Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

Lifelong learning and the knowledge economy: those that know and those that do not—the discourse of the European Union

This article is based on a textual analysis of European Commission documents that, from 1993 to 2006, construct the discourses of lifelong learning and the knowledge economy. Exploring an apparent conceptual laxity, it finds absolute consistency in the construction of two categories of learner: the high knowledge‐skilled learner (graduate/postgraduate) for the knowledge economy, and the low knowledge‐skilled learner located in (or beyond) the knowledge society. The low knowledge‐skilled learners are not only those at risk, they are increasingly constructed as the risk. The analysis suggests that the binary classification is initially classed and raced—and only then is it gendered. In contrast, labour market studies of the knowledge economy, providing either gendered or national data, obscure the vital cross‐cutting matrix of social class, ‘race’ and age. The article advocates further studies of lifelong learning practices and labour market data based on finely‐crossed analyses of social class, poverty, age and race.     

The design process for PLATO®

Conclusion The final product, PLATOMath Problem Solving, was the result of many months of design, iteration, learner trials, and hard work. Each activity has enough depth to promote exploration, but enough coaching to provide structure and establish a strategic-level dialog with the learner. The result is courseware that truly promotes strategic approaches to math problem solving. For more information, visit us atwww.plato.com. You might also be interested in the following : Hall, E. P., Gott, S. P., & Pokorny, R. A. (1995). Aprocedural guide to cognitive task analysis: The PARI methodology. Report AL/HR-TR-1995-0108. Brooks Air Force Base, TX: Human Resources Directorate. Foshay, R., & Kirkley, J. (1998).Principles for teaching problem solving. Technical Paper #4. Bloomington, MN: PLATO Learning, Inc.     

Exploring pre-service student-teachers' commitment to community engagement in the second year of training

This preliminary study examines the utility of the Community Engagement Commitment Scale and reports on factors that predict student-teachers' commitment to community engagement in their second year of study at a South African university. Independent variables included Dalbert's personal belief in a just world, an adaptation of Rusbult et al.'s original predictors of Commitment level (Satisfaction, Quality of alternatives, Investment size) and a new factor, Meaningfulness. The latter factor is based theoretically on the work of Lieberman and Heine, Proulx and Vohs and was included to study the extent to which identity-expression could be related to Commitment level. We administered the Community Engagement Commitment Scale on a non-random sample of 414 student-teachers in their second year of study, who all participated in compulsory community engagement project. The results support a four-factor model of Commitment with Meaningfulness, Quality of alternatives and Investment as predictors of Commitment level. Surprisingly, all the Satisfaction items loaded with the Commitment level items on one factor. Together, Meaningfulness, Quality of alternatives and Investment accounted for approximately two-thirds of the variance in Commitment level. Significant (p?Meaningfulness and Commitment level. The results of the study suggest the importance of studying meaningfulness as an additional predictor of Commitment level in community engagement contexts.     

Inter-level Scaffolding and Sequences of Representational Activities in Teaching a Chemical System with Graphical Simulations

Chemistry knowledge can be represented at macro-, micro- and symbolic levels, and learning a chemistry topic requires students to engage in multiple representational activities. This study focused on scaffolding for inter-level connection-making in learning chemistry knowledge with graphical simulations. We also tested whether different sequences of representational activities produced different student learning outcomes in learning a chemistry topic. A sample of 129 seventh graders participated in this study. In a simulation-based environment, participants completed three representational activities to learn several ideal gas law concepts. We conducted a 2 × 3 factorial design experiment. We compared two scaffolding conditions: (1) the inter-level scaffolding condition in which participants received inter-level questions and experienced the dynamic link function in the simulation-based environment and (2) the intra-level scaffolding condition in which participants received intra-level questions and did not experience the dynamic link function. We also compared three different sequences of representational activities: macro-symbolic-micro, micro-symbolic-macro and symbolic-micro-macro. For the scaffolding variable, we found that the inter-level scaffolding condition produced significantly better performance in both knowledge comprehension and application, compared to the intra-level scaffolding condition. For the sequence variable, we found that the macro-symbolic-micro sequence produced significantly better knowledge comprehension performance than the other two sequences; however, it did not benefit knowledge application performance. There was a trend that the treatment group who experienced inter-level scaffolding and the micro-symbolic-macro sequence achieved the best knowledge application performance.     

Linking logo,levels and language in mathematics

In his book Mindstorms (1980) Papert discusses Turtle Geometry as ego-syntonic or fitting the ways of thinking of the child as a geometric knowledge builder. The van Hiele theory (Wirzup, 1976) looks at geometric knowledge building as occuring through a necessary sequence of levels. Thus if Turtle Geometry is ego-syntonic it would seem that one could apply these van Hiele levels to better describe and understand this form of geometry. It is the first purpose of this paper to provide such an analysis.Using Logo to do Turtle Geometry requires that a child (or learner at any level) do geometry through the deliberate use of language. Thus one ought also to be able to relate theoretically levels of language use to Turtle Geometry. It is a second purpose of this paper to relate a language use framework suggested by the work of Frye (1982) to the language activities of Turtle Geometry.     

