An In-Depth Design of Introductory Calculus

Response paper for the session "Course Design"
Certificate in University Teaching Program - University of Waterloo

In this essay, I plan to investigate course design alternatives for introductory Calculus, often called "Calculus I" in various science and engineering disciplines. It is a course upon which virtually all university mathematics courses are built, therefore its thorough design for maximal effectiveness is of utmost importance. Mathematics is a primarily cumulative subject, thus out of consideration for the learner, the design of a math course must be carried out with great care. Despite the rational nature of the field of mathematics, it does not necessarily imply ease of reasonable course design. The "elementary" nature of the course for many professors, in itself can imply carelessness. In this essay, I will attempt to extensively detail some pitfalls of this course's design, as well as make constructive practical suggestions which I believe would improve in general the prevalent methodologies in use. It is a course I have been involved in teaching several times, and I feel confident in sharing my predispositions still under development.

I believe that the recurring issues with first-year Calculus courses spring from two main sources of pitfalls: (1) the "toolset" nature of the material, and (2) its elementary nature for the instructor. These two sources may branch out into an avalanche of consequences, that may fall upon the unlucky learner at any time, without them realizing in time, or even later, its adverse affects on their early mathematical foundations. Naturally the learner is mostly helpless, and must rely on the instructor's wisdom. I believe a wise instructor remains a student when s/he enters the classroom to teach, regardless of where s/he might be in terms of career. This is clearly difficult to do so, and is perhaps the main challenge of teaching, to see the material fresh and anew every time one teaches it. Then and only then, may one's teaching become fully didactic. This is imperative with such a foundational course, as a didactic structure transfers into the mind of the learner, in building their own inner picture of the material. Therefore via attempting to remain a "learner in progress", the instructor consciously must shred the "elementary feeling" inherent in the task of teaching Calculus I.

As far as the first pitfall stated above, it is somewhat more striking and therefore easier to identify. First let us confess that Calculus is indeed a toolset, but at the same time it is an interconnected living breathing theory, with a historical past and a future. In my opinion, a well-designed course incorporates a permeating veil of rhetoric in the material, which subconsciously brings the learner to these realizations, essentially showing the beauty and relevance of Calculus. This imperative factor in the design has its own consequences, not the least of which is student motivation. The mindset of all the students in a class can be taken on an ideal course, by an instructor who projects their own appreciation for the theory. These emotional ties may be coupled with rational reasoning for the theory's relevance, with the vast spectrum of theories and applications which hinge upon it. This double parallel subconscious reasoning should enter the learner's mind sufficiently, to induce a seductive motivational effect. Among all the distractions of a freshman's college life, such an attempt at seduction seems quite reasonable, the ends thereby justifying the means.

The above stated points are meant to assist the instructor in taking an ideal philosophical approach to teaching an introductory Calculus course. The actual course design however must be guided by the intended learning outcomes. As stated in the pre-workshop worksheet, by the end of my Calculus course the students should be able to...
• provide an overview of elementary Calculus.
• state the definition of a function, and derive its domain, range and inverse.
• calculate the limits of a wide variety of functions.
• explain and apply differentiation and integration techniques.
• elaborate on the conceptual meaning of the learned techniques.
• model and solve mathematical and real-world problems using Calculus.
I shall elaborate below on the often ineffective ways of attaining these ILO's, and suggest some alternative approaches, in light of Fink's taxonomy.

By "providing an overview" I mean the ability to rise to a global perspective above the material and seeing its inner multilevel connections. This is not a skill that comes easily, and it might even be one of the highest-level skills attainable, which could be why many students tend to lack it, and most instructors fail to encourage it. It ties in directly with the first pitfall above, as an overtly toolset-style teaching works against the development of a global perspective. Therefore, the instructor must consciously focus on this element, in order to counter-balance. It ties in with the idea of "conceptual understanding" under foundational knowledge in Fink's taxonomy. Indeed, the goal must be to develop a complete understanding of elementary calculus, as a foundation for future subjects. On the other hand, this ILO also has an "integration" aspect as per Fink, since connecting ideas is essential to being able to having an oversight.

Stating the definition of a function, and working with its domain, range, or inverse are basic elements of the toolset of Calculus. As such, they represent foundational knowledge that serves as skills for proper performance in cumulative topics building upon them. The pitfall here is that the instructor may forget to remain a student, and may rush through these imperative ideas, without the full comprehension of students. I have observed this multiple times as a student and as a teaching assistant. Clearly, this ILO may be addressed with a sufficiently thorough treatment by a conscientious instructor.

