Characteristics of Effective Teaching of Mathematics: An Evidential Synthesis

Characteristics of Effective Teaching of Mathematics: An Evidential Synthesis Glenda Anthony Margaret Walshaw Massey University g.j.anthony@massey.ac.nz Massey University m.a.walshaw@massey.ac.nz Paper presented at the American Educational Research Association, Chicago, April 2007 Worldwide, policy makers are placing increasing demands on schools to use effective, research-based practices. In New Zealand a collaborative knowledge building strategy The Iterative Best Evidence Synthesis Programme has been implemented at policy level. This paper outlines the findings from the mathematics Best Evidence Synthesis focused on identifying characteristics of pedagogical approaches that facilitate learning for diverse learners in the school sectors. In examining the links between pedagogical practices and a range of social and academic student outcomes we draw on the histories, cultures, language and practices for the New Zealand context and comparable international contexts. Our synthesis reinforces the complexity of teaching, suggesting that effective teaching occasions learning within a complex nested system involving communities, schools and classrooms. Introduction Mathematics, it is widely understood, plays a key role in shaping how individuals deal with the various spheres of private, social and civil life. Recent mathematics initiatives have endorsed this belief by calling for changed classroom communities that shift teaching and learning away from an emphasis on learning rules for manipulating symbols. New initiatives like Principles and Standards for School Mathematics (PSSM) (National Council of Teachers of Mathematics, 2000) are now focused on developing communities of practice, actively engaged with mathematics. Pedagogy within such communities is at the heart of this paper. In particular, we explore the role that the teacher plays in developing mathematical capability and disposition within an effective learning community. We were interested in what research has found about the sorts of pedagogies that contribute to desirable outcomes for students. Research (e.g., Lampert & Blunk, 1998; Richardson, 2001) has shown that teachers who create effective communities in classrooms have done so through their belief in the rights of all students to have access to education in a broad sense—understanding of the big ideas of curriculum and an appreciation of its value and application in everyday life. Identifying effective pedagogical practices that are consequent on that belief is, however, a recent research endeavour. Notwithstanding that such research is still in its formative stages, the consensus is that identifying effective mathematics pedagogy is not simply a matter of applying generic pedagogical approaches and cures (Ball, Lubienski, & Mewborn, 2001; Shulman, 1986; Stein, 2001). Our review is timely, given that for many students, mathematics is a series of hurdles and challenges—a task met with continued failure and seeming irrelevance. Today, just as in past decades, many students do not succeed with mathematics, are disaffected by it and continually confront obstacles to engage with the subject. Recent analyses of international test data (e.g., the Programme for International Student Assessment, OECD, 2004) confirm a trend of systematic mathematics underachievement amongst particular groups of students. These data point to the dilemma teachers, schools and policy-makers face, confronted with the realization that schools will cater for increasingly diverse groups of learners. These changing demographics require a wider understanding of what pedagogy might look like in the classroom in order for it to be effective for all students. Our comprehensive and critical review explores pedagogy in mathematics classrooms that has been shown to enhance desirable outcomes for diverse students. Arguably, beyond the classroom, the immediate professional community has a marked effect on teacher effectiveness and hence on learner outcomes. For example, a number of researchers (see McClain & Cobb, 2004; Millett, Brown, & Askew, 2004) have demonstrated that principals and lead mathematics teachers are key players in the development of classroom practice. Crucially, what is done in classrooms can be attributed in no small way to the material and personal resources provided by others in the school. Other researchers (e.g., Clarke, 2001; Sheldon & Epstein, 2005) have found that effective and sustainable relationships between the home, community, and school, significantly influence classroom teachers’ enthusiasm for and success with curriculum. Findings, like these, that point to shared responsibilities and mutual investment in students’ well-being, serve to underwrite our discussion on effective pedagogy. In reporting on the work undertaken on effective pedagogy, we have drawn on these findings to conceptualize teaching as nested within an evolving network of systems. The system itself functions like an ecology, in which the activities of the students and the teacher, as well as the school community, the home, the processes involving the mandated curriculum and education-at-large, are constituted mutually through their interactions with each other. From a bottom-up vantage point, the classroom assumes importance and it is this aspect of the system that remains the focus of this report. Method of locating and assembling data We draw on findings from the Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration [BES] (Anthony & Walshaw, 2007). The synthesis is part of the Iterative Best Evidence Synthesis (BES) Programme, established by the Ministry of Education in New Zealand to deepen understanding from the research literature of what works in education for diverse learners. Our task was to determine what the literature says about quality mathematics teaching for diverse students. Whilst our focus was initially on the New Zealand literature, we later drew heavily on relevant work reported in other English-speaking countries. Our search involved making use of a range of search engines to allow us to pay attention to “all forms of research evidence regardless of methodological paradigms and ideological rectitude” (Luke & Hogan, 2006, p. 174). We also drew on personal networks to help us locate publications in academic journals, theses and projects, as well as other scholarly work, across sectors with a focus on mathematics in schools or centers worldwide. The search took into account relevant publications within the general education literature and within specialist educational areas. Putting that search strategy into place allowed us to access a large literature base. In total the number of sourced 2 references about effective pedagogical practices and student outcomes was slightly greater than 1100. In our search we were looking at pedagogy that made links with student outcomes, both academic and social. In particular, we took the National Research Council’s (2001) understanding of mathematical proficiency as: • conceptual understanding: comprehension of mathematical concepts, operations, and relations; • procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; • strategic competence: the ability to formulate, represent, and solve mathematical problems; • adaptive reasoning: ability for logical thought, reflection, explanation, and justification; and • productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. We added to these academic outcomes a range of other outcomes that relate to affect, behaviour, communication, and participation. The outcomes include: • a sense of cultural identity and citizenship; • a sense of belonging; • contribution; • well-being; • exploration; and • commonly held values, such as respect for others, tolerance; fairness, caring, diligence, non-racist behaviour, and generosity. In our first pass through the literature we noted that many studies offered specific explanations of student outcomes yet failed to draw conclusive evidence about how those outcomes related to specific teaching practices. On the other hand, some studies provided detailed explanations of pedagogical practice yet made unsubstantiated claims about, or provided only inferential evidence for, how those practices connected with student outcomes. These were not particularly useful to us precisely because our focus was on studies that not only offered detailed descriptions of pedagogy and outcomes but were also able to provide rigorous explanations for a close association between a pedagogical practice and particular outcomes. Assessments about the quality of research depended on the nature of the knowledge claims made, and the degree of explanatory coherence between those claims and the evidence provided for the context in question. In the end 660 references found their way into the BES report. Included are research reports of empirical studies, from very small single site settings to large scale longitudinal experimental studies. This range of studies characterizes our focus here on the development of effective classroom communities. What came to the fore were a number of critical aspects of pedagogical practice. These included: (a) creating community, (b) discourse, (c) instructional tasks, and (d) tools and representations. We use this thematic cluster to organize the results section following. Serving as a point of discussion, each theme provides insight into definitions of effective domain-specific pedagogy in mathematics classrooms. 3 Creating a supportive environment Teachers who create effective classroom communities truly care about student engagement (Noddings, 1995; Palmer, 1998). They demonstrate their caring in their relations with their students, by establishing a classroom space that is hospitable as well as intellectually ‘charged.’ What is important is that they do more than the current ‘politeness’ culture demands: they work hard to find out what helps and what hinders students’ learning. Because they work hard at enhancing students’ capacity to think, to reason, to communicate, and to critique what is said and done in class, teachers take pains to steer students away from developing dependent relationships. Put simply, caring teachers work at developing interrelationships that open up spaces for students to develop their own mathematical identities (Hackenberg, 2005). Students develop their mathematical identities in response to the care they receive within the classroom. For example, mathematics teaching in kura kaupapa embraces the concepts, practices, and beliefs of te ao Māori (the Māori world). Teachers in kura kaupapa Māori firmly believe that “affectionate nurturing breeds happy hearts and lithesome spirits and, thereby, warm and caring people” (Ministry of Education, 2000, p. 23). They maintain that such nurturing in a caring learning environment will contribute to positive life-long futures. Mutual responsibilities are created in a caring, supportive environment as older children care for younger ones and assist in their learning activities. Studies documented in the BES provide substantial evidence that caring teachers are those who identify, recognise, respect and value the mathematics of diverse cultural groups. Caring about students from diverse cultural backgrounds requires teachers to ‘move closer’ to their students, which carries with it the implication of reciprocity—that teachers and students have something to learn from each other. For example, Angier and Povey (1999) demonstrated that student academic and social outcomes in a Year 10 mathematics classroom were greatly enhanced by the inclusive pedagogy of mathematics that the teacher had established. This was a culture that did not minimise individuals’ experiences; nor were collective experiences downplayed. Participation in this classroom went hand in hand with students’ responsibility for themselves and for their own learning. The way in which students take responsibility for their relationship with mathematics is significantly influenced by what practices are validated in the classroom. Whitenack, Knipping, and Kim (2001) report how a teacher communicated the value of student effort and knowledge generated in individual, paired or whole-class activity. The teacher used students’ ideas to shape instruction and to enhance particular mathematical understanding in the classroom. Bartholomew (2003), however, found that teachers do not always value student contributions equally. She found that mathematics teachers in the study valued the experiences and contributions of top-stream students more highly than the experiences of other students. This evaluation was communicated to students in a range of subtle ways. In their New Zealand Progress at School study, Nash and Harker (2002) illustrate how profoundly inequitable pedagogical attention can affect students. They found that teachers who distribute their attention differentially tend to offer less encouragement to students who they have stereotyped as ‘not mathematical’. One student in their study said: “Like when you ask the teachers you think, you feel like you don’t know, you’re dumb. So it stops you from asking the teachers, yeah, so you just try to hide back, don’t worry about it. Everyday you don’t understand, you just don’t want to tell the teacher” (p. 180). The same inclination to hold back from asking the teacher was expressed by secondary school students in a study by Anthony (1996): 4 Some of the time I don’t understand the stuff enough in mathematics to answer questions ‘cause I’ll probably get it wrong. I only answer questions if I know the answers. (Jane, p. 40) Brooks and Brooks (1993) have observed that students’ unwillingness to answer a teacher’s questions (unless they are confident that they already know the sought-after response) is a direct consequence of the teacher’s questioning techniques. “When asking students questions, most teachers seek not to enable students to think through intricate issues, but to discover whether students know the ‘right’ answers” (p. 7). The caring teacher, on the other hand, constructs more equitable relationships within the classroom. Effective