Chapter 2: The Literature Review
2.2 Effective Mathematics teaching: What is it?
2.2.1 Effective Teaching: What makes the Difference? Difference?
‘Some teachers are better than others, and the difference is significant’
(Wiliam 2016: 29)
This enduring debate of whether great teachers are born or are made has led to over five decades of research (Wiliam 2016). Findings suggest that whilst
natural inclination (or talent) toward teaching may matter a great deal in the beginning, this influence becomes insignificant over time by the ‘effects of practice’ (Wiliam 2016: 242). Regardless of any natural talent or otherwise,
‘extensive deliberate practice is still needed to become highly accomplished in teaching’ (Berliner 2001: 465). Engaging in this type of ‘deliberate practice’ sets expert practitioners apart from the rest (Wiliam 2016: 4). ‘Deliberate practice’, according to Ericsson (Ericsson and Pool, 2016) a professor of psychology, involves stepping outside your comfort zone and developing capabilities beyond your current skill-base. Deliberate practice, therefore, can often be
uncomfortable and not particularly enjoyable – you have to, metaphorically, fall-down a lot of times. You have to move beyond the comfort zone to the learning zone (Colvin 2009). This can be distinguished from the more regular type of practice which simply involves repeating or doing more of something that has already been mastered. Ericsson’s research was the basis of the oft quoted Gladwell’s (2008) ‘10 000-hour rule’ but according to Ericsson just doing more
of the same is not enough, ‘deliberate practice’ is key. However simply being motivated to engage in deliberate practice is insufficient to engender change;
well-defined goals, strategies and direct instruction are also required to achieve expertise (Ericsson and Pool 2016), as too is reflection (Stobart 2014).
Interestingly, Ericsson makes the point that committing to this type of deliberate practice - to achieve expert status - is all consuming and may not be the best course of action for the majority; he suggests there is nothing inherently wrong with being average, ‘for much of what we do in life, it’s perfectly fine to reach a middling level of performance and just leave it like that… but there is one of very important thing to understand here: once you have reached this
satisfactory skill level and automated your performance…you have stopped improving’ (2016: 12).
Stigler and Hiebert (1999) also promote the investment in deliberate or
purposeful practice, and in the ‘development of effective methods for teaching’;
it is this, and not the ‘identification and recruitment of talented individuals’, that will result in long-term ‘improvement in teaching’ (Stigler and Hiebert 1999:
133).
Considering the distinction between teaching effectiveness and teacher
effectiveness, Darling-Hammond (2014) draws our attention to ‘teacher quality and teaching quality’ (2014: 7). Teacher quality might be thought of as the
‘bundle of personal traits, skills, and understanding an individual brings to teaching’ (2014: 7), including subject skills and knowledge along with a set of teacher beliefs, attitudes and behaviours, which help students succeed.
Teaching quality refers to ‘strong instruction that enables a wide range of
students to learn’ (Darling-Hammond 2014: 7); this type of teaching successfully meeting the needs of the subject curriculum and the needs of the students ‘in a
particular context’ (Darling-Hammond 2014: 7). Teaching quality can be seen to be part of teacher quality, in terms of it being what a teacher would do, but the overall quality of teaching is heavily influenced by the context in which it is cited and by factors ‘aside from what the teacher knows and can do’
(Darling-Hammond 2014: 7). In difficult circumstances or with limited resources, the quality of teaching will be undermined even for strong teachers. Conversely less effective teachers can be boosted by enhanced surroundings, support-systems and reputable resources. In short, strong ‘teacher quality may heighten the probability of effective teaching’, but with due consideration given to context ‘it does not guarantee it’ (Darling-Hammond 2014: 7).
Darling-Hammond (2014) conjures up the idea of an effective education system being built of interconnecting cogs, encompassing teacher quality, teaching quality, conducive learning environments and context - all co-operating in an intricate and indeterminate way. The complexities of separating these cogs, to disconnect the process (of teaching) from the people (the teacher) suggest that:
‘it is difficult, and perhaps impossible, to entirely disentangle teacher quality from teaching quality’ (Wiliam 2016: 30).
