Internalising Symptoms and Working Memory As Predictors Of Mathematical Attainment Trajectories Across The Primary–Secondary Education Transition

Children of parents with a degree are almost a year of schooling ahead in maths by the age 11 , according to research from Sussex University

Greater parental education is the strongest predictor of maths attainment and faster future growth for children moving into secondary school even after adjusting for their intelligence (IQ).

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      string(381) "... a powerful way of enabling learners to engage with mathematical ideas, provided that teachers ensure that learners understand the links between the manipulatives and the mathematical ideas they represent. The effect on student mathematical attainment of using one textbook scheme rather than another is very small, although the choice of ..."
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      string(5701) "The headline finding is that far from hindering young maths pupils, using calculators in maths lessons can improve pupils’ calculation and problem-solving skills. The report stresses that, for calculators to be helpful, teachers need to teach pupils how to use them in a thoughtful and considered way. The review finds is that calculators can actually boost pupils’ fluency and understanding of maths – but that to do so, teachers should ensure they are used in a considered and thoughtful way, particularly with younger students. It showed that pupils who had used calculators systematically throughout primary school had a greater understanding of and fluency with arithmetic. However, the report added that primary school pupils should use calculators regularly but not every day. And it said secondary school students should be allowed more frequent access to calculators so they can make decisions about when and when not to use them. Key Findings: Calculator use does not in general hinder students’ skills in arithmetic. When calculators are used as an integral part of testing and teaching, their use appears to have a positive effect on students’ calculation skills. Calculator use has a small positive impact on problem solving. The evidence suggests that primary students should not use calculators every day, but secondary students should have more frequent unrestricted access to calculators. As with any strategy, it matters how teachers and students use calculators. When integrated into the teaching of mental and other calculation approaches, calculators can be very effective for developing non-calculator computation skills; students become better at arithmetic in general and are likely to self-regulate their use of calculators, consequently making less (but better) use of them. Feedback has a large effect on learning, but the range of effects is wide and a proportion of studies show negative effects. The effect of formative assessment is more modest, but is more effective when teachers receive professional development or feedback is delivered through computer-assisted instruction. Collaborative Learning (CL) has a positive effect on attainment and attitude for all students, although the effects are larger at secondary.Discussion is a key element of mathematics teaching and learning. However, there is limited evidence concerning the effectiveness of different approaches to improving the quality of discussion in mathematics classrooms. The available evidence suggests that teachers need to structure and orchestrate discussion, scaffold learners’ contributions, and develop their own listening skills. Explicit instruction encompasses a wide array of teacher-led strategies, including direct instruction (DI). There is evidence that structured teacher-led approaches can raise mathematics attainment by a sizeable amount. DI may be particularly beneficial for students with learning difficulties in mathematics. Inquiry-based learning (IBL) and similar approaches involve posing mathematical problems for learners to solve without teaching a solution method beforehand. Guided discovery can be more enjoyable and memorable than merely being told, and IBL has the potential to enable learners to develop generic mathematical skills, which are important for life and the workplace. However, mathematical exploration can exert a heavy cognitive load, which may interfere with efficient learning. Peer and cross-age tutoring appear to be beneficial for tutors, tutees and teachers and involve little monetary cost, potentially freeing up the teacher to implement other interventions. Students’ misconceptions arise naturally over time as a result of their attempts to make sense of their growing mathematical experience. Generally, misconceptions are the result of overgeneralisation from within a restricted range of situations. Misconceptions should be viewed positively as evidence of students’ sense making. Exploration and discussion can reveal to students the limits of applicability associated with the misconception, leading to more powerful and extendable conceptions that will aid students’ subsequent mathematical development. Teaching thinking skills, metacognition and self-regulation can be effective in mathematics. Technology provides powerful tools for representing and teaching mathematical ideas. However, as with tasks and textbooks, how teachers use technology with learners is critical. Concrete manipulatives can be a powerful way of enabling learners to engage with mathematical ideas, provided that teachers ensure that learners understand the links between the manipulatives and the mathematical ideas they represent. The effect on student mathematical attainment of using one textbook scheme rather than another is very small, although the choice of a textbook will have an impact on what, when and how mathematics is taught. Learners generally find algebra difficult because of its abstract and symbolic nature and because of the underlying structural features, which are difficult to operate with. This is especially the case if learners experience the subject as a collection of arbitrary rules and procedures, which they then misremember or misapply. Learners benefit when attention is given both to procedural and to conceptual teaching approaches, through both explicit teaching and opportunities for problem-based learning.Number and numeric relations are central to mathematics. Teaching should enable learners to develop a range of mental and other calculation methods. Quick and efficient retrieval of number"
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Leadership

