Before I came to study a Post Graduate Certificate (PGCE) Mathematics course at University College London Institute of Education (UCL IOE), I had been working as an Academic Tutor at a behavioural centre, linked to a mainstream secondary school for the past 7 months. Students placed here had either learning difficulties or behaviour issues experienced in the classrooms. During my time here, it became apparent that the challenges I faced in not just the planning and teaching lessons, marking work and dealing with behaviour but trying to articulate an explanation to the students’ questions on why topics like geometry and algebra serves a purpose and relevance in their day-to-day lives and future endeavours. I felt at the time the mathematics curriculum that I taught to 11-16 year olds should be of some use to their learning. I recall one of my students who wanted to pursue a career in construction, claimed he had no need of maths for his job. Mathematics lessons for him placed too much emphasis on topics such as algebra and trigonometry which were of no interest, and this meant that even when they were learning relevant topics such as fractions and ratios he was still unwilling to work/‘keep the’ interest. Having said that, there should be a place for more advanced mathematical learning. Students with an aptitude for mathematics should not be held back merely because others have no interest in it, but have a whole new field of application open up for more advanced concepts. Einstein once quoted ‘Teaching should be such that what is offered is perceived as a valuable gift and not as a hard duty’. Mathematics lessons should be equipping students with useful skills that they can take into later life, not putting them off with topics only of interest to the mathematically minded. I was keen to discover and develop the skills and knowledge of how I could achieve this better in the classrooms hence my motivation to embark on a PGCE course at UCL IOE.
Although I consider that a teacher’s mathematical knowledge is an important ingredient for teaching, and while a teacher needs to be able to do the mathematics required for the curricular level being taught, I have started to critically evaluate that this may not be sufficient to ensure pupil progress and stimulate their interest. But to consider the pedagogy required to successfully teach it: What is mathematics education? What is good (maths) teaching? Does mathematics education serve a different purpose to different people?
On one our first Subject Studies session on the PGCE course, Welcome to PGCE Maths (UCL IOE, 2018), we were introduced to fellow student teachers by being numbered from 1-30, and via series of exercises we partnered up with our corresponding thoughts. The exercises helped us realise the beginnings of misunderstandings that can occur amongst students in a mathematics lesson. For example, in one of the exercises we were asked to Introduce yourself to your double, does one pair up with their double 10 or one pair up with their mirror number 5? Such choice of language as Skemp (1976) illustrates can create a faux amis situation, in which words which sound the same, or very alike, but whose meanings are different. This in effect can cause inconvenient mistakes that is considered possibly one of the reasons for the difficulties in mathematics education today. A thought I thought lesser of when considering a career in teaching. It made me appreciate the challenges that teachers face daily, not only the choice of language and words when teaching the Mathematics Curriculum but also the pedagogy to successfully teach it.
On another session, we were introduced to polygons with a circle number placed at each vertex and a square number in between so that each square number is the sum of the two adjacent circle numbers, known as an Arithmagon problem (UCL IOE, 2018) – Figure 1. Although finding the missing square numbers in a triangular arithmagon when the circle numbers are given is relatively straightforward (involving addition). Finding the circle numbers however, when the square numbers are given will requires a number of logical reasoning before we can derive answers.
The exercise leads us to consider how we might approach a response to the individual or to the whole to the class? How as teachers do we manage the process of teaching for learning?
