Ellie

Readings: Department of Education Tasmania (2007). Teaching for numeracy: What might a dedicated mathematics time involve? Retrieved February 15, 2008, from http://www.ltag.education.tas.gov.au/focus/beingnumerate/Dedicatedmathstime.pdf

__Teaching for numeracy, what might a dedicated mathematics time involve? __ - The //Essential Learnings// Framework is based on sound research about how students learn best and appropriate teaching strategies to develop deep understandings of essential ideas. Mathematical ideas are central to becoming numerate, a key outcome for all Tasmanian students. - The //Essential Learnings// Framework therefore provides us with opportunities to examine the curriculum and to ask questions about our practices such as:  • ‘What are the essential ideas I need to focus on?’ • ‘What aspects of the mathematics curriculum may now be redundant?’ • ‘How am I explicitly focusing on the development of thinking and communicating?’ • ‘What areas might I need to give increased focus to?’ • ‘How might I be clearer about what I want students to understand and be able to do?’ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How can I teach for connections in students learning?’ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">‘How can I best assess what students understand?’ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">‘What is best taught in dedicated mathematics teaching time and what is best dealt with in transdisciplinary/ inquiry based studies?’ <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">During a dedicated teaching time teachers will consider issues such as: <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What do I know about the students in my class and their current understandings? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What are the important ideas in mathematics which students require in order to develop numerate behaviours? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What does research tell me about effective teaching for numeracy? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How might I teach for the development of students as mathematical thinkers? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How does what I am about to teach link to what has gone before and how can I make these connections clear? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What is the purpose of the learning experience? How do I make this clear to students? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What assessment procedures will I plan for? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How will I know that students understand the ideas I am dealing with? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What opportunities am I providing for students to become mathematical communicators using a range of communication styles? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How can I incorporate technology in my teaching? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How will I cater for the differing needs within my class? <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">- <span style="font-family: "Arial","sans-serif";">Brown and Askew’s UK study //Effective Teachers of Numeracy// (1997) concluded that the most effective teachers were those described as ‘connectionist’. These teachers were characterised by approaches which //Attempted to connect with both the children’s ideas and between different// //mathematical ideas and representations. This meant giving students time to// //develop, explain and negotiate their ideas.// <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">The connectionist teachers also believed: __ <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">That learning about mathematical concepts and the ability to apply these concepts are learnt alongside each other <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">That the connections between mathematical ideas need to be acknowledged in teaching <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">That numeracy teaching is based on dialogue between teacher and pupils to explore understandings <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">That pupils have strategies for calculating but the teacher has responsibility for helping them refine their methods <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Pupil misunderstandings need to be recognised, made explicit and worked on. <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">- <span style="font-family: "Arial","sans-serif";">Less effective teachers were described as transmission teachers who: ‘//asked low level factual questions to move the lesson on in the desired direction rather// //than expose children’s ideas. Such teachers demonstrated specific procedures children// //were expected to follow’// rather than exposing and working with children’s strategies. <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">- <span style="font-family: "Arial","sans-serif";">The Leverhulme Numeracy Research Project (1997-2000) has distilled a list of eighteen key characteristics of effective teaching for numeracy. These teachers were those who: <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Challenge children to think mathematically <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Expose and relater to children’s existing knowledge <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Develop significant mathematics eg strategies and generalisations <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Develop connections between mathematical ideas and between mathematics and the real world <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Stimulate children’s interest, curiosity, excitement and sustain engagement <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Permit access to mathematics and tasks for all pupils <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Have integrity of mathematics and context <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Have consistency between tasks and objectives <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Use symbols, diagrams and apparatus, not for window dressing or for objects in themselves but to communicate, represent and or provide good models of thinking <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Involve a range of models of expression <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Encourage the development of more sophisticated strategies <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Focus on the mathematics not just on work or getting answers <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Allow for sharing of methods and value the contributions of children <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Show teacher working with children (use of ‘we’) <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Focus on reasoning not just answers (not cued elucidation) <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Accept and work with children’s errors. