The work reported in this thesis involved the exploration of 12 year old children's conceptions of inheritance. Results derived from interviews and from videotaped recordings of small group discussions, indicate that children's conceptions of inheritance are well developed before they are formally taught these notions in school. A series of open-ended problem solving tasks were designed to elicit student's conceptions and to facilitate group discussion. Results suggest that children's conceptions of inherited characteristics are heterogeneous and 'organized' in a highly flexible way. A research model, based on the notion of conceptual ecosystems, was developed to provide a framework for data analysis. Features of conceptual ecosystems are described. It is suggested that such systems are characterised by their 'openness', adaptiveness and resilience. Results suggest that the heterogeneity, flexibility and fluctuating character of such ecosystems confer on conceptions the ability both to transform well and to resist change well. The study describes how these characteristics of resilience and adaptiveness are displayed in the cognitive and social interactions of individual students. A theory of conceptual change is advanced which considers learning as a series of continuous qualitative changes made by the learner to existing personal conceptions. The significance of these 'metatransitions' is discussed in the light of existing teaching and learning strategies. It is suggested that conceptual change can be facilitated by helping students to make their existing conceptions explicit, co-active and interactive within conceptual ecosystems. The social and cognitive consequences of conceptual conflict, disagreement and consensus are described. It is proposed that an ecosystemic view of children's conceptions may help explain and overcome the difficulties experienced by students when they try to reconcile scientific concepts with their existing conceptions.

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Initiatives in many countries to improve learner performances in mathematics in poor communities have been described as largely unsuccessful mainly due to their cursory treatment of curriculum alignment. Empirical evidence has shown that in high achieving countries the notion of coherence was strongly anchored in cognitively demanding mathematics programs. The view that underpins this study is that a cognitively demanding and coherent mathematics curriculum has potential to level the playing field for the poor and less privileged learners. In South Africa beyond 1994, little has been done to understand the potential of such coherent curriculum in the context of the NCSM. This study examined the levels of cognitive demand and alignment between the written, tested and taught NCSM. The study adopted Critical Theory as its underlying paradigm and used a multiple case study approach. Wilson and Bertenthal’s (2005) dimensions of curriculum coherence provided the theoretical framework while Webb’s (2002) categorical coherence criterion together with Porter’s (2004) Cognitive Demand tools were used to analyse curriculum and assessment documents. Classroom observations of lesson sequences were analysed following Businskas’ (2008) model of forms of mathematical connections since connections of different types form the bases for high cognitive demand (Porter, 2002). The results indicated that higher order cognitive skills and processes are emphasized consistently in the new curriculum documents. However, in the 2008 examination papers the first examinations of the new FET curriculum, lower order cognitive skills and processes appeared to be emphasized, a finding supported by Umalusi (2009) and Edwards (2010). Classroom observations pointed to teachers focusing more on rote learning of both concepts and procedures and less on procedural and conceptual understanding. Given the widespread evidence of the tested curriculum impacting on the taught curriculum, this study suggests that this lack of alignment between the advocated curriculum on one hand, the tested and the taught curricula on the other, needs to be investigated further for it endangers the teaching and learning of higher order cognitive skills and processes in the FET mathematics classrooms for the poor and less privileged. Broader evidence suggests that this would work against efforts towards supporting the upward mobility of poor children in the labour market.

The nature of geometric tasks that students engage with in classrooms influences the development of their geometric thinking. Although mathematics standards emphasize formal proofs and mathematical reasoning skills, geometric tasks in classrooms remain focused on students’ abilities to recall mathematical facts and use simple procedures rather than conceptual understanding. In order to facilitate students’ high-level mathematical thinking, teachers need to provide sufficient opportunities for students to engage in cognitively demanding mathematical tasks. The use of dynamic geometry software (DGS) in classrooms facilitates conceptual understanding of geometric proofs. The black box approach is a new type of task in which students interact with pre-constructed figures to explore mathematical relationships by dragging and measuring geometric objects. This approach is challenging to students because it “requires a link between the spatial or visual approach and the theoretical one” (Hollebrands, Laborde, & Sträßer, 2008, p. 172). This study examined how preservice secondary mathematics teachers make choices or create geometric tasks using DGS in terms of cognitive demand levels and how the black box approach influences the way preservice teachers conceptualize their roles in their lesson designs. Three preservice secondary mathematics teachers who took a semester-long mathematics teaching course participated in this qualitative case study. Data include two lesson plans, before and after instructions for geometric DGS tasks, pre- and post-interview transcripts, electronic files of geometric tasks, and reflection papers from each participant. The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2009) was used to characterize mathematical tasks with respect to level of cognitive demand. A Variety of geometric task types using DGS was introduced to the participants (Galindo, 1998). The dragging modalities framework (Arzarello, Olivero, Paola, & Robutti, 2002; Baccaglini-Frank & Mariotti, 2010) was employed to emphasize the cognitive demand of geometric tasks using DGS. The PURIA model situated the participants’ conceptualized roles in technology use (Beaudin & Bowers, 1997; Zbiek & Hollebrands, 2008). Findings showed that the preservice teachers only employed geometric construction types on low level geometric DGS tasks, which relied on technological step-by-step procedures students would follow in order to arrive at the same results. The preservice teachers transformed those low level tasks into high level tasks by preparing DGS tasks in advance in accordance with the black box approach and by encouraging students to explore the tasks by posing appropriate questions. However, as soon as they prepared high level DGS tasks with deductive proofs, low level procedure-based tasks followed in their lesson planning. The participants showed positive attitudes towards using DGS to prepare high level geometric tasks that differ from textbook-like procedural tasks. Major factors influencing preservice teachers’ preparation of high level tasks included teachers’ knowledge of mathematics, pedagogy, and technology, as well as ways of using curriculum resources and teachers’ abilities to set appropriate lesson goals. Findings of this investigation can provide guidelines for integrating DGS in designing high level geometric tasks for teacher educators, researchers, and textbook publishers.