Is Knowledge Insertion Desirable?

In this article, John Tillson discusses the conditions under which what he calls “knowledge insertion” would be desirable for the one who has knowledge inserted. He argues that making use of knowledge insertion would not be cost free; in particular, it would come at the price of relationship goods realized through teacher–learner relationships, and of the achievement of learning, at least for the knowledge inserted. Despite these costs, though, Tillson concludes that knowledge insertion would most often be desirable since the losses could usually be more than compensated by the knowledge gained or would matter little so long as one realizes the relevant relationship and achievements goods to a satisfactory level at some point in one's life.     

Risky forms of knowledge: configuring pedagogical practices and their excesses in a sexuality education programme in South Africa

Within the context of sexuality education as an HIV prevention strategy, much attention has been given to what content should be taught and the effectiveness of that content in achieving desired goals. While some research has problematised how curricular content is understood or taken-up, what remains largely unquestioned is a pedagogical imaginary of knowledge as a fixed and stable object that can be transmitted from teacher to learner. This paper builds on feminist readings of pedagogy, and in particular the work of Elizabeth Ellsworth, to interrogate what might happen within sexuality education by thinking about knowledge as continually being made. Drawing on data from a year-long ethnographic research study conducted with loveLife, a national HIV prevention programme for young people in South Africa, the paper problematises the perceived boundaries of what is taught and explicitly engages the pedagogical approach as a constitutive part of what can/should/must be known as well as what kind of relation is offered to that knowledge. The question then is how sexuality education might be re-articulated to engage with ongoing and power-laden struggles within configurations of knowledge, and how a pedagogical approach might open those configurations to ways of knowing, and being known, that are more just for more people.     

A pragmatic epistemology for free‐choice learning

A critical review of the epistemological foundations of free‐choice learning (FCL) theory was undertaken to evaluate how this theory treats knowledge, whatever importance we might attach to it. It is argued here that free‐choice learning has great promise yet would benefit from theoretical adjustments that modify Vygotsky’s learning theory by using Dewey’s pragmatic epistemological theories. It is suggested that the concept of intramental knowledge in free‐choice learning needs to be grounded on Dewey’s pragmatic conceptions of knowledge, in order to valorise the learner’s own knowledge, and their volitional activity in choosing their beliefs, and to provide a more robust foundation for the conceptualization of knowledge. It is further argued that the notion of intermental knowledge needs significant adjustment, as it seems implausible that knowledge actually exists between people. However, in order to keep the focus of free‐choice learning on the importance of social exchanges in learner construction of knowledge, Dewey’s compatible notion of the importance of the learning exchange is proposed as a substitute for intermental knowledge. It is also suggested that there is a need for additional guidance to educators on how most effectively and sensitively to mediate learner values and interests, given that educators working with learners have interests, aims, and values of their own. I suggest a solution based on pragmatic epistemological and learning theories.     

What are the affective and social outcomes for low-achieving students within an inclusive school in Hong Kong?

In response to several major curriculum reforms in Hong Kong since 2000, schools were required to initiate policies that catered for learner diversity. As well academic achievement, the reforms also emphasized the affective and social outcomes of the learning experience. A whole-school approach to learner diversity includes students with low academic achievement and mild disabilities, with schools being free to develop policies and practices inline with the needs of the students, teachers, and parents. Against the Chinese cultural context, where high academic achievement is highly valued, this study describes the affective and social outcomes of students in one Hong Kong primary school that streamed students according to achievement level. Four hundred and ten students across Primary 3–5 responded to four instruments measuring academic self-concept, alienation from school, teacher–students relationship, and social integration. The responses showed that, when studying the regular curriculum, students in the low-achievement group reported greater levels of alienation and lowest levels of social integration compared with students with the highest levels of achievement. On the other hand, low-achieving students who studied a differentiated curriculum reported affective and social outcomes more consistent with the responses from the high-achieving students. The results suggest that there is a mismatch between the aims and practice of the educational policy reflecting learner diversity and the cultural expectations of both parents and students.     

Seizing the Opportunity to Create Uncertainty in Learning Mathematics

The paper is a reflective account of the design and implementation of mathematical tasks that evoke uncertainty for the learner. Three types of uncertainty associated with mathematical tasks are discussed and illustrated: competing claims, unknown path or questionable conclusion, and non-readily verifiable outcomes. One task is presented in depth, pointing to the dynamic nature of task design, and the added value stimulated by the uncertainty component entailed in the task in terms of mathematical and pedagogical musing.     

Instructional Guidance and Realism of Manipulatives Influence Preschool Children's Mathematics Learning

Educators often use manipulatives when teaching mathematics because manipulatives are assumed to promote learning. However, research indicates that instructional variables impact the effectiveness of manipulatives. In this article, the authors consider the relations between two instructional characteristics: (a) level of instructional guidance and (b) perceptual qualities of manipulatives. Results from the randomized experiment with preschoolers (N = 72) suggest that learning is improved when instruction is conducted with high levels of instructional guidance and is impacted by the perceptual qualities of manipulatives. Perceptually rich manipulatives decreased learner performance on outcomes associated with conceptual knowledge and improved performance on transfer of learning. In addition, transfer was positively affected by perceptually rich manipulatives when low levels of instructional guidance were present.     

