title: The development of multiplicative thinking and proportional reasoning: Models of conceptual learning and transfer

authors: Wright, Vincent John

subjects: proportional reasoning

abstract: This thesis considers the development of multiplicative thinking and proportional reasoning from two perspectives. Firstly, it examines the research literature on progressions in conceptual understanding to create a Hypothetical Learning Trajectory (HLT). Secondly, it surveys modern views of how transfer by learners occurs in and between situations, contrasting object views of abstraction with knowledge in pieces views.
Case studies of six students aged 11-13 years illustrate conceptual changes that occur during the course of a school year. The students are involved in a design experiment in which I (the researcher) co-teach with the classroom teacher. The students represent a mix of gender, ethnicity and level of achievement. Comparison of the HLT with the actual learning trajectory for each student establishes its validity as a generic growth path.
Examination of the data suggests that two models of learning and by inference, transfer, describe the conceptual development of the students. There is consideration of students’ use of anticipated actions on physical and imaged embodiments as objects of thought with a focus on the significance of object creation for conceptual growth, and the encapsulation, completeness and contextual detachment of objects. 
There is broad consistency in students’ progress through the phases of the HTL within each sub-construct though the developmental patterns of individual are variable and temporal alignment across the sub-constructs does not uniformly hold. Some consistency of order effect in concept development is noted. Discussion on the limitations of the HTL includes the difference between knowledge types from a pedagogical perspective, absence of significant model-representation-situation transfer, and order relations in conceptual development.
Considerable situational variation occurs as students solve problems that involve applications of the same concepts. Partial construction of concepts is common. This was true of all learners, irrespective of level of achievement. High-achieving students more readily anticipate actions and trust these anticipations as objects of thought than middle and low achievers. The data supports knowledge in pieces views of conceptual development. Complexity for learners in observing affordances in situations, and in co-ordinating the fine-grained knowledge required, explains the difficulty of transfer. While supporting the anticipation of action as significant from a learning perspective the research suggests that expertise in applying concepts involves a process of noticing similarity across contextually bound situations and cueing appropriate knowledge resources.

text: The development of multiplicative thinking and proportional reasoning: Models of conceptual learning and transfer This thesis considers the development of multiplicative thinking and proportional reasoning from two perspectives. Firstly, it examines the research literature on progressions in conceptual understanding to create a Hypothetical Learning Trajectory (HLT). Secondly, it surveys modern views of how transfer by learners occurs in and between situations, contrasting object views of abstraction with knowledge in pieces views.
Case studies of six students aged 11-13 years illustrate conceptual changes that occur during the course of a school year. The students are involved in a design experiment in which I (the researcher) co-teach with the classroom teacher. The students represent a mix of gender, ethnicity and level of achievement. Comparison of the HLT with the actual learning trajectory for each student establishes its validity as a generic growth path.
Examination of the data suggests that two models of learning and by inference, transfer, describe the conceptual development of the students. There is consideration of students’ use of anticipated actions on physical and imaged embodiments as objects of thought with a focus on the significance of object creation for conceptual growth, and the encapsulation, completeness and contextual detachment of objects. 
There is broad consistency in students’ progress through the phases of the HTL within each sub-construct though the developmental patterns of individual are variable and temporal alignment across the sub-constructs does not uniformly hold. Some consistency of order effect in concept development is noted. Discussion on the limitations of the HTL includes the difference between knowledge types from a pedagogical perspective, absence of significant model-representation-situation transfer, and order relations in conceptual development.
Considerable situational variation occurs as students solve problems that involve applications of the same concepts. Partial construction of concepts is common. This was true of all learners, irrespective of level of achievement. High-achieving students more readily anticipate actions and trust these anticipations as objects of thought than middle and low achievers. The data supports knowledge in pieces views of conceptual development. Complexity for learners in observing affordances in situations, and in co-ordinating the fine-grained knowledge required, explains the difficulty of transfer. While supporting the anticipation of action as significant from a learning perspective the research suggests that expertise in applying concepts involves a process of noticing similarity across contextually bound situations and cueing appropriate knowledge resources.