Calculating the limits of functions, can become a stale and mechanical topic, if presented without the intricacies giving its beauty. It can quite easily be reduced to a set of calculation rules and dogmatic methods. This reduction is certainly not the challenge of an educator, but quite the opposite. Without going into too much detail, one may build towards this intended learning outcome, by discussing the nature of the "limit" in the more rigorous sense. This would justify and establish its relevance, giving it sufficient weight in the mind of the learner. Otherwise, one may take this seemingly simple concept for granted. Such a somewhat more precise discussion, would lay the foundations for the eventual introduction of the concepts of continuity and differentiability. Thus making the effort to introduce the approach early on, shall pay off later to the learner's benefit. In Fink's taxonomy, this may be considered a way of motivation, by projecting the exciting implications into the future.

The next learning outcome is tied to the one following it for a good reason: to once again avoid the potential reduction of the material to mechanical calculations. Understanding the conceptual undercurrents of the topics of differentiation and integration, in and of itself is sufficient to motivating students towards exploring these theories created by scientists over two hundred years ago. The idea of time-traveling to walk alongside these pioneering explorers of the world of mathematics, may awaken the more romantic side of a learner, thereby serving as motivation for learning. I am speaking from personal experience, that such motivation can be deeply effective and long-lasting. It is precisely the "human dimension" of caring to develop feelings, interests, and values in Fink's taxonomy, which these thoughts of mine converge towards.

Reaching the level of ability to independently model and solve real-world problems, is clearly an expected goal for learners in applied disciplines. It is also clear, that the maturity and readiness for independence comes gradually, over the span of several courses, not only introductory Calculus. Yet, I do believe that the instructor must make steps towards and assist the development of learner maturity - again, this is one of the intricacies of teaching that often goes ignored, however essential. These steps could be gradually built up from smaller exercise-level problems, to the level of term projects in later terms. In Fink's model, it invokes the full spectrum of implications of concept "application", such as critical thinking and the ability to make project-directing decisions. It is primarily a way of "learning how to learn" - or even to research - independently via intriguing applicational problems. Inducing independence in students may thus be the most challenging yet most important aspect of teaching, matching the magnitude of its relevance.

In my essay I have reflected on my personal beliefs and strategies for designing an ideal introductory Calculus course. The elementary nature of the material is never to be confused with the significance of this contemplation. Indeed I have outlined two major pitfalls such confusion may lead to, as well as analyzed the intended learning outcomes which should guide the process of course design. I found this discussion all-in-all valuable and considerably self-reflective, hopefully increasing the effectiveness of my future endeavors.

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Presentations as a Type of Formative Assessment

Response paper for the session "Assessing Student Learning"
Certificate in University Teaching Program - University of Waterloo

We may have all experienced the elevating feeling of having a great idea inspired from thin air, and suddenly wishing to share it with someone. This must be an ancient human characteristic indeed, one that may have driven the evolution of the modern human by a significant factor. The feeling of “sharing through words” not only makes us feel good for “giving” to others, but can cause us to self-reflect on our own thoughts on a deeper level, as well as give other people a chance to comment on our ideas, thereby potentially improving them. In fact in a more organized and developed manner, this is the fundamental idea behind “presentations” and the response to them, either by the audience or an instructor, in which case it is a type of formative assessment. In this essay, I wish to take a closer look at presentations and their assessment as a means for student learning.

I believe that the primary most important outcome of a presentation, is the effect it has on the audience. The effect can be considered in varying degrees of weighting to be the amount of information retained by a listener, or the influential effect it has on a person, in the short or long run. Certain effects may be qualitatively measured, while some are hazier matters of opinion. What I believe to be certain is that every presentation must have a set of clear objectives - primary, secondary, and else. Objectives assist in the overall structuring and branching of a presentation, while their clear statement in the beginning may help guide the presentation itself. Indeed the structure is the linking bridge from the stated objectives to the observable outcomes. Thus an effective communicational channel opened by the presenter, must have a trait of clarity and didactic structure, serving as the master plan of a master manipulator, which any great presenter must ultimately become.