Improving Mathematics in Key Stages Two and Three: Evidence Review

28 Mar 2018

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      string(368) "... positive impact on pupil outcomes. ■ Support pupils to make a successful transition between primary and secondary school ■ There is a large dip in mathematical attainment and attitudes towards maths as childrenmove from primary to secondary school in England. For example, one large national study ofprimary attainment in England ..."
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      string(426) "Leaving school with a good GCSE in maths is a prerequisite for progressing into quality jobs, apprenticeships, and further education. Yet too many of our young people do not make the grade and, as a result, risk social and economic exclusion. This report by the Education Endowment Foundation gives clear and actionable guidance for senior leaders and teachers in primary and secondary schools.
Leadership Briefing 12.03 (135)"
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      string(2718) "the intervention should be guided by effective assessment ofpupils’ individual strengths and weaknesses. ■ The easiest way to identify high-quality interventions is to look for those that have beenrigorously evaluated and have had a positive impact on pupil outcomes. ■ Support pupils to make a successful transition between primary and secondary school ■ There is a large dip in mathematical attainment and attitudes towards maths as childrenmove from primary to secondary school in England. For example, one large national study ofprimary attainment in England found that, at the end of Year 7 – a full year after the transitionto secondary school – pupils’ performance on a test of primary numeracy was below theirperformance at the end of Year 6. ■ It is clear that schools should be concerned about supporting pupils to make an effective Leadership BriefingLeadership Briefing – www.teachingtimes.comReports Summaries – a weeklyservice of essential summaries,in-depth reports and criticalanalysis that’s new and relevantto education, linked to originalsource documents.transition. Unfortunately, there is very little evidence concerning the effectiveness ofparticular interventions that specifically address this dip.Key Principles To Consider: ■ Continuing Professional Development (CPD) will be an important component ofimplementation and is key to raising the quality of teaching and teacher knowledge. Asummary of the best available evidence regarding CPD is available from the TeacherDevelopment Trust. ■ These recommendations do not provide a ‘one size fits all’ solution. It is important toconsider the delicate balance between implementing the recommendations faithfullyand applying them appropriately in a school’s particular context. Implementing therecommendations effectively will therefore require careful consideration of context as wellas sound professional judgement. ■ It is important to consider the precise detail provided beneath the headlinerecommendations. For example, schools should not use Recommendation 7 to justifythe purchase of lots of interventions. Rather, it should provoke thought about the mostappropriate interventions to buy. ■ Inevitably, change takes time, and we recommend taking at least two terms to plan,develop, and pilot strategies on a small scale before rolling out new practices acrossthe school. Gather support for change across the school and set aside regular timethroughout the year to focus on this project and review progress.WEB LINKImproving Maths in Key Stages Two & Threehttps://educationendowmentfoundation.org.uk/public/files/Publications/Campaigns/Maths/KS2_KS3_Maths_Guidance_2017.pdf"
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Leadership

