Response paper for the session "Interactive Teaching Activities"

Certificate in University Teaching Program - University of Waterloo

A commonly observed phenomenon in undergraduate mathematical education, is the lack of motivating conceptual instruction. All too often, students are lectured on techniques of problem-solving, without a parallel development of their conceptual understanding. This lack of depth often continues on to graduate-level classes, for the sake of rational efficiency. The method developed by Robert Lee Moore, is a form of mathematical activity spanning a single class event to potentially entire terms. In a sense, it may be viewed as ideal for the conceptual development of students in science disciplines, primarily in mathematics. This paper will consider its hypothetical application in a first-year Calculus setting onwards of Pure and Applied Mathematics students.

First-year Calculus for a student can be a time of great transitions with unusual concepts, formulas, and methods, that many instructors seek to hard-wire in students, to establish a foundational toolset for later courses. Most instructors achieve just that, storming by underlying concepts and philosophies that may only occur to the most contemplative students, in such a fast-paced setting. Likely most professors would cite lack of time, size of classes, and efficiency of information propagation. On the other hand, elementary Calculus could easily become an ideal platform for activizing students, and involving them in the creative development and learning of the material. Bringing to the surface and upholding students' inherent mathematical acumen, must be the goal of any program which honestly wishes to raise able mathematicians and future researchers.

Indeed, most students accepted to university, possess the mathematical background and potential to develop the Calculus of Newton, Leibniz, and Cauchy - however outrageous this claim may sound. Encouraging them to do so in a classroom setting, may potentially turn some into quite capable and confident mathematicians once they become researchers themselves. The suggested alternative activity would hard-wire deep research and collaboration behaviours they stumble upon themselves, instead of the shallow toolsets of technique they are mostly provided. Thus my argument that follows for using the Moore Method in first-year Calculus classrooms, is primarily intended for students who may become researchers one day, in the fields of Pure or Applied Mathematics.

I will hereby outline the overall class as well as the classroom activity itself, which the entire longspan method would reduce to. The Moore Method has numerous existing interpretations, branching off from the original central idea behind it, and this is reflected in the way the actual classroom activity may be carried out. Often the interpretation reflects the course it is applied to, so what follows is my own interpretation, as I would potentially apply it.

The central idea of the Moore Method is to involve a class of students in the development of historical theories of mathematics and science, from the ground up. Students are generally given a set of basic knowledge, often in the form of definitions and axioms, providing a foundation for the relevant initial questions and directions. In regards to elementary first-year calculus, the initial concepts provided to the students might entail a vague idea of limits, continuity, and tangent / area determination, which they would be first tasked with refining. The refinement process itself for these basic definitions may last 1-2 classes - to be determined experimentally. Then as a syllabus, the task of putting the definitions in practice, such as for the evaluation of specific limits, then later determining continuity and differentiation of functions could follow. While solving problems they invent for themselves together, the students may encounter cases that will imply the invention of the Squeeze Theorem, the Intermediate Value Theorem, or even L'Hopital's Rule - not missing out on relevant ideas. The same implications via applications would follow for differentiation and integration, as well as their inverse relationship, which the students would be likely to discover. Specific formulas and techniques of integration, would be developed by the students, for the sake of creating a toolset for themselves, so that they can crack the nuts found in problems they encounter in their exploration. Instructor involvement would amount strictly to asking occasional motivational questions that may blow the students' sail in the right direction. Otherwise, students would be expected to create and collaborate independently, both during and after class, representing their "assignments". Soon, such a Calculus class may have the effect of occupying the students' minds 24/7, preparing them for the analogous experience they may encounter as researchers in their future careers.

The classroom activity itself would possess a loose yet consistently coherent structure, with explicit rules of conduct such as free speech balanced with mutual respect, stated in advance. These explicit rules may serve to instill and induce principles of integrity, and serving as foundations for engagement in future research. Furthermore, the instructor would provide the initial definitions, motivational ideas and questions, to ignite both group discussion and individual efforts, in a sentiment of "collaborative competition". The incentives being inherent mathematical interest, the thrill of building something together, or even peer pressure to perform.