As far as the specific skillsets - implicit or explicit - required by the intended outcomes of presentations, they all revolve around the idea of effective communication, which I define for myself as the ability of opening a communicational channel, maintaining it, and then applying it for the channeling of information to the audience. Clarity not only in structure but in mere speech itself is obviously a must for such a channel to open, in addition to adjusting the proper sound volume to the situation, and maintaining the continuity of speech. Eye contact may serve as a means to keep the channel open, in addition to actually opening one potentially. Subconscious signals sent via facial and postural expressions, serve to enhance communication. For a presenter, they provide a chance to take their self-expression to the next level, creating an audio-visual orchestra of signals with varying intonations and perceptibility. According to certain theories of the birth of human languages, initially we all communicated via sign languages, which may be why certain regional cultures - such as the Mediterranean - have still retained them to the present day.

In addition to the skills of speech, the speaker's proficiency in the material can influence the presentation to a major extent. It not only gives the impression of knowledgeability, but raises the level of self-confidence in the speaker. Self-confidence in turn may influence a wide range of other factors mentioned above, such as continuity of speech and clarity of structure. Self-confidence is radiant. It shines from the presenter on the audience, and in turn influences the level of trust and confidence the audience has in the verity of the communicated information.

An important dilemma to be decided prior to assessment, is the weighting of the relevance of feedback by either the instructor or the peer students in the audience, as well as the sequential method of communication of the formative assessment. I believe the ideal resolution of this to be as follows: peer assessment should have temporal precedence, while instructor feedback should serve as the deciding factor. Most students may sub/consciously agree that the instructor has a right to serve as a presiding “judge” having the final say. On the other hand, peer opinion must be upheld as valuable for its inherent spectral characteristic, providing a wide range of opinions to consider. This semi-hierarchical relationship must serve as the definitive template for any specific explicit assessment, to ensure the relevance and reliability of the overall reflection on the presenter.

In considering the efficiency of student presentations and their peer discussion as a formative assessment in terms of student and instructor time, one must identify the given base constraints, such as class size, class time, and meeting frequency. Clearly, such constraints require advance planning on the part of the instructor, in order to ensure the smooth flow of class activities, by minimizing the unnecessary pressures of the constraints. Thus considering time limitations, presentations may or may not be considered efficient, depending on the situation. The point of “presenting” however is not necessarily efficiency, but rather that of creating an atmosphere of open communication, by finding and maintaining the right channel for this experience. Therefore the student must be aware of this fully, so that s/he can focus on the more relevant aspects of the presentation and its subsequent interactive assessment.

One may consider alternative forms of assessment, other than the suggested weighted instructor-peer partition. These can range anywhere between the absolutely autocratic, relying on solely the instructor's criteria, to the entirely laissez-faire, whereby peer feedback is the main mode of formative assessment. We may clearly find virtues at both ends of the spectrum. However, a more weighted approach can minimize the weaknesses and maximize the effectiveness of the assessment. The advantages of the autocratic method, is the reliability of expert opinion by the instructor, and the projection of security on the classroom due to it. Peer feedback however, can increase the students' involvement in the betterment of one-another's presentation skills. Some students may be more receptive to their peers’ opinions than an instructor’s, since they are on an equal hierarchical level. Thus the quest of finding an ideal middle way for a specific situation, may result in a more optimal assessment strategy.

The verbal formative assessment matches quite well the desired outcomes and required skills mentioned above. A communicative feedback approach fits appropriately into the format of presentations, and may be viewed as their natural extension. The marking scheme or specific criteria should in my opinion be reserved for the instructor, who is to give a more rigorous and analytical assessment. The audience of students however must be allowed to engage in free conversational feedback with the presenter, for an optimal active learning outcome of all students.

In summary, I have discussed “presentations” as serving the assessment of students of various topics. I have considered a number of analytical questions to observe my ideas in this regard. I found this endeavor to be quite valuable in its self-reflective potential, making me think deeper about the conceptual approach to this type of assessment.

Self-Reflection for Identifying my Ideal Learning Experience

Response paper for the session "Understanding the Learner"
Certificate in University Teaching Program - University of Waterloo

A commonly voiced belief is that the learning environment greatly influences the learning experience of students. In fact, mostly the teacher is the one to blame, when students' expectations are not met in regards to their imagined ideal of classroom instruction. In this essay, I will reason that learning is a bipolar experience, greatly dependent on both the student and the teacher. Via an acute self-reflective analysis, I will identify my own learning characteristics in the context of two very different learning experiences - one negative the other positive. Meanwhile, I shall keep in consideration the potential experiences of other types of students, based on my retrospective observations. I will begin with a brief description of myself as a learner, to provide a working foundation for the subsequent analysis of my experiences.

