Investigating Formative Assessment: Exploring the Impact on the Self-Efficacy and Motivation of Mathematics Students

This is a study of the implementation and impact of formative assessment strategies on the motivation and self-efficacy of secondary school mathematics students. An explanatory sequential mixed methods design was implemented where quantitative and qualitative data were collected and analyzed sequentially in 2 different phases. The first phase involved quantitative data from student questionnaires and the second phase involved qualitative data from individual student and teacher interviews. The findings of the study suggest that formative assessment is implemented in practice in diverse ways and is a process where the strategies are interconnected. Teachers experience difficulty in incorporating peer and self-assessment and perceive a need for exemplars. Key factors described as influencing implementation include teaching philosophies, interpretation of ministry documents, teachers’ experiences, leadership in administration and department, teacher collaboration, misconceptions of teachers, and student understanding of formative assessment. Findings suggest that overall, formative assessment positively impacts student motivation and self-efficacy, because feedback is provided which offers encouragement and recognition by highlighting the progress that has been made and what steps need to be taken to improve. However, students are impacted differently with some considerations including how students perceive mistakes and if they fear judgement. Additionally, the impact of formative assessment is influenced by the connection between self-efficacy and motivation, namely how well a student is doing is a source of both concepts.

Teachers' professional learning within communities of mathematics-for-teaching practice

Ontario bansho is an emergent mathematics instructional strategy used by teachers working within communities of practice that has been deemed to have a transformational effect on teachers' professional learning of mathematics. This study sought to answer the following question: How does teachers' implementation of Ontario bansho within their communities of practice inform their professional learning process concerning mathematics-for-teaching? Two other key questions also guided the study: What processes support teachers' professional learning of content-for-teaching? What conditions support teachers' professional learning of content-for-teaching? The study followed an interpretive phenomenological approach to collect data using a purposive sampling of teachers as participants. The researcher conducted interviews and followed an interpretive approach to data analysis to investigate how teachers construct meaning and create interpretations through their social interactions. The study developed a model of professional learning made up of 3 processes, informing with resources, engaging with students, and visualizing and schematizing in which the participants engaged and 2 conditions, ownership and community that supported the 3 processes. The 3 processes occur in ways that are complex, recursive, nonpredictable, and contextual. This model provides a framework for facilitators and leaders to plan for effective, content-relevant professional learning by placing teachers, students, and their learning at the heart of professional learning.

The use of explicit algorithms and episodic context to teach subtraction to students with learning problems in mathematics /

This research attempted to address the question of the role of explicit algorithms and episodic contexts in the acquisition of computational procedures for regrouping in subtraction. Three groups of students having difficulty learning to subtract with regrouping were taught procedures for doing so through either an explicit algorithm, an episodic content or an examples approach. It was hypothesized that the use of an explicit algorithm represented in a flow chart format would facilitate the acquisition and retention of specific procedural steps relative to the other two conditions. On the other hand, the use of paragraph stories to create episodic content was expected to facilitate the retrieval of algorithms, particularly in a mixed presentation format. The subjects were tested on similar, near, and far transfer questions over a four-day period. Near and far transfer algorithms were also introduced on Day Two. The results suggested that both explicit and episodic context facilitate performance on questions requiring subtraction with regrouping. However, the differential effects of these two approaches on near and far transfer questions were not as easy to identify. Explicit algorithms may facilitate the acquisition of specific procedural steps while at the same time inhibiting the application of such steps to transfer questions. Similarly, the value of episodic context in cuing the retrieval of an algorithm may be limited by the ability of a subject to identify and classify a new question as an exemplar of a particular episodically deflned problem type or category. The implications of these findings in relation to the procedures employed in the teaching of Mathematics to students with learning problems are discussed in detail.

Exploring the impact of a hands-on intervention on grade 7 girls and their feelings about self, mathematics, science, technology and related careers

Forty-five 12- and 13-year-old females attending Grade 7 in North York, Ontario were randomly selected from a group of 100 females who had volunteered to participate in a oneday hands-on workshop called It's Your Choice at Seneca College. The goals of this intervention were to broaden the career horizons of these students and to help them realize the need to continue mathematics and science through high school in order to keep occupational options unlimited. The young women were given a pre- and post-attitude survey to provide background information. In the month following participation in the workshop the students were interviewed in small groups (S students per group) to discover their perceptions of the impact of the workshop. The interviews revealed that participants felt that after the workshop their feelings of self-confidence increased, specifically with respect to working with their hands. Participants felt more aware of the usefulness and importance of the study of mathematics, science and technology, They also felt that It's Your Choice increased their interest in careers in these domains and helped them to see that these careers are viable choices for females. The interviews also revealed that many of the participants felt that in this society their roles and their choices were influenced and probably limited by the fact that they are female.

