Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction

Mathematics instruction contains two conflicting demands: on the one hand, the demand for economical efficiency and for well-developed motorways and, on the other hand, the demand that pupils should investigate and discover for themselves and have the freedom to pave their own ways. It is argued that tasks with a certain richness and quality offer some steps towards a constructive handling of this dilemma. The author tries to develop some properties of powerful tasks and to sketch the structure and philosophy of one concrete system of powerful tasks for the concept of angle. The main part of this paper presents seven examples of powerful tasks: five from the system of tasks and two with regard to pupils' working with 2-D-graphic systems. The construction of powerful tasks is viewed as a valuable contribution to bringing the theory and practice of mathematics education closer together.     

Cognitive development,genetics problem solving,and genetics instruction: A critical review

Recent studies have analyzed the cognitive demands of solving problems in genetics, focusing primarily on the Piagetian schemas of combinations, proportions, and probability. Based on data from these primarily correlational studies, some authors have argued for the elimination of classical genetics from the high school curriculum. The critical review of the literature presented in this article reaffirms that formal-operational thought is conducive to successful genetics problem solving. The weight of the evidence to date, however, does not support the position that formal operational thought is strictly required for solving typical genetics problems. Arguments are therefore presented in support of the inclusion of genetics and genetics problem solving in high school biology. Implications of this analysis for the selection of appropriate content, problems, and instructional techniques for genetics instruction for nonformal students are presented.     

Individualizing instruction in higher education: A review

The salient characteristics of a variety of approaches to individualizing instruction are described and their respective merits with regard to higher education discussed. The review includes: programmed instruction, computer-assisted instruction and -management, information retrieval systems, audio-tutorial and modular instruction, contingency management and contracting, and personalized and individually prescribed instruction. While these approaches differ from each other in some respects, they all share an overriding concern for individual differences among students and seek to better adapt instruction to the learner.A section on general issues regarding the degree of individualisation, the role of the professor, institutional support for improving instruction, the role of the student, content and method, evaluation, and the future of individualized instruction concludes the review.Formerly, Centre for Learning and Development, McGill University, Montreal, Quebec, Canada.     

Modular instruction in higher education: A review

The principles and purposes of modular instruction (MI), its advantages for both students and instructors, and a comparison between the conventional and modular approach are presented. Separate sections deal with implementation and management of MI and include a discussion of evaluation and cost. Several examples of modular formats in use at North American universities are described.Present evidence suggests that MI meets the needs of today's students more adequately than traditional instruction both with respect to the quality of learning and the content. However, certain problems may arise in implementing MI. These are presented from the perspective of the student, instructor, and administrator. Given its emphasis on individualized learning and its adaptability to large numbers of students, MI has emerged as one of the most promising alternatives in higher education today.     

Learning from instruction: the case of mathematics

Starting from a brief analysis of adaptive competence in mathematics, this article describes a series of research-based characteristics of the kind of learning processes that should be elicited in students to facilitate and support in them the progressive acquisition of such competence. Four major characteristics are discussed in some detail: learning is constructive, self-regulated, situated or contextual, and collaborative. A rather new approach to transfer of learning is then presented in which transfer is conceived as the preparation for future learning. Throughout the article it is argued that, notwithstanding the progress made in research on learning from instruction, numerous and complex issues and problems remain for continued inquiry.     

Examining pedagogical approaches and types of mathematics knowledge in educational games: A meta-analysis and critical review

This meta-analysis systematically reviewed math game studies published between 2010 and 2020 and evaluated them with respect to a) the type of pedogeological foundations inherent in games using Kebritchi and Hirumi's (2008) framework, b) the type of mathematics knowledge they facilitated (Bisanz & LeFevre, 1990; Rittle-Johnson, 2017), and c) their effect on math learning. Only 23 out of 26 studies used games based on a clear pedagogical approach and many studies measured multiple knowledge types. A direct instructional approach was most often used in games to target factual knowledge and resulted in an overall medium sized effect (g = 0.58), whereas procedural and conceptual knowledge were used by games using three types of pedagogical approach: experiential, discovery, and constructivist approaches but with mixed effect sizes. Overall, behaviorally oriented pedagogies are still dominant in math games and the effectiveness of each pedagogical approach varies as a function of knowledge type.     

Technology-enhanced mathematics instruction: A second-order meta-analysis of 30 years of research

It is important to assess the cumulative effects of technology on student achievement captured in the last 30 years of technologyenhanced mathematics instruction. Synthesizing the thousands of articles and gray literature on this subject is necessary but would require a considerable commitment of academic resources. A second-order metaanalysis or meta-analysis of meta-analyses is an alternative that is reasonable and effective. Thus, a second-order meta-analysis of 19 prior meta-analyses with minimum overlap between primary studies was conducted. The results represent 663 primary studies (approximately 141,733 participants) and 1,263 effect sizes. The random effects' mean effect size of .38 was statistically significantly different from zero. The results provide a historical and contextualized summary of 30 years of meta-analytic research, which supports meta-analytic thinking and better interpretation of future effect sizes. Results indicate that technology function and study quality are major contributors to effect size variation. Specifically, computation enhancement technologies were most effective, while studies that examine combinations of enhancements were least effective. Implications for technology-enhanced mathematics instruction and meta-analytic research are provided.     

