The work of steering instruction toward the mathematical point: a decomposition of teaching practice
Volume 49 Number 5, October 2012; Pages 935–970
Teaching to the 'mathematical point' means to focus mathematical teaching on core instructional goals: it means being clear on those goals and on how students may be given a sound opportunity to learn the content being covered. Failure to teach to the mathematical point distorts the learning process, leading, for example, to superficial coverage of curricula or to teaching that engages students without improving their knowledge. A study has used evidence from 17 preservice teachers' instructional practices to identify and illustrate different elements in teaching to the mathematical point. The participants were undergraduates in a teacher-education program at a US university. Evidence was obtained from lesson observations and pre- and post-test interviews with the participants. Analysis of the evidence distilled seven elements of teaching to the point. The article reviews each element, providing examples of successful and failed implementation in each case. The first element was attending to and managing multiple purposes. Often, maths teachers have 'non-mathematical' purposes that are ultimately intended as indirect supports to the learning of mathematical content. Such purposes include promoting student engagement, relieving students' anxiety about maths, and 'gaining access to their thinking' via questioning. Care must be taken that non-mathematical elements do not distort, eclipse or dilute the mathematical goals. The second element, closely related, is spending instructional time on mathematical work. Time spent on organisational tasks needs to be efficient. So does time spent on contextualising the task, eg the use of foods in the classroom to illustrate numbers and fractions. It is useful to achieve multiple mathematical or non-mathematical tasks at once. For example, explaining bar graphs may be a mathematical end in itself while also facilitating another mathematical topic; transition between tasks, a necessary organisational procedure, also offers a chance for refreshing physical movement. Third is spending instructional time on the intended mathematics – mathematics pitched at a suitable level of challenge. Teachers may be tempted to reduce the inherent cognitive demands of a task through leading questions, through overly supportive representations during set-up, or by failing to take students beyond familiar examples, eg decimal currency for the understanding of decimal fractions. The point can also be lost if it is illustrated through the use of excessively complicated tasks. The fourth element, closely related, is making sure students are doing the mathematical work rather than having it done for them by the teacher. Fifthly, the teacher develops and maintains a mathematical storyline. Activities need to be sequenced in ways that bring out their underlying mathematical coherence, and the connection between one activity and the next also needs to be made clear. The sixth feature is opening up and emphasising key ideas, eg by deliberate repetition of key mathematical terms, or through otherwise-unnecessary language, eg using the term 'positive 43' rather than just 43, to emphasise number sign. However, such intentional redundancy can be difficult to maintain because it is unnatural and may be difficult to manage when multiple ideas are being covered. The final element is keeping a focus on meaning, eg by highlighting the mathematically relevant aspects of a representation.
Key Learning AreasMathematics
Subject HeadingsTeaching and learning
Dreams to reality: closing the reading achievement gap with a focus on fluency
Volume 17 Number 3, October 2012; Pages 16–19
A team of Griffith University researchers report on their work with two school clusters in a literacy innovation partnership project 2009–2011. In 2011 the project involved 133 classroom teachers and over 3,000 students in 12 schools. The schools were based in culturally diverse, low-SES areas of Brisbane. The schools' students had been identified as having among the lowest NAPLAN test scores in the area. A commonly used reading comprehension test identified the students as approximately three to four years behind expectations. The aim of the project was to develop teachers' knowledge of reading, in order to customise their teaching to the varied needs of their students. Discussions with teachers and principals revealed that students had little or no interest in reading or even listening to books read aloud. They rarely borrowed from the school library, the collection of which was in any case out of date and shabby. Teachers expressed a need to learn more about how to help their students read fluently. Professional learning was therefore based around fluency instruction. Fluency embraces accuracy, with students becoming effortless and accurate word readers; speed, or reading at a rate conducive to thinking; and prosody. Prosody involves reading with suitable phrasing and expression, out loud or as rendered within the reader's mind; vocal expression that extends the meaning of the text, through variations in volume, pitch or speed; and natural phrasing, all of which requires comprehension of the text. The project staff focused on the strategy of repeated oral reading, which research supports as a means to engage and motivate students. They called on the educators to teach students fluent reading through a combination of methods. As well as providing explicit instruction, teachers modelled good oral reading and scaffolded students' oral reading, for example through choral, paired or taped reading. Students were given ample opportunity to read. Another method was the use of think-alouds to help students 'chunk' texts into meaningful phrases and show them how different ways of reading a text can change its meanings. Students quickly became enthusiastic about the new approach. In 2011 student cohorts in the project managed significant, and in some cases dramatic, gains in both NAPLAN scores and the abovementioned tests in reading comprehension. It should be noted, however, that reading fluency does not always go with good comprehension, and fluency instruction does not substitute effectively for thorough instruction in reading comprehension.
