## Knowledge Quartet Framework

This is an image that we use to indicate our conceptualization of and the connections between the four dimensions in the Knowledge Quartet.

This is an image that we use to indicate our conceptualization of and the connections between the four dimensions in the Knowledge Quartet.

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…

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…

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…

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…

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.

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…

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…

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…