The impact of integrated programming on student attitude and achievement in grade 9 academic mathematics and science

This research studioo the effect of integrated instruction in mathematics and~ science on student achievement in and attitude towards both mathematics and science. A group of grade 9 academic students received instruction in both science and mathematics in an integrated program specifically developed for the purposes of the research. This group was compared to a control group that had received science and mathematics instruction in a traditional, nonintegrated program. The findings showed that in all measures of attitude, there was no significant difference between the students who participated in the integrated science and mathematics program and those who participated in a traditional science and mathematics program. The findings also revealed that integration did improve achievement on some of the measures used. The performance on mathematics open-ended problem-solving tasks improved after participation in the integrated program, suggesting that the integrated students were better able to apply their understanding of mathematics in a real-life context. The performance on the final science exam was also improved for the integrated group. Improvement was not noted on the other measures, which included EQAO scores and laboratory practical tasks. These results raise the issue of the suitability of the instruments used to gauge both achievement and attitude. The accuracy and suitability of traditional measures of achievement are considered. It is argued that they should not necessarily be used as the measure of the value of integrated instruction in a science and mathematics classroom.

What nourishes the spirit of adolescents in the classroom? : a qualitative study into the gateways of adolescents' souls /

This thesis investigated the question, "What nourishes the adolescent spirit in the classroom?" Action research was conducted by the teacher of 16 grade 8 participants. By undergoing Kessler's (2000) "Passages Program," the students participated in 6 sessions which exposed them to a holistic intrapersonal curriculum. Student journal responses were documented after each session. The action researcher also kept journal reflections after each session. Upon completion of the 6 sessions, a postinterview was conducted which posed the research question. The research found that the adolescent spirit gets nourished through encouragement by the teacher, peers, and parents. This increases their competency, which increases their confidence, and ultimately their self-esteem goes up, which affects their selfconcept. In addition, the role of the teacher permeates every aspect of what nourishes the adolescent spirit in the classroom. In addition to the encouragement of the student, how a teacher teaches plays a vital role. A holistic approach to teaching provided the best atmosphere for the adolescent. It promoted creativity and choice, which stimulated the spirit of the adolescent. By working from a holistic philosophy, the teacher/action researcher created an environment conducive to teaching the whole person, which ultimately nourished the participants' spirit. The research highlighted that in order for this type of environment to exist the teacher needs to make a conscious and deliberate effort to look within and develop their inner self before they can begin to promote this type of classroom for the adolescents they teach. When teachers and students develop an inner life together, they can begin to work in harmony to achieve an atmosphere where the teaching and learning environment becomes one seamless transaction. Only then can one's whole potential be realized.

Completing the picture: the effects of video game practice on verbal and performance I.Q. scores in high functioning children

The strength and nature of the video game practice effect on tests of visual and perceptual skills were examined using high functioning Grades Four and Five students who had been tested with the WISC-R .for the purpose of gifted identification and placement. The control group, who did not own and .play video games on a sustained basis, and the experimental group, who did own a video game system and had some mastery of video games, including the -Nintendo game, "Tetris", were each composed of 18 juniorg:r;-ade students and were chosen from pre-existing conditions. The experimental group corresponded to the control group in terms of age, sex, and community. Data on the Verbal and Performance I.Q. Scores were· collected for both groups and the author was interested in the difference between the Verbal and Performance Scores within each group, anticipating a P > V outcome for the experimental group. The results showed a significant P > V difference in the experimental, video game playing group, as expected, but no significant difference between the Performance $cores of the control and experimental groups. The results, thus, indicated lower Verbal I.Q. Scores in the experimental group relat'ive to 'the control group.' The study conclu~ed that information about a sUbject's video game experience and "learhing style pref~rence is important for a clear interpretation of the Verbal and Performance I.Q. Scores of the WISC-R. Although the time spent on video game play may, 'indeed, increase P~rformance Scores relative to Verbal Scores for an individual, the possibilities exist that the time borrowed and spent away from language based activities may retard verbal growth and/or that the cognitive style associated with some Performance I.Q.subtests may have a negative effect on the approach to the tasks on the Verbal I.Q. Scale. The study also discussed the possibility that exposure to ,the video game experience, in pre-puberty, can provide spatial instruction which will result in improved spatial skills. strong spatial skills have been linked to improved performance and preference in mathematics, science, and engineering and it was suggested that appropriate video game play might be a way to involve girls more in the fields of mathematics and science.