Vid undervisning i matematik är det vanligt att undervisningen i stor utsträckning baseras på ett läromedel, både för lärarens teoribeskrivningar och vid elevernas enskilda arbete med uppgiftslösning. Det är därför viktigt att uppgifterna i läromedlet är utformade så att eleven kan utveckla de kunskapersom läroplanen föreskriver. I Sverige, liksom i flera andra länder i framförallt Europa och USA, beskrivs dessa matematiska kunskaper i termer av olika typer av förmågor. Elevers färdigheter i en viss förmåga utvecklas då eleven utför uppgifter där den specifika förmågan aktiveras på en nivå som innebär en kognitiv utmaning. Den önskvärda nivån på den kognitiva utmaningen beror på den enskilda elevens förkunskaper och förutsättningar. För att möjliggöra att alla elever skall kunna utveckla samtliga de efterfrågade förmågorna är det nödvändigt att läromedlet innehåller uppgifter med varierande nivå av kognitiv utmaning där samtliga förmågor aktiveras. Utredningar och studier från myndigheter och forskare antyder att undervisningen saknar den önskvärda variationen utan till stor del uppehåller sig vid procedurkunskap och tillämpningar av algoritmer. I syfte att undersöka variationen i nivå på kognitiv utmaning undersöker detta examensarbeteuppgifter inom området andragradsekvationer ur tre olika svenskspråkiga läromedel. Nivån på den kognitiva utmaningen definieras i denna studie som antalet olika former av svårigheter en uppgift innehåller. Den ökade kognitiva utmaningen består således i att fler förmågor aktiveras i samma uppgift. Analysen visar en majoritet av uppgifterna i samtliga läromedel har en låg nivå av kognitiv utmaning som inte erbjuder möjlighet för eleven att utveckla samtliga matematiska förmågor. Dessutomär antalet uppgifter av hög kognitiv nivå låg, så även vid ett aktivt urval av uppgifter av läraren är begränsas möjligheten till variation av läromedlen. Detta resultat är i linje med resultat från tidigareforskning. / When teaching mathematics, it is common that classroom activities to a large extent follow a textbook, both for the teacher’s presentations and for the students’ individual work on tasks. It is therefore important that the textbook is structured in a way that adheres to the requirements of the curricula. In Sweden, as well as in several other countries in primarily Europe and the US, mathematical knowledge is expressed in the form of different kind of capabilities or competencies. Students’ level of knowledge of a specific competence evolves from performing tasks with a cognitive demand to activate that competence. The desirable level of cognitive demand depends on the individual student’s ability and prior knowledge. To enable all students to develop all the required competencies, it is important that the textbook contains tasks with a varying level of cognitive demand where every competency is activated. Reports and studies from authorities and scientist indicate that classroom instruction often lack the desirable variation and instead focus on procedural knowledge and application of algorithms to a large degree. With the aim to study the variation of the level of cognitive demand this degree project investigates tasks on the subject of quadratic equations from three different upper secondary school textbooks written in Swedish. The level of cognitive demand is in this study defined as the number of different types ofdifficulties a task contains. A higher level of cognitive demand is thus interpreted as more competencies being activated in a single task. The analysis shows that majority of the tasks in all the textbooks have a low level of cognitive demand that do not offer opportunities for the student to evolve all the required competencies. In addition, the number of tasks with a high level of cognitive demand is low, restricting the possibility for the teacher to actively choose appropriate tasks from the textbook alone. This result is in line with earlier similar studies.