The context for teaching is clearly significant: Effective teachers in one setting are not necessarily so in another (Berliner 2001, Wiliam 2016,
Darling-Hammond 2014). The success enjoyed by Finnish teachers in Finland would not be easily emulated by Finnish teachers elsewhere, as the context would be compromised (Sahlberg 2011). This suggests the system, the context, the curriculum, the students, the support networks, the resources, the teacher and the teaching interact in such a way that enables successful learning. Burghes (2015) suggests our description of effective education should be based on learner-focused criteria, with definitions such as ‘effective teaching and learning
is that which motivates and engages the learner to progress mathematically’
(Burghes 2015); with processes that have ‘the power to evoke a mathematical response’ from the learner (Fletcher 1964: 1). And this indeed appears to be the direction of travel for Ofsted with a greater focus on pupil learning and outcomes, and a move away from grading teachers for individual lessons;
(GOV.UK 2018b: 12). An adjustment of the focus, to learning instead of teaching, provides the rational for explaining why a raft of different teaching approaches and styles can appear to function differently in different
circumstances. In Finland, for example, the cultural expectations may be
sufficient to ensure student engagement and motivation for most of the students for most of the time. In England, to overcome deep-seated and educationally-damaging attitudes towards learning mathematics (Smith 2017b), inspirational teaching may play a more significant role; the most successful teachers often described as those who ‘motivate and inspire’ their students (Porkess et al 2011: 96). In the Ofsted School Inspection Handbook (GOV.UK 2018b) references to a curriculum which ‘inspires’ (2018b: 47) links outstanding teaching with being ‘inspirational’ (2018b: 67). Sammons et al (2014) too, suggest that, in the UK context, ‘there may be links between characteristics of inspiring and effective practice’ (2014: 5). But Sammons et al (2014) go on to point out that there is an ‘important distinction between inspiring and effective teaching’ in terms of outcomes and the evaluation thereof. Traditionally the evaluation of ‘effective teaching’ is based on student outcomes in terms of performance – which can be crudely measured by way of examinations. But the question of how ‘inspiring teaching’ can be observed and measured is more complex (Sammons et al 2014). Indeed, the impact of being inspired at school, may not be realized until many years hence (Hattie, 2012). Whether all inspiring
teachers are effective, and whether all effective teachers are inspiring, is another interesting conundrum. According to Sammons et al (2014), there is belief amongst teachers that being inspiring and being effective are ‘two related and mutually dependent aspects of teaching’ (2014: 29) and there is evidence to suggest the ‘two concepts are complimentary’ (2014: 29). Being effective appears to be an ‘important and necessary pre-requisite’ to becoming an inspirational practitioner but, in addition, inspirational teachers are also genuinely interested in their students’ well-being, display ‘a high degree of engagement with their students’, and place a ‘strong emphasis on making learning enjoyable and engaging’ (Sammons et al 2014: 33). In essence, inspiring teachers demonstrate strong social and inter-personal skills and a commitment to, and a liking for, their students (Sammons 2014, Smithers and Hill 2006).
The ongoing difficulty in defining great teaching, may best be summarised by the much used quote from Gates (2013): ‘Unfortunately, it seems the field doesn’t have a clear view of what characterizes good teaching.’ With effective practice seemingly dependent on multifarious factors, it is perhaps not
surprising an agreed definition of which, remains elusive. The report by Coe et al (2014) for the Sutton Trust explores ‘What makes great teaching?’ and acknowledges that defining ‘effective teaching’ is no easy task. The definition settled upon in their report suggests effective teaching to be ‘that which leads to improved student achievement using outcomes that matter to their future
success’ (2014: 2). This then poses further questions: What outcomes ‘matter’
and what do we mean by ‘future success’? The current Chief Inspector of Ofsted, Spielman (2017), believes outcomes that matter are linked to society’s regard concerning the body of knowledge we wish to impart to the next
generation, so children can ‘flourish in the future’ (GOV.UK 2017: 2). And what we believe to be culturally important and opt to impart, constitutes the
curriculum we choose to design. Agreeing upon valid methods for measuring such future outcomes is not easy. Spielman (GOV.UK 2017) has recently indicated that exam results alone are insufficient to ascertain whether students have ‘received rich and full knowledge from the curriculum’ (2017: 2). Hattie (2015) also reports that it is engagement with school (and the number of years of education), not examination grades, that leads to the better outcomes for individuals in later life. Coe et al (2014) also argue that ‘enhanced student outcomes’ need not be limited to student academic attainment, but should also include whatever is valued in education (2014: 11). This leads us back to the question Spielman so recently posed: ‘What do we understand to be the real substance of education?’ (GOV.UK 2017: 1). Starting with the question of why education is important, Wiliam (2016: 8) has drawn together four broad
categories outlining the purpose of education and these are described in Table 2.5.
1. Personal empowerment - enabling young people to take greater control over their lives
2. Transmission of culture - passing on the “Great things that have been thought and said” from one generation to the next
3. Preparation for citizenship - preparing young people to take active role in society and to make a difference in the world
4. Preparation for work - ensuring that young people are able to find fulfilling and rewarding employment
Table 2.5 Four Broad Categories Outlining the Purpose of Education
Wiliam (2016) suggests it is the category regarding ‘preparation for work’ which may require the most attention, not because it is the most important but
because demands from the world of work are accelerating faster than the improvements in education; education needs to ‘race’ to keep up (Wiliam 2016:
11). Cuoco et al (1996) agrees and suggests the aims for mathematics
education should evolve with students tasked to ‘learn and adopt some of the ways that mathematicians think [original emphasis]’ (1996: 376), rather than continue the traditional classroom provision mired in properties and procedures.