Women’s Literary Club of St. Catharines Fonds

The Women's Literary Club of St. Catharines was founded in 1892 by a local author, Emma Harvey (Mrs. J.G.) Currie (1829-1913) and held its last official meeting on February 19, 1994. The Club developed, flourished and eventually waned. After more than one hundred successful years, the last members deposited the Club's archives at Brock University for the benefit of researchers, scholars and the larger community. The ‘object of the Club’ was established as “the promotion of literary pursuits.” The Club was a non-profit social organization composed of predominantly white, upper middle class women from the St. Catharines and surrounding areas. Club meetings were traditionally held fortnightly from March to December each year. The last meeting of the year was a celebration of their Club anniversary. The early meetings of the Club include papers presented and music performed by Club members. The literary pursuits that would dominate the agendas for the entire life of the Club reflected an interest in selected authors, national and local history, classical history, musical performances and current cultural and newsworthy events. For example in 1893 a typical meeting agendas would contain papers on Henry Wadsworth Longfellow, Hawaii, Brook Farm, Miss Louisa May Alcott and “Education of Women 100 years Ago.” Within the first year of the Club’s existence, detailed minute books became the norm and an annual agenda or program developed. The WLC collection contains a near complete set of meeting minutes from 1892 until 1995 and a comprehensive collection of yearly programs from 1983-1967 which members took great care to publish each year. Mrs. Currie brought together a group of women with a shared interest in literature and history, who wanted to pursue that interest in a formal and structured manner. She was well educated and influenced at an early age by her tutor and mentor William Kirby, local historian, writer and newspaper editor from Niagara-on-the-Lake. While Currie’s private education influenced her love of literature and history, the Club movement of the 1890’s offered a more public forum for her to share knowledge and learning with other women. Mrs. Currie was the wife of St. Catharines lawyer, James G. Currie, who also served as a Member of Parliament for the county of Lincoln. Mrs. W.H. McClive, who was also married to a St. Catharines lawyer, worked closely with Currie and they began research into the possibility of a literary Club in St. Catharines. Currie corresponded with a variety of literary Clubs across North America before she and Mrs.McClive tagged onto the momentum of the Club movement and published “A Clarion call for Women of St. Catharines To Form a Literary Club” in the local paper The St. Catharines Evening Journal. in 1892 and asked like Clubs to publish the news of their new Club. The early years of the WLC set the foundation of how the Club meetings and events would unfold for the next 80 plus years. Photos and minutes from the first ten years reveal an excitement and interest in organized Club outings. One particular event, an annual pilgrimage to the homestead of Laura Secord, became a yearly celebration for the Club. Club President, Mrs. Currie’s own personal work on Laura Secord amplified the Club’s interest in the ‘heroine of 1812’ and she allocated the profits from her publication on Secord in order to create a commemorative plaque/monument in the name of Laura Secord. The Club celebrated this event with a regular pilgrimage to this site. The connection felt by Club members and this memorial would continue until the Club’s last meetings. The majority of members in the early years were of the upper middle classes in the growing city of St. Catharines. Many of the charter members were the wives of merchants, business men, lawyers, doctors, even a hatter. Furthermore, the position of president was most often held by a woman with a comprehensive list of interests. This is particularly the case in Isabel Brighty McComb (1876-1941). Brighty who became a member in 1903, became Club president in 1932 and stayed in her post until her death in 1941. Similar to Mrs. Currie, Brighty was a local historian and published 2 booklets on local history. Her obituary indicates her position in the community as an author and involved community member committed to lifetime memberships in the Imperial Order of Daughters of Empire, I.O.D.E., the National Organization of Women, N.O.W. and the United Empire Loyalist Society, as well as the WLC. She was a locally known ‘teacher of elocution’ and a devoted researcher of Upper Canadian history. In a Club scrapbook dedicated to her, the biographical sketch illustrates the professionalism surrounding Brighty. There is very little personal history mentioned and the focus is on her literary works, her published essay, booklets and poetry. This professional focus, evident in both her obituary and the scrapbook, illustrate the diversity of these women, especially in their roles outside of the home. The WLC collection contains a vast array of essay, lectures clippings and scrapbooks from past meetings. Organized predominantly by topic or author, the folders and scrapbooks offer a substantial amount of research opportunity in the literary history of Canada. The dates, scope of topics and authors covered offer historians an exciting opportunity to examine the consumption of particular literary trends, artists and topics within the context of a midsized industrial city in English Canada. This is especially important because the agenda adhered to by the Club was bent on promoting, discussing and reviewing predominantly Canadian material. By connecting when and what these women were studying, scholars many gain a better understanding of the broader consumption and appreciation of literary and social trends of Canadian women outside of publishing and institutional records. Furthermore, because the agendas were set by and for these women, outside of the constructs of an institutionalized canon or agenda, they offer a fresh and on the ground examination of literary consumption over an extensive length of time.

