Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

Beliefs,Practical Knowledge,and Context: A Longitudinal Study of a Beginning Biology Teacher's 5E Unit

The purpose of this three‐year case study was to understand how a beginning biology teacher (Alice) designed and taught a 5E unit on natural selection, how the unit changed when she took a position in a different school district, and why the changes occurred. We examined Alice's developing beliefs about science teaching and learning, practical knowledge, and perceptions of school context in relation to the 5E unit. Data sources consisted of interviews, classroom observations, and lesson materials. We found that Alice placed more emphasis on the explore phase, less emphasis on the engage and explain phases, and removed the elaborate phase over time. Alice's beliefs about science teaching and learning acted as a filter for making sense of practical knowledge and perceptions of context. Although her beliefs were student centered, they aligned with discovery learning in which little intervention from the teacher is required. We discuss how her beliefs, practical knowledge, and perceptions of context explained the changes in her practice. This study sheds insight into the nature of beliefs and how they relate to the 5E lesson phases, as well as the different lenses for viewing the 5E instructional model. Implications for science teacher preparation and induction programs are discussed.     

Use of invented algorithms by second graders in a reform mathematics curriculum

Second-grade students in three schools were individually tested on multidigit addition and subtraction problems and solution procedures observed. The schools were all using a reform mathematics curriculum (UCSMP) with an emphasis on problem solving in broader mathematical contexts. Both contextualized and bare computation problems were included in these interviews. On all but one problem, more students used a mental procedure than used the standard written algorithms, and both methods were used with about the same degree of accuracy. Although the standard school algorithm was the only written algorithm used, a number of different mental procedures were employed by students, and choice appeared to be influenced by characteristics of the problems (magnitude of the numbers or the need for regrouping). Major differences between the three schools were found, which are linked to instruction.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Engineering design and the development of knowledge for teaching among preservice science teachers

The goal of this article is to inform professional understanding regarding preservice science teachers’ knowledge of engineering and the engineering design process. Originating as a conceptual study of the appropriateness of “knowledge as design” as a framework for conducting science teacher education to support learning related to engineering design, the findings are informed by an ongoing research project. Perkins’s theory encapsulates knowledge as design within four complementary components of the nature of design. When using the structure of Perkins’s theory as a framework for analysis of data gathered from preservice teachers conducting engineering activities within an instructional methods course for secondary science, a concurrence between teacher knowledge development and the theory emerged. Initially, the individuals, who were participants in the research, were unfamiliar with engineering as a component of science teaching and expressed a lack of knowledge of engineering. The emergence of connections between Perkins’s theory of knowledge as design and knowledge development for teaching were found when examining preservice teachers’ development of creative and systematic thinking skills within the context of engineering design activities as well as examination of their knowledge of the application of science to problem‐solving situations.     

Learning to teach science in urban schools: Context as content

The purpose of this phenomenological study was to explore how science teachers who persisted in urban schools interpreted and responded to the unique features of urban educational contexts. With 17 alumni who taught in metropolitan areas across seven states, the Science Educators for Urban Schools (SEUS) program provided a research setting that offered a unique view of science teachers’ development of knowledge of urban education contexts. Data sources included narratives of teaching experiences from interviews and open‐ended survey items. Findings were interpreted in light of context knowledge for urban educational settings. Findings indicated that science teaching in urban contexts was impacted by the education policy context, notably through accountability policies that narrowed and marginalized science instruction; community context, evident in teacher efforts to make science more relevant to students; and school contexts, notability their ability to creatively adjust for resource deficiencies and continue their own professional growth. Participants utilized this context knowledge to transform student opportunities to learn science. The study suggests that future science education research and teacher preparation efforts would benefit from further attention to the unique elements of urban contexts, specifically the out of classroom contexts that shape science teaching and learning.     

协调区间值决策形式背景的知识约简

知识约简是概念格理论的核心问题之一.主要讨论协调区间值决策形式背景的知识约简问题.首先从经典的协调决策形式背景出发,定义了协调区间值决策形式背景,同时给出了协调区间值决策形式背景上决策保序集的定义和判定定理,并进一步阐明了决策保序集和协调集之间的关系,然后通过定义辨识矩阵,给出了协调区间值决策形式背景的属性约简方法.     

Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course

In this research report we examine knowledge other than content knowledge needed by a mathematician in his first use of an inquiry-oriented curriculum for teaching an undergraduate course in differential equations. Collaboratively, the mathematician and two mathematics education researchers identified the challenges faced by the mathematician as he began to adopt reform-minded teaching practices. Our analysis reveals that responding to those challenges entailed formulating and addressing particular instructional goals, previously unfamiliar to the instructor. From a cognitive analytical perspective, we argue that the instructor's knowledge — or lack of knowledge — influenced his ability to set and accomplish his instructional goals as he planned for, reflected on, and enacted instruction. By studying the teaching practices of a professional mathematician, we identify forms of knowledge apart from mathematical content knowledge that are essential to reform-oriented teaching, and we highlight how knowledge acquired through more traditional instructional practices may fail to support research-based forms of student-centered teaching.     

Maple explorations,perfect numbers,and Mersenne primes

Some examples from different areas of mathematics are explored to give a working knowledge of the computer algebra system Maple. Perfect numbers and Mersenne primes, which have fascinated people for a very long time and continue to do so, are studied using Maple and some questions are posed that still await answers.     

Teaching alternative mathematical procedures

The psychological theory of transfer of learning is examined as an aid in teaching mathematics to students. In particular, Ellis’ variety‐of‐tasks principle of transfer of learning embodying alternative procedures is scrutinized for its beneficial application to two categories of students. The subtraction‐of‐mixed‐numbers‐with‐regrouping procedure is used in different alternative forms as an example of the applicability of the principle with students having differing degrees of comprehension. It was concluded that the use of alternative procedures was a viable teaching strategy and should be developed for all standard algorithms used in the solution of mathematical problems.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Classroom note: Computational and conceptual understanding of the connections among standard deviations,z-scores,and normal distributions

A structure for learning the connections among standard deviations, z-scores, and normal distributions is presented. The components of this structure are classified into intuitive or previously learned conceptual knowledge, computational knowledge, and formalized conceptual knowledge.     

Catalan numbers,q-Catalan numbers and hypergeometric series

Catalan numbers are examined in the context of hypergeometric series. We are thus able to produce new and simple q-analogs related to the theory of partitions.     

What do teachers need? Math and special education teacher educators’ perceptions of essential teacher knowledge and experience

Special education and mathematics education are becoming increasingly intertwined in inclusive classrooms. However, research and practice in these two fields are not always aligned. We discuss, in the context of extant research on pedagogical theory, concepts of access, and the findings of an exploratory study, how these two education sub-fields view teacher expertise. Teacher educators (from math and special education) were asked to rank the importance of different types of expertise for effectively posing purposeful mathematical questions. The groups differed significantly in their rankings of the importance of knowing individual students and general teaching experience. There were also notable differences between the groups’ rankings of the importance of knowing the needs of students with disabilities and mathematical content knowledge. The possible reasons for this are discussed, along with suggestions for improving professional collaboration.     

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

Catalan numbers,binary trees,and pointed pseudotriangulations

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations.     

Two properties of condition numbers for convex programs via implicitly defined barrier functions

We study two issues on condition numbers for convex programs: one has to do with the growth of the condition numbers of the linear equations arising in interior-point algorithms; the other deals with solving conic systems and estimating their distance to infeasibility.?These two issues share a common ground: the key tool for their development is a simple, novel perspective based on implicitly-defined barrier functions. This tool has potential use in optimization beyond the context of condition numbers. Received: October 2000 / Accepted: October 2001?Published online March 27, 2002     

How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

Generalized harmonic numbers with Riordan arrays

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.     

A method for exact computation with rational numbers

An easily programmed method is presented for solving N linear equations in N unknowns exactly for the rational answers, given that all coefficients and constants appearing in the equations are rational numbers. The rational answers are deduced from floating point approximations to the answers obtained by any of the standard solution algorithms. Criteria are given for determining for a particular set of equations the floating point precision needed.