Online discussion compensates for suboptimal timing of supportive information presentation in a digitally supported learning environment

This study used a sequential set-up to investigate the consecutive effects of timing of supportive information presentation (information before vs. information during the learning task clusters) in interactive digital learning materials (IDLMs) and type of collaboration (personal discussion vs. online discussion) in computer-supported collaborative learning (CSCL) on student knowledge construction. Students (N = 87) were first randomly assigned to the two information presentation conditions to work individually on a case-based assignment in IDLM. Students who received information during learning task clusters tended to show better results on knowledge construction than those who received information only before each cluster. The students within the two separate information presentation conditions were then randomly assigned to pairs to discuss the outcomes of their assignments under either the personal discussion or online discussion condition in CSCL. When supportive information had been presented before each learning task cluster, online discussion led to better results than personal discussion. When supportive information had been presented during the learning task clusters, however, the online and personal discussion conditions had no differential effect on knowledge construction. Online discussion in CSCL appeared to compensate for suboptimal timing of presentation of supportive information before the learning task clusters in IDLM.     

Testing the instructional fit hypothesis: the case of self-explanation prompts

Cognitive science principles should have implications for the design of effective learning environments. The self-explanation principle was chosen for the current work because it has developed significantly over the last 20 years. Early formulations hypothesized that self-explanation facilitated inference generation to supply missing information about a concept or target skill, whereas later work hypothesized that self-explanation facilitated mental-model revision (Chi, Handbook of research on conceptual change, 2000). To better understand the complex relationship between prior knowledge, cognitive processing, and changes to a learner’s representation, two classes of self-explanation prompts (gap-filling and mental-model revision) were tested in the domain of physics problem solving. Prompts designed to focus the learner on gap-filling led to greater learning and a reduction in the amount of tutoring assistance required to solve physics problems. The results are interpreted as support for the instructional fit hypothesis—the idea that the efficacy of instruction is contingent on the match between the cognitive processing that the instruction elicits, how those processes modify the underlying knowledge representations for the task, and the utility of those representations for the task or problem.     

Web‐delivered,problem‐based learning in organisational behaviour: a new form of CAOS

Recent calls have been made for a fundamental reorientation in higher education from a curriculum that simply transmits information (learner as receiver) to one that encourages students actively to construct their own knowledge and practice using new skills and concepts (learner as constructor). In response, the Case Analysis in Organisational Situations (CAOS) assessment tool was developed for use in teaching organisational behaviour. It follows the principles of problem‐based learning and group work, using an unfolding case design. A pre‐test (n=365), immediate post‐test (n=323) and delayed post‐test (n=129) design was implemented to evaluate the effectiveness of this new learning tool. The assessment of student approaches to learning revealed that students were more likely to adopt a deep approach to learning at the end of CAOS than at the beginning. In general, the results show that CAOS is a useful web‐based tool that promotes learning in organisational behaviour. The benefits of CAOS and some suggested improvements to the existing CAOS format are discussed.     

Selecting learning tasks: Effects of adaptation and shared control on learning efficiency and task involvement

Complex skill acquisition by performing authentic learning tasks is constrained by limited working memory capacity [Baddeley, A. D. (1992). Working memory. Science, 255, 556–559]. To prevent cognitive overload, task difficulty and support of each newly selected learning task can be adapted to the learner’s competence level and perceived task load, either by some external agent, the learner herself, or both. Health sciences students (N = 55) participated in a study using a 2 × 2 factorial design with the factors adaptation (present or absent) and control over task-selection (program control or shared control). As hypothesized, adaptation led to more efficient learning; that is, higher learning outcomes combined with less effort invested in performing the learning tasks. Shared control over task-selection led to higher task involvement, that is, higher learning outcomes combined with more effort directly invested in learning. Adaptation also produced greater task involvement.     

Beyond linearity: Using IRT-scaled level models to describe the relation between prior proportional reasoning skills and fraction learning outcomes

Previous research on the role of prior skills like proportional reasoning skills for the development of mathematical concepts offers conclusions such as “more (prior skills) is better (for later learning).” Insights, which prior skill level goes along with which level of learning outcomes, may advance the understanding of the development of mathematical concepts. An exploratory approach is presented based on level models to describe the relation between symbolic proportional reasoning skills and fraction outcomes beyond linearity. Analyses draw on samples of German fourth to sixth graders from a scaling (2017, N = 325, 54.8% female) and longitudinal study (2018/2019, N = 436, 42.7% female). Particularly mastering natural and internal rational ratios in proportional reasoning seems relevant for successful fraction learning.     

元认知意识与英语阅读

基于对60名英语学习者的日记分析,发现许多学习者在英语阅读过程中学习者的主体知识、任务知识和策略知识。三个方面的元认知知识均有所认识。讨论了这些发现对英语教学的启示,认为教师应该帮助学习者提高英语阅读方面的元认知意识,从而达到培养学习者自主学习能力的目的。