Promoting pupils’ conceptual learning is one overarching goal that teachers might have in order to help pupils understand mathematics in a way that goes beyond just following a technique to reach an answer. The objective to support pupils’ conceptual mathematics learning can be accomplished through evidence-based practice where teachers learn with and from each other. There is strong research underpinning teacher learning and reflection that is focused on the classroom (MacGilchrist et al., 2004). In my experience of teaching mathematics, I have observed that despite pupils having sufficient awareness of angles in their daily spatial relationships, the formal concept of angles is a severe barrier for them.
Why are angles difficult?
The concept of angles has several facets and can be difficult for pupils to understand (Crompton, 2015). I believe that one of the main issues stems from a lack of understanding of the dynamic character of angles. Pupils experience angles in dynamic contexts in everyday life, but in mathematics classrooms, they are still taught about angles in static form (i.e. as immovable models). According to research, teaching solely in the static domain makes it difficult for pupils to grasp the idea and encourages them to develop unfavourable alternate conceptions that they frequently take over to more advanced stages of learning. (Mitchelmore, 1998; Burns and Clements, 2000).
The concept of angles as measures of turn is important to how pupils view angles. This needs to be taught explicitly so that pupils understand angle measurements and rotation in both domains. This is reinforced by the work of Mitchelmore (1998) who found that when pupils’ knowledge of angles is limited to within the static domain, they do not develop a deeper, conceptual understanding. For example, pupils often struggle to translate their knowledge of real-life situations such as the movement of a door; a pair of scissors; a swing or the hands of a clock, to their learning of basic angle facts in school. They are not able to recognise that the angles in these dynamic settings as essentially the same as what they have learnt in the classroom.
Aside from problems with the dynamic character of angles, I have also identified two other areas where pupils struggle in the learning of angles: the ability to conceive of angle rotation as a continuous, measurable quantity, and the need for a support mechanism or schema to solve multi-layered angle problems.
Pupils’ introduction to angles
Understanding complex problems involving angles requires pupils to have a solid understanding of the many facets of the angle idea. In my experience, I have found that when pupils first encounter angles in secondary school, the teaching of angles does not always emphasise the dynamic element i.e. seeing angles as a measure of a turn or the turn being the relationship between two lines. Hence, pupils tend to struggle with the idea of angles in relation to movement and the interpretation of this movement.
Unfortunately, the opportunity for pupils to identify angles in various physical contexts is rarely explored. Consider, for example, why children do not easily connect the time on a clock to angle types [e.g. 3 o’clock as a representation of a right angle]. This disconnect is reinforced by responses to the survey for this research. A sample of 25 pupils from a Year 9 class (14 year olds) were asked to pick out the odd one out from five groups of four words and numbers from a list that are associated with angles. Below is an example from the survey.
- 50 degrees; b) a swing; c) left angle; d) protractor
The rationale was to ascertain if pupils could associate dynamic objects and movement (swing) to angles and thus be able to correctly pick out the odd item (left angle). In this anonymous survey conducted at the start of the project, pupils were also asked to give their definitions of an angle. Interestingly, only a few pupils attempted a definition or provided one that was close to accurate. Most of their definitions mentioned “measurement”, “degree” and “size” but omitted important words like “rotation” and “turn”. Mitchelmore (1998) argues that the gap between the understanding of angles as both dynamic and static has given rise to misconceptions around the definition of angles. It was interesting to note that the majority of pupils in the survey were also unable to make a link between dynamic objects (a pair of scissors, for example) and their learning of angles, showing a disconnected knowledge base between school and everyday experiences.
Identifying the problem
In my context, a large mixed comprehensive school serving a significantly deprived area in the West Midlands, there was general agreement among teaching staff that angles as a topic area in the maths curriculum was challenging for the majority of our pupils. The decision to focus on angles was due partially to an overwhelmingly poor performance in multi-step angles questions from a summative test. Following from this, an analysis of data using test scores from this specific cohort of pupils was undertaken. The data looked at the number of pupils who attempted two key angles questions from the test and the number of marks scored for each question. In one of the questions, only 34 pupils out of 142 who took this test scored any marks, representing less than a quarter of the total number of pupils and by all accounts, a low score rate. As teachers, we also collectively drew from our own various experiences of teaching angles to bolster our position to focus on this area. This is in alignment with MacGilchrist et al.’s (2004) assertion that teachers bring invaluable knowledge and experience to a process of change.