teachers use a range of organizational processes to enhance students’ thinking and to engage them more fully in the creation of mathematical knowledge. Studies undertaken by Barnes (2005) and by Sfard and Kieran (2001) have shown that within the classroom all students need some time alone to think and work quietly away from the demands of a group. This line of research has also revealed that reliance on classroom grouping by ability (e.g., Boaler, Wiliam, & Brown, 2000; Zevenbergen, 2005) may have detrimental effects on the development of a mathematical disposition. Effective teachers establish organisational structures with a view towards their potential for enhancing students’ mathematical identity, constantly monitoring, reflecting upon, and making necessary changes to, those arrangements on the basis of their inclusiveness and effectiveness for the classroom community. Pedagogy that is inclusive demands careful attention to students’ articulation of ideas. Lubienski (2002), as teacher-researcher, focused on the inclusive aspects of classroom dialogue when she compared the learning experiences of students of diverse socioeconomic status (SES) in a seventh grade classroom. She reported that higher SES students believed that the patterns of interaction and discourse established within the classroom helped them learn other ways of thinking about ideas. The discussions helped them reflect, clarify, and modify their own thinking, and construct convincing arguments. However, in Lubienski’s study, the lower SES students were reluctant to contribute because they lacked confidence in their ability. They claimed that the wide range of ideas contributed in the discussions confused their efforts to produce correct answers. Their difficulty in distinguishing between mathematically appropriate solutions and nonsensical solutions influenced their decisions to give up trying. Pedagogy, in Lubienski’s analysis, tended to privilege the ways of being and doing of high SES students. Teachers who truly care about promoting inclusive relationships also promote mathematical thinking and reasoning. Research, however, documents many cases (e.g., Anderson, 2003; Bergqvist, 2005) in which teachers tended to underestimate their students’ reasoning ability: they believed that only a few students in a class were able to use higher-level reasoning in mathematics. In contract Watson (2002), in her landmark study with low-attaining students, found that, teachers believed that students want to learn in a ‘togetherness’ environment; that students’ questions should propel teaching and learning; and that teaching should foster an awareness of learning. Subscribing to a proficiency agenda, teachers in this study believed that teaching should not offer students simplified tasks, but should challenge them and provide support for them to task risks. Teachers did “not dwell simply on the positive aspects of behaviour, motivation or attitudes, although those would play a part”; their pedagogical practices recognized and emphasized the thinking skills which students exhibited and offered “opportunity for these to be used [by all students] to learn mainstream curriculum mathematical concepts” (p. 473). 5 Engagement in mathematical discussion There is now a large body of empirical and theoretical evidence that demonstrates the beneficial effects of participating in mathematical dialogue within the classroom (e.g., Fraivillig, Murphy, & Fuson, 1999; Goos, 2004; Wood, Williams, & McNeal, 2006). Such an approach involves significantly more than developing a respectful, trusting and nonthreatening climate for discussion and problem solving. It involves socialising students into a larger mathematical world that honours standards of reasoning and rules of practice (Popkewitz, 1988). Teachers who facilitate student participation and elicit student contributions, and who invite students to listen to one another, respect one another and themselves, accept different viewpoints, and engage in an exchange of thinking and perspectives, are teachers who exemplify the hallmarks of sound pedagogical practice (Yackel & Cobb, 1996). However, it is a major challenge to make classroom discourse an integral part of an overall strategy of teaching and learning (Lampert & Blunk, 1998). Fraivillig and colleagues (1999) reported on how a teacher of Year 1/2 classroom rose to the challenge. What was particularly effective was the way the teacher sustained the discussions. She developed a sensitivity about when to ‘step in and out’ (Lampert & Blunk) of the classroom interactions and had learned how to resolve competing student claims and address misunderstanding or confusion (theirs and hers). For their part, the students listened to others’ ideas and debate to establish common meanings. In short, they participated in a ‘microcosm of mathematical practice’ (Schoenfeld, 1992), learning how to appropriate mathematical ideas, language and methods and how to become apprentice mathematicians. Knowing when to ‘step in’ is important for teachers focused on making a difference to students’ learning. Turner and colleagues (1998, 2002) found that what distinguished highinvolvement Year 5 and 6 classrooms was the engagement of the teachers in forms of instruction that allowed them to ‘step in’ at significant moments during classroom discussions. In particular, the teachers negotiated meaning through ‘telling’ tailored to students’ current understandings. They shared and then transferred responsibility so that students could attain greater autonomy. They also tended to foster motivation by sparking curiosity and by supporting students’ goals. In these classrooms, telling was followed by a pedagogical action that had the express intent of finding out students’ understandings and interpretations of the given information. Engagement in effective classroom discussion demonstrates control over the specialized discourse (Gee & Clinton, 2000). However, the specialized language of mathematics can be problematic for learners. Particular words, grammar, and vocabulary used in school mathematics can hinder access to the meaning sought and the objective for a given lesson. Words, phrases, and terms can take on completely different meanings from those that they have in the everyday context. In particular, mathematical language presents certain tensions in multilingual classrooms. Adler (2001), Khisty (1995) and Moschkovich (1999), for example, have all explored the teacher’s role within these contexts. Neville-Barton and Barton (2005) looked at these tensions as experienced by Chinese Mandarin-speaking students in New Zealand schools. Their investigation focused on the difficulties that could be attributable to limited proficiency with the English language. It also sought to identify language features that might create difficulties for students. Two tests were administered, seven weeks apart. In each, one half of the students sat the English version and the other half sat the Mandarin version, ensuring that each student experienced 6 both versions. There was a noticeable difference