Abstracts

How mathematics counts

It is no longer acceptable to have large numbers of students dropping out of education, so students’ continuing struggle with mathematics, one of the major obstacles to student retention in the USA, poses a significant social challenge. In general, school teaching emphasises one element of maths, computation, at the expense of another, interpretation, ie the assignment of meaning. Students learn best when the two elements are combined. For example, people tend to think about fractions and proportions in relation to concrete instances, but the treatment of fractions in school maths usually proceeds from formal definitions that are distant from people's experience. These definitions are themselves presented through a ‘confusing array of approaches’. Knowledge of algebra is now widely required for school graduation in the USA, but the enforcement of this policy has often led to only ‘lip service’ adoption by schools and systems, leading to the need for remedial algebra at tertiary level, and has also contributed to high drop out rates, especially amongst disadvantaged ethic groups. These approaches to maths teaching are not meeting social needs. Employers now demand not just technical skills but the ‘ability to synthesise information, make sound assumptions, capitalize on ambiguity, and explain their reasoning'. They need to be able to communicate quantitative information effectively. Students therefore need skills in verbal expression, including the ability to write ‘complete sentences and coherent paragraphs’ to convey meanings of data, tables, graphs and formulae. To learn these skills they should have extensive experience of using maths in other subject areas, eg interpreting data on global warming in science classes, or explaining voting systems in civics classes. Teachers of all academic and vocational subjects should collaborate to ensure numerical concepts are covered in their classes. School maths also needs to be contextualised. In contrast to the current emphasis on highly abstract concepts ‘illustrated with oversimplified template exercises’ students should be taught ‘sophisticated realistic problems that require only simple skills’, such as the interpretation of ratios and percentages, that can be applied to many interesting, relevant, complex and data-rich topics such as global warming and petrol pricing.

What can we do about achievement disparities?

In the USA attempts have been made to improve the maths learning of low-SES students through the use of problem solving and other measures designed to stimulate thinking skills. While taking part in one such project, the author noted that these methods disproportionately benefited students from professional or middle class backgrounds, an observation supported by research in Britain and the USA. She found that disadvantaged students tended to lack confidence to explore ideas and seemed to expect direction; they were confused rather than stimulated by uncertainty and debate over maths; and were more distracted than other students by irrelevant details of problems posed in real world settings. In general, disadvantaged children tend to experience more direction from their parents, whose own work is routinised and heavily directed, and lacking in the elements of questioning, experimentation and play enjoyed in many professional and middle class occupations. A number of measures are needed to help low-SES children overcome these disadvantages and to ‘wean students away’ from direct instruction. Teaching should highlight the need to grasp meanings rather than just facts. Curriculums developed along these lines by the US National Science Foundation are effective in teaching reasoning and problem solving as well as basic skills such as computation. Low-SES students can be focussed on the educationally relevant aspects of mathematical problems in real world settings, rather than incidental details, by the use of focused questions, whole-class discussion or writing exercises designed to summarise a lesson or recap the last one, and by journal prompts to explain key definitions and formulas. Educators also need to identify which topics are most difficult for disadvantaged students, using detailed school level assessment data. Educators also need to recognise that higher-SES parents are more skilled in following the ‘unwritten rules’ about how to have their children placed with the best teachers and courses. Educators should compensate by keeping procedures transparent, by offering advice to low-SES parents, or by assigning low-SES students to the most helpful placements before other students. Low-SES parents also need explicit advice on how decisions about maths courses in the early years impact on future study and career.

