Combining theories in research in mathematics teacher education

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.     

How fragile is consolidated knowledge?: Ben's comparisons of infinite sets

This article builds on two previous ones in which we presented the processes of construction and consolidation of one student's knowledge structures about comparisons of infinite sets, according to a recently proposed theory of abstraction. In the present article, we show that under slight variations of context, knowledge structures that have apparently been well-consolidated may become inactive and subordinate to more primitive ones.     

Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations.     

Consistencies and inconsistencies in students' solutions to algebraic ‘single-value’ inequalities

This paper describes students’ solutions to a commonly taught and not commonly taught inequality. The findings showed students’ difficulties. Participants implicitly and explicitly exhibited two intuitive beliefs: inequalities must result in inequalities and solving inequalities and equations are the same process. Following the analysis of students’ written solutions, individual interviews were conducted that gave a better insight into their reasoning and provided some ideas for teaching. The concluding section of the paper offers relevant educational implications.     

Ben's consolidation of knowledge structures about infinite sets

According to a recently proposed model for processes of abstraction in context, the construction of a new structure is to be followed by a consolidation phase. In this paper, we develop an empirically based, theoretical analysis of consolidation that emerges from a sequence of interviews about the comparison of infinite sets with a talented student. We take for granted that construction has occurred in the first interview and analyze the second one. Our analysis shows that consolidation can be identified by means of the psychological and cognitive characteristics of self-evidence, confidence, immediacy, flexibility and awareness. We also found three modes of thinking conducive to consolidation, one related to problem solving, one to reflective activity and an intermediate one.     

Stereospecific mono- and difluorination of the C7-bridge of norbornenes

Fluorinated norbornenes are very desirable monomers in the semiconductor and high-temperature polyimide industries. We describe herein a synthetic strategy for the stereospecific mono- or difluorination of the C₇-carbon in norbornene systems beginning with 7-ketonadic anhydride 1. In particular, anti-7-fluoro methyl diester 4 and its 7,7-difluoronadic analog 7 can be prepared from 1 in 3 or 4 steps: saponification, reduction (for 4), esterification, fluorination with DAST. In addition, anti-7-fluoro-syn-7-fluoromethylnadic diester 16 is obtained from epoxide 14, and dimethyl 7,7-difluorobicyclo[2.2.2]oct-5-ene-2,3-dicarboxylate (17) from ketone 15. Anchimeric assistance of the norbornene double bond guides the introduction of attacking fluoride anions stereospecifically anti to the olefinic linkage.     

Students' strategies and difficulties: the case of algebraic inequalities

This article describes a study related to secondary school students' ways of thinking about linear, quadratic, rational and square-root inequalities. Findings show that using graphic representations of parabolas when solving rational and quadratic inequalities usually yielded correct solutions. Difficulties arose when students failed to reject the excluded values or chose inappropriate, logical connectives. The most prevalent source of difficulties was inappropriate analogies between equations and inequalities. The article concludes with some suggested educational implications.     

1,1‐Diamino‐2,2‐dinitroethylenes are always zwitterions

The nitration of tetraiodoethylene (7) yields 1,1‐diiodo‐2,2‐dinitroethylene (8). The latter reacts with alkylamines 9 or alkyldiamines 11 to give the corresponding acyclic 1,1‐diamino‐2,2‐dinitroethylenes 10 or their cyclic analogs 12, respectively. On the basis of liquid and solid‐state ¹³C and ¹⁵N NMR data, x‐ray analysis and ab initio calculations, we suggest that the title compounds are always zwitterionic and that the C_A–C_N bond is not a true double bond. Copyright © 2012 John Wiley & Sons, Ltd.     

A three-term recurrence relation for accurate evaluation of transition probabilities of the simple birth-and-death process

BIT Numerical Mathematics - The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over...     

Prospective teachers’ reactions to Right-or-Wrong tasks: The case of derivatives of absolute value functions

This paper illustrates the role of a Thinking-about-Derivatives task in identifying learners’ derivative conceptions and for promoting their critical thinking about derivatives of absolute value functions. The task included three parts: Define the derivative of a function f(x) at x = x_0,Solve-if-Possible the derivative of f(x) = |x| at x = 2 and at x = 0, and evaluate the correctness of suggested solutions in a Right-or-Wrong part. Three prospective teachers, Noa, Anat and Daniel were individually interviewed when solving the task. We found that while the participants correctly solved the Define part, they exhibited some erroneous images in the Solve-if-Possible part, and their work on the Right-or-Wrong part contributed to their critical thinking about functions and derivatives. All three participants expressed their appreciation of their work on the Right-or-Wrong part of the task.