in their performances on the two versions. On average, the students were disadvantaged in the English test by 15%. What created problems for them was the syntax of mathematical discourse. In particular, prepositions, word order, and interpretation of difficulties arising out of the contexts. Vocabulary did not appear to disadvantage the students to the same extent. Importantly, Neville-Barton and Barton found that the teachers of the students in their study had not been aware of some of the student misunderstandings. Like the students in the study undertaken by Neville-Barton and Barton (2005), students from Sāmoa and Tonga, in Latu’s (2005) research, had difficulty with syntax. Word problems involving mathematical implication and logical structures such as conditionals and negation were a particular issue for students from senior mathematics classes. They also found technical vocabulary, rather than general vocabulary, to be problematic. Latu noted that English words are sometimes phonetically translated into Pasifika languages to express mathematical ideas when no suitable vocabulary is available in the home language. The same point was made by Fasi (1999) in his study with Tongan students. Concepts such as ‘absolute value’, ‘standard deviation’, and ‘simultaneous equations’ and comparative terms like ‘very likely’, ‘probable’, and ‘almost certain’ have no equivalent in Tongan culture, while some English words, such as ‘sikuea’ (square), have multiple Tongan equivalents. The suggestion is that special courses in English mathematical discourse be delivered with the express intent of connecting the underlying meaning of a concept in English with the students’ home language. Students, other than those from multilingual backgrounds, also have difficulties with mathematical language. Sullivan, Zevenbergen, and Mousley (2003) found that students with a familiarity of standard English (usually students from middle-class homes) had greater access to school mathematics. As the teachers in their study said, the students were able to ‘crack the code’ of the language being spoken. One teacher of students from nonEnglish-speaking backgrounds made the point about meanings of words: “[Y]ou need to reinforce: ‘Tell me what I mean when I say estimating?’ or ‘Where are some things that you estimate?’ Ground it in their world because for a child for whom English is not their first language, if there are numbers they’ll be right, but if you say ‘estimating’ they won’t have a clue what that might mean” (p. 118). Competency with mathematical language involves more than technical vocabulary. It also encompasses the way it is used within mathematical argumentation. O’Connor and Michaels (1996) have highlighted the importance of shaping mathematical argumentation by fostering students’ involvement in taking and defending a particular position against the claims of other students. They point out that this instructional process depends upon the skilful orchestration of classroom discussion by the teacher. The skill “provides a site for aligning students with each other and with the content of the academic work while simultaneously socializing them into particular ways of speaking and thinking” (p. 65). As straightforward as it might seem from the framework, socializing students’ mathematical thinking and speaking is, in fact, a highly complex activity (Taylor & Cox, 1997). It is complex because teachers and students are “negotiating more than conceptual differences…they are building an understanding of what it means to think and speak mathematically” (Meyer & Turner, 2002, p. 19). Yackel and Cobb (1996) reported from their research that building that understanding requires the teacher to first construct the norms for what constitutes a mathematically acceptable, different, sophisticated, efficient, or elegant explanation. These are the norms that were found to regulate the content and 7 direction of mathematical argumentation and govern the learning opportunities and ownership of knowledge made available within the classroom. Fraivillig and colleagues (1999) observed teachers who did more than sustain discussion–they moved conversations in mathematically enriching ways, they clarified mathematical conventions and they arbitrated between competing conjectures. In short, they picked up on the critical moments in discursive interactions and took learning forward. In another study, Stein, Grover, and Henningsen (1996) report on the importance of a sustained press from the teacher for justifications, explanations, and meaning. This factor was frequently accompanied by the teacher’s or capable student’s modelling of competent performance—often in the format of a class presentation of a solution. In many cases, a press for understanding resulted in successive presentations that illustrated multiple ways of approaching a problem. The press for understanding is an aspect of quality mathematics pedagogical practice highlighted by many researchers. Morrone, Harkness, D'Ambrosio, and Caulfield (2004) provide us with examples of this practice: (1) “So in this situation how did you come up with 18/27 and 18/30?” (2) “When can you add the way we’re adding, using the traditional algorithm, finding the common denominator? When does that make sense?” (p. 33). When a teacher “presses a student to elaborate on an idea, attempts to encourage students to make their reasoning explicit, or follows up on a student’s answer or question with encouragement to think more deeply” (p. 29), the teacher is getting a grip on what the student actually knows and is providing an incentive for the student to enrich that knowledge. Franke and Kazemi (2001) make the important claim that an effective teacher tries to delve into the minds of students by noticing and listening carefully to what students have to say. Yackel, Cobb, and Wood (1990) provide evidence to substantiate the claim. They report on the ways in which one Year 2 teacher listened to, reflected upon, and learned from her students’ mathematical reasoning while they were involved in a discussion on relationships between numbers. Analyses of the discussion revealed that her mathematical subject knowledge and her focus on listening, observing, and questioning for understanding and clarification greatly enhanced her understanding of students’ thinking. Numerous studies of classroom discourse highlight the importance of teacher knowledge. The findings of studies undertaken by Ball and Bass (2000) and many others (e.g., Hill, Rowan, & Ball, 2005; Kilpatrick, Swafford, & Findell, 2001; Ma, 1999; Warfield, 2001) signal that teachers must have sound content knowledge if they are to access the conceptual understandings that students are articulating in their methods. The teacher must make good sense of the mathematics involved to help move students towards more sophisticated and mathematically grounded (Fraivillig et al., 1999; Schifter, 2001). Creating Opportunities for Learning with Mathematical Tasks In advancing our understanding of what effective pedagogy looks like, it was important to consider