A critical analysis of university examinations in mathematics

Part I of this paper introduced a procedure for assessing the questions set in University mathematics examinations to see what skills were needed for their solution. This part of the paper discusses the results obtained when the procedure was applied to a sample of some 1400 questions set during 1976 in ten British Universities. Although there is an unavoidable subjective element, the method is a way of focussing rational argument.     

A critical analysis of university examinations in mathematics

In many Universities, mathematics examinations use a timed, closed-book format. The paper discusses what areas of ability are being assessed in these examinations, and to what extent the procedure might foster possible educational objectives. We describe a list of ten qualities by which a mathematics question can be graded in order to see what skills are needed for its solution, and some examples are discussed, as well as the background system in which the procedures are set.Part II of the paper (to follow) will discuss the results obtained from a study which applied this procedure to a large sample of questions set in examinations in ten British Universities.     

Teacher lust: reconstructing the construct for mathematics instruction

Two collegiate mathematics courses for prospective elementary and middle grades teachers provide the context for the examination of Mary Boole’s construct of teacher lust. Through the use of classroom observations and instructor interviews, the author presents a refined conception of teacher lust. Two working aspects of the construct were identified: (1) enacted teacher lust; an observable action that may remove an opportunity for students to think about or engage in mathematics for themselves; and (2) experienced teacher lust; an internal impulse to act in the manner described. Empirical examples of each facet, differences between conscious and unconscious interactions with teacher lust, along with potential antecedents are discussed.     

A profile of mathematics instruction time in Irish second level schools

Similar to counties such as the UK and Netherlands, second level schools in Ireland are free to decide how to allocate instruction time between curriculum subjects. This results in variations between the quantum of time allocated to teaching mathematics in different schools and between different class groups within the same school. This quantitative study builds a profile for both lower second level (Junior Cycle) and upper second level (Senior Cycle) mathematics instruction time in Ireland. The results of the study highlight that although the proportion of time is on par with the OECD average, there are many issues of concern regarding instruction time in Ireland. These include the short length of the school year, experimentation with new teaching approaches, the high number of subjects studied, variations between individual school and class group allocations and the number of classes that do not take place.     

Two readiness measures as predictors of first and third-grade reading achievement

Multiple-regression procedures were used to assess the effectiveness of the ABC Inventory and the Metropolitan Readiness Test (MRT) in predicting first and third-grade reading achievement. Sex and chronological age were included in the first-grade analysis (N = 62) and the first-grade PMA intelligence test score was added to the equation in predicting third-grade reading (N = 65). MRT performance accounted for 11% of the variance in first-grade SRA reading scores (R = .34). In predicting third-grade reading, the MRT accounted for 26% of the variance and the PMA IQ scores accounted for an additional 6% (final multiple R = .57). No other predictor made a significant contribution to explaining variance in first or third-grade reading achievement.     

Using critical incident reflections: a self-study as a mathematics teacher educator

In this paper, I report the results of a 4-year study into how my students learn to become mathematics teachers during the combined 15-week methods and field placement course I teach. At the start of most weekly methods class meetings, groups of three or four students reported their critical incidents to each other, and then chose one incident to report to the whole class. Each student then submitted a written report of 10 critical incidents for grading. At the end of each semester, I administered a questionnaire about the usefulness of each of the elements of the course. Using these two main data sources, along with my own personal reflections on the course, this paper answers the question “What are the critical incidents preservice teachers encounter during their field experience, and what do they learn about teaching for understanding through reflecting on those critical incidents?” My analysis of the students’ incident reports found that the issues raised focused on four main areas: teaching and classroom management; student factors such as pre-requisite knowledge, understanding, resistance and motivation; issues concerning relationships with colleagues, students and parents; and school organizational issues such as policies and access to resources. Their learning about teaching for understanding focused on three broad areas: the conditions necessary to teach for understanding; facilitators of teaching for understanding; and barriers to teaching for understanding. The paper concludes with a discussion of the lessons learned about fostering reflection in novice teachers, and suggestions for further research.     

略谈数学教学中创新意识的培养

培养学生具有“创新意识”是中学数学教学的重要目标和一条重要原则。这里的“创新意识”不同于“创新能力”；这里的“创新”也不同于数学研究中的“创新”。它的基本含义是，使学生具有举一反三的能力，穷本溯源的精神和敢于探索的勇气。培养学生的数学创新意识，对教师的教育思想和教学方法提出了新的挑战，文章对此提出了一些有益的建议。     

Understanding and procedural knowledge in mathematics instruction 1

Using ideas developed recently in the theory of understanding language, an analysis is given of the process of understanding mathematical concepts. An analysis also is given of procedural knowledge in mathematics, based on recently developed theories of problem solving including production systems and automaton models. A suggestion is made as to how these theories might be synthesized to provide an analysis of performance of mathematical skills with understanding. Two examples of such performance are described, in the form of theories that are realized as computer programs, and there is discussion of ways in which these theories succeed and fail in representing procedural knowledge with understanding.     

When mathematics and statistics collide in assessment tasks

Because the disciplines of mathematics and statistics are naturally intertwined, designing assessment questions that disentangle mathematical and statistical reasoning can be challenging. We explore the writing statistics assessment tasks that take into consideration potential mathematical reasoning they may inadvertently activate.