Connection

Dimensions, Connection
This category binds together certain choices and decisions that are made for the more or less discrete parts of mathematical content. It concerns the coherence of the planning or teaching displayed across an episode, lesson or series of lessons. Our conception of coherence includes the sequencing of topics of instruction within and between lessons, including the ordering of tasks and exercises which reflect deliberations and choices entailing both knowledge of structural connections within mathematics and an awareness of the relative cognitive demands of different topics and task contingency
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MCP: Scenario 1

Making connections between procedures
Scenario: Christiana teaching division by multiples of ten Country: Cyprus Grade (student age): Year 5 (age 10-11) Contributed by: Marilena Petrou Context – national, curricular, professional, other Christiana was a final year university student in a 4 year teacher prapeartion programme. To become a teacher in Cyprus you take part in a 4 year university degree in elementary education. Students are trained to teach all subjects and with the completion of the programme they are considered qualified teachers. During the last year of the progrmme students choose an area of specilisation and take part in a school based teaching experience.  Christiana was in her last years of her training and she chose to specialise in mathematics. Mathematics was one of her favourite subjects and she had a very positive attitude…
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MCP: Scenario 2

Making connections between procedures
Scenario: Jess teaching about the relationship between multiplication and division within the context of a lesson about solving word problems Country: UK Grade (student age): Year 5 (age 9-10) Contributed by: Fay Turner, University of Cambridge, UK Context – national, curricular, professional, other Jess had completed a one year graduate teacher programme the previous year.  The lesson took place in the second term of her first year of teaching.  The curriculum guidance for England at this time (2006) gave an objective that stated pupils in year 5 should ‘understand the effect of and relationship between the four operations, and the principles (not the names) of the arithmetic laws as they apply to multiplication’.  The main objective for this lesson was ‘Choose and use appropriate number operations and appropriate ways of…
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MCP: Scenario 3

Making connections between procedures
Scenario: Lucy teaching trigonometric ratios Country: UK Grade (student age): Year 10 (age 14-15) Contributed by: Anne Thwaites, University of Cambridge, UK Context – national, curricular, professional, other The National Curriculum for mathematics in England introduces trigonometric relationships in Key Stage 4 (years 10 and 11, pupil age 14-16). Lucy was reviewing the idea of a trigonometric ratio with her class, before moving on to discuss how to calculate the size of an angle given the lengths of two sides. Lucy 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 Lucy, a graduate student-teacher, was teaching in an open-entry school (pupil age 11-18) in a small town in the UK. The school divides each year into…
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MCC: Scenario 1

Making connections between concepts
Scenario: Bella discussing connections Country: UK Grade (student age): Trainee teacher Contributed by: Gwen Ineson, Brunel University, UK Context –professional Bella is at the end of her one year post-graduate teacher training course.  She has completed nineteen weeks of sustained school experience and her most recent experience was in a year 1 class of five and six year olds.  She was engaging in a discussion about her mathematical subject knowledge for teaching and how she would respond to pupils encountering difficulty with various numerical problems.  Questions included subtraction of a two digit number from a three digit number (234 – 48), multiplication of two digit numbers (52 x 34), multiplication of decimals (3.4 x 4.9) and division of fractions (1 ¾ ÷ ½) . The transcript included below is an…
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AC: Scenario 1

Anticipation of complexity
Scenario: Find 5/3 when 9 is the whole Country: Norway Grade (student age): Year 5 (age 10-11) Contributed by: Ove Gunnar Drageset, University of Tromsoe, Norway Context The teacher has 16 years of teaching experience. The lesson is from fifth grade (students aged 10-11) in the middle of the year. The students have only met fractions like ½ and ¼ used in everyday situations prior to fifth grade. 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 competence goals regarding fractions after grade seven focuses on calculations using positive and negative fractions, and placing fractions on a number line. The following excerpt is from one of the first lessons on fractions, and the…
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AC: Scenario 2

Anticipation of complexity
Scenario: Chloe teaching strategies for subtracting 9, 11, 19 and 21 Country: UK Grade (student age): Year 1/2 (age 5-7) Contributed by: Fay Turner, University of Cambridge, UK Context – national, curricular, professional, other Chloe was a student teacher in the final term of a one year postgraduate teacher education programme.  The lesson took place during her final school placement.  In planning this lesson Chloe referred to curriculum guidance in the National Numeracy Strategy framework (DfEE, 1999) in place at that time.  A mental calculation strategy suggested for Year 2 in this guidance was ‘add/subtract 9 or 11: add subtract 10 and adjust by 1.  Begin to add/subtract 19 or 21: add/subtract 20 and adjust by 1’.  Chloe was following up a lesson in which her class had used these…
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AC: Scenario 3

Anticipation of complexity
Scenario: Nóirín teaching fraction concepts Country: Ireland Grade (student age): 5th Class (age 10-11) Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland Context – national, curricular, professional, other Nóirín is an undergraduate student in her final year in a teacher education programme engaged in a Learning to Teach Mathematics Using Lesson Study elective course. This lesson was a ‘reteaching’ of a lesson which the group had amended following discussion of its first iteration. The initial lesson in a middle-class school was supported by a worksheet with six equal sized circles representing pizzas to be divided among a group of friends. On this occasion, in a designated disadvantaged setting, it was decided to use the same ‘problems’, but half the page was left blank so children could represent the problem…
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DS- Scenario 1

Decisions about sequencing
Scenario: Hans teaching fractions greater than one Country: Norway Grade (student age): Year 5 (age 11-12) Contributed by: Bodil Kleve, Oslo and Akershus University College of Applied Sciences Context – national, curricular, professional, other In our present curriculum LK 06, competence aims for the subject are presented after year 2, year 4, year 7 and year 10. This lesson is from year five. With regard to fractions, competence aims after year 4 do not include any. After year 7 fractions are included in Competence aims for numbers and algebra: “The aim for the education is that the pupil shall be able to describe the place value system for decimal numbers, calculate with positive and negative whole numbers, decimal numbers, fractions and percentages, and place them on the real number line…
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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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