Drawing upon the writing literature and the task-based language teaching literature, the study examined two cognitive complexity dimensions of L2 writing tasks: rhetorical task varying in reasoning demand and topic familiarity varying in the amount of direct knowledge of topics. Four rhetorical tasks were studied: narrative, expository, expo-argumentative, and argumentative tasks. Three topic familiarity tasks were investigated: personal-familiar, impersonal-familiar, and impersonal-less familiar tasks. Specifically, the study looked into the effects of these two cognitive complexity dimensions on L2 writing quality scores, their effects on complexity, accuracy, and fluency (CAF) of L2 production, and the predictive power of the CAF features on L2 writing scores for each task. Three hundred and seventy five Chinese university EFL students participated in the study, and each student wrote on one of the six writing tasks used to study the cognitive complexity dimensions. The essays were rated by trained raters using a holistic scale. Thirteen CAF measures were used, and the measures were all automated through computer tools. One-way ANOVA tests revealed that neither rhetorical task nor topic familiarity had an effect on the L2 writing scores. One-way MANOVA tests showed that neither rhetorical task nor topic familiarity had an effect on accuracy and fluency of the L2 writing, but that the argumentative essays were significantly more complex in global syntactic complexity features than the essays on the other rhetorical tasks, and the essays on the less familiar topic were significantly less complex in lexical features than the essays on the more familiar topics. All-possible subsets regression analyses revealed that the CAF features explained approximately half of the variance in the writing scores across the tasks and that writing fluency was the most important CAF predictor for five tasks. Lexical sophistication was however the most important CAF predictor for the argumentative task. The regression analyses further showed that the best regression models for the narrative task were distinct from the ones for the expository and argumentative types of tasks, and the best models for the personal-familiar task were distinct from the ones for the impersonal tasks.

Promoting Mathematical Understanding Through Open-Ended Tasks; Experiences of an Eighth-Grade Gifted Geometry Class by Carol H. Taylor Gifted students of mathematics served through acceleration often lack the opportunities to engage in challenging, complex investigations involving higher-level thinking. This purpose of this study was to examine the ways mathematically gifted students think about and do mathematics creatively as indicators of deep understanding through collaborative work on four open-ended tasks with high-level cognitive demand. The study focused on the mathematical thinking involved in students’ construction of mathematical understanding through the social interaction of group problem solving. This case study used ethnographic methodology within a social constructivist frame with gifted education and sociocultural contextual influences. Participants were 15 gifted students in an 8th-grade gifted geometry class. Data collection included field notes, student artifacts, student journal entries, audio recordings, and reflections. Transcribed audio recordings were segmented (Tesch, 1990) into phases of interaction, coded by function, then coded by levels of exhibited mathematical thinking from observable cognitive actions (Dreyfus, Hershkowitz, & Schwarz, 2001; Williams, 2000; Wood, Williams, & McNeal, 2006), and analyzed for maintenance or decline of high-level cognitive demand (Stein, Smith, Henningsen, & Silver, 2000). Interpretive data analysis was connected to data analysis of transcribed recordings. Results indicated social interaction among students enabled them to talk through the mathematics to understand mathematical concepts and relationships, to construct more complex meaning, and exhibit mathematical creativity, inventiveness, flexibility, and originality. Students consistently exhibited these characteristics indicating mathematical thinking at the levels of building-with analyzing, building-with synthetic-analyzing, building-with evaluative-analyzing, constructing synthesizing, and occasionally constructing evaluating (Dreyfus et al., 2001; Williams, 2000; Wood et al., 2006). The results of the study support the claim of a relationship between mathematical giftedness and the ability to abstract and generalize (Sriraman, 2003), provide evidence that given the opportunity, students can construct deep mathematical understanding, and indicate the importance of social interaction in the construction of knowledge. This study adds to the body of knowledge needed in research on gifted education, problem solving, small-group interaction, mathematical thinking, and mathematical understanding, through empirically assessed classroom practice (Friedman-Nima et al., 2005; Good, Mulryan, & McCaslin, 1992; Hiebert & Carpenter, 1992; Lester & Kehle, 2003; Phillipson, 2007; Wood, Williams, & McNeal, 2006).