Improving Maths in Key Stages Two & Three

8 Nov 2017

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      string(366) "... at all ages. After Lipman’s early experiment with fifth graders, numerous assessments were made in America which showed significant gains in reading comprehension and mathematical attainment, as well as in general reasoning skills. Trials have produced similar results in different countries, including a full scale evaluation of year one ..."
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      string(5609) "more expert judgement. In this sense, scientific inquiry is only one among many forms of inquiry that are to be found in the crafts and arts, in the humanities and professions, and indeed wherever human beings are engaged in making, saying, or doing.”This linking of enquiry with critical thinking, and beyond that with understanding and judgement, may well be stretching the limits of common usage, but it is not wildly out of keeping with Peirce’s original coinage. Understanding and judgement must surely be prime aims of education, and the role of critical enquiry in attaining them can hardly be underestimated.The linking of the idea with that of ‘community’ points again to the fact that a healthy democracy depends upon a sense of wide community in the hearts of its citizens. And a child’s immediate community can play a significant role in their personal and intellectual development.Dewey was not the only one in his day to recognise the social aspects of how – and why – we think. George Herbert MeadGeorge Herbert Mead, an American psychologist who published Mind, Self and Society in 1934, wrote an article as early as 1910 maintaining that: “The consciousness of self depends primarily upon social relations.”Long before the modern fashion for mediation, he was agreeing with Dewey that: “Instruction should be an interchange of experience in which the child brings his experience to be interpreted by the parent or teacher. This recognises that education is interchange of ideas, is conversation – belongs to a universe of discourse.”He summarises his own article as follows: “I have sought to indicate that the process of schooling in its barest form cannot be successfully studied by a scientific psychology unless that psychology is social, ie unless it recognises that the processes of acquiring knowledge, of giving attention, of evaluating in emotional terms must be studied in their relation to selves in a social consciousness. So far as education is concerned, the child does not become social by learning. He must be social in order to learn.”How close the Soviet philospher Lev Vygotskii’s cultural- historical theory (see Teaching Thinking issue 1) came to fulfilling Mead’s particular vision of a successful scientific psychology is a matter of some debate at the edges. But at its core it does recognise the importance to the learning child of the surrounding culture and community with special emphasis on the role of speech in conveying socially determined meaning.In developing the theory and practice of a ‘community of enquiry’, Lipman turned to the Vygotskiian ‘social’ model of cognitive development, and celebrated the extent to which children in such a community taught each other to think and feel better through internalising each other’s speech patterns and values.Not every speech pattern of children is conducive to better thinking and feeling in their peers, and Lipman recognised that every community of enquiry needed to develop an appreciation of good, critical, and creative patterns of questioning and reasoning. In the early stages of developing such an appreciation, the teacher has a guiding role to play, encouraging patterns of speech such as “I agree (or disagree) with ..”, or “Could you explain …”, or “For example…”In the longer term the best thing a teacher can do is allow the children to grow in responsibility themselves – to ask questions that open up enquiry and allow pupils to develop arguments, experiments, and evaluations of their own.One way of encouraging the growth of independent responsibility and learning is to let the children realise that you are as interested in their questions as they should be in yours.There must something to stimulate the children’s thinking – generally a story, read aloud so it is a shared experience – from which their interests, puzzles, or concerns can be drawn and formulated into questions.Once an enquiry is under way, various experiences, data, and perspectives are aired and processed, with a frequent call for reasons to justify particular interpretations.New interpretations or resolutions are being created all the time, both publicly and privately, and the community moves towards public evaluation of the most prominent of these.That is an ideal pattern of discursive development, not one that is always attained. But it has been witnessed in enough classrooms doing philosophical enquiry to be sure that it is attainable, even by very young children.There is evidence from trials that the community of enquiry process assists cognitive development at all ages. After Lipman’s early experiment with fifth graders, numerous assessments were made in America which showed significant gains in reading comprehension and mathematical attainment, as well as in general reasoning skills. Trials have produced similar results in different countries, including a full scale evaluation of year one children in 18 schools in Wales.In the UK teachers who use a community of enquiry find the approach carries over into other parts of the timetable: children ask more and better questions, and engage with each other’s thinking more critically and creatively. This has been remarked upon in at least two OFSTED reports.Perhaps this is not surprising. The opportunities for empirical exploration at school are bounded by many physical constraints. But the opportunities for conceptualand philosophical exploration are virtually boundless. More and more teachers are appreciating that such exploration pays dividends –"
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Creative Teaching & Learning