Teachers' mathematics knowledge for teaching and school context

Scholars have debated the extent to which formal ‘paper qualifications’ indicate teachers’ content knowledge for teaching. Teachers require both ‘common’ content knowledge, as required by anyone working in that subject’s field, and ‘specialised’ teaching knowledge, ie content knowledge specific to the teaching profession within a subject discipline. A recent study testing teachers’ specialised and common knowledge of mathematics has shown that teachers with appropriate specialised and common content knowledge are unequally distributed in schools in the USA. The study focused on mathematics content knowledge as it is believed to play ‘a key gatekeeping role for academic and occupational advancement in the United States’ and disparities in mathematics teacher quality are therefore seen as severely damaging in terms of educational equity. Teachers attending the California Mathematics Professional Development Institutes were administered multiple choice tests examining both common mathematics knowledge, such as how to multiply 35 by 25, and specialised mathematics knowledge, such as how to examine unfamiliar methods for solving 35 x 25. Results showed that in comparison with white students, African American and Hispanic students were both disadvantaged in terms of teacher content knowledge. The percentage of students eligible for a school lunch in a school – a common indicator of a school’s socioeconomic status – was also significantly related to teachers’ scores on the test, with lower scores associated with poorer schools. Affluent schools were observed to employ very few low-performing teachers, and high-poverty schools, while often employing numerous highly knowledgeable teachers, also had significantly more poorly performing teachers. The study concluded that students more in need of high-quality maths instruction had teachers who ‘are less prepared to deliver it’, and advocated more equitable methods of distributing highly knowledgeable teachers. However, this study showed only slight differences between the knowledge of teachers at affluent and impoverished schools. Because the study used a non-random sample, further studies are needed to confirm or challenge these identified patterns.

Mathematics in Singapore

Two articles in Educational Leadership: 'Learning from Singapore math' and 'Singapore Math: simple or complex?' consider Singapore’s national system for teaching mathematics. Singapore’s poor performance in international maths exams in the early 1990s drove concerted efforts to improve students’ mathematical achievement. As a result, each element of Singapore’s education system, including the maths framework, national standards and teacher preparation programs, is now explicitly and deliberately aimed at clear and common goals. The success of such efforts can be observed in recent Trends in International Mathematics and Science Studies (TIMSS) results, which have ranked Singapore students first in the world in maths proficiency for the last three years. These results have led researchers from the USA to investigate Singapore maths education. Recent studies show that Singaporean textbooks are narrower in focus, covering far fewer topics per grade level in far greater depth. This in-depth coverage is believed to promote rapid mastery of essential skills, which reduces the need to reteach skills, particularly with low-achieving students. Another common characteristic of Singaporean textbooks is their use of multiple models in explaining a concept. For example, a problem such as 5 x 6 might be accompanied by an illustrated figure noting '5x6=6x5' and by six columns of five blocks. This 'bar model' technique is familiar to students in Singapore, who first encounter it in simple addition and subtraction problems in Grade 3, and gradually develop the representation until Grade 6 where it serves as a basis for algebraic equations. Because this one graphic representation is used consistently, students know how to apply it to many complex problems. The simple graphic representations greatly benefit students with weak reading and maths skills, and English language learners benefit from the range of synonyms for ‘add’ and ‘subtract’ typically used in examples. Singapore texts also contain rich, multi-step problems which require students to apply a range of skills and develop wider and deeper mathematical understandings. By comparison, problems typical of US textbooks are routine and simplistic. However, both articles warn against uniformly adopting Singaporean maths techniques and textbooks, promoting instead careful consideration of the Singapore Math ‘program’ and incorporating its essential features.

Mathematical knowledge for teaching fraction multiplication

A study in the USA has examined the approach used by two teachers during lessons on fraction multiplication in Grade 6 mathematics classes. The classes were analysed to determine how the teachers organised their conceptual knowledge, and to see how this knowledge was used in explaining the material and responding to student questions. The study also distinguished between the teachers' knowledge of mathematical content and their knowledge of how to teach content, known as ‘pedagogical content knowledge’ or ‘knowledge for teaching’, which includes understanding students’ thinking, anticipating the difficulties they may have, and knowing how to make content accessible to students. For the purposes of the study each teacher used Connected Mathematics Project materials, chosen because they used drawings to visually represent fractions as length or area quantities. The tasks called for the ability to think on multiple levels at once: to understand, for example, that the concept of 1/5 can also be represented as 3/15 and 20/100, a concept known as recursive partitioning. Classroom observations, videotapes and interviews with teachers and students pointed to a limitation in teachers' reasoning with more than two levels of units. One teacher’s ability to work with three levels of units, 'although sufficient to infer numeric methods, was not sufficiently flexible to respond quickly and securely to the variety of ways that her students might assemble such structures during lessons’. In other words, her explanations of the material used three levels of units but her answers to students' questions did not. The other teacher appeared to possess adequate knowledge of the concepts involved, but did not express an understanding of more than two levels when either explaining concepts to students or answering their questions. In a positive finding, both teachers expressed clearly to their students that there was more than one way of doing the activities. These included folding strips of paper into fractions and dividing up a picture of a rectangular cake in different ways. In general the results pointed to a need to develop the cognitive models that underlie teachers’ knowledge for teaching so they are able to adapt more quickly and flexibly to the needs of their students.