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Effective classroom teaching strategies in the middle years would include: __ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Regular and systematic use of open ended questions, games, authentic problems and extended investigations to enhance students’ mathematical understanding and Capacity to apply what they know <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Teaching strategies focused on connections and strategies for making connections <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Students actively engaged in conversations and texts that encourage them to reflect on their learning and justify and explain their thinking <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Special attention given to the literacy aspects of mathematical texts and <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">representations <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Learning activities designed or chosen appropriate to learners’ needs and interests and use a balance of teacher-directed and student-directed approaches <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Opportunities provided for meaningful and enjoyable practice of essential knowledge and skills <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Four key principles of the National Numeracy Strategy to keep in mind: __ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">A dedicated mathematics lesson each day for every class <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Direct teaching and interactive oral work with the whole class and in groups <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">An emphasis on mental calculation <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Controlled differentiation during group work, with all children doing work linked to a common mathematical topic. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">The daily mathematics lesson is structured in three parts. __ <span style="font-family: "Arial","sans-serif";">- <span style="font-family: "Arial","sans-serif";">Although the lessons will vary, they will always include these clearly identifiable parts: <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">An oral and mental warm-up of about 10 minutes, involving work with the whole class. This will sometimes, but not always, relate to the main teaching activity <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">A main teaching activity of 30 to 40 minutes, involving work with the whole class, groups, pairs or individual children <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">A plenary session of about 10 minutes with the whole class. <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">

Munro, J. Diagnosing learning difficulties in maths learning. Retrieved February 10, 2010 from http://www.edfac.unimelb.edu.au/eldi/selage/documents/MLDT-DIagnosingmaths.pdf

__<span style="font-family: "Arial","sans-serif";">Diagnosing learning difficulties in maths learning __ <span style="font-family: "Arial","sans-serif";">- <span style="font-family: "Arial","sans-serif";">To diagnose maths difficulties, look at  <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Math difficulty ? <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Math the student knows. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Math disabilities //__<span style="font-family: "Arial","sans-serif";">Describing maths performance __// <span style="font-family: "Arial","sans-serif";"> //<span style="font-family: "Arial","sans-serif";">To collect the samples of maths //<span style="font-family: "Arial","sans-serif";">for later analysis, keep a record of  <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Which test to use? __ //<span style="font-family: "Arial","sans-serif";">1. Diagnostic Mathematical Tasks //<span style="font-family: "Arial","sans-serif";">: Survey Tests for Junior Grades Prep to 6 (Schleiger, 1980; Gough, 1990). <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Each test covers skills grouped into two parts: __ **<span style="font-family: "Arial","sans-serif";">Part A: **<span style="font-family: "Arial","sans-serif";"> Pure Number; numeration, place value, counting, recalling number facts, the four operations, number sentences and fractions. //<span style="font-family: "Arial","sans-serif";">KeyMath Revised //<span style="font-family: "Arial","sans-serif";">(Connolly, 1988) comprises scales in three areas: <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">Each subtest covers 3 or 4 'domains'. __<span style="font-family: "Arial","sans-serif";">Limitations of KeyMath __ //<span style="font-family: "Arial","sans-serif";">4. ACER Tests of Basic Skills : Aspects of Numeracy //<span style="font-family: "Arial","sans-serif";">(McQueen & Doig, 1997) sample <span style="font-family: "Arial","sans-serif";">number, space and measurement at the upper primary levels at three levels, A, B and C.  <span style="font-family: "Arial","sans-serif";">Number is assessed by having students: <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">- //__<span style="font-family: "Arial","sans-serif";">Analysing mathematics performance __//**//<span style="font-family: "Arial","sans-serif";">. //**<span style="font-family: "Arial","sans-serif";">To complete any maths task, pupils need to  <span style="font-family: "Arial","sans-serif";">• think about the ideas in particular ways; to visualise and say them, model them link them with what is known, question them. <span style="font-family: "Arial","sans-serif";">- //<span style="font-family: "Arial","sans-serif";">On task attentional and attitudinal behaviours towards maths. //<span style="font-family: "Arial","sans-serif";">Learners' attitudes are displayed in how they behave or react to maths situations, //Learners attitudes to maths// The types of information you might examine include: <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">What they believe about maths, values, what maths is like, its value and functions. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Feel about maths. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Self-concept as learners, successful as learners. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Believe about how to do math. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">interest //__<span style="font-family: "Arial","sans-serif";">Explaining maths performance __//__<span style="font-family: "Arial","sans-serif";">; ____<span style="font-family: "Arial","sans-serif";">how well can learners: __ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Symbolic codes <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Recall the names of symbols <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Verbal versus nonverbal information <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Store symbolism in STM. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Retain sufficient information in auditory short term memory. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Retain spatial information in short term memory <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Analytically vs synthetically <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Comprehend part-whole concepts both for sequential properties and Equivalence properties <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Sequence quantities, recognise a sequence <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Dimensional thinking <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Process spatial-numerical information <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Keep track of two categories or attributes at once, refer to two categories <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Represent and perform mental actions of adding and taking away in different contexts <span style="font-family: "Arial","sans-serif";"> //__<span style="font-family: "Arial","sans-serif";">Referral data __// <span style="font-family: "Arial","sans-serif";">Information from the learner's teacher and school: <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Past approaches and present approach to mathematics instruction <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Learner prerequisites for learning to read, such as  <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Visual capacities. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Oral language and communication strategies, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Cognitive strategies <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Appropriate task-organisational strategies <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Previous experiences of the world and of mathematics. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Attitudes to mathematics; positive attitude to mathematics and to self as a mathematics student. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">General learning in the classroom, eg learning difficulties in other areas. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Information from learners. This includes: __ <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How they see maths; what learners do when they learn maths, how they feel about making <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Mistakes, to re-read, to guess at words they don't recognise immediately. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">How they see maths as a discipline, its values, how it can be enjoyable, interesting, useful. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Their emotional response to mathematics; whether they feel frustration, anxiety <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Their self-concept as mathematics students. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Information from the child's parents: __ <span style="font-family: "Arial","sans-serif";"> <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Maths held by the learner, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Need to be forced to do maths tasks more than other homework tasks <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Give up on maths tasks more quickly than other tasks, or whether they spend inordinate <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Periods of time trying to do maths, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Achieved the appropriate developmental milestones in language and in other areas, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Indicate an ongoing dislike of, and frustration towards, maths. <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">view of maths held by parents, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Have the learner's siblings shown an interest in maths? Have there been other maths <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Models in the child's home <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Did the learner's parents find maths hard to learn <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Has parent assistance in the past clashed with approaches being used at school <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">the learner's developmental history. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">Parents can provide information about: __ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Learners achieved appropriate developmental milestones <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Learners displayed intermittent hearing loss <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">When learners started and stopped the use of egocentric speech <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">The types of play in which learners engaged. <span style="font-family: "Arial","sans-serif";"> __<span style="font-family: "Arial","sans-serif";">General referral information; __ <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">When the teacher or parent first became aware of the child's maths problem, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Earlier steps taken to help the child, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Learning difficulties in other areas, <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Involvement of other professionals in the child's learning history such as  <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Sensory impairment <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Psychologists <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Medical <span style="font-family: "Arial","sans-serif";">• <span style="font-family: "Arial","sans-serif";">Motor development
 * <span style="font-family: "Arial","sans-serif";">Tasks, conditions, learner's output,
 * <span style="font-family: "Arial","sans-serif";">Time taken.
 * <span style="font-family: "Arial","sans-serif";">Learners say as they work through tasks.
 * <span style="font-family: "Arial","sans-serif";">Learners' level of stress and any behavioural indicators of this.
 * <span style="font-family: "Arial","sans-serif";">Maths ability changes when learners are cued to operate in particular ways.
 * <span style="font-family: "Arial","sans-serif";">Part B: **<span style="font-family: "Arial","sans-serif";"> Applied Number; length, time, money, space, capacity, mass, statistics and graphs.
 * <span style="font-family: "Arial","sans-serif";">Basic Concepts; Numeration, Rational Numbers and Geometry
 * <span style="font-family: "Arial","sans-serif";">Operations; Addition, Subtraction, Multiplication, Division and Mental Computations
 * <span style="font-family: "Arial","sans-serif";">Applications; Measurement, Time & Money, Estimation, Interpreting Data, Problem Solving.
 * <span style="font-family: "Arial","sans-serif";">Usefulness of the norms for Australian conditions.
 * <span style="font-family: "Arial","sans-serif";">The language used by many of the items is comparatively complex.
 * 1) //<span style="font-family: "Arial","sans-serif";">Progressive Achievement Tests in Mathematics //<span style="font-family: "Arial","sans-serif";">measures knowledge for grade levels 3-5, 5-8 and 6-9 (Levels A, B and C respectively) in the areas of:
 * <span style="font-family: "Arial","sans-serif";">Number
 * <span style="font-family: "Arial","sans-serif";">Computations
 * <span style="font-family: "Arial","sans-serif";">Fractions
 * <span style="font-family: "Arial","sans-serif";">Statistics +graphs
 * <span style="font-family: "Arial","sans-serif";">measurement + money
 * <span style="font-family: "Arial","sans-serif";">Spatial relations
 * <span style="font-family: "Arial","sans-serif";">Relations + functions •
 * <span style="font-family: "Arial","sans-serif";">Logic
 * <span style="font-family: "Arial","sans-serif";">Sets.
