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: Land, water and tourism in Aitutaki, Cook Islands

authors: Cowling, Wendy E.

subjects: Cook Islands

abstract: Most island governments in the South Pacific have placed a great deal of faith in a continued annual increase in numbers of tourists as a guarantee of national economic growth. This is particularly the case in the Cook Islands. The response by local, mostly small-scale entrepreneurs has been to invest in the building of tourist accommodation, not only on the main island of Rarotonga, but also on the island of Aitutaki. Until recently the piecemeal development of accommodation on that island has occurred with little reference to the impact on the environment.

text: Land, water and tourism in Aitutaki, Cook Islands Most island governments in the South Pacific have placed a great deal of faith in a continued annual increase in numbers of tourists as a guarantee of national economic growth. This is particularly the case in the Cook Islands. The response by local, mostly small-scale entrepreneurs has been to invest in the building of tourist accommodation, not only on the main island of Rarotonga, but also on the island of Aitutaki. Until recently the piecemeal development of accommodation on that island has occurred with little reference to the impact on the environment.

year: 2004

title: Early French views of the Waikato and its inhabitants (1840-54)

authors: Jennings, William

abstract: This paper will investigate how the cultural background of these Frenchmen informed their descriptions of Waikato Maori and the Waikato environment. It will use the newly-published ten-volume collected correspondence of the pioneer Marists in the Pacific,the Lettres refues d'Odanie, as its principal source.

text: Early French views of the Waikato and its inhabitants (1840-54) This paper will investigate how the cultural background of these Frenchmen informed their descriptions of Waikato Maori and the Waikato environment. It will use the newly-published ten-volume collected correspondence of the pioneer Marists in the Pacific,the Lettres refues d'Odanie, as its principal source.

year: 2011