Power or purpose? Some critical reflections on future school leadership

School leaders of the future need to be aware of the changing climate of education and, in particular, the new challenges and issues this raises. Managing social relationships and developing trust must begin to take priority over administrative powers such as staff hiring, firing and pay distribution: ‘The power associated with future school principalship will not be something controlling or dominating but something enabling and developmental.’ Future principals may need to minimise these administrative powers in order to gain trust and an equal footing with staff, laying the foundations for collaborative decision making. Creation of a unique and positive school culture must also be a priority. Individual school cultures are often overlooked in general educational reforms, but these reforms can be most effectively implemented in a collaborative culture based on support, trust, introspection, appreciation and an open sense of purpose. This means that school leaders of the future will need to focus on the cultivation of trust, on persuasive communication and on the empowerment of teachers. Moral authority will replace traditional hierarchical forms of authority, and leadership will become increasingly a process of influence grounded in interaction.

Assessing in the social domain: the whole-school assessment of practical and personal abilities

‘Whole school assessment’ offers promise as a way to measure personal and social abilities. This approach has been evaluated in a trial involving ten schools, 110 teachers and 350 students across Australia. The assessment process used in the trail requires ‘no new or different work for teachers and students’, but draws instead on teachers’ ‘global impressions’ of individual students within existing contexts. Participating teachers entered their assessments of each of their individual students on spreadsheets. A software program integrated these assessments into a relational database. An ‘overall assessor’ uses the database to compare different teachers’ evaluations of the various competencies of a given student. Analysis of data from the trial found that teachers covering different subject areas were usually in fairly close agreement about the social abilities of a given student. This finding contrasts with the arguments made by ‘critics of the key competencies’. The data also indicated that teachers can clearly distinguish general social qualities of the student from qualities that are either subject specific or specific to their own relations with the student. Participating teachers found the process efficient, since the information they entered ‘in two or three minutes’ could then be integrated into whole school reports.

Power lines: power and management in the education system

The author argues that education in Victoria has taken significant steps backward at times over the past 30 years. In the early 1990s the then Government made approximately 7,000 teachers redundant, raised teaching workloads and class sizes and encouraged a shift to contract work. Devolution of authority to schools has meant, in practice, a larger administrative workload for principals, much of which has passed to teachers. Through devolution the Government has also abandoned the economy of scale available within a large system. Teachers’ real wages have fallen significantly since the 1970s. Surveys show that teachers work about 50 hours per week, which means working more hours over a year than someone with four weeks’ holiday on a 38-hour week. The language used to support such changes has lowered teachers’ standing in the community. For example teachers’ former influence in educational decision making has been represented as ‘provider capture’. Funding for public schools has been described as ‘throwing more money at education’ in contrast to sympathetic terminology applied to funding for non-government schools. There have been positive changes over the last eight years. They include ‘lifting the ban on teachers speaking out on education’, restoring teachers’ participation in principal selection, and major capital works. However, much more is needed to ‘re-professionalise’ teaching, including a ‘full frontal assault’ on the language used to justify retrograde changes. The author describes his experiences at three schools, in roles including timetabling subject coordination and 'sub-school coordination'.

At the crossroads?