the ‘what’ of learning: any opportunities to learn are influenced by what is made available to, and required of, learners. The selection of quality instructional tasks is critical. Tasks influence how learners come to think about, develop, use, and make sense of mathematics: … the cumulative effect of students’ experience with instructional tasks is students’ implicit development of ideas about the nature of mathematics—about whether mathematics is something 8 they personally can make sense of, and how long and how hard they should have to work to do so. (Stein, Smith, & Henningsen, 2000, p. 11) In the school sectors, tasks are the primary means through which teachers introduce important mathematical ideas and provide opportunities for learners to engage in mathematical practices. Whilst the research provides evidence that tasks can justifiably vary in purpose or format, it is clear that tasks should all share some commonality: they should be problematic for the learner and leave a mathematical ‘learning residue’ (Davis, 1992)—something of mathematical value to the learner. The research is clear that effective teaching at all levels ensures that mathematical tasks are not simply ‘fillers’ but require the solving of genuine mathematical problems. For teachers, the tension to develop students’ sense of mathematical well-being alongside their sense of social well-being is very real (see Bills & Husbands, 2005). Too often tasks can slip into being busy or fun type activities. For example, Cahnamann and Remillard (2002) and Rubick’s (2000) research on statistics lessons noted that the students were able to complete tasks that focused on counting data sets rather than the intended exploration of relationships within data. Instances of displaced learning are also evidenced in studies of group activities—co-operative tasks or mathematical field trips for example—that have been insufficiently structured to engage students with mathematical ideas (e.g., Higgins, 1997; Stein, 2001). Teacher attempts to make mathematics interesting appeared to be at the expense of accuracy and meaning. In contrast, students who engage in meaningful mathematical tasks are potentially able to treat tasks as problematic. To engage in problem-based tasks, students must impose meaning, make decisions about what to do and how to do it, and interpret the reasonableness of their actions and solutions (Holton, Spicer, Thomas, & Young, 1996). For those research studies that demonstrated students’ high task engagement and mathematical thinking, exploration of the structure of mathematics was foremost in task design and implementation; mathematical thinking was “woven into the daily fabric of instruction” (Blanton & Kaput, 2005, p. 440). De Geest, Watson, and Prestage’s (2003) Improving Attainment in Mathematics Project attributed improvements in students’ mathematical attainment to teachers and learners focusing on the development of ways to think with, and about, key ideas in mathematics. In accord with findings about effective teaching in the primary context (e.g., Askew, Brown, Rhodes, Johnson, & Wiliam, 1997; Mulligan, Mitchelmore, & Prescott, 2005), De Geest and colleagues noted the importance of becoming “intimately attuned to the ways in which mathematics is internally connected” (p. 306). Large-scale empirical studies of educational change in the U.S. also link significant achievement gains to changes in classroom practices centered on inquiry-based problem-solving approaches (e.g., Balfanz, MacIver, and Byrnes, 2006; Swanson & Stevenson, 2002; Thompson & Senk, 2001). Effective task design is also informed by recent research concerning task variation and the use of ‘example spaces’ (Watson & Mason, 2005), modelling (Lesh & Doerr, 2003) and open-end tasks (Zevenbergen, 2001). For example, Watters, English, and Mahoney (2004) demonstrated how use of extended modelling problems provided opportunities for learners to engage in a range of mathematical processes and develop mathematical understanding. Because the modelling activities in the study were designed for small-group work, they also provided opportunities for developing collaborative problem-solving skills and important metacognitive skills that enabled students to distinguish between personal and task knowledge and to know when and how to apply each during problem solution. 9 The relationship between the task and the learner is an important factor in successful task engagement (Turner & Meyer, 2004). Quality tasks both maintain their mathematical integrity and purpose, and link to learners’ prior knowledge and experiences. Watson and DeGeest (2005) found that effective teachers in their Improving Attainment in Mathematics Project [IAMP] focused planning for instruction on their students’ current mathematical competencies and interests. Other studies (e.g., Askew, 2004; Groves, & Doig, 2004) have found that effective teachers use students’ thinking and experiences to make appropriate choices regarding the difficulty level and degree of task explicitness. Ongoing assessment of students’ reasoning—assessment for learning—enables teachers to continually adapt learning goals and instruction. Sullivan, Mousley, and Zevenbergen’s (2004) research on responsive task differentiation provided promising results for participation of diverse learners in the classroom community. Situating tasks in contexts—be they real or imaginary settings—can provide a learning situation that is experientially real for students (Gravemeijer, 1997; van den HeuvelPanhuizen, 2005). Watson (2004) advocates that tasks should be seen as ‘realistic’ not because they relate to any particular everyday context, but because they make students think in ‘real’ ways. Watson noted that students in the IAMP were usefully motivated and intrigued by tasks that exemplified the ‘power’ of mathematics. When contextualizing tasks, however, researchers (e.g., Cooper & Dunne, 2000; Lubienski, 2000; Sullivan, Zevenbergen, & Mousley., 2002) caution that the link to improved learning outcomes is fragile. Researchers (e.g., Boaler, 2006; Sullivan et al., 2004) have found that successful practices make explicit those hidden aspects of pedagogy that can inhibit students’ participation in open-ended contextually-base tasks. Quality tasks need to present suitable levels of challenge if the learner is to gain a sense of control and develop valuable mathematical learning and dispositions. Mathematical tasks that are problematic and offer an appropriate degree of challenge have high cognitive value. In contrast, tasks that are too easy or too hard have limited cognitive value (Henningsen & Stein, 1997; Williams, 2002). Whilst we know that providing learners with the opportunity to work on complex tasks—as opposed to a series of simple tasks devolved from a complex task—is crucial for stimulating their mathematical reasoning and building durable mathematical knowledge (Francisco & Maher, 2005; Stein et al. 1996), we also have research evidence that documents groups of students being provided with less than optimum opportunities to engage