Memoirs of a Mentor: Using Creativity for Counselors in Coping With the Loss of a Mentor

This article was inspired by my (S.S.) own personal loss. My mentor passed away during spring break of my 2nd year postgraduate school after a short battle with systemic lupus. I remember the deep sadness that I felt when it became apparent that she was coming home from the hospital for the last time. No words can describe the emotions; she had helped me through the toughest times in my academic life. How would I ever get the type of mentorship she provided again? She was there when I almost quit as a young student, back when my anger still got the best of me. She talked me down from the edge so many times; I never expected to be on this journey without her.

I dedicate this article to her and mentors like her. Equally, I dedicate this article to mentees who have lost their mentors. I offer my story (in italicized font) in the hopes that it will help others who are dealing with a similar loss. In this article, we attempt to illuminate the true power of mentorship, honor the significance of the relationship between mentor and mentee, and provide a tool useful to anyone who has lost their guide. I share my story in gratitude for my own mentor; I am so thankful that she was a part of my journey and that I can pass on to others the patience she had with me.     

The mental representation of integers: an abstract-to-concrete shift in the understanding of mathematical concepts

Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of pairs of integers. Experiment 1 demonstrated an inverse distance effect: When comparing numbers across the zero boundary, people are faster when the numbers are near together (e.g., −1 vs. 2) than when they are far apart (e.g., −1 vs. 7). This result conflicts with a straightforward symbolic or analog magnitude representation of integers. We therefore propose an analog-x hypothesis: Mastering a new symbol system restructures the existing magnitude representation to encode its unique properties. We instantiate analog-x in a reflection model: The mental negative number line is a reflection of the positive number line. Experiment 2 replicated the inverse distance effect and corroborated the model. Experiment 3 confirmed a developmental prediction: Children, who have yet to restructure their magnitude representation to include negative magnitudes, use rules to compare negative numbers. Taken together, the experiments suggest an abstract-to-concrete shift: Symbolic manipulation can transform an existing magnitude representation so that it incorporates additional perceptual-motor structure, in this case symmetry about a boundary. We conclude with a second symbolic-magnitude model that instantiates analog-x using a feature-based representation, and that begins to explain the restructuring process.     

Contexts as switchmen: The variable effects of formal and informal instructional strategies on student achievement

This study presents an exploratory theoretical framework that analyzes the extent by which the contexts of classroom instruction mediate the effects of instructional strategies on achievement. The study first presents the two main tasks that teachers face in classrooms-teaching, and controlling the social order. The study then elaborates on formal and informal instructional strategies that teachers use to manage these tasks. The main foci of the study is to show that the effects of these different instructional strategies on student achievements vary by grade level. The results suggest that the contexts of instruction determine the magnitude and direction of the effects of instructional strategies on achievement. Positive effects on student achievement appear when a functional congruence occurs between teachers' instructional strategies and the contexts wherein they are used. The analyses suggest that changes in the cognitive, social, and institutional contexts of classrooms may change the relative efficacy of specific instructional strategies. Consequently, researchers should pay more attention to the changes in the sociology of classrooms during students' school careers. This study was supported by a post-doctoral fellowship from the Fulbright Foundation and by a small research grant from The Spencer Foundation. My wholehearted appreciation is given to the support I received from the Foundations. I am especially grateful to Charles Bidwell from the University of Chicago, who encouraged me to pursue this study during my post-doctoral visit in 1994–95. Discussions with Benjamin Wright and his warm support have been more than essential. The hospitality and the academic environment at the NORC and the University of Chicago have done wonders to the fruition of ideas advanced in this study. Yechezkel Dar and Ruth Butler made suggestions to the improvement of this article. Daniel Shalem and Rivka Berman assisted with editorial as well as substantive comments. The reviewers ofSocial Psychology of Education insightfully suggested ways to improve the final version.     

Some problems concerning language and physics

We discuss three problems concerning the use of formal languages in theoretical physics: (i) the definability of time and spacetime in classical physical theories; (ii) how to cope with indistinguishable elementary particles in quantum mechanics without labeling them; and (iii) how to get a formal picture of quantum states jumping.     

Deontic action logic, atomic boolean algebras and fault-tolerance

We introduce a deontic action logic and its axiomatization. This logic has some useful properties (soundness, completeness, compactness and decidability), extending the properties usually associated with such logics. Though the propositional version of the logic is quite expressive, we augment it with temporal operators, and we outline an axiomatic system for this more expressive framework. An important characteristic of this deontic action logic is that we use boolean combinators on actions, and, because of finiteness restrictions, the generated boolean algebra is atomic, which is a crucial point in proving the completeness of the axiomatic system. As our main goal is to use this logic for reasoning about fault-tolerant systems, we provide a complete example of a simple application, with an attempt at formalization of some concepts usually associated with fault-tolerance.     

Automatic verification of multi-agent systems by model checking via ordered binary decision diagrams

We present a methodology for the verification of multi-agent systems, whose properties are specified by means of a modal logic that includes a temporal, an epistemic, and a modal operator to reason about correct behaviour of agents. The verification technique relies on model checking via ordered binary decision diagrams. We present an implementation and report on experimental results for two scenarios: the bit transmission problem with faults and the protocol of the dining cryptographers.     

Paraconsistent Informational Logic

We introduce a Paraconsistent Informational Logic that formalizes the idea of conjectures which are acceptable as to the quality and the variety of the information that they convey with respect to a given theory T, even if they are classically inconsistent with T. The work constitutes an extension of a previously developed Informational Logic for classical frameworks, where a new notion of logical entropy measure H on formulas and on proofs plays a central role.     

Formal mentoring programs: The relationship of program design and support to mentors’ perceptions of benefits and costs

This study investigates the relationship of formal mentoring program design elements (i.e., voluntary participation, input to matching, and effectiveness of training) and management support to the benefits and costs perceived by formal mentors. Data were collected from 97 formal mentors from a Midwestern financial institution. Multiple regressions were performed controlling for time as a mentor in the program, hours spent mentoring, and number of protégés. Voluntary mentor participation was positively related to perceiving rewarding experiences and negatively related to being more trouble than it was worth. Input to the matching process was negatively related to nepotism, and perceptions of training effectiveness were positively related to generativity. Finally, perceived management support for the program was positively related to rewarding experience and recognition, and negatively related to generativity and bad reflection. Three supplemental group interviews were conducted to further explore some of the survey findings. Directions for future research and implications for formal workplace mentoring programs as well as mentoring programs in cross-disciplinary contexts are discussed.     

Hilbert's epsilon as an operator of indefinite committed choice

Hilbert and Bernays avoided overspecification of Hilbert's ε-operator. They axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. After briefly reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε simplifies proof search and is natural in the sense that it mirrors some cases of referential interpretation of indefinite articles in natural language.