on enacted lessons', ZDM, 46(5), pp. 767-780.
Tait-McCutcheon, S., Drake, M. & Sherley, B. (2011) 'From direct instruction to active construction:
teaching and learning basic facts', Mathematics Education Research Journal, 23(3), pp. 321-345.
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Feb 2015
Towers, J. (2002) 'Blocking the growth of mathematical understanding: a challenge for
teaching', Mathematics Education Research Journal, 14(2), pp. 121-132.
Yelland N.J. & Masters J.E. (1997) 'Learning mathematics with technology: young children’s
understanding of paths and measurement', Mathematics Education Research Journal, 9(1), pp. 8399.
Appendix a – post-test (pre-test consists of q1, 3, 5 only)
1 What is an angle?
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2 Decide whether each angle is acute, obtuse, reflex, 90˚
or 180˚. Hence estimate the size of each angle
3 Identify 5 places/objects where you might see
angles
4 Find the missing angle in each triangle
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
5 Mark the smallest angle with an ‘S’ and the
largest angle with an ‘L’
6 When you add all the angles in a triangle, what
value would you expect to get? Are there any
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
cases where you might get a different answer?
Why?
7 Find the area of these triangles
8 Measure the following angles, using a protractor
l
Appendix b - transcription of focus group
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
Me
P1
P2
P3
Me
P4
Me
P4
Me
P4
Me
P1
P5
P2
Me
P5
Me
P5
P2
P1
P5
P4
Me
P2
Me
P4
Me
P3
P1
P4
P1
P3
Me
P5
Me
P1
P5
Me
P5
P2
P5
P2
Me
Many voices
Me
P5
P2
P5
P4
Feb 2015
So these were two of the answers given to the question, ‘which of these angles is the
smallest and which is the largest?’
Do you two want to some over here so you can see clearly?
Yes that’s obviously yes
I think that one
Yes that one
So which do you think is the smallest angle?
That one’s mine!
Is it?
Yes
Do you agree with what you’ve written there?
Yes
Is he right? Is that the smallest angle?
Yes. No!
That’s the largest, isn’t it?
Yes, because it’s reflex
That’s what I’m wondering. How would you know?
On a protractor, if it goes that way [left] isn’t it over ninety degrees?
Is that how it works?
Yeah
Because you’re measuring it on a point, and it would be like whoop [large arc]
That’s wrong
No, it is the largest
Oh that isn’t my work, it’s too neat
Okay, we’re talking about measuring angles. What is an angle?
An angle is a thing you measure with a protractor. An angle is how open the lines are
Okay. So which one is more open then? This one, or this one?
That one
And which is the least open one?
This one!
This one [same as P3]
No it isn’t. It’s open but that one’s open less
Oh yeah, that one
Yes, that one’s too wide
What kind of angle is that one?
Right angle
Right angle. What kind of angle is that one?
Obtuse
Yes, obtuse.
Which is bigger? Obtuse or right angle?
Obtuse
Right angle
Right angle is 90 degrees and obtuse is above 90 degrees
Oh okay then, obtuse is bigger then
Does that make sense?
Yes
So you’re saying that this is the smallest one, and this is the largest one? Is that right?
No that’s the smallest
It is right
No that’s the largest because it’s over 180 degrees because it’s more than a straight line
[reference to an acute angle in quadrant four position]
That’s not right, P5. If it was reflex, then that means the thing would be on the outside. So
that means that one’s the smallest because it’s on the inside. If it was reflex it would be on
the outside and then it would be the biggest
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P5
Me
Oh I forgot about that thing, I get it now. So it is right then?
Yes. That was a good explanation, thank you P4.
P2/P5/P3
P5
P1/P3
P1
Me
P2
P1
P2
P1
Me
P1
P5
Me
P5
Me
Many voices
P3
Me
P1
Right, another question! What kind of triangle is this?
Equilateral!
Because it has all the same
Yeah
All the angles and sides
Okay. What kind of triangle is that one?
Oh that’s my one! It’s equilateral
Isn’t it isosceles?