Students should be encouraged to create, invent, conjecture and experiment – and be given the tools to ‘understand, and even make, mathematics that does not yet exist’ (Cuoco et al 1996: 376). Finland is currently implementing radical reform to its school curriculum, with changes ‘rooted in the realisation that as the world changes so, too, should education’ (Smith, N. 2017). In an uncertain world with ‘a future that as adults we can’t imagine’ (Smith, N. 2017), Finland is attempting to enhance the education experience and future-proof their next generation.
Many teachers and educators will say that they ‘know’ when they see good teaching, or that they can spot a good teacher. Fenstermacher and Richardson agree: ‘Perhaps we cannot define quality teaching, but we know it when we see it’ (2000: 2). The idea of ‘quality’ suggests that teaching can be of high as well as low quality. Again, definitions differ as to what high and low quality
represents. Many however are based on the educational effectiveness paradigm (Seidel and Shavelson 2007) in which high quality teaching
represents everything which leads to positive effects on student outcomes – this mainly being student achievement. Fenstermacher and Richardson (2000) and Berliner (2001) have, however, emphasized that quality is more than successful teaching, and make distinctions which illustrate the difference between ‘good’
and ‘successful’ teaching (2000: 6). ‘Successful’ teaching has an emphasis on what is taught, and is linked to student achievement (for example, teaching-to-the-test with the primary goal being exam performance), whilst ‘good’ teaching
has an emphasis on how ‘it’ is taught, employing teaching styles sensitive to the issues of inclusion and morality and can be seen as being more rooted in
achieving higher levels of motivation to learn, ‘higher feelings of self-efficacy’
and ‘deeper, rather than surface understanding of the subject matter’ (Berliner 2001: 470). Good teaching can be considered to be rooted in the ‘task sense of teaching’ and successful teaching in the ‘achievement sense’ of teaching
(Fenstermacher and Richardson 2000: 7). The idea of quality teaching being neatly comprised of both ‘successful’ and ‘good’ teaching is an appealing one, it is nevertheless ‘fraught with complexities’ (Fenstermacher and Richardson 2000: 7). It is the quality of learning which must be considered, and distinctions can again be made between ‘good’ and ‘successful’: ‘good’ when a learner engages with tasks; ‘successful’ when a learner also succeeds at them (Fenstermacher and Richardson 2000: 8). Hattie (2003) agrees, suggesting students’ motivation and engagement ‘account for about 50% of the variance’ in terms of attainment (2003: 1). Fenstermacher and Richardson (2000) suggest the quality of learning depends on the four factors outlined in Table 2.6, only one of which pertains to teaching.
1. Willingness and effort by the learner
2. A social surround (of peer, family, community, and culture) supportive of teaching and learning
3. Sufficient facilities, time and resources (opportunities) to accomplish the learning that is sought
4. Good teaching
Table 2.6 Four Factors for Quality of Learning (Fenstermacher and Richardson 2000: 8)
Using this model, Fenstermacher and Richardson (2000) suggest ‘quality
teaching’ is most likely to arise when all four of the above features combine and, to improve learning, policy initiatives could address any or all of the above four factors for learning. In contrast, traditional policy makers have viewed individual teachers ‘as the sole responsible agents for the quality of educational
processes’ (Kyriakides et al 2013: 130). The idea that improvement in teaching alone will result in improvement in student learning is a causal connection that Fenstermacher and Richardson (2000) believe to be ‘naïve’ and ‘wrongheaded’
(2000: 10). If it were so, learning successes could be replicated in almost any circumstances, including in differing socio-economic and cultural contexts. This is clearly not the case: As opportunities, facilities, resources and support
systems differ, so do student outcomes (Campbell et al 2004, Gorad 2017, Darling-Hammond 2014, Berliner 2001, Smith 2017b).
Reflecting the first factor outlined in Table 2.6 above, Slater et al (2009) report
‘pupil effectiveness’ as being ‘the single most important influence’ on test outcomes with ‘teacher effectiveness’ exerting about only ‘one quarter’ the effect (2009: 641). However, Slater et al (2009) qualify this by highlighting that a
‘teacher’s effectiveness influences the GCSE performance of…around 30 [students] per class’; ‘Hence there is greater leverage for the teacher’s effectiveness to matter’ (2009:641). Highlighting the gap in GCSE points
between ‘a poor and non-poor student to be 6.08 GCSE points’ (where 1 grade differential = 1 point), Slater et al (2009) calculate that if the poor student was taught by ‘good (75th percentile teachers)’ across all subjects and the non-poor student taught by ‘poor (25th percentile teachers)’, this gap could close by 3.4 points (2009: 641). And Slater et al (2009) suggest that choices surrounding teacher deployment could play a significant role in ‘alleviating unequal outcomes’ (2009: 641).