Subject HeadingsPrimary education
Culturally relevant science teaching in middle school
Volume 47 Number 6, November 2012; Page 1106– 1134
The article reviews literature on culturally relevant science teaching, noting particularly the work of AC Barton. It also reviews literature on the related concept of culturally relevant pedagogy, drawing on the work of G Ladson-Billings, (1995a, 1995b and 2006) and noting its close parallels to Barton's ideas. Both Barton and Ladson-Billings stress the importance of academic success in science for minority students, and note the barriers to that success. These barriers include lack of familiarity with scientific vocabulary and lack of background knowledge of basic items such as fresh fruit and vegetables that are commonly used to illustrate scientific ideas. More generally these students' backgrounds leave them ill-equipped to learn through traditional Western scientific instruction, which tends to treat concepts abstractly, in isolation from social context. Educators seeking to overcome this problem are liable to fall into either of two errors. One error is a failure to engage with mainstream scientific instruction, falling back instead on 'feel good' teaching that is culturally sensitive, engages with students' cultural backgrounds and instills pride, but does not pose adequate levels of academic challenge. The other, opposite error is encouraging students to disconnect entirely from their cultural heritage in order to access mainstream instruction. Teachers may overcome this 'false binary' by drawing on students' cultural backgrounds and community knowledge to illustrate a scientific concept. For example, one teacher used the example of a tortilla with young Latino students to illustrate the workings of a simple machine. Another important dimension of culturally relevant pedagogy is the cultivation of socio-political consciousness. This involves a critique of the way in which cultural and political factors shape the role of science in society. The issue of socio-political consciousness has been described as 'the most difficult and surprising revelation for teachers and teacher educators', as it calls upon them to move beyond a neutral, apolitical stance towards science 'and actively challenge oppression'. One way that educators may meet this challenge is to teach about prominent scientists from minority backgrounds, noting any examples of racism they have experienced. A related approach is to introduce the topic of racist behaviour exhibited by certain prominent scientists. The balance of the article describes one teacher's use of a science fiction short story to introduce discussion of racism and bias.
Key Learning AreasScience
Subject HeadingsScience teaching
Social life and customs
Educating the educators
November 2012; Pages 20–23
The discussion paper Great Teaching, Inspired Learning, recently launched by the NSW Government, proposed the imposition of minimum ATAR scores for entrants to teacher education courses. The article considers issues surrounding this policy proposal, and reports comments of a range of educational leaders. The President of the Australian Primary Principals Association, Norm Hart, agrees with the NSW Government that 'the ATAR score should be raised', and also calls on universities to take responsibility to improve the capabilities of any preservice teachers who have been admitted to courses with low ATAR scores. He also argues the need to recognise qualities other than ATAR scores as criteria for admission into these courses. These criteria include an understanding of pedagogy and a capacity for collaborative work. The calibre of applicants to teaching courses will be influenced by teacher pay and opportunities for career progression, while retention in the profession requires solid support in the early years of teaching. Greg Craven, Vice-Chancellor of the Australian Catholic University, argues for more recognition of achievements in Australian education, such as the number of students now entering university, and reduction in class sizes. He notes that ATAR scores correlate more strongly with students' socio-economic status than with their subsequent success at university. ATAR scores above 80 are good predictors of subsequent success, but their reliability declines sharply for candidates at middle levels of academic achievement, 'where most of the action is in terms of expanded university participation'. He also notes that the 'usefulness' of the ATAR score declines even further after the completion of students' first year at university. ATAR scores do not allow adequately for the disadvantages students inherit from their backgrounds, nor do they allow for positive qualities such as 'ethics, vocation or dedication'. The President of the Australian Secondary Principal's Association, Sheree Vertigan, also stresses the need to consider applicants' capacity to relate well to other people and cope with change. She raises concerns at the negative example of educational reformers in the USA, 'particularly those that blame teachers, specifically teacher unions' for problems in education. The article also reports opinions of education recruitment manager, Steve Whittington, and in relation to VET, the opinions of instructional designer and trainer, Sandy Welton.
Subject HeadingsTeacher training
The two purposes of teacher evaluation
Volume 70 Number 3, November 2012; Pages 14–19
Teacher evaluation may serve two distinct purposes: to develop teachers' professional practice or to measure the quality of that practice. Different systems of evaluation are used for each purpose. The criteria used within each system also differ significantly, and these differences become evident when three broad characteristics of teacher evaluation are compared. The first characteristic of a system for teacher development is that it is both comprehensive and specific. It is comprehensive in that it includes all the elements of practice that research has linked to student improvement. It is specific in that it distinguishes these elements at a granular level. The article includes a table setting out 41 such elements, grouped under three categories. Routines involve communicating learning goals and establishing and maintaining rules and procedures. Content strategies are used for introducing content and for helping students deepen their understanding of it and applying it. Strategies enacted on the spot are those prepared for use when needed in the classroom. Of the 41 elements in these categories, only 15 apply to systems for measurement of teacher performance. The reason for the difference is that some of these elements of teacher performance – for example, classroom management and the building of student-teacher relationships – only improve student achievement up to a certain point. A second reason is that some elements, such as the use of computer games, correlate with improved student achievement, but are not necessary components of good teaching. The second general characteristic of a system for teacher improvement is that progress on each of these elements is measured on a developmental scale: not using, beginning, developing, applying and innovating. The scales used to measure teacher performance, such as RATE, offer little guidance for the development of teacher practice. The third general characteristic of a system for teacher improvement is that it acknowledges and rewards growth. For purposes of measurement, teachers can be awarded 'status' scores, recognising the level of proficiency they have achieved, alongside their growth scores. Measurement and development are both important aspects of teacher evaluation, but a smaller set of criteria apply to measurement.
