Stage One illuminated for me the severity of the problem for those pupils disaffected with learning algebra in secondary mathematics classrooms, discussed by Ruthven et al (2011);
see Section 1.3.2. I initiated a formal introductory meeting with both teachers in the school’s staffroom. Neither had seen nor used Grid Algebra software. Its withdrawal from the school’s system suggested that the resource may have been considered inappropriate for learning algebra. The Mathematics department had relinquished a computer suite within three years of use with a previous Year 8 group (see Section 1.4.1). This development suggested under-use of the facilities. However, the software was located in the departmental cupboard and re- loaded onto computers prior to this study. Teachers 1E and 2E described participating pupils as “cooperative, ready to share, willing to step up and try, working well together”. I attended all mathematics lessons for the class over 12 weeks punctuated by: a half-term break, a HOD- Science interview, two Bank Holidays, a national teachers’ strike, and end-year
examinations. I will discuss in Section 4.1 my use of written work, observations, discussions and focused interviews with participants. I witnessed apparent undermined accessibility of algebra I outlined in Sections 1.2.2 and 3.1.3. Pupils had very limited opportunities to value collaborative learning and experience serious consideration for their thinking (Ruthven et al, 2011).
The lack of universal consensus on what learning activity is considered ‘algebraic’ (Hewitt, 2011; Van Amerom, 2003) caught my attention based on strategic decisions of pupils solving word problems. I was puzzled by specialist teachers disputing the relevance of Grid Algebra
in algebra; it stimulated my reading more about approaches to school algebra. Pupils seemed willing to discover new ways of working while teachers hesitated. Minimal ICT use seemed aligned to teacher-directed instruction; teachers reinforced individual pupil learning. Pupils’ silence when required to offer immediate responses in whole-class discussions suggested they exercised agency to deflect any labelling and judgment of their learning abilities. A change in learning strategy had revealed startling insecurity of several pupils’ number sense. Higher expectations of effort can motivate engagement in mathematics, as pupils demonstrated when presented with GCSE-level questions. Agency, affect and mathematical thinking took on new dimensions. I attributed the apparent disinterest in pupils’ kinaesthetic Grid Algebra use to stimulate learning-based talk to teachers accustomed to controlling pupil thoughts or actions in mathematics. Teacher hegemony can restrict positive learning experiences; conditions, opportunities and resources may render lessons dreary and isolating (Nardi and Steward, 2003). Pupils’ hatred of self-checking work indicated ‘learned helplessness’ (Dweck, 2000) or possibly low expectations and limitations imposed by curriculum or examination boards. Teachers have been known to set bars of attainment at the minimum. I have often heard a
teacher telling a pupil something to the effect that “Grade B in mathematics is brilliant! You only need to get C in GCSEs!” It is hardly surprising that some researchers (Brown et al, 2008; Nardi and Steward, 2003; Boaler et al, 2002) described secondary mathematics classrooms as landscapes riddled with quiet disaffection. Some researchers (Healy et al, 2001; Bednaz et al, 1996) argued that favoured curriculum options inform pupils’ strategic competence. Thus, diminished access to algebraic concepts can be attributed to pedagogic decisions to a greater extent than to pupils’ mathematical abilities or the nature of algebra. I propose further investigation into the defining of algebra as ‘generalised arithmetic’ in the UK National Curriculum; it seems many teachers construe the emphasis as evaluating numerical solutions and finding patterns (Hewitt, 1992). Bell (1996) argued that ‘generalising’ may direct pupils’ attention to procedural learning.
To a large extent, this ‘exploratory’ stage reaffirmed the heightened need to acknowledge that pupils actively construct knowledge despite being ‘taught’ by teachers. Bruner (1961) argued for teachers to inspire pupil learning rather than transmitting knowledge. The UK pilot study highlighted vital benefits to learning afforded by software increasing classroom interactions, namely peer collaboration and teacher intervention. It strengthened my conviction concerning schooling as an inherently social process overseen by the teaching fraternity that provides and manages variety for pupils in mathematics lessons. Combined ICT and non-ICT resources in classroom environments can be an enabling factor for formative assessment and ‘transfer’ of learning that may occur when pupils encounter ‘difficult’ concepts. Classroom contexts can encourage pupils to either be helpless or to learn mathematics with understanding. Positive learning behaviour reinforced my thinking that affective aspects can influence mathematical learning. Pupils can welcome appropriate and challenging targets; they engage and work well when encouraged, and take responsibility for their learning when provided with opportunities to act. Otherwise, pupils are less likely to ‘author’ high self-expectations; many see no need to increase effort in learning activity, hence low pupil participation. I contend that teachers need not burden pupils with their own fears and inadequacies; neither should teachers bow to external pressures to the detriment of positive learning experiences. These findings convinced me to amend my research design and questions to take into account the effect of using ICT on the teachers and on teaching in lessons. I reflected on my own role supporting teachers
making sense of ICT-enhanced pupils’ algebraic learning and developing TPACK as teacher ‘learning’. I set out to try this intervention in a different setting in the next stage as follows.