Building collaboration between professionals in health and education through interdisciplinary training

A recent study has evaluated the effectiveness of a training program for professionals who work with language-delayed children. Learning Language and Loving It (LLLI), developed by the Hanen Centre in Canada, is a workshop-based program that develops proficiency in a number of teaching strategies including ‘Wait and listen’, ‘Follow the child’s lead’ and ‘Use a variety of questions’. Sixteen teachers, special needs assistants and health professionals, all of whom worked with children with mild to severe language difficulties, took part in the workshops. The study concentrated on these professionals’ attitudes, perception of their own skills, and actual differences in use of communication strategies as measured by a number of video-feedback sessions. Video-feedback sessions consisted of a five-minute videotaping session followed by 30 minutes of individual feedback on the participant's use of the program's strategies. The program itself consisted of eight 2 ½-hour workshops over a period of ten weeks, plus a follow-up workshop three months later. Results for attitude change were measured by questionnaire and showed significantly more positive attitudes towards incorporating speech and language therapy in the classroom after completion of the program. Participants’ evaluations of their own skills, including ‘recognising nonverbal communication in children’ and ‘identifying speech and language goals in the classroom’, were also significantly improved. Attitudes towards collaboration were unchanged, possibly because they were at the ceiling before the program. Participants’ actual use of child-focused interaction strategies also improved considerably. However, on average the participants' use of these strategies still needed improvement after the workshops. Participants were satisfied with the program and many commented that it had had a clear impact on their own classroom teaching.

The birth to school study: evidence on the effectiveness of PEEP, an early intervention for children at risk of educational under-achievement

The Birth to School Study (BTSS) is a longitudinal evaluation of the family-focused intervention program Peers Early Education Partnership (PEEP). PEEP is targeted at disadvantaged communities at risk of educational underachievement. It aims to promote literacy, numeracy and self-esteem from birth to school commencement. PEEP sessions are voluntary, are offered to parents on a weekly basis and consist of discussions and literacy support activities such as songs, rhymes and book sharing. The BTSS examined the long-term success of PEEP with regard to parenting views and children’s educational outcomes. The BTSS was conducted between 1998 and 2004 and consisted of yearly interviews, the first a few weeks after the birth of the child. Interviews included a large battery of tests designed to measure parental distress, parent–child interaction, pleasure in parenting and child achievement in pre-literacy and pre-numeracy tasks. A group of 301 children in a PEEP catchment area in Oxford were matched with a comparison group of 303 children in a similar area nearby. The comparison group was not in a catchment area and was not exposed to PEEP. The study found significant differences in several of the parental outcomes at age one and two, for example in quality of the care-giving environment. However, these differences levelled out at the older age levels. Outcomes for children were mixed, with the comparison group showing greater achievement in terms of absolute scores at all ages. This result was attributed to the pre-existing advantages of children in the comparison group. When children’s progress over time was calculated, the PEEP group showed significant improvement in a cluster of literacy outcomes between the ages of two and five. This result was evident even for families who were in the catchment area but were not involved directly in the program, and suggests that early literacy programs may have a wide-reaching positive impact on their target communities.

Delicate balance

English language learners form a growing proportion of the student population in the USA. These students face the 'twin challenges' of learning academic content while at the same time developing their English language ability. Studies suggest that while social English takes only around one year to learn, learning the academic language of school textbooks can take five to seven years. This means that classroom teachers can no longer assume native-like proficiency, and will increasingly need to balance the content of their curriculum with language learning. English language skills vary considerably even among native speakers, so this adjustment has the potential to benefit all students. Integrating language learning in the mainstream curriculum will involve a shift in how ESL teaching is viewed. As part of this shift, teachers will need to make sure the content they are teaching is accessible for English language learners. This is often known as ‘sheltering’ instruction. Sheltering instruction involves a greater awareness of the language already embedded in the curriculum and a more conscious use of language in the classroom. Outside the classroom, collaboration between ESL teachers and classroom teachers must be a priority. This collaboration will include time for joint lesson planning between ESL and mainstream teachers, as well as the sharing of language proficiency data for individual students. Classroom teachers will then be able to tailor their lessons most effectively to both English language learners and native English speakers.