 * <span style="font-family: "Arial","sans-serif";">Apply arithmetic operations to whole numbers
 * <span style="font-family: "Arial","sans-serif";">Manipulate counting patterns
 * <span style="font-family: "Arial","sans-serif";">Arrange numbers in order
 * <span style="font-family: "Arial","sans-serif";">Manipulate place value
 * <span style="font-family: "Arial","sans-serif";">Select the appropriate operation to solve word problems
 * <span style="font-family: "Arial","sans-serif";">Read number sentences
 * <span style="font-family: "Arial","sans-serif";">Read decimal numbers
 * <span style="font-family: "Arial","sans-serif";">Add and subtract decimals, add two fractions
 * <span style="font-family: "Arial","sans-serif";">Sequence decimals and fractions
 * 1) //<span style="font-family: "Arial","sans-serif";">Teacher made mathematics tasks //<span style="font-family: "Arial","sans-serif";">

Chinn, S. (2004). Thinking styles in mathematics (ch. 4). In The Trouble with Mathematics: a Practical Guide to Helping Learners with Numeracy Difficulties (pp. 59-75). RoutledgeFalmer.

__<span style="font-family: "Arial","sans-serif";">Thinking styles in mathematics __ <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">Curriculum must encourage flexible thinking. Presumably this is to encourage good problem solving skills to complement good computational skills. <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">'Inchworm' and 'grasshopper'. Grasshoppers are holistic, intuitive and resist documenting methods. Inchworms are formulaic, procedural, sequential and need to document. <span style="font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif"; font-size: 11pt;">- <span style="color: black; font-family: "Arial","sans-serif"; font-size: 11pt;">Thinking style will influence how a learner uses numbers and the operations (+, -, x, -:-). <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">Inchworms see numbers and the symbols for operations literally. In the example below, 98 is seen as just 98, not as a number very close to 100. Indeed if you ask an inchworm to adjust 98 to an easier number they may not relate to the question and if they do try to answer they may well say 96. They will go into subtraction mode applying the subtraction rules automatically, probably with little or no understanding of the maths behind the procedure. <span style="font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">The grasshopper will use his good sense of number values and the interrelationship of operations. <span style="color: black; font-family: "Arial","sans-serif";"> · <span style="color: black; font-family: "Arial","sans-serif";">The 97 will be rounded up to 100 (by 3); · <span style="color: black; font-family: "Arial","sans-serif";">The (simple) subtraction 340 -100 gives 240; · <span style="color: black; font-family: "Arial","sans-serif";">The grasshopper knows that this intermediate answer is smaller than the correct answer (by 3); · <span style="color: black; font-family: "Arial","sans-serif";">Adding 3 takes him to the correct answer of 243. <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">The load on short term memory is less. There is less need to visualise the process in your mind. The method uses good awareness of number values. <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">An inchworm methodically calculates, starting at the top of the list <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">A grasshopper looks at the same question and writes £90. The teacher asks 'Where is your working out?' //'Didn't do any'// 'So how did you do it?' //'Just knew.'// 'No working, no marks.' <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">It is possible to survive maths as an inchworm, though there are a number of essential skills needed to make this an effective style, for example a good memory for sequential information. It is less likely that a grasshopper will survive school maths, especially at higher levels where documentation is essential, but it is likely he will be successful at 'life maths'. <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">inchworm biased, when the text is italic, the outcome is more favourable to a grasshopper thinking style. Underlined text is not thinking style specific. <span style="color: black; font-family: "Arial","sans-serif";"> · //<span style="color: black; font-family: "Arial","sans-serif";">have a sense of the size of a number and where it fits in the number system; //<span style="color: black; font-family: "Arial","sans-serif";"> · <span style="color: black; font-family: "Arial","sans-serif";">know by heart number facts such as number bonds //(l0),// multiplication tables, //doubles and halves;// · //<span style="color: black; font-family: "Arial","sans-serif";">use what they know by heart to figure out answers mentally; //<span style="color: black; font-family: "Arial","sans-serif";"> · //<span style="color: black; font-family: "Arial","sans-serif";">judge whether their answers are reasonable and have strategies for checking them where necessary; //<span style="color: black; font-family: "Arial","sans-serif";"> · <span style="color: black; font-family: "Arial","sans-serif";">recognise when it is appropriate to use a calculator and be able to do so effectively; · //<span style="color: black; font-family: "Arial","sans-serif";">make sense of number problems, including non-routine problems, //<span style="color: black; font-family: "Arial","sans-serif";">and recognise the operations needed to solve them. <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">Classified as grasshopper include: · <span style="color: black; font-family: "Arial","sans-serif";">In Year 5 (continue) to add/subtract 9, 19,29 ... or **11,21,31.