in genuine mathematical tasks and associated practices. In England, teachers reported the use of explanatory or investigative methods with ‘able’ students and ‘show and tell’ with ‘less able’ students. Houssart (2001) found that teachers of higher-streamed classes showed more enthusiasm for investigative tasks that encouraged creativity: There has to be an element of challenge about it ... they want to be tested in what they’re doing and not feel they’re doing something babyish or below them. (Graham) Challenge, especially with the top set ... Probably, had I had the lower set, the challenge bit would … be far lower down, until they got the basics in obviously. (John) Similarly, only a small number of the 162 Australian primary school teachers surveyed by Anderson (2003) indicated that all students could learn by doing open-ended and unfamiliar problems on a regular basis: It’s safer—children feel more comfortable if they’re not made to think. I realise this is cynical—but for many children with low IQs and poor/non existent English language skills, the concept of problem solving is alien. Also it takes up too much time and there is great pressure to “get through” 10 the curricula. So whilst in theory I acknowledge the potential of problem solving, in reality with some clientele it’s too hard. (p. 76) These studies and others challenge pedagogical practices based on simplification and repetition for low-achieving students. Watson (2002) and Watson and De Geest (2005) provide evidence of enhanced instructional practices that support the mathematical thinking of students previously identified as low attainers. Based on their belief that these students were entitled to access mathematics, teachers in the Improving Attainment in Mathematics Project chose not to simplify mathematical activities. They planned tasks that encouraged links with previous learning and were responsive to students’ responses. Working with students identified as learning disabled, Behrend (2003) and Thornton, Langrall, and Jones (1997) found that, given the opportunity, these students successfully engaged with rich and meaningful problem tasks. Task challenge is also crucial for academically gifted students. Diezmann and Watters (2004) report on teachers who successfully increased challenge by way of task problematisation. Without changing the mathematical focus, a task can be problematised by methods such as inserting obstacles to the solutions, removing some information, or requiring students to use particular representations or develop generalisations. Diezmann and Watters found that problematizing, adapting, and enriching regular curriculum tasks provided underachieving gifted students with the opportunity to oscillate between regular activities and more challenging activities according to their capability, confidence, and motivation. Designing quality tasks is not the end of the matter. Without effective pedagogy we know that high quality tasks can fail to achieve their desired purpose. Stein et al. (1996) study of task implementation within secondary schools found that the higher the task demands in the set-up phase, the less likely it was that the task would be carried out faithfully during the implementation phase. Watson (2002) reported that teaching mathematics to low-attaining students in secondary school “often involves simplification of the mathematics until it becomes a sequence of small smooth steps which can be easily traversed” (p. 462). Frequently teachers took the student through the chain of reasoning and students merely filled in the gaps with the arithmetical answer, or low-level recall of facts. This ‘path smoothing’, it was found, did not lead to sustained learning precisely because the strategy deliberately reduced a problem to what the learner could already do— with minimal opportunity for cognitive processing. Researchers (e.g., Anthony, 1996; Stein et al., 1996; Turner & Meyer, 2004) have consistently identified a range of factors that contribute to the lowering of task demands: inappropriate challenge, a shift in focus from understanding to correctness or completeness, inappropriate allocation of time, relaxing of accountability requirements, and a lack of alignment between the task and students’ prior knowledge, interest, and motivation. Multiple ways of assisting students’ to maintain high-level engagement in mathematical tasks has similarly been identified in a range of research studies. In 64% of the tasks in Stein et al.’s (1996) study that remained high-level, a sustained press for justifications, explanations, and meaning, as evidenced by teacher questions, comments, and feedback, was a major contributing factor. This factor was frequently accompanied by the modelling of competent performance by the teacher or by a capable student—often in the format of a class presentation of a solution. Scaffolding (Anghileri, 2006) and opportunities to engage in meaningful practice activities, where the goal is to achieve understanding with fluency are also important (Watson & Mason, 2005). For example, Watson and De Geest (2005) found that students in the IAMP were assisted to make 11 progress when they were given explicit guidance about ‘what’ they needed to remember and supported with strategies to assist them to remember. Tools and Representations In the previous sections we have seen that quality teaching is able to capitalize on students’ prior knowledge, interests, and thinking to support their development of increasingly sophisticated forms of mathematical reasoning. An important way in which the teacher can take account both of students’ current competencies and interest and their long term learning goals is by introducing judiciously chosen tools and representations (Cobb, 2007). Used effectively, tools and representations—artefacts—offer spaces to help organise mathematical thinking (Askew, 2004). Numerous studies (e.g., Blanton & Kaput, 2005) demonstrate that the choice of tools students can access make a difference to their achievement. In mathematics the opportunity to access non-linguistic representation is particularly important; inscriptions in the form of notations, graphical, pictorial, tabular, and geometric representations abound. For example, Chick, Pfannkuch, and Watson (2005) illustrated how telling stories with graphical representations supported young students’ development of statistical thinking. Other researchers (e.g., Bremigan, 2005; Diezmann, 2002) have documented students’ use of diagrams. In these and other studies that promoted the use of representations, teachers played a critical role, particularly in understanding how the tools can act as a springboard for discussion and for structuring mathematical knowledge (McClain, Cobb, & Gravemeijer, 2000). Teachers who foster students’ mathematical development make continual inferences about the way their students ‘see’ the mathematical concepts embodied in the artefacts used. Reliable inferences, however, can only be made from appropriate external representational choices (English & Goldin, 2002). Ball (1993) and Lampert (1989), among others, have found that effective teachers select and construct artefacts that their students can relate to and have the intellectual resources to make sense of. In challenging an over-reliance on adult contrived equipment researchers contend that representational contexts need to be real or at least imaginable; be varied; relate to real problems to solve; be sensitive to cultural, gender and racial norms and not exclude any group of students; and allow the making of models (Sullivan et al. 2003) The work of the Realist Mathematics Education program (e.g., Gravemeijer, 1997) has shown that through a process of generalising and formalising, meaningful equipment gradually takes on a form of its own and contributes to the shaping of mathematical reasoning. Authentic situations that use artefacts to provide a bridge between the mathematics and the situation can occasion effective learning experiences. Lowrie (2004) found that children involved in a planning activity (costing and scheduling a family excursion to a theme park) were assisted to ‘make sense’ of the task through the use of brochures, menus, bus timetables, and photographs. Students were observed to extend, adapt, and revise mathematical ideas. They readily established their own sense of authenticity by aligning the problem with their personal experiences and understandings. Significantly, some of the children who were not considered ‘mathematically capable’ invented more powerful ideas than those who did not see the task as an open-ended challenge. Whilst research (see Thornton et al., 1997) has shown that tools can provide effective compensatory support for students with learning disabilities, there is also plenty of evidence to suggest that manipulatives, in particular, are sometimes utilized inappropriately with low-achieving students. Observations of low-achieving students in 12 Baxter, Woodward, and Olson’s (2001) study in elementary schools revealed that whilst in some classes manipulatives were a distracter, in others they provided a conceptual scaffold. In three of the five classrooms, manipulatives became the focus rather than a means for thinking about mathematical ideas. A distinctive feature of instruction for those teachers who engaged target students in mathematical thinking was the way they used a variety of representations of a concept prior to the use of the manipulative specified in the curriculum. For example, in a geometry lesson, parallel lines were represented by a range of arm movements, lengths of string were used to create angles, calculators were used, and finally representations were transferred to geoboards. Moreover, all students worked with a wide array of geometric terms, building conceptual understandings of key mathematical ideas, such as ‘parallel’, rather than memorising a list of definitions generated by the teacher. The mathematical textbook, together with the worked example, are examples of often taken-for-granted tools (Goos, 1999). A group of research studies point to ways that effective teachers make use of these tools. Pirie and Martin (2000) found that teachers can actively support students to fold back and ‘collect’ by overt modelling of collecting when working examples, by promotion of writing about one’s understanding, by assistance with reading texts, and through students’ discussion and direction intervention—for example, reminding students of a particular technique in order to allow them to make progress in the building of a new concept. Discursive practices of mathematical inquiry, we have seen earlier, are a hallmark of effective pedagogic practice. Tools provide an effective way for students to communicate their thinking. For example, Hatano and Inagaki (1998) describe an instructional episode involving first grade children. Students familiar with join–separate problems were presented with the problem: There are 12 boys and 8 girls. How many more boys than girls are there? Most of the children answered correctly, but one child insisted that subtraction could not be used because it was impossible to subtract girls from boys. None of the students who had answered correctly was able to argue persuasively against this assertion. It was only after the students physically modelled the situation that they realised that finding the difference was a matter of subtracting the 8 boys who could hold hands with girls from the 12 boys. In recent times, there has been intensive interest in research that links learner outcomes with pedagogies that utilize new technologies. Studies have shown that technological tools, like other conceptual mediators, can act as catalysts for classroom collaboration, independent enquiry, shared knowledge, and mathematical engagement. For example, Arnold (2004) found that algebraic tools available on a computer not only offer mathematical insight but also make students’ tacit mathematical understanding public. Likewise, Goos, Renshaw, Galbraith, and Geiger (2000) provided evidence that the graphics calculator can be a catalyst for discursive interactions focused on mathematical thinking that simultaneously support personal (small-group) and public (whole-class) knowledge production. Providing opportunities for mathematical exploration, technological tools can also serve to increase the relevance and accessibility of mathematical practices for learners. Vincent (2003), in a study of the use of dynamic software, found that where students worked in pairs on an exploratory task there were improvements in students’ arguments— their ability to connect conjecturing with proving increased. Nason, Woodruff, and Lesh (2002) report on a study in which groups of students developed spreadsheet models to record quality of life in a number of Canadian cities. As part of their study, the researchers 13 explored the potential of the computer to stimulate collaborative student efforts. As a result of public and critical scrutiny of their ideas, the students learned about mathematical efficiency and organising information for presentation. The computer became a mediator not only for building personal knowledge but also for the development of learning at the interpersonal level. It did this by occasioning interactions within and between student groups in the classroom. Yelland (2005) also noted the impact of ICT on the community of young learners. The children in her study articulated their enjoyment of this project work in a variety of ways: they liked working with their friends, choosing what to do, and using computers to ‘find out stuff’ on the Internet and to make (PowerPoint) presentations and movies. Whatever the technologies in use, research has found that messages conveyed by teachers’ words and actions are of paramount importance in influencing the way in which technological tools are used by students. For example, Pierce, Herbert, and Giri (2004) found that where teachers continued to privilege the high value of done-by-hand algebraic manipulations, students more likely perceived that CAS offered insufficient advantages over a graphics calculator to warrant the time and