Figure 1 Figure 2
Although the problem is equivalent to solving three simultaneous linear equations in three unknowns (Figure 2), it is not necessary for students to use algebra to make sense of the structure. Given the opportunity, this exercise can provide an interesting context for students to make predictions and conjectures to develop their powers of mathematical reasoning. Traditionally, teachers would have approached the problem in a prescriptive method, where the methods are explained to learners one step at a time, possibly via a memorised rule: mnemonics or songs. In this example, the teacher could have introduced a memorised rule such as, orally saying, “Add the two near (clenching the fists and brought together), subtract the (square) number opposite (arms go apart) and divide by two (halving action). This approach as Swan (2015, p. 3) argues is “mechanistic… imitating a routine or procedure without any depth of thought” or as “rules without reasons” (Skemp, 1976, p. 2), where learners would have derived at the answer via teacher’s prescriptive method rather than at their own level of understanding. This potentially risks a mathematics lesson becoming monotonous and tiresome for learners and in turn made me re-evaluate what constitute a good maths teacher, not just assessing our performance, the children’s behaviour, the subject content of the lesson or what the intended learning is, but also being able to strike the balance for managing both proscriptive method, where learners are challenged and arrive at understanding through discussion and prescriptive method, where learners watch, listen and imitate until fluency is attained, without ruining what could be considered a rich problem for solving activities like the Arithmagon activity .
that is to have an ability to explain a topic in a number of different ways. Each student has a particular preferred method for learning, even if they are not aware of it. Skemp (1976) identifies learning theory points to different types of understanding – relational and instrumental. Instrumental understanding is more algorithmic; the pupil will learn a sequence of steps for solving a particular problem, and will simply apply this to any similar problem, ending up with the right answer, but with little or no understanding of how or why the method works. Relational understanding, on the other hand, refers to a much wider level of comprehension; the pupil can see why a method works, has some grasp of the mechanics behind the steps and can apply their wider knowledge to related problems which a purely instrumental knowledge would be inadequate for. At GCSE level and below, there is little advantage to the pupil in gaining a relational understanding rather than an instrumental one in terms of exam results, and so a teacher should be aware of these two approaches and make sure that the way they teach allows for both instrumental learning and relational understanding. In practice this usually manifests itself as solving a series of example problems by a given sequence of steps, but also explaining the reasoning behind each at each stage.
This argument is further supported by Hiebert’s (1999) findings demonstrates that once students have memorised and practiced procedures that they do not understand, they tend to have less motivation to understand the meaning or the reasoning behind them, which I can resonate from my own experience of teaching and learning mathematics . At one of the lessons I observed during the first school placement, the session was revising on dividing fractions, where learners recited a predetermined rule, keep flip change (KFC) method for dividing fractions. When I probed them why they multiplied the reciprocal of second fraction to the first fraction, they responded, “It’s what we were taught” This made me reminisce my own learning of mathematics . I did not question my teacher at the time why it was the case and how it worked, but ‘accepted’ that it was a method to divide fractions without ploughing deeper how it worked. When I followed up with the teacher later, they responded “The learners had worked on this on another session, but when they need to recall for exam or generally as means for quick calculation, it is the easiest go-to method to calculate the answer” and while the teacher who taught them could have approached a questioning style that checks their understanding or act as a guide to support and stimulate them, it seems apparent time constraints of the classroom learning and the Curriculum demands means that there is not always the opportunity for learners to “actively explore, discuss, debate and gradually come to understand” (Swan, 2015, p. 3).
On one of the course’s literature reading, we were introduced to the works of Shratz (2006) where he proposed that “every new innovation or initiative, teachers go through a sequence of changes characterised by four states of awareness from unconscious incompetence competence where a teacher is unaware of the limitations of their knowledge” striving towards to unconscious competence where a teacher has internalised new competencies. It highlights that as a teacher develops professionally that more sophisticated levels of reflection are attained. Although I consider it is important to have the subject knowledge in order to teach the subject, but that it is also closely linked to pedagogical knowledge in order to teach it successfully. For example, having an awareness of common misconceptions and by way of looking at them, we can forge links and connections between mathematical ideas and have the flexibility that comes from seeing alternative ways of looking at the same idea or problem are essential for effective teaching.