The context, and the humans involved in the transaction between teaching and learning, is significant (Coe et al, 2014); Coe et al’s (2014) analysis highlights this, with measures of teacher effectiveness, in terms of student gains, ‘only moderately stable from year to year and class to class’ (2014: 4). A successful teacher in one context will not necessarily be so in another (Campbell et al 2004). Berliner (2001) also believes context to be a significant factor when considering expertise, and one which is often overlooked; the ethos, conditions and climate of an institution can ‘powerfully affect’ teachers’ perceptions and practices (2001: 466).
Ultimately teaching is about human connections and relationships and it
appears that ‘the behaviour of effective teachers and less effective teachers are not easily characterised; much depends on the particular way that teachers and classes as people relate together’ (Coe et al 2014: 12). Education is an
‘essentially human activity’ based on communication, enthusiasm and
knowledge (Porkess et al 2011: 96). The Schleicher report (2012) highlights the significance of these human relationships with much global agreement
regarding important structures and attributes for 21st century learning environments. These attributes are summarised in Table 2.7, below.
• Encouraging engagement
• Ensuring learning is social and collaborative
• Acknowledging students’ motivations and the significance of emotional well-being
• Be demanding of every student without overloading
• Promoting connections across activities and subjects
• Using assessment that emphasize formative feedback
Table 2.7 Human Relationships Key for 21st Century Learning Environments
The above list suggests effective teaching requires empathy, and that enhanced learning takes place in a social, collaborative, emotionally intelligent
environment. A theory, based on the premise that effective teaching is largely concerned with human-relationships, suggests those attracted to exploring mathematical structures are less likely to be interested in connecting with people (Porkess et al 2011, Smithers and Robinson 2013). Smithers and Robinson (2013, iii) report: ‘Relatively few physics and mathematics graduates are attracted to teaching because the pleasures of these impersonal subjects are so different from working with children day-in, day-out’. Teaching is perceived as a career which largely offers intrinsic rewards, such as job
satisfaction and relationships, and See (2004) suggests that mathematics and science students are more likely to value extrinsic factors, such as salary and status, over intrinsic values. This reflects Smithers and Hill (1989) study which found that mathematics students were more motivated by extrinsic rewards.
Social science and arts students, on the other hand, are more likely to be people-oriented and hence more likely to be attracted to teaching (See 2004).
In the continuing quest to establish what effective teaching actually is, Coe et al (2014) believe, all too often, the wrong research questions have been pursued.
Various spurious findings have then been espoused which have subsequently been debunked, including the promotion of one teaching style above another, the advocating of large group teaching, or small group teaching or individual teaching. The issue of quality appears to be less about the shape or style of teaching and more about ‘the quality of thought and effort that can occur within these structures’ (Good and Biddle 1988: 116).
Coe at al (2014: 2) sifts six ‘components of great teaching’ from contemporary research; the evidence of which illustrates varying levels of impact on student
outcomes. The list outlined in Table 2.8, is offered as an effective practice
‘starter kit’ with the suggestion that all will feature in the tool-kit of the most effective practitioners:
Component Impact on student
outcomes
1. Pedagogical Content knowledge (PCK) Strong
2. Quality of instruction Strong
3. Classroom climate Moderate
4. Classroom management Moderate
5. Teacher beliefs Some
6. Professional behaviours Some
Table 2.8 Six Components of Great Teaching
‘Pedagogical Content Knowledge’ (PCK), highlighted above as having a strong impact on student outcomes by Coe et al (2014), is now a common phrase but was first coined by Shulman (1986: 9) to describe the teacher knowledge that links subject content and pedagogy and the interweaving of the two; this is discussed in detail in 2.3.2. Blending of subject knowledge and pedagogy leads to the understanding of why something is so: ‘the teacher need not only
understand that something is so; the teacher must further understand why it is so’ (Shulman 1986: 9). In a direct refute to Shaw’s (1903) oft repeated ‘He who can does. He who cannot, teaches’, Shulman (1986) aligns himself with
Aristotle’s view, that those able to teach by translating and communicating their own knowledge, demonstrate greater depths of understanding. This is
paraphrased by Shulman (1986: 14): ‘Those who can, do. Those who
understand, teach.’ This is undoubtedly true but it can be argued that advanced industrial nations, like, the UK, the United States, and Germany, have typically
not encouraged their ‘best brains to go into teaching; they want them to be at the cutting edge of research, innovation, creativity and wealth generation.’
(Smithers and Robinson 2013: 58). Now, with the UK set to leave the EU, and
(Smithers and Robinson 2013: 58). Now, with the UK set to leave the EU, and