In addition to the aforementioned benefits of a Moore-based activity which shape its very format, we may recognize an even more fundamental characteristic in the method. By placing the learning process in a creative social setting, the intructor immerses the students in a primordial soup which they are used to functioning in, and from which they may naturally evolve as maturing mathematicians. Thus each student would find their natural function in the micro-social environment of the classroom, paralleling their social preferences in real life. Thus collaborators would collaborate, thinkers would contemplate, leaders would emerge, tasks would be divided, global minds would present the vista, and sequential ones the process to explore it. All would find their own role in such Stone Age conditions, in which we are genetically programmed to rely on one another and contribute as we can.

Much of the above is hypothetical to me, and based on pure rationale, versus any prior implementation. To me, the above is an idealization of a mathematical learning activity I would have always wished to be a part of. I have been observing the inefficiency of the classical lecture-tutorial settings, ever since I became a university student, and I find this to be a solution. Though I realize that the Moore method may not be for all students. The initial selection of the students must be based on whether they find the challenge appealing, and not based on academic standing. The worst possible outcome for an aspiring mathematician, would be realizing that they are not fit for doing research, which is still better than a realization in graduate school or real life. On the other hand, many who are unsure initially, may gain self-confidence for the long run. I personally had doubts about my own research abilities as an undergraduate, projecting an unpredictable dark cloud into my future. I believe that all students should be given an opportunity to put themselves to the test, in classroom activites of this kind. Furthermore, the professor may serve as a mentor in this setting - accompanying, guiding, and nurturing potential talent.

The benefits of such an adventure, even if experimental, undoubtedly outweigh the risks. As the Ancient Greek aphorism of virtue recommends "Know thyself!", so should mathematicians learn as much about themselves as possible early on. What better way to do so, than a setting that mimics what they are apt to experience later on in real life. This is akin to teaching children to swim in shallow water so that they would be ready for the deep, while throwing them in deep water at once may prove fatal. Is it not the responsibility of experienced researchers and educators to prepare students for the deep water, besides striving to enhance their overall conceptual understanding?

Certificate in University Teaching Program - University of Waterloo

A commonly observed phenomenon in undergraduate mathematical education, is the lack of motivating conceptual instruction. All too often, students are lectured on techniques of problem-solving, without a parallel development of their conceptual understanding. This lack of depth often continues on to graduate-level classes, for the sake of rational efficiency. The method developed by Robert Lee Moore, is a form of mathematical activity spanning a single class event to potentially entire terms. In a sense, it may be viewed as ideal for the conceptual development of students in science disciplines, primarily in mathematics. This paper will consider its hypothetical application in a first-year Calculus setting onwards of Pure and Applied Mathematics students.

First-year Calculus for a student can be a time of great transitions with unusual concepts, formulas, and methods, that many instructors seek to hard-wire in students, to establish a foundational toolset for later courses. Most instructors achieve just that, storming by underlying concepts and philosophies that may only occur to the most contemplative students, in such a fast-paced setting. Likely most professors would cite lack of time, size of classes, and efficiency of information propagation. On the other hand, elementary Calculus could easily become an ideal platform for activizing students, and involving them in the creative development and learning of the material. Bringing to the surface and upholding students' inherent mathematical acumen, must be the goal of any program which honestly wishes to raise able mathematicians and future researchers.

Indeed, most students accepted to university, possess the mathematical background and potential to develop the Calculus of Newton, Leibniz, and Cauchy - however outrageous this claim may sound. Encouraging them to do so in a classroom setting, may potentially turn some into quite capable and confident mathematicians once they become researchers themselves. The suggested alternative activity would hard-wire deep research and collaboration behaviours they stumble upon themselves, instead of the shallow toolsets of technique they are mostly provided. Thus my argument that follows for using the Moore Method in first-year Calculus classrooms, is primarily intended for students who may become researchers one day, in the fields of Pure or Applied Mathematics.

I will hereby outline the overall class as well as the classroom activity itself, which the entire longspan method would reduce to. The Moore Method has numerous existing interpretations, branching off from the original central idea behind it, and this is reflected in the way the actual classroom activity may be carried out. Often the interpretation reflects the course it is applied to, so what follows is my own interpretation, as I would potentially apply it.