cognitive effort required to become effective users of this new technology. Ball and Stacey (2005) suggest that teachers should share decision making about technology-based approaches with their classes and have the students monitor their own underuse or overuse of technology. These researchers argued that the use of CAS technology needs to be accompanied by the development of algebraic insight. When students see an algebraic expression, they should think about what they already know about the symbols used, the structure and key features of the expression, and possibly its graph before they move further into the question. Pierce and Stacey (2004) found that when teachers routinely demonstrated this initial step in class, it was likely to become a habit for their students. Pedagogical practices associated with access to resources will be increasingly affected as teachers move to incorporate technology-based presentations and web-based facilitation of learning (Heid, 2005). McHardy (2006) used action research to investigate the utilisation of PowerPoint presentations and an email discussion group with senior mathematics classes. Students in her study reported that the email contact and availability of online resources supported their learning by increasing their access to information and by giving them a more flexible work environment and greater opportunities to practise. Despite these positive signposts of enhanced student outcomes, we need to continue to monitor the affective and social aspects of student use of new technologies. Current research documents students’ mixed views when evaluating the impact of technologies on their mathematics learning (Goos & Cretchley, 2004). Whatever the form of tool or representation, quality teaching requires careful consideration of the purpose of its use, how it will be valued, and whether the outcomes are justified by the learner investment required. Conclusions The synthesis has highlighted the complexity of teaching as an activity. Quality teaching is not simply the fact of ‘knowing your subject’, or the condition of ‘being born a teacher’. By nesting teaching within a systems network, we cannot claim that teaching causes student outcomes. But if student outcomes are not caused by teaching practices, they can at least be occasioned by those practices. They are occasioned by a complex web of relationships around which knowledge production and exchange revolve (Tower & 14 Davis, 2002). This synthesis offers insights from research about how that occasioning might take place. Patterns about teaching have emerged that have enabled us to foreground ways of doing and being that mark out an effective pedagogical practice. Each aspect, of course, constitutes but one piece of evidence and must be read as accounting for only one variable, amongst many, within the teaching nested system. Taking all these aspects together allows us to envisage what quality teaching might look like. We found that within classrooms, teachers facilitate learning for diverse learners by truly caring about student engagement. Research has found that effective teachers demonstrate their caring by establishing learning spaces that are hospitable as well as academically ‘charged.’ They work at developing interrelationships that create spaces for students to develop their mathematical and cultural identities. Teachers who care work hard to find out what helps and what hinders students’ learning. They have high yet realistic expectations about enhancing students’ capacity to think, reason, communicate, reflect upon and critique their own practice and they provide students opportunities to ask why the class is doing certain things and with what effect. At the same time, research quite clearly demonstrates that pedagogy focused solely on the development of a trusting climate and on listening to students’ ideas, does not get to the heart of what mathematics teaching truly entails. Classroom work is made more enriching when classroom discussion involves co-construction of mathematical knowledge through the respectful exchange of ideas. When teachers work at developing inclusive partnerships using structural arrangements that benefit all students, they ensure that the ideas put forward are, or become, commensurate with mathematical convention and curricular goals. Effective teaching for diverse students demands teacher knowledge. Studies have shown that what teachers do in classrooms is very much dependent on what they know and believe about mathematics and what they understand about the teaching and learning of mathematics. Successful teaching of mathematics involves a teacher with both the intention and the effect to assist pupils to make sense of mathematical topics. A teacher with the intention of developing student understanding will not necessarily produce the desired effect. Unless teachers make good sense of the mathematical ideas, they will not develop the flexibility to spot opportunities and points of entry that would prompt solutions towards more sophisticated and mathematically grounded notions. Sound teacher knowledge is a prerequisite for accessing students’ conceptual understandings and for deciding where those understandings might be heading. It is also critical for accessing and adapting resources to bring the mathematics to the fore. Studies provide conclusive evidence that teaching that is effective is able to bridge students’ intuitive understandings and the mathematical understandings sanctioned by the world-at-large. Consistently emphasized in research is the fact that teaching is a process involving analysis, critical thinking, and problem solving. Language, too, plays a central role. The teacher who has the best interests of learners at heart ensures that the home language of students in multilingual classroom environments connects with the underlying meaning of mathematical concepts and technical terms. The responsibility for the distinguishing between terms and phrases and sensitizing their particular nuances weighs heavily with the teacher. Teachers who make a difference are focused on shaping the development of novice mathematicians who speak the precise and generalizable language of mathematics. Effective teaching at all levels ensures that mathematical tasks are not simply ‘time fillers’ but require the solution to a genuine mathematical problem. For all students the ‘what’ that they do is integral to their learning. The ‘what’ is the result of sustained 15 integration of planned and spontaneous learning opportunities made by the teacher. It will be planned from many factors, some determined by the individual student’s knowledge and experiences, and others mediated by the pedagogical affordances and constraints, and the participation norms of the classroom. Research in this area has found that tasks that allow students to access important mathematical concepts and relationships, to investigate mathematical structure, and to use techniques and notations appropriately, are often employed over sustained periods of time. 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