Philosophy For Children

9 Mar 2001

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      string(393) "... pupils participate more fully in classroom discussionsl	There are more ‘authentic’ questions asked by teachers and pupilsl	Pupils make better progress in the acquisition of mathematical conceptsl	Mathematical levels of attainment are higher than predicted for these pupilsAn important contribution to success would be the production of clear guidance for ..."
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      string(5662) "exploration of a philosophical questionraised by pupils in response to a stimulus.A P4C session is usually 50 minutes to an hour long andundertaken once a week or once a fortnight. The stimulusmight be a story, poem, picture or video clip. Part of thefacilitator’s skill is to help theparticipants decide on a question‘you really have to think about’.The question draws them intoreflection, reasoning and furtherquestion making. The benefitsof this activity are noticedacross the curriculum and inthe social behaviour of pupilsin and around school. Enquirytechniques can be adapted foruse in all curriculum areas.Research into P4CPotential supporters of an oracyand maths network will wantto establish evidence of theeffectiveness of P4C. Researchfrom Clackmannanshire LA andDundee University into P4Cprovides some examples. Asummary of the results, whichare in line with the type ofmodified behaviour that anoracy for mathematics networkwould want to promote, is asfollows:l	The whole population of children [doing p4c] gained onaverage 6.5 standard points on a measure of cognitiveabilities after 16 months of weekly enquiry. (Tricky andTopping 2007a)l	Pupils increased their level of participation in classroomdiscussion by half as much again following 6 months ofweekly enquiry. (Tricky and Topping 2007b)l	The frequency with which pupils supported their viewswith reasons doubled over a 6 month period. (Tricky andTopping 2007b)l	Teachers doubled their use of open-ended questionsover a six month period. (Tricky and Topping 2007b)l	When pupils left primary school they did not haveany further enquiry opportunities yet their improvedcognitive abilities were sustained two years intosecondary school.  (Tricky and Topping 2007b)l	Pupils and teachers perceived significant gains incommunication, confidence, concentration, participationand social behaviour following 6 months of enquiry.(Tricky and Topping 2006 and 2007c)Oracy for mathematics activitiesA starting point for collaboration on a P4C approach anddevelopment within a mathematical and scientific context 56maths and oracy   www.teachthinking.co.ukTeaching Thinking&Creativity  Vol 8:4will be to create a database of appropriate starters andstimuli. I have begun such a collection below and inviteothers to add to it. What follows are a sample of activitiesdrawn from different sources, some of which are ‘pure P4C’stimulus activities. Others can be viewed as open-endedmathematical tasks and starting points which could beundertaken more productively if the classes are alreadyengaged in P4C. That is because they will be used to askingquestions and  exploring possibilitiesStory books: Some examples of story books which can leadto philosophical enquiry within a mathematical context arereferenced below:l	Allen, P.(1980) Mr Archimedes’ Bath. Puffin Books,Londonl	Foreman, M. (1972) Dinosaurs and all that Rubbish. PuffinBooks, Londonl	Wormell, C. (2003) Two Frogs. Red Fox, LondonPossible tasks for an oracy for mathematics networkIf a network of the kind I am proposing came about, then members would have to decide on a programme ofactivities. Below are some suggestions about what to do and how to evaluative the outcomes.l	Undertake initial training in P4C and begin to use in the classroom (within appropriate contexts). Establish abaseline score for pupils to be involved. Explore ready-made starting points which have mathematical andscientific themes.l	Begin to experiment with the P4C approach within mathematics (and science lessons) where appropriateusing some of the materials already developed. Provide developmental feedback on these.l	Meet to develop further starting points and to explore linkage with First Steps in Maths, Let’s think, and CAMEmaterials.l	Conduct classroom trials of the above and collaborate with other network members as appropriate. Re-assess pupils involved in the programme.l	Meet to produce final guidance documentation and evaluation of the programme.Success CriteriaProjects are unlikely to be supported unless clear success criteria for evaluations (using control groups) areestablished at the outset. Success of p4c groups might be assumed if:l	 Interactions are longer and pupils participate more fully in classroom discussionsl	There are more ‘authentic’ questions asked by teachers and pupilsl	Pupils make better progress in the acquisition of mathematical conceptsl	Mathematical levels of attainment are higher than predicted for these pupilsAn important contribution to success would be the production of clear guidance for use in schools onstimulating dialogue and discussion leading to improved attainment in mathematics.Proposed evaluation toolsAlong with success criteria the network will be expected to indicate the tools and instruments it will use tomeasure progress. I suggest some possibilities below.l	Appropriate pupil tracking using National Curriculum assessment levels and CAT scores where available.l	The ‘Myself as a Learner’ scale.l	A sample of classroom interaction observations.l	Views of teachers and pupils gauged by short interviews and teacher research diaries.57Teaching Thinking&Creativity  Vol 8:4www.teachthinking.co.uk   maths and oracyStorywise activities: Storywise (Karin Murris and JoannaHaynes 2000) is a publication to support teachers dophilosophy with children in primary schools using picturebooks. It contains various starting points for philosophicalenquiry including some with a numerical theme. There is anexploration of the concept of ‘number’, and an invitationto consider what life would be like without"
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Creative Teaching & Learning

Talking mathematically

16 Mar 2008

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