Given that the motivation and objective was to improve outcomes for pupils by changing the way we teach angles, a mastery approach was adopted to address this issue. The concept of ‘teaching for mastery’ refers to the practices used in classrooms to increase pupils’ chances of deeper understanding of mathematics (NCETM, 2022). In practice, the idea is to explore a topic at greater depth such that learning is firmly understood and embedded. With support from the NCETM Maths hub, a national network that is coordinated by the NCETM to support excellent practice, an evaluation of research studies on angle pedagogy was conducted.
This knowledge was shared and discussed across the faculty. The scheme of learning was also scrutinised to find where changes could be implemented to reflect a different way of teaching that was more effective. The intention from the outset Iwas to refine our practice by enacting the following agreed approaches:
- Explicit teaching of correct definition using Frayer modelling
- Discuss real-life examples of angles including their dynamic nature
- Demonstrate angle rotation and measurement
- Introduction of goal-free questions to solve more complex problems.
In the revamped approach, the Frayer Model (mentioned above) was used to introduce the concept of angles, giving a clear definition based on the idea of an angle as a measure of turn. The Frayer Model is a graphic organiser originally developed for vocabulary learning but has been adapted and is now increasingly used in maths teaching and other disciplines. It is divided into four distinct headings, namely: definition; characteristics; examples and non-examples. We used this template to set out the main features of angles, and allowed pupils to explain why the features they identified were examples or non-examples. Examples offer an instance of resemblance, whereas non-examples offer contrast. A typical non-example would be two straight lines that do not intersect to form an angle, for example.
This activity sought to help pupils avoid misconceptions by correctly identifying angles based on an angle’s definition and characteristics. Discussions of real-life application of angles were also incorporated in lessons to make learning meaningful and engaging. Pupils were encouraged to research the different angles they encountered in their daily experiences both in static and dynamic domains. Angle rotation was taught to show the changes in angle size and how this movement or turn relates to two intersecting lines.
These changes led to a redesign of schemes of learning to incorporate the agreed themes, as well as supporting colleagues to implement them in their lessons. After agreed timelines, continuing professional development (CPD) was delivered using a focus group of students to reflect on how this work was influencing pupil progress and how to improve on this for a larger cohort of pupils.
Research studies found that when students employ dynamic actions for angles, they create schemas that apply in other circumstances. Because of this, they develop a deeper understanding that encourages the use of more precise angle measurements and links between ideas (Clements and Burns, 2000). Just as pupils may use body movements and physical objects to describe and understand angles, goal-free questions were recast as schema for our pupils to develop their ability to solve more complex problems involving angles. In Craig Barton’s (2018) book, How I wish I’d Taught Maths, he highlights the benefits of using goal-free questions to focus pupils’ thinking (Barton, 2018). Goal-free questions serve as schema to help pupils approach angle problems in a manner that is structured and avoids ‘attempting to juggle all the possible sub-steps’ towards the goal or final answer. (Barton, 2018, p.162).
A recent study actually found that an undue focus on the answer can be reductive (Johnson et al., 2022). This usually occurs when teachers place more value on procedural fluency than the key mathematical concept in the learning. In the goal-free scenario, the teacher can choose to conceal the original question and design one of their own that encourages pupils to attempt the aspects they know in small steps. Goal-free questions also have cognitive benefits as they help pupils avoid cognitive overload by de-emphasising the goal. This allows for some flexibility and takes away some of the pressure that pupils often encounter when faced with multi-step questions, therefore freeing up working memory to complete the sub-steps that lead towards the final answer (Barton, 2018).
Although this is ongoing work, in terms of impact, pupils have described their learning of angles using themes suggested in this research in the following words: “interesting”; ”useful”; ”challenging”; ”good”; ”alright”, suggesting a positive shift in our practice. There is also evidence from pupils’ classwork that goal-free questions are being tackled and the work is reflecting a good understanding of the concepts taught. In the meantime, teachers report being more confident in their teaching of angles using the agreed themes, and are very positive about using goal-free’ questions in their lessons.
It is crucial that pupils have the opportunity to develop a meaningful idea of angles through connecting formal learning with applied domains. The majority of angles in practical circumstances are dynamic, which also makes studying them more interesting and engaging. This research offers a pedagogic strategy that is practical, relevant and effective in the teaching of angles. The findings presented have key implications for teachers to harness and develop their collective professional learning to promote pupils’ conceptual learning in mathematics.