No it’s equilateral because all of the sides are the same
Oh yeah! And the angles are the same
So what’s the definition of an equilateral triangle?
And angle [sic] that has all the same sides
And all the same angles
Lovely. No what kind of triangle is this one?
That’s my one! It’s isosceles. It says it there!
It does say isosceles, but do you agree with that?
Yes
Because it has two different angles
Okay. What is the third angle?
Oh it isn’t isosceles it has a right angle
Me
P5
Me
P2
P1
Me
P3
P4
P5
P2
P5
P1
P4
Me
Many voices
P3
P1
Me
P1
P2
P3
P4
P5
P1
Me
P1
Feb 2015
But two angles are the same, and two lengths are the same.
I agree that it’s isosceles, because, like P1 just said, two angles are the same size and two
sides are the same length. But what about this angle? What does that tell us?
It’s a right angled triangle
But it’s also an isosceles triangle?
Oh it’s a right angled isosceles triangle!
That was easy
Okay let’s look at one more. What kind of triangle is that?
Scalene
Yes, scalene
It has a right angle
So it isn’t scalene then
It’s a right angled triangle and a scalene triangle. So it’s a right angled scalene triangle?
No it’s the same as the other one
It isn’t the same, because look [points at angles] 40, 50, 90. They aren’t the same. It is kind
of the same. But it isn’t because that one has two same angles
So the first one was isosceles. Is this one isosceles?
No/Scalene!
Right angled scalene triangle
Yes
Okay, great.
Someone in your class said to me the other day that, ‘any triangle, however big or wide it is,
however long the sides are, the angles add up to 180’
That’s wrong
No it’s correct
Yes it’s correct
It is wrong
If it isn’t 180 then it won’t be a triangle. Unless it’s irregular.
Actually that is right
What’s an irregular triangle?
It can’t be any angle. Not absolutely any angle.
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Angles and Triangles: Identifying and Overcoming Common Misconceptions
Me
P1
P4
Me
P4
P3
Me
Many voices
Me
P4
P5
Me
P5
Me
P5
P2
Me
Many voices
P1
P5
P4
Me
P1
P2
Me
P5
Me
P2
P3
P5
Me
P5
Me
P4
P5
Me
Feb 2015
Why can’t it be any angle?
It can’t be 180 plus 180 plus 180. So it can’t be any size
And it can’t be 90, 90, 90 either
Yes those are valid points. But if I drew a triangle here. And this angle was 110, this one was
40. Can you automatically tell be what the third angle must be?
20. No 30!
Yes 30!
Do you all agree that angle’s 30 degrees?
yes
So if this side was 100 and this side was 200 and this side was 13. Would you still agree that
the missing angle was 30?
Does it matter or does it not matter what length the sides are?
It doesn’t matter
But it’s already over 180. So it’s wrong!
If I give you this triangle here, and I tell you that this side is 17cm and this side is 15cm and
this side is 14 cm. And that this angle is 40 degrees, and this angle is 6o, can you tell me
what angle that is?
It’s different to the other two angles
How big is it?
80 degrees
Yes, 80 degrees
So you can tell me what it is, right?
Yes!
Why would it be 80?
Because all together it’s 180. The other two are 100 so 80 is left
100 plus 80 is 180
And what do angles in a triangle add up to?
Oh yeah, 180!
180!
Is that true for all triangles?
What about an irregular triangle?
Can you draw me an irregular triangle please?
I can! It’s like an irregular polygon [draws a shape with curved edges]
That can’t be a triangle, it doesn’t have three corners
Miss, can I draw one?
Sure, go ahead
Like this [draws a quadrilateral, like an arrow head]
What is the definition of a triangle?
It has to have three corners. That has four!
It has to have three corners and three sides. This doesn’t!
Right, so it isn’t a triangle.
I see where you’re coming from though, and it’s a good idea. Because that shape isn’t a
triangle, its angles won’t add up to 180.
Thank you guys, this conversation has been very useful. Let’s get back to class.
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