## Making the connection: research and teaching in undergraduate mathematics educationDirected toward instructors but also useful to administrators and curriculum specialists, this collection of 23 articles focuses on improving undergraduate courses at the introductory through advanced levels. Beginning with research on how students think mathematically, contributors explain how to set strong foundations for beginning calculus by developing a rich conception of variables, rethinking change, building reasoning abilities and supporting the concept of accumulation. Contributors describe how students develop notions about infinity, limit and divisibility, covering the theory and design for the instruction of concepts, and describe the process of learning about proving theorems, starting with overcoming students' difficulties with proofs, and move to cross-cutting topics such as interacting with students, using definitions and examples as well as appropriate technology, and acquiring knowledge, assumptions and problem-solving behaviors in teaching. This works as a reader as well as a course text and a foundation for self-study. |

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### Contents

On Developing a Rich Conception of Variable | 3 |

Rethinking Change | 15 |

Foundational Reasoning Abilities that Promote Coherence in Students Function Understanding | 27 |

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abstract abstract algebra accumulation functions actor-oriented algebra angle bisector approach approximation arguments ask students awareness calculus calculus students Carlson chapter classroom cognitive concept image conjectures construct construct proofs context cosets course covariational curriculum definite integral definitions described develop difficulties discussion Dubinsky Educational Studies equation example experience Figure formal graduate students graph Harel Hazzan help students heuristics infinite infinity instructional integral interviews intuitive L'Hopital's rule learners learning limit mathematical induction mathematical proof mathematicians Mathematics Education multiple Oehrtman particular patterns pedagogical precalculus prime problem solving programs proof scheme prove quantities question quotient groups ramp rate of change reasoning representation Research in Mathematics Riemann sums role Schoenfeld secant lines sense sequence situations slope solution specific strategies structure student thinking Studies in Mathematics symbols tasks teachers teaching techniques theorem topic transfer triangle undergraduate mathematics understanding values variable Zandieh Zazkis