Foundation

Dimensions, Foundation
This category consists of knowledge, beliefs and understanding acquired in the academy, in preparation (intentionally or otherwise) for their role in the classroom. Such knowledge and beliefs inform pedagogical choices and strategies in a fundamental way. The key components of this theoretical background are: knowledge and understanding of mathematics per se and knowledge of significanttracts of the literature and thinking which has resulted from systematic enquiry into the teaching and learning of mathematics. The beliefs component relates to convictions held, and values espoused, by prospective teachers. Such beliefs typically concern different philosophical positions regarding the nature of mathematical knowledge, the purposes of mathematics education, and the conditions under which pupils will best learn mathematics
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Theoretical underpinning of pedagogy

Theoretical underpinning of pedagogy
One aspect of Mathematical Knowledge for Teaching is a teacher’s use of a theoretical foundation to guide instructional decisions, rather than relying on imitation of another teacher or trial and error. Therefore, it is important for teachers to know factors that are significant in the teaching and learning of mathematics (Rowland et al., 2009; Ball, Lubienski & Mewborn, 2001). Descriptors/Aspects of TUP (Theoretical Underpinning of Pedagogy) "Strong" examples: Draws on knowledge of well-established results in mathematics education research regarding the way pupils learn the topic in hand to underpin the planning and/or delivery of the lesson (whether this is implicit through lesson observation or revealed explicitly later through post-observation interview). Draws on knowledge of well-established results in mathematics education research regarding pupils’ misconceptions to underpin the planning and/or delivery of…
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TUP: Scenario 1

Theoretical underpinning of pedagogy
Scenario: John teaching properties of 3-D shapes Country: UK Grade (student age): Year 9 (age 13-14) Contributed by: Anne Thwaites, University of Cambridge, UK Context – national, curricular, professional, other The National Curriculum for mathematics in England includes properties of 3D shapes in Key Stage 3 (years 7-9, pupil age 11-14) including surface area and volume of 3D shapes based on prisms. John had introduced the idea of surface area and volume of 3D shapes in a previous lesson and was moving on to consider the properties of a cylinder in this lesson. John was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of his one-year teacher preparation. Scenario John was teaching in an open entry secondary school (pupil age 11-18) in…
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TUP: Scenario 2

Theoretical underpinning of pedagogy
Scenario: Heidi revising percentages Country: UK Grade (student age): Year 8 (age 12-13) Contributed by: Anne Thwaites, University of Cambridge, UK Context – national, curricular, professional, other The National Curriculum for mathematics in England includes work on fractions, decimals and percentages in Key Stage 3 (years 7-9, pupil age 11-14). Heidi was revising the four operations with fractions, before moving on to discuss some word problems related to percentages. Having completed a mathematics degree, which included an optional mathematics education element, Heidi was a graduate on a pre-service training course. The lesson took place in a school-based placement towards the end of her one-year teacher preparation. Scenario Heidi, a graduate student-teacher, was teaching in an open entry secondary school (pupil age 11-18) in a village in the UK. The school…
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AP: Scenario 1

Awareness of purpose
Scenario: Finding three fractions that adds up to one whole Country: Norway Grade (student age): Grade 7 (age 12-13) Contributed by: Ove Gunnar Drageset, University of Tromsoe, Norway Context The teacher has five years teaching experience. The lesson is halfway through grade seven. The competence goals in the Norwegian curriculum are not formulated for each year, but are given after the completion of grade 4, 7 and 10. The students have only met fractions like ½ and ¼ used in everyday situations prior to fifth grade. The competence goals regarding fractions after grade seven focuses on calculations using positive and negative fractions, and placing fractions on a number line. In this case the teacher asks the students in plenary of suggestions to a task that many found difficult. The task…
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AP: Scenario 2

Awareness of purpose
Scenario: Franca solving a story problem with the students. Country: Italy Grade (student age): Year 4 (age 9-10) Contributed by: Marco Bardelli, University of Padova, Italy Context – public primary school, national curriculum The Italian National Curriculum emphasizes the development of students’ number sense through realistic mathematical problems where the estimate of numbers can be performed. The mental calculation has taken great importance  and the students should manage the control of their reasoning.  Moreover the National Curriculum underlines that students should be able to solve mathematical problems with different strategies. Franca is teaching volume measures to her pupils. She has already explained the litre and its submultiples as a measure of capacity. In this lesson she wants to show her pupils how to use mental calculation in a story problem…
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AP: Scenario 3

Awareness of purpose
Scenario: Jim teaching construction of bar charts from tally charts Country: UK Grade (student age): Year 6 (age 9-11) Contributed by: Fay Turner, University of Cambridge, UK Context – national, curricular, professional, other Jim was a student teacher in the final term of a postgraduate teacher education course. During his final school placement, he was teaching a unit about data handling to a Year 6 class.  In a previous lesson he had shown them how to make tally charts to record the frequency of different phenomena. One of the displays in the classroom consisted of a number of different bar charts that the pupils had made previously.  At this time, curriculum guidance for mathematics in England suggested that pupils should be able to use tally charts by Year 4 and…
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TUP: Scenario 3

Theoretical underpinning of pedagogy
Scenario: Amy teaching a lesson about counting Country: UK Grade (student age): Reception/Kindergarten (age 4-5) Contributed by: Fay Turner, University of Cambridge, UK Context – national, curricular, professional, other This lesson took place during the second half of Amy’s first term teaching in a large city primary school.  It was a lesson with Amy’s Reception class about counting.  The national curriculum for England at this time (2006) put great emphasis in the early years on counting.  Two key objectives for pupils in their first formal year of school were that they should be able to: Say and use the number names in order in familiar contexts Count reliably up to ten everyday objects By the end of Year 1, pupils were expected to count up to 20 objects and to…
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