year: 2011

title: 'Not too calm'

authors: Lodge, Martin

abstract: At the time of his death in September 1999, music historian John Mansfield Thomson left a number of projects unfinished. One of these projects was an autobiography. In the course of my duties as executor of his literary estate, I subsequently discovered the beginnings of autobiographical writing on John's computer hard drive, and it was possible to include these fragments in the 2003 book John Mansfield Thomson: Notes towards a biography'.

text: 'Not too calm' At the time of his death in September 1999, music historian John Mansfield Thomson left a number of projects unfinished. One of these projects was an autobiography. In the course of my duties as executor of his literary estate, I subsequently discovered the beginnings of autobiographical writing on John's computer hard drive, and it was possible to include these fragments in the 2003 book John Mansfield Thomson: Notes towards a biography'.

year: 2005

title: Intersecting knowledge worlds: Describing the cultural interfaces within a socioscientific issue

authors: Wagner, Brent

subjects: cultural interface

abstract: Increasing numbers of science teachers in Aotearoa New Zealand are exploring ways to incorporate indigenous mātauranga Māori (Māori knowledge and ways of knowing) into their science programmes. This article presents findings from a study that examined how senior biology students in two English-medium secondary schools incorporated mātauranga Māori alongside biological science knowledge in their discussions about the use of 1080 poison to control mammalian predators. From a social constructivist ontology and interpretivist mode of inquiry using qualitative methodology, a triadic epistemic agency conceptual framework was developed inductively and used to analyse students’ written scripts for their national school leaving qualification (NCEA), and transcriptions of individual interviews. The study took place prior to the introduction of equal status to mātauranga Māori within the New Zealand Curriculum. Deductive analysis using the Triadic identified a group of ten students who drew from both knowledge systems, with varying degrees of knowledge integration. In addition to detailed representations of the theoretical space of the locale of the learner, the findings also identified three distinctive cultural interfaces occupied by the students: conflicted, parallel, and, connected. The findings underscore the importance of enabling diverse knowledge systems in biology classes to converge. Specifically, the incorporation of mātauranga Māori alongside biological science knowledge encourages a critical stance towards knowledge claims, supports pluralist knowledge use among culturally and linguistically diverse learners and promotes critical engagement with socioscientific issues (SSI) in senior secondary school biology.

text: Intersecting knowledge worlds: Describing the cultural interfaces within a socioscientific issue Increasing numbers of science teachers in Aotearoa New Zealand are exploring ways to incorporate indigenous mātauranga Māori (Māori knowledge and ways of knowing) into their science programmes. This article presents findings from a study that examined how senior biology students in two English-medium secondary schools incorporated mātauranga Māori alongside biological science knowledge in their discussions about the use of 1080 poison to control mammalian predators. From a social constructivist ontology and interpretivist mode of inquiry using qualitative methodology, a triadic epistemic agency conceptual framework was developed inductively and used to analyse students’ written scripts for their national school leaving qualification (NCEA), and transcriptions of individual interviews. The study took place prior to the introduction of equal status to mātauranga Māori within the New Zealand Curriculum. Deductive analysis using the Triadic identified a group of ten students who drew from both knowledge systems, with varying degrees of knowledge integration. In addition to detailed representations of the theoretical space of the locale of the learner, the findings also identified three distinctive cultural interfaces occupied by the students: conflicted, parallel, and, connected. The findings underscore the importance of enabling diverse knowledge systems in biology classes to converge. Specifically, the incorporation of mātauranga Māori alongside biological science knowledge encourages a critical stance towards knowledge claims, supports pluralist knowledge use among culturally and linguistically diverse learners and promotes critical engagement with socioscientific issues (SSI) in senior secondary school biology.

year: 2026