The von Neumann Minimax Theorem and its relatives and a study of externality in on-line auctions

This work consists of a theoretical part and an experimental one. The first part provides a simple treatment of the celebrated von Neumann minimax theorem as formulated by Nikaid6 and Sion. It also discusses its relationships with fundamental theorems of convex analysis. The second part is about externality in sponsored search auctions. It shows that in these auctions, advertisers have externality effects on each other which influence their bidding behavior. It proposes Hal R.Varian model and shows how adding externality to this model will affect its properties. In order to have a better understanding of the interaction among advertisers in on-line auctions, it studies the structure of the Google advertisements networ.k and shows that it is a small-world scale-free network.

Response curves of deterministic and probabilistic cellular automata in one and two dimensions

One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules.

Acyclic 5-Choosability of Planar Graphs Without Adjacent Short Cycles

The conjecture claiming that every planar graph is acyclic 5-choosable[Borodin et al., 2002] has been verified for several restricted classes of planargraphs. Recently, O. V. Borodin and A. O. Ivanova, [Journal of Graph Theory,68(2), October 2011, 169-176], have shown that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3<=j<=5 if i=3 and 4<=j<=6 if i=4. We improve the above mentioned result and prove that every planar graph without an i-cycle adjacent to a j-cycle with3<=j<=5 if i=3 and 4<=j<=5 if i=4 is acyclically 5-choosable.

Edge-choosability of Planar Graphs

According to the List Colouring Conjecture, if G is a multigraph then χ' (G)=χl' (G) . In this thesis, we discuss a relaxed version of this conjecture that every simple graph G is edge-(∆ + 1)-choosable as by Vizing’s Theorem ∆(G) ≤χ' (G)≤∆(G) + 1. We prove that if G is a planar graph without 7-cycles with ∆(G)≠5,6 , or without adjacent 4-cycles with ∆(G)≠5, or with no 3-cycles adjacent to 5-cycles, then G is edge-(∆ + 1)-choosable.

Backbone Colouring of Graphs

Consider an undirected graph G and a subgraph of G, H. A q-backbone k-colouring of (G,H) is a mapping f: V(G) {1, 2, ..., k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least q. The minimum number k for which there is a backbone k-colouring of (G,H) is the backbone chromatic number, BBCq(G,H). It has been proved that backbone k-colouring of (G,T) is at most 4 if G is a connected C4-free planar graph or non-bipartite C5-free planar graph or Cj-free, j∈{6,7,8} planar graph without adjacent triangles. In this thesis we improve the results mentioned above and prove that 2-backbone k-colouring of any connected planar graphs without adjacent triangles is at most 4 by using a discharging method. In the second part of this thesis we further improve these results by proving that for any graph G with χ(G) ≥ 4, BBC(G,T) = χ(G). In fact, we prove the stronger result that a backbone tree T in G exists, such that ∀ uv ∈ T, |f(u)-f(v)|=2 or |f(u)-f(v)| ≥ k-2, k = χ(G). For the case that G is a planar graph, according to Four Colour Theorem, χ(G) = 4; so, BBC(G,T) = 4.

Optimal and Robust Designs of Step-stress Accelerated Life Testing Experiments for Proportional Hazards Models

Accelerated life testing (ALT) is widely used to obtain reliability information about a product within a limited time frame. The Cox s proportional hazards (PH) model is often utilized for reliability prediction. My master thesis research focuses on designing accelerated life testing experiments for reliability estimation. We consider multiple step-stress ALT plans with censoring. The optimal stress levels and times of changing the stress levels are investigated. We discuss the optimal designs under three optimality criteria. They are D-, A- and Q-optimal designs. We note that the classical designs are optimal only if the model assumed is correct. Due to the nature of prediction made from ALT experimental data, attained under the stress levels higher than the normal condition, extrapolation is encountered. In such case, the assumed model cannot be tested. Therefore, for possible imprecision in the assumed PH model, the method of construction for robust designs is also explored.