Let's define 'learning disabilities' at last

Earlier this year, a Year 11 student in Victoria was awarded $80,000 to compensate for a lack of classroom support for her severe language and learning disabilities. Such lack of support for people with learning disabilities comes partly from confusion over terminology. In Australia, unlike the US and UK, there is no official definition of ‘learning disability’ or ‘learning difficulty’. These broad terms tend to be used interchangeably to refer to several groups, including students with intellectual disabilities, students with English as a second language, and those who have specific reading difficulties without problems in other academic areas (dyslexia). Australia’s lack of a precise definition is partly due to a reluctance to apply negative diagnostic labels; however, a diagnosis is often the first step to self-awareness. Self-awareness in turn leads to the development of effective coping strategies, a strong predictor of life success for the learning disabled. An established definition would form the basis for increased understanding and help to close the gap in educational policy and funding for these students.

Usefulness of cognitive intervention programs for socioemotional and behaviour problems in children with learning disabilities

A pilot study in Belgium has examined the effects of a cognitive intervention program for children with concurrent learning disabilities and behavioural problems. Previous programs focusing on social skills had achieved only modest results. In contrast, this study evaluated a program that aimed to improve cognitive skills and measured the flow-on effect this had on children’s behaviour. Feuerstein’s Instrumental Enrichment Programme (FIE) was used, an intervention program that emphasises the teacher’s role as a flexible and student-oriented mediator who ‘helps the child to organise and translate the world’. The study looked at twenty-four 11–13-year-olds in a Belgian special school. A control group of 24 students, matched with the experimental group on education, age, IQ and gender, did not receive FIE lessons. The intervention was short-term and consisted of hour-long lessons given twice a week for seven weeks. Sessions specifically addressed socially-oriented cognitive skills such as considering other people’s perspectives and solving social problems. Results were mixed. Some participants reported substantial personal changes, including greater confidence, more friends and more stable mood, while others did not feel the intervention was effective. Statistical analysis showed substantial improvement in the children’s ability to think hypothetically, in their understanding of complex humour and in their perception of emotions. Improvement on other cognitive tasks was non-significant. There was no statistically significant change in either internalising (anxious, depressed and withdrawn) or externalising (aggressive and rebellious) behaviour. This may be due to the short length of the intervention, which was probably not sufficient to address deep-seated and entrenched behavioural problems. Qualitative data collected at interviews showed a more promising picture, suggesting that a majority of children and teachers saw improvements in social relations and self-regulation. The study provides a base for further development of programs that aim to teach metacognitive skills with a view to improving behavioural problems.

Changing classroom practice at Key Stage 2: the impact of New Labour's national strategies

The National Literacy Strategy (NLS) and National Numeracy Strategy (NNS) were implemented by the British Government in the late 1990s, with the effect of introducing daily literacy and numeracy hours into British primary schools. The strategies also specified class organisation and teaching methods in these areas with the aim of improving learning outcomes and teacher accountability. A recent study has found that these policies effected more change in pedagogy than was achieved in the previous two decades. Over four years, the Association of Teachers and Lecturers observed and interviewed 188 teachers in 50 schools, with the intention of forming a composite picture of the effects of the NLS and NNS on teachers’ approaches and attitudes to teaching. In the research interviews, teachers were highly critical of the government’s imposition of the strategies, which they saw as a public declaration of the Government’s lack of trust in teachers’ professionalism. However, the strategies themselves were viewed positively in retrospect. Teachers commented that the strategies promoted ‘continuity of teaching and learning in literacy and numeracy’ across all schools, gave structure to the curriculum through learning objectives, and provided ideas and resources for teachers. Researchers compared the data gained from interviews and classroom observations to data from an equivalent study conducted from 1992–1994 and found that teaching practices had changed dramatically. Whole-class learning, including the use of higher-order questioning, extended responses from students and increased student engagement in classes, was observed in 94 per cent of classes, compared with only 50 per cent in the 1992–study. The strategies have also led to the widespread practice of planning lesson objectives and explicitly sharing these with students, and the re-organisation of class seating arrangements to better suit whole-class practices. The success of the NLS and NNS is significant because it represents an exception to the trend whereby teachers’ ownership and recognition of a policy mediates its success. However, the forced implementation of the strategy also had negative effects on teacher morale, and restricted teachers in their ability to exercise creative practices. The subsequent introduction of the Primary National Strategy went some way to reversing these effects.