The central idea of the Moore Method is to involve a class of students in the development of historical theories of mathematics and science, from the ground up. Students are generally given a set of basic knowledge, often in the form of definitions and axioms, providing a foundation for the relevant initial questions and directions. In regards to elementary first-year calculus, the initial concepts provided to the students might entail a vague idea of limits, continuity, and tangent / area determination, which they would be first tasked with refining. The refinement process itself for these basic definitions may last 1-2 classes - to be determined experimentally. Then as a syllabus, the task of putting the definitions in practice, such as for the evaluation of specific limits, then later determining continuity and differentiation of functions could follow. While solving problems they invent for themselves together, the students may encounter cases that will imply the invention of the Squeeze Theorem, the Intermediate Value Theorem, or even L'Hopital's Rule - not missing out on relevant ideas. The same implications via applications would follow for differentiation and integration, as well as their inverse relationship, which the students would be likely to discover. Specific formulas and techniques of integration, would be developed by the students, for the sake of creating a toolset for themselves, so that they can crack the nuts found in problems they encounter in their exploration. Instructor involvement would amount strictly to asking occasional motivational questions that may blow the students' sail in the right direction. Otherwise, students would be expected to create and collaborate independently, both during and after class, representing their "assignments". Soon, such a Calculus class may have the effect of occupying the students' minds 24/7, preparing them for the analogous experience they may encounter as researchers in their future careers.

The classroom activity itself would possess a loose yet consistently coherent structure, with explicit rules of conduct such as free speech balanced with mutual respect, stated in advance. These explicit rules may serve to instill and induce principles of integrity, and serving as foundations for engagement in future research. Furthermore, the instructor would provide the initial definitions, motivational ideas and questions, to ignite both group discussion and individual efforts, in a sentiment of "collaborative competition". The incentives being inherent mathematical interest, the thrill of building something together, or even peer pressure to perform.

In addition to the aforementioned benefits of a Moore-based activity which shape its very format, we may recognize an even more fundamental characteristic in the method. By placing the learning process in a creative social setting, the intructor immerses the students in a primordial soup which they are used to functioning in, and from which they may naturally evolve as maturing mathematicians. Thus each student would find their natural function in the micro-social environment of the classroom, paralleling their social preferences in real life. Thus collaborators would collaborate, thinkers would contemplate, leaders would emerge, tasks would be divided, global minds would present the vista, and sequential ones the process to explore it. All would find their own role in such Stone Age conditions, in which we are genetically programmed to rely on one another and contribute as we can.

Much of the above is hypothetical to me, and based on pure rationale, versus any prior implementation. To me, the above is an idealization of a mathematical learning activity I would have always wished to be a part of. I have been observing the inefficiency of the classical lecture-tutorial settings, ever since I became a university student, and I find this to be a solution. Though I realize that the Moore method may not be for all students. The initial selection of the students must be based on whether they find the challenge appealing, and not based on academic standing. The worst possible outcome for an aspiring mathematician, would be realizing that they are not fit for doing research, which is still better than a realization in graduate school or real life. On the other hand, many who are unsure initially, may gain self-confidence for the long run. I personally had doubts about my own research abilities as an undergraduate, projecting an unpredictable dark cloud into my future. I believe that all students should be given an opportunity to put themselves to the test, in classroom activites of this kind. Furthermore, the professor may serve as a mentor in this setting - accompanying, guiding, and nurturing potential talent.

The benefits of such an adventure, even if experimental, undoubtedly outweigh the risks. As the Ancient Greek aphorism of virtue recommends "Know thyself!", so should mathematicians learn as much about themselves as possible early on. What better way to do so, than a setting that mimics what they are apt to experience later on in real life. This is akin to teaching children to swim in shallow water so that they would be ready for the deep, while throwing them in deep water at once may prove fatal. Is it not the responsibility of experienced researchers and educators to prepare students for the deep water, besides striving to enhance their overall conceptual understanding?