As we explored the development of the mathematics curriculum over the past 35 years, it is prevalent on a number of reports identified that there is a need to renew the focus on approach to problem solving. Gove (2013) outlines that reform of these key subjects (English and maths) was, ‘a matter of pressing urgency’: “The new mathematics GCSE will demand deeper and broader mathematical understanding. It will provide all students with greater coverage of key areas such as ratio, proportion and rates of change and require them to apply their knowledge and reasoning to provide clear mathematical arguments. It will focus on ensuring that every student master the fundamental mathematics that is required for further education and future careers . It will provide greater challenge for the most able students by thoroughly testing their understanding of the mathematical knowledge needed for higher-level study and careers in mathematics, the sciences and computing. …” Prompting exam boards like OCR (2015) to revise their exam questions “There will be an increased emphasis on problem solving, often requiring multi-step solutions (not prescribed in the paper) with less emphasis on rote learning .”
Previously, questions like shown in Figure 3 were not perceived as complex. There might have been step-by-step instructions and in two parts:
(a) i. Write an expression for the perimeter of the triangle (1)
ii. Fully simplify the expression (1)
(b) Find an expression for the length of the side of the square …
In order to stimulate mathematical thinking that can arise from engagement in open problems and investigations, Foster (1999) values that “Good questioning techniques” are a “fundamental tool of effective teachers. I witnessed this in a probability lesson where I observed the teacher encouraging leaners to discover different ways of describing the number of red balls contained in a bag that was mixed with other coloured balls, through a series of questioning to find their own reasoning. This allowed learners to recollect their knowledge of conversion between percentages (50%), decimal (0.5) and fractions ( 1/2 ). Students participated enthusiastically but what struck me is the teacher’s capability given the choice of language to encourage learners to describe by choice of words rather than seeking numbers . Additionally, the teacher ensured the learning of students were present by having classroom management strategies in place where teacher always reminded the class/individuals of the school’s behaviour policy and effectively in control of their behaviour to ensure minimal disruption to the class, simultaneously using a number of praises to uphold the interest, which I admire in my own review of what sort of teacher I want to come when it comes to teaching.
Although Foster (1999) identified that 93% of teacher questions are “lower order” knowledge-based questions focusing on recall of facts (Daines, 1986).” One teacher could argue that it is in their best intention to provide quality education for learners. I found reading Van Oers’ (2014) on ‘scaffolding in mathematical education’ is closely linked to the works of Bloom (1994) – see figure 4. Where scaffolding represents an intentional relationship between the teacher and the student, where the teacher aims to provide support to the learner in such a way the learn can use their existing expertise but is supposed with elements not yet mastered independently. Which in the above observation the teacher has done to stimulate the mathematical thinking of learners to stimulate their learnin g.
Although, Way (2008) pointed out that “Teachers’ instincts often tell them that they should use investigational mathematics more often in their teaching but are sometimes disappointed with the outcomes when they try it.” In my brief teaching experience, I have already faced a number of whole-group scenarios where I completed an activity or explanation and asked “Show of hands who understands this? Keep your hands up if you can convince the class how you would do this” Only to find a couple of children kept their hands up. Possibly, there are two reasons for this : one is that the children are inexperienced in this approach and find it difficult to accept responsibility for the decision-making required and need a lot of practise to develop organised or systematic approaches. The other reason is that the teachers have yet to develop a questioning style that guides, supports and stimulates the children without removing the responsibility for problem-solving process from the children .
On another occasion, a number of student teachers and I had the opportunity to conduct a mathematical problem to two classes of students (one class being in year 8 and on another session, class being in year 7), where each class was split into groups of 6-8 students. Working in pairs, we would spend about 20 minutes with one group and carrousel around to another group to work on the same problem.
During the first session, we introduced The Prime Numbers activity (UCL IOE, 2018, p. 2) – see below, to year 8 students and observed how they responded.
Students noticed the age gap between each child in the example was two years, I then observed some students writing down all the possible ages that the youngest to the oldest could be over the next 20 years. At the time I did not interfere to question whether even numbers are prime numbers. Observing others as they recollected what a prime number is and agreed that only even prime number was two and proceeded to write down only the odd integers whether or not they were prime. I was keen to scaffold the students’ mathematical thinking based on the four main categories (Badham, 1994) identified by a serious of
1. Starter questions – taking the form of open-ended questions, which focus the students’ thinking in a general direction and give them a starting point. Example: How could you sort these…?
