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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 Page 1 of 29 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 Page 2 of 29 1 2 3 3 3 4 5 6 6 7 7 8 9 9 9 10 10 11 12 12 13 14 15 18 19 20 23 24 24 27 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 Page 3 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 4 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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). Page 5 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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). Page 6 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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’ Page 7 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 (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. Page 8 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 9 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 10 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 11 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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). Page 12 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 13 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 14 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 15 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 16 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 17 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 ‘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 Page 18 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 19 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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 Page 20 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions 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, Page 21 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions 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. Page 22 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions Feb 2015 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. Page 23 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions 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? Page 24 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions 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 Page 25 of 29 Feb 2015 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 Page 26 of 29 Feb 2015 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 Page 27 of 29 Feb 2015 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 Page 28 of 29 Angles and Triangles: Identifying and Overcoming Common Misconceptions 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. Page 29 of 29 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. Page 30 of 29