Subject HeadingsTeacher evaluation
The potential of peer review
Volume 70 Number 3, November 2012; Pages 20–25
Peer assistance and review (PAR) by teachers is now being used in a number of school districts in the USA, as a means to evaluate teachers' performance and establish the support they need. Peer evaluators can free school leaders' time, contribute subject specialist knowledge to the review process, introduce a teacher's perspective, and help teachers take more control over their profession. It has been challenged, however, as encroaching on school leaders' prerogatives. Teachers may be seen as biased toward peers or unwilling to make hard decisions, or lacking the authority and credibility for evaluations. The article reports on the PAR process that is now being used in a small number of US school districts. Peer reviewers, or consulting teachers, leave the classroom for three to five years to take up evaluation and support 'case loads' of 15–20 teachers: usually novices, but sometimes teachers deemed in need of improvement. The reviewers provide intensive individualised help over several months, following which they formally evaluate each teacher's performance. They then submit a report on the teacher to a joint management-union committee overseeing the program. In some cases this reviewer's report is accompanied by the principal's evaluation, but in each case the reviewer's report is the key document in the review process. The authors, from the Project on the Next Generation of Teachers, studied seven PAR projects in 2007–08, examined relevant documents and interviewed participants and stakeholders. The authors found that fully implemented PAR programs 'retained more novice teachers and dismissed more underperforming teachers' than comparable school districts. Approximately one third of underperforming teachers improved enough to meet district standards, while the others were dismissed. The dismissal process 'went smoothly' owing to close observance of due process and also as a result of the very exacting and transparent procedure by which reviewers were appointed.
Subject HeadingsTeacher evaluation
United States of America (USA)
Interleaving helps students distinguish among similar concepts
Volume 24 Number 3, September 2012; Pages 355–367
Interleaving is a teaching method that involves the dispersal and intermixing of concepts, terms and principles to be learned, 'so that a question on one concept is followed by a question on a different concept'. Interleaving contrasts to blocking where students are exposed to a group of questions, assignments or practice problems involving the same type of problem. Interleaving differs from the concept of spacing, in that spacing describes the dispersed scheduling of exposures to a single concept, while interleaving describes the dispersed scheduling of exposures to multiple concepts. There is evidence to support the value of interleaving as a learning technique. It has been found to be particularly useful in teaching students to discriminate between closely related concepts, and such fine distinctions of meaning often appear in the study of languages, sciences and maths. The article examines the benefits of interleaving for the study of mathematics. For example, the addition and the multiplication of fractions require different procedures for their solution; when these two types of problems are interleaved, students must first identify what type of problem it is, before they can select the correct procedure for solving it. When students are confronted by blocked questions, by contrast, they are already aware of the type of problem they face, and thus how it is to be solved. In this sense blocking provides scaffolding: it may be helpful when students are first exposed to a concept or procedure, but may limit learning thereafter. Interleaving can be implemented without changing the curriculum: it can be adopted simply by rearranging examples, questions or problems. However, evidence to support interleaving is largely experimental at this stage: it needs further testing in the classroom. A further reason for caution is that the level of challenge imposed by interleaving may generate resistance from students, rendering it ineffective, 'like bad-tasting cough syrup' which children refuse to use.
Key Learning AreasMathematics
Subject HeadingsTeaching and learning
Educating primary teachers to teach physical education
Volume 18 Number 3, October 2012; Pages 275–286
The article offers an overview of research into physical education (PE) instruction undertaken by generalist primary teachers. Primary school PE aims to develop children's motor skills and physical competencies, along with social and affective skills and behaviours, and lay the groundwork of lifelong patterns of healthy physical activity. International research indicates that primary PE is usually taught by generalist teachers, but also indicates that PE is usually of poor quality. The causes of this problem include teachers' limited subject knowledge and PE pedagogy, due in turn to insufficient preparation in teacher education courses, and limited professional learning opportunities. Students commonly participate in whole-class games, offering limited engagement time for individual students and frequently transitioning to free play; this problem is notably less common in classes taught by PE specialists. The sport education model of curriculum and instruction appears to offer more rounded experience for students, combining games with an opportunity for students to 'coordinate and manage their sporting experiences'. Some studies highlight the importance of teachers' own early experience of PE for their later teaching practices. These remembered experiences were commonly of embarrassment, physical injury, and gender equity issues. Such experiences need to be confronted in teacher education courses.
Key Learning AreasHealth and Physical Education
Subject HeadingsPhysical education
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