2. Questions to stimulate mathematical thinking – to assist students to focus on particular strategies and help them to see patterns and relationships. This aids the formation of developing a conceptual thinking. The questions can serve as a prompt when children become ‘stuck’. Example: Can you group these … in some way?
3. Assessment questions – encourage students to explain what they are doing or how they arrived at a solution. Allowing me to see how the children are thinking, what they understand and what level they are operating at. Example: How did you find that out?
4. Final discussion questions – these questions draw together the efforts of the group and prompt sharing and comparison of strategies and solutions. This is considered a vital phase in the mathematical thinking processes, providing further opportunity for reflection and realisation of mathematical ideas and relationships and encourages children to evaluate their work.
Examples: Do you think we have found the best solution ?
I found during the year 7s sessions, they preferred to work as a group compared to the year 8s mainly worked pairs. This is probably because they were still familiarising themselves with what a prime number was. On both sessions, I found that communicating with both groups, the children was fond of engaging activities where they were allowed to discuss and share their
Despite these ideas, reading Watson’s (2008, p. 8) article, I am still assessing whether I be ‘ready’ to teach good mathematics equally be a good maths teacher, and vice versa?, “It would be possible to become qualified and never have to think through a single aspect of school mathematics from scratch”, and “to achieve and demonstrate every guideline, every list item, every criterion of every whole-school initiative and still not really work hard on how to help your students understand mathematics better than their existing expectations, and better than the school’s grade predictions .”
Though participants have echoed Cockcroft (1982, para 242) in recognising that it is not possible to define a single ‘best practice’ in mathematics teaching. There are many different types of learning, and a wide range of teaching methods will need to be deployed, appropriate to the learners and the particular learning outcomes desired.
Bloom, B., 1994. Bloom’s Taxonomy in Mathematics Education. [Online]
Available at: https://moodle-1819.ucl.ac.uk/pluginfile.php/718700/mod_book/chapter/11277/Blooms%20Taxonomy%20in%20Mathematics%20Education.pdf
Cockcroft, W., n.d. Mathematics Counts, London: Her Majesty’s Stationery Office.
Foster, C., 2014. www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Arithmagons.pdf. Arithmagons, p. 57–59.
French, D., n.d. Subject Knowledge and Pedagogical Knowledge. [Online]
Available at: www.maths.manchester.ac.uk/~avb/pdf/DougFrenchSubjectKnowledge.pdf
Skemp, R., 1976. Relational understanding and instrumental understanding. Mathematics Teaching, 77, pp. 20-26..
Swan, M., 2015. Improving learning in mathematics: challenges and strategies. [Online]
Available at: www.ncetm.org.uk/public/files/224/improving_learning_in_mathematicsi.pdf
UCL IOE, 2018. Ten Problems to Introduce. [Online]
Available at: https://moodle-1819.ucl.ac.uk/pluginfile.php/718695/mod_folder/content/0/Ten%20problems%20to%20introduce.docx
UCL IOE, 2018. Welcome to PGCE Maths. [Online]
Available at: https://moodle-1819.ucl.ac.uk/mod/folder/view.php?id=533437
UCL IOE, 2018. What is School Mathematics?. [Online]
Available at: https://moodle-1819.ucl.ac.uk/mod/folder/view.php?id=533438
Van Oers, B., 2014. Scaffolding in Mathematics Education. In: Lerman S. (eds) Encyclopedia of Mathematics Education. , s.l.: Springer, Dordrecht.
Watson, A., 2008. Fragments and coherence: Anne Watson’s opening address to the joint Easter conference. Mathematics Teaching 210, pp. 8-11.
Way, J., 2008. Using Questioning to Stimulate Mathematical Thinking. Australian Primary Mathematics Classroom, 13(3), pp. 22-27.
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