Contingency

Dimensions, Contingency
This category concerns classroom events that are almost impossible to plan for. In commonplace language it is the ability to ‘think on one’s feet’. In particular, the readiness to respond to children’s ideas and a consequent preparedness, when appropriate, to deviate from an agenda set out when the lesson was prepared. A constructivist view of learning provides a valuable perspective on children’s contributions within lessons. To put aside such indications, or simply to ignore them or dismiss them as ‘wrong’, can be construed as a lack of interest in what it is that that child (and possibly others) have come to know as a consequence, in part, of the teacher’s teaching. However, Brown and Wragg (1993) observe that “our capacity to listen diminishes with anxiety” (p. 20). Uncertainty about the…
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Responding to students' ideas

Responding to students' ideas
Responding to Students’ Ideas (RSI)   This code includes the ability to make cogent, reasoned and well-informed responses to unanticipated ideas or suggestions from students These teachers’ responses are to students’ contributions to the (mathematical) development of the lesson. These contributions are typically oral, but could be written. Our analysis of the data available to us identifies three sub-types of triggers in this category: student’s response to a question from the teacher; student’s spontaneous response to an activity or discussion; student’s  incorrect answer - to a question, or as a contribution to a discussion. Furthermore, our data show that the teacher’s response is one of three kinds: to ignore; to acknowledge but put aside; to acknowledge and incorporate.
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RSI: Scenario 1

Responding to students' ideas
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 The scenario described below comes from a lesson taught by Elsa. Elsa, a final year university student in a teacher preparation programme, was teching problem solving using Schema-Based Instruction with a…
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RSI: Scenario 2

Responding to students' ideas
Scenario: Róisín teaching equivalence of fractions Country: Ireland Grade (student age): 3rd class (age 8-9) Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland Context – national, curricular, professional, other The Irish primary curriculum proposes that “the child should be enabled to identify fractions and equivalent forms of fractions with denominators 2, 4, 8 and 10” first among a list of six objectives for children in third class. Róisín, who was a second year Bachelor of Education student, with a relatively strong background in mathematics devised an ambitious lesson to develop an understanding of equivalent fractions with this class of 21 girls in the third week of her spring teaching placement. The lesson lasted 40 minutes. Scenario Róisín set the lesson in a pizzeria, with five friends sharing a “ten…
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RSI: Scenario 3

Responding to students' ideas
Scenario: Christiana teaching average Country: Cyprus Grade (student age): Year 6 (age 11-12) Contributed by: Marilena Petrou Context – national, curricular, professional, other Christiana was a final year university student in a teacher praparation programme , Teacher training in Cyprus involves taking part in a 4 year university degree in elementary education. Teachers are trained to teach all subjects of the school curriculum and during their last year the take part in a school based placement where they teach a series of lessons in all subjects throughout the year. With the completion of the programme the teachers are qualified to teach. The lesson took place in a school-based placement towards the end of her four-year teacher preparation. Scenario Christiana, was teaching mean with a Year 6 (pupils age 11-12) class.…
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RSI: Scenario 4

Responding to students' ideas
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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RSI: Scenario 5

Responding to students' ideas
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. Heidi 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 Heidi, a graduate student-teacher, was teaching in an open entry school (pupil age 11-18) in a village in the UK. The school divides the year group into maths sets (by ability) and Heidi was teaching one of…
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Deviation from lesson agenda

Deviation from lesson agenda
Contingency, Deviation from Agenda Descriptors / aspects of DA Good examples: When the teacher displays deeper subject matter knowledge to enhance pupils’ understanding When the teacher displays deeper subject matter knowledge in taking a pupil’s remark as a starting point for deeper enquiry Spending time in questioning /probing pupils in order to find out why a pupil comes up with a wrong answer Deviation characterized by conceptual focus Presentation of the concept in a new way Illuminating with an everyday example Bad examples: Letting a pupil hijack the lesson which may result in less mathematical focus When the DA is “privatized” – the lesson is deviated in direction of only one (or a few) pupil(s) Only repeating previous lesson content Displaying mathematical uncertainty
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Teacher insight

Teacher insight
TI ‘Good’ examples Teacher stops to reflect-in- action and changes tack (example or representation being used) Explains that perhaps something else might work better & explains why Says, ‘Gosh, I hadn’t thought of that’ or says ‘Let’s retrace our steps’ & perhaps work backwards Realising that children are constructing the mathematical ideas and something that sounds ‘half baked ‘ means that they are in a ZPD where teacher can help with a scaffolding question or two. Probing questions to try to elicit where child’s thinking is coming from; I’m not sure about this but it could point to a stance or openness to teacher insight ?? Bad examples Never mind what (Mammy/Daddy or Anyone Else) says,this is the way I want you to do it. That’s wrong! When its not,…
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Responding to (un)availability of tools and resources

Responding to the (un)availability of tools and resources
Good examples:   Draws on alternative knowledge resources, and/or makes significant epistemological accommodation in response to lack of intended technology or resource. Draws on alternative knowledge resources, and/or makes significant epistemological accommodation in response to availability of unplanned for technology or resource. Finds an alternative means to explain a concept or demonstrate a procedure in the intended way following failure of technology or lack of expected resource.   Bad examples:   Unplanned technology/resource presents a representation that does not support understanding of the concepts or procedures being taught. Examples used are not appropriate for developing understanding when used in relation to the unplanned representation. Does not alter planned lesson script, including predetermined examples, to accommodate the new technology or resource.  
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