What makes a counterexample exemplary?

Zazkis, Rina; Chernoff, Egan J.

doi:10.1007/s10649-007-9110-4

Cited by 82 publications

(66 citation statements)

References 16 publications

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“…Another approach discussed by Pedemonte (2007Pedemonte ( , 2008 employs the construct of structural distance, and she argues for an abductive step in the structurant argumentation in order to assist transition by decreasing the gap between argumentation and proof. Another proposition is that pivotal, bridging or counterexamples could assist students with proof ideas (Stylianides and Stylianides 2007;Zazkis and Chernoff 2008). A potential benefit of a counterexample is to produce cognitive conflict in the student, while a pivotal example is designed to create a turning point in the learner's cognitive perception.…”

Section: Proof and Provingmentioning

confidence: 99%

“…Counterexamples may also foster deductive reasoning, since deductions are made by building models and looking for counterexamples. For Zazkis and Chernoff (2008) a counterexample is a mathematical concept, while a pivotal example is a pedagogical concept, which is within, but pushing the boundaries of the set of examples students have experienced. The role of examples also arose in research by Weber and Mejia-Ramos (2011) on proof reading by mathematicians.…”

Section: Proof and Provingmentioning

confidence: 99%

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Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

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This report1 from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics. This revealed a multi-faceted web of cognitive, curricular and pedagogical issues both within and across the mathematical topics above. In addition we conducted an international survey of those engaged in teaching in university mathematics departments. Specifically, we aimed to elicit perspectives on: what topics are taught, and how, in the early parts of university-level mathematical studies; whether the transition should be smooth; student preparedness for university mathematics studies; and, what university departments do to assist those with limited preparedness. We present a summary of the survey results from 79 respondents from 21 countries.

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Section: Proof and Provingmentioning

confidence: 99%

Section: Proof and Provingmentioning

confidence: 99%

Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

View full text Add to dashboard Cite

show abstract

“…If the denominator is not the difference between the two independent variables of the numerator, then it does not belong to '( ) f a ." Extending examples can be used to generate or describe examples and to define the possible variations of the category, while the variation itself falls within a range of permissible changes [13] and creates and resolves cognitive conflict [17]. These observations and examples are relevant from the perspective of the theory of variation, for the comparison of variance and invariance is important to prepare students to grasp the characteristics of mathematical objects [18,19].…”

Section: Extending Examplesmentioning

confidence: 99%

Teachers’ Choice and Use of Examples in Teaching Derivatives

Huang¹

2017

EDUCATION

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Students' understanding of a mathematical idea depends upon the examples used by their teachers. Thus, the key responsibility of a math teacher is to select examples that provide the best learning opportunities for students, as well as present the examples in the way most suitable to promote learning. The study reported here is part of a larger research program addressing the role and nature of examples in mathematical thinking, learning, and teaching. The main goal of this part of the study is to characterize mathematics teachers' choice and use of examples in teaching derivatives. This study provided evidence of the application of instructional examples to teach derivatives in calculus courses.

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“…Zazkis and Chernoff argued that pivotal counterexamples can be addressed in an instructional learning design to anticipate a state of which students confront with contradictive conceptions [10]. A counterexample is said to be pivotal for students when the example leads a "turning point" to their conceptions, i.e., the example causes a dissonance to the wrong or lack of understanding of students.…”

Section: The Problem Solvings On the Destructive Situations 1) Pivomentioning

confidence: 99%

Evoking Cognitive Conflict and Intellectual Awareness of Students: A study in the topic of combinatorics

Djadir¹,

Dinar²,

Arwadi³

2017

Proceedings of the 2nd International Conference on Education, Science, and Technology (ICEST 2017)

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Abstract-The present study was a part of a broader research aimed at designing instructional learning which can evoke students' conflict cognitive and intellectual awareness. The specific purposes of this study were reviewing factors which can evoke conflict cognitive and intellectual awareness of students and designing a basis model to create an instructional learning. Literature studies were applied in two main parts. Firstly, the study reviewed the stages of conflict cognitive process and intellectual awareness including their destructive factors. The second, the study reviewed several strategies in solving these problems. The findings suggest that there are some factors which can hamper the occurrence of cognitive conflict and intellectual awareness of which one of them is mathematics anxiety. These factors can be overcome by inducing and applying pivotal counterexample, problem solving strategy, and realistic mathematics education to the basic model of learning instructional design.

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What makes a counterexample exemplary?

Cited by 82 publications

References 16 publications

Key Mathematical Concepts in the Transition from Secondary School to University

Key Mathematical Concepts in the Transition from Secondary School to University

Teachers’ Choice and Use of Examples in Teaching Derivatives

Evoking Cognitive Conflict and Intellectual Awareness of Students: A study in the topic of combinatorics

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