** ... by adding or subtracting 10, 20, 30 ... then adjusting. A Year 5 pupil is taught to 'see' · <span style="color: black; font-family: "Arial","sans-serif";">1.5 + 1.6 as double 1.5 plus 0.1. · <span style="color: black; font-family: "Arial","sans-serif";">In Years 5 and 6 pupils are asked to find percentages by halving and quartering and halving again, as in finding 12.5% of £36 000 by halving three times and in finding 75% of £300 by halving to get 50%, halving again to get 25% and adding to obtain 75%. This is inter-relating numbers, building up and breaking down numbers. <span style="font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">Classified as inchworm include; · <span style="color: black; font-family: "Arial","sans-serif";">Knowing facts by heart in Year 1. · <span style="color: black; font-family: "Arial","sans-serif";">In Year 2 find a small difference between a pair of numbers using counting. · <span style="color: black; font-family: "Arial","sans-serif";">Using number lines for addition in Year 3. · <span style="color: black; font-family: "Arial","sans-serif";">In Year 4 using standard written methods for short multiplication. · <span style="color: black; font-family: "Arial","sans-serif";">Using standard written methods for addition and subtraction in Years 5 and 6. <span style="font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">Eire, the Mathematics Curriculum of 1999 is structured to 'enable the teacher to cater for individual differences in ability, previous learning and learning style.' While in Hong Kong, the curriculum includes: <span style="color: black; font-family: "Arial","sans-serif";">- //<span style="color: black; font-family: "Arial","sans-serif";">Fostering general abilities and skills. //<span style="color: black; font-family: "Arial","sans-serif";">It is important that students need to develop their capabilities to learn how to learn, to think logically and creatively, to develop and use knowledge, to analyse and solve problems, to access information and process it effectively and to communicate with others so that they can meet the challenges that confront them now and in the future. <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">many learners can be taught flexible thinking, but, inevitably there will be those who are exceptions. There will be those whose thinking style is so fixed that they can only be taught in that dominant style. <span style="color: black; font-family: "Arial","sans-serif";">- <span style="color: black; font-family: "Arial","sans-serif";">The ethos of the classroom is another key factor. If learners are encouraged to explore different methods and their efforts are praised and appreciated (children are adept at spotting false praise) then they will generate a learning culture of flexible thinking. <span style="color: black; font-family: "Arial","sans-serif";">Two key grasshopper skills an inchworm should adopt: <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">1. <span style="color: black; font-family: "Arial","sans-serif";">Inter-relating numbers, for example, seeing 9 as 1 less than 10, seeing 5 as half of 10. <span style="color: black; font-family: "Arial","sans-serif";">2. <span style="color: black; font-family: "Arial","sans-serif";">Overviewing any problem, for example reading to the end before starting or getting a feel of what the answer may be. <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">Two key inchworm skills a grasshopper should adopt: <span style="color: black; font-family: "Arial","sans-serif";"> <span style="color: black; font-family: "Arial","sans-serif";">1. <span style="color: black; font-family: "Arial","sans-serif";">Explaining their methods. <span style="color: black; font-family: "Arial","sans-serif";">2. <span style="color: black; font-family: "Arial","sans-serif";">Documenting their methods.
 * <span style="font-family: "Arial","sans-serif";">calculate accurately and efficiently, both //mentally// and with paper and pen, //drawing on a range of calculation strategies;//
 * <span style="font-family: "Arial","sans-serif";">explain their methods and reasoning using correct mathematical terms;
 * <span style="font-family: "Arial","sans-serif";">In Year I pupils are encouraged to explore all the pairs of numbers which add to 6.
 * <span style="font-family: "Arial","sans-serif";">In Year 2 the mental addition 16 + 7 is shown as 16 + 4 + 3and22 -7is 22-2-5 = 15.
 * <span style="font-family: "Arial","sans-serif";">In Year 3 the four times-table facts are obtained by doubling the two timestable facts.
 * <span style="font-family: "Arial","sans-serif";">In Year 4 pupils practise responding rapidly to oral or written questions, //explaining the strategy used.// (Sadly there is still this need to do maths quickly, which is not so good for many dyslexics.)