RAT: Scenario 2

Responding to the (un)availability of tools and resources
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 solutions of quadratic equations in Key Stage 4 (years 10-11, pupil age 14-16). These pupils were near the end of year 9 and John had been working on graphical representation of quadratic expressions in a previous lesson and then on ‘completing the square’ [i.e. expressing in an equivalent form ], as specified in the 16-18 pupil age curriculum. 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…
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RCA: Scenario 1

Recognition of conceptual appropriateness
Scenario: Find all fractions larger than 1/2 Country: Norway Grade (student age): Year 6 (age 11-12) Contributed by: Ove Gunnar Drageset, University of Tromsoe, Norway Context The teacher has 20 years of teaching experience. The lesson is at grade 6 well into the second half of the year. 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.   The task is to find all fractions larger than ½ from a pink sky (see…
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CUE: Scenario 3

Choice of examples
Scenario: Solving problems using Schema-Based Instruction Country: Cyprus Grade (student age): Year 5 (age 10-11) Contributed by: Marilena Petrou Context – national, curricular, professional, other Schema-Based Instruction aims to develop students’ understanding of the basic relations found in arithmetic word problems (Marshall, 1995). Students are taught to map features of word problems onto problem schemata. Figure 1 illustrates the four schemas that are included in mathematics textbooks in Cyprus used to describe the semantic relations found in story problems, namely, ‘change’, ‘group’, ‘compare’, and ‘vary’ Figure 1: Schema- problems   Scenario Elsa, a final year university student in a teacher preparation programme, was teching problem solving using Schema-Based Instruction with a Year 5 (pupils age 10-11) class. The lesson took place in a school-based placement towards the end of her…
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CUE: Scenario 4

Choice of examples
Scenario: Naomi teaching multiplication Country: UK Grade (student age): Year 4 (age 8-9) Contributed by: Ray Huntley, Brunel University, UK Context – national, curricular, professional, other Naomi was an undergraduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her three-year teacher preparation. She was working from lesson plans from the Hamilton Trust, a UK based group which provides training and curriculum materials in English, mathematics and science, including detailed unit plans for mathematics with objectives, teaching ideas, resources and so on. Scenario In her lesson, Naomi sets out a number of examples so that children can use a grid method to multiply a 2-digit number by a single digit. The starter example is 34 x 2 which is presented in grid form so…
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CUE: Scenario 5

Choice of examples
Scenario: Suzy teaching multiplication methods Country: UK Grade (student age): Year 5 (age 9-10) Contributed by: Ray Huntley, Brunel University, UK Context – national, curricular, professional, other Suzy was reviewing ideas about multiplication of tens and hundreds with her class, before moving on to discuss other examples which extended the method into thousands. Suzy was a final year undergraduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her three-year teacher preparation. Scenario Suzy, an undergraduate student-teacher, was reviewing multiplication methods with a Year 5 (pupil age 9-10) class. She was teaching the children to multiply using the ‘grid method’, and it is the third lesson of a sequence of lessons on multiplication that is described here. It opens with a worked example, 34 x…
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ATB: Scenario 1

Adherence to textbook
Scenario: Holly teaching probability Country: USA Grade (student age): Grade 4 (age 9-10) Contributed by: Tracy Weston, The University of Alabama, USA Context –curricular, professional, other As is the case in the United States, North Carolina has state mandated goals and objectives for mathematics instruction at each grade level.  The North Carolina Standard Course of Study (NCSCOS) indicates that in grade 4 students are to “maintain” skills in permutation and combinations from Grade 3, which state “the learner will understand and use simple probability concepts,” which are further explained as determining the number of permutations and combinations of up to three items and solving probability problems using permutations and combinations. (NCSOS). This school used the EnVisions curriculum, as mandated by the district (local authority).  Students in this school were not…
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ATB: Scenario 2

Adherence to textbook
Scenario: Eoin revising perimeter and area concepts Country: Ireland Grade (student age): 6th class (age 11-12 years) Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland Context – national, curricular, professional, other While espousing a reform approach with the mandated development of mathematics process skills, the Irish primary mathematics curriculum is relatively specific in that content to be taught is divided into five mathematical strands (number, algebra, measures, shape and space and data and chance). These are further divided into strand units with lists of content objectives (followed by very brief exemplars) specified for each strand for each class level over the eight years of Irish primary schooling. Mathematics textbook series from one of three different publishing companies are currently available in Irish schools and these are ubiquitous in Irish…
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RAT: Scenario 3

Responding to the (un)availability of tools and resources
Scenario: Chloe teaching a mental arithmetic strategy Country: UK Grade (student age): Year 1/2 (age 5-7) Contributed by: Tim Rowland, University of Cambridge, UK Context – national, curricular, professional, other At the time when this lesson was recorded, mathematics in primary schools was prescribed by a National Numeracy Strategy and a related curriculum framework. In the early years, in keeping with Dutch RME principles, arithmetic emphasises children’s mental methods, and introduces a number of related strategies. The focus for this lesson is subtracting near-multiples of 10 (specifically, 9, 11, 19, 21) by subtracting 10 (or 20) and then adjusting by 1. Chloë was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her one-year teacher preparation. Scenario The objectives of the lesson…
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COP: Scenario 1

Concentration on procedures
Scenario: Maeve discussing the use of formal standard algorithms Country: UK Grade (student age): Pre-service teacher Contributed by: Gwen Ineson, Brunel University, UK Context – professional The session described below took place in week five of a one year postgraduate teacher training programme.  Trainees had not completed either of their sustained school experiences but they had undertaken an observation period of two weeks in primary classrooms as preparation for the course.  They had received 14 hours of mathematics input, which included an overview of the national curriculum, planning and assessment as well as subject knowledge relating to calculation, measures and shape. A feature of the course is the strong emphasis on a range of calculation approaches for the number operations. It would be useful to explain what is in the…
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COP: Scenario 2

Concentration on procedures
Scenario: Eleanor discussing approaches to helping pupils solve a multiplication problem Country: UK Grade (student age): Primary Contributed by: Gwen Ineson, Brunel University, UK Context – professional Eleanor was at the end of her one year graduate pre-service teacher training course and had completed all of her sustained school experience.  She was interviewed at the end of her programme about her approaches to calculation and about her approaches to teaching calculation.  The scenario below involves the multiplication of 2-digit numbers. Scenario Eleanor was asked what her approach would be to help pupils working on multiplication of 2-digit numbers.  The specific examples are 52 x 34 and 3.4 x 4.9. Interviewer:     What strategies would you use to teach your pupils to solve the following problem: 52 multiplied by 34?   Eleanor:…
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