Some scholars have criticized the treatment of proportionality in middle-school textbooks, but these criticisms seem to be based on informal knowledge of the content of textbooks rather than on a detailed curriculum analysis. Thus, a curriculum analysis related to proportionality was needed. To investigate the treatment of proportionality in current middle-school textbooks, nine such books were analyzed. Sixth-, seventh-, and eighth-grade textbooks from three series were used: ConnectedMathematics2 (CMP), Glencoe's Math Connects, and the University of Chicago School Mathematics Project (UCSMP). Lessons with a focus on proportionality were selected from four content areas: algebra, data analysis/probability, geometry/measurement, and rational numbers. Within each lesson, tasks (activities, examples, and exercises) related to proportionality were coded along five dimensions: content area, problem type, solution strategy, presence or absence of a visual representation, and whether the task contained material regarding the characteristics of proportionality. For activities and exercises, the level of cognitive demand was also noted. Results indicate that proportionality is more of a focus in sixth and seventh-grade textbooks than in eighth-grade textbooks. The CMP and UCSMP series focused on algebra in eighth grade rather than proportionality. In all of the sixth-grade textbooks, and some of the seventh- and eighth-grade books, proportionality was presented primarily through the rational number content area. Two problem types described in the research literature, ratio comparison and missing value, were extensively found. However, qualitative proportional problems were virtually absent from the textbooks in this study. Other problem types (alternate form and function rule), not described in the literature, were also found. Differences were found between the solution strategies suggested in the three textbook series. Formal proportions are used earlier and more frequently in the Math Connects series than in the other two. In the CMP series, students are more likely to use manipulatives. The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2000) was used to measure the level of cognitive demand. The level of cognitive demand differed among textbook series with the CMP series having the highest level of cognitive demand and the Math Connects series having the lowest.

Younger adults who are walking and doing additional tasks at the same time may not realize if their performance suffers, putting some at greater risk for injury and impairment during certain tasks. This thesis has addressed this confound by developing a divided attention paradigm focusing on discrete and continuous demand manipulations. The work assessed in motorcognitive processing changes with cerebral and behavioral monitoring of over-ground walking with or without cognitive tasks. Participants (n = 19, 18-35 years, 13 females) were asked to walk at their usual pace [usual walking condition (SM)], walk at their usual pace while performing a cognitive task [dual-task condition (DT)] as well as conduct a cognitive task while standing [single cognitive condition (SC)]. All participants conducted two discrete [simple response time (SRT) & go-no-go (GNG)] and two continuous cognitive tasks [N-back (NBK) & double number sequence (DNS)] of increasing demand. The study revealed significant brain and behavior interactions during the most demanding continuous cognitive task, the DNS. The findings demonstrated lower accuracy rates, slower walk speeds as well as greater cerebral oxygenation in DNS DT in comparison to single task conditions. With increasing cognitive demands and tasks, there were longer response times, as well as lower accuracy rates. The behavioral findings were qualified by marginally significant interactions in a 2 x 4 RM ANOVA between SC-DT task and demand for accuracy rate [F (3, 54) = 2.66, p = 0.06, η2 =.13], significant interactions in response time [F (2, 36) = 4.1, p = 0.026, η2 =.18] as well as significant SM-DT task and demand findings for walk speed [F (3, 54) =5.3, p = 0.003, η2 =.23]. The 2 x 2 x 4 RM ANOVA revealed significant HbO2 interactions between walking tasks (single and dual), hemisphere and demand [F (3, 54) = 5.730, p = 0.002, η2 =.24] in the DNS only. The data suggests that greater demand manipulations with continuous cognitive tasks may be sensitive to both prefrontal cortex (PFC) and behavioral assessments in younger adults (YA). Further validation of the discrete-continuous demand paradigm in motor studies may provide a basis for cognitive assessment with applications in motor learning, cognitive training, aging and more.

High cognitive demand (HCD) tasks can help students develop a deeper understanding of mathematics. Teachers need interventions that encourage students to engage in HCD activities. Small-group discourse provides HCD opportunities for students while solving mathematics problems. Discourse can take place after students solve problems individually (reflective) or in groups as students solve problems (exploratory). This study looks at the relationship between these two types of small-group discourse and student-enacted cognitive demand. This study looks at how students engage with tasks that were designed at four different cognitive demand levels using Webb’s depth of knowledge (DOK) framework. Ninety-seven grade 5 students from four different classrooms were grouped in small groups of two or three students to solve two sets of mathematics problems on operations with fractions and decimals. Each class engaged in Reflective Discourse after solving one set and engage in Exploratory Discourse while solving the other set. To help understand any order effects, half the classes used Reflective Discourse with Set 1 while the other half used Exploratory Discourse with Set 1. Then, they switched for Set 2, so that whoever used Reflective Discourse with Set 1 used Exploratory Discourse with Set 2 and vice versa. The researcher analyzed whether there were patterns in levels of cognitive demand and quality of the discussion when students engaged in each type of discourse for math problems at four different levels. First, the researcher looked at any numerical differences between the intended cognitive demand of the problems and how students engaged with the problems using frequency tables, heat maps, and statistical analyses. Next, the researcher looked at differences in student actions and the way they talked about the math problems. Findings showed that both Reflective and Exploratory Discourse can be used by teachers to promote high student-enacted levels of cognitive demand. Results also showed that a supportive environment, such as the environment created by Reflective Discourse, can help support typically struggling students. Finally, this research reinforced the importance of dissonance in prompting students to engage with the tasks at higher levels of cognitive demand.