Currently I am a graduate student working on my PhD in Applied Mathematics, primarily occupied with doing research, which is quite different from my undergraduate mindset, when studying was my primary focus. Therefore much of this discussion will be a travel back in time, though my main characteristics as a learner have not changed, only become less elaborate and refined. When I was an undergraduate in Hungary at the Eötvös Loránd University, I was literally like a sportsman of the mind - constantly monitoring myself for what shape I was in. This was of utmost necessity so that I could allocate my time and efforts to the most appropriate times for a particularly or less demanding subject. I monitored my diet, breathing, everything - it was a bit insane, but the tasks at hand required it. I was also very independent as a learner. I did not value having to sit through lectures, where I was superimposed with a "sequential" learning experience in time, in which I was unable to flip back and forth as I would do by myself with a book, to connect formerly learned concepts. I am what I can now properly categorize as an absolutely "global" learner. Even in doing research, I feel more like a bird flying over the savannah than an antelope galloping through it. Whenever I notice something interesting, I zoom in somewhat, then zoom out, or zoom in more if I find it to be of value, guided by my intuitive inner compass. Just as a bird, I am a great lover of freedom, and being restrained in any way, whether by a non-liberating lecture or else, can be very disheartening to me. I feel lucky to have been endowed with a considerately adaptive supervisor who understands me by instinct, and is willing to act as a companion, versus a guide.

Considering the above, the ideal learning condition for me is most likely an environment in which I am given a chance to contemplate the material with freedom, allowing me to internalize it in a gradual process of digestion, in which I reassemble the buffered information packets into a new didactic whole within myself, via a highly independent process. Is this possible through any kind of lecture or tutorial format? Does not my own sense of independence make the reception of any external learning projection impossible? Sometimes it does feel like I have this impediment. In rare occasions however, I encounter a learning environment in which sequential spoonfeeding is not the primary mode, and which I find highly enlightening and motivational, activizing my neurons on all levels.

A specific negative experience was my Partial Differential Equations (PDE) course as an undergraduate. It was undoubtedly a large bulk of information we had to internalize, or perhaps only seemed like it? The professor acted like a sage on the stage throughout the lectures, speaking his mind like a radio, and scribbling away on the board, without any consideration for our understanding. Certainly contrary to a multilevel learning experience, that would have sent off sparks in the minds of various types of learners. I personally felt being given no chance to reflect on the material, the pace was too fast and uninterruptibly monotonic, which made me feel disconnected, from both the professor and the material. I was given no opportunity to make a connection either, as it felt like the instructor had no idea there was even an audience. I was not motivated in any way to learn the material, and suffered greatly when I "had to" in the exam period.

I believe a good teacher not only connects with the audience, but monitors their level of understanding throughout a lecture, and sets a fluctuating pace accordingly. Before the process of information flow is even initiated however, the professor must set a base tone for the rest of the lecture in some way, much like a painter might paint a base color for a painting to be laid out upon. The base tone might involve some motivational words of wisdom on the nature and relevance of the material, its historical importance, or their own enthusiasm for the topic. These may serve as an extension of a hand to help students on board for a journey of sailing the waters of knowledge. Most of all, teachers must primarily know themselves and their own motivations, so that they may accompany students on their own journey of learning, whether as captain or companion. In a classroom, a teacher must above all create a multilevel learning experience, in which learners of all kinds shall thrive.

A specific positive experience was my Functional Analysis course as an undergraduate, which I remember as my favorite one of all time. Unlike the PDE course in which I never got past learning the toolset, here I internalized it from the notes quite naturally and largely independently, since I was eager to apply it in assignments given at the tutorial. The tutorial was also held by a professor who I have had for years in Analysis, and found his approach reliable and respectable. His opinion and marking represented to me the unquestionable law carved into stone. In other words I truly trusted him, regardless of him being a difficult personality, who did not think much of "applied" mathematician students. So clearly, I had an incentive to prove myself in attempting to solve the rather difficult problems given by him. Indeed week after week I handed in each assignment in full completion, even though a vast majority of students sabotaged the tutorial by missing it and not doing the homework - the general consensus was that this was a teacher to be disliked. Yet, I saw him as a representative of mathematical rigour from the past, and his very character projected reliability and wisdom. So even though I was a largely independent learner, his character still had an influence on my motivation, captivating my interest as if by undetectable magic. I was not alone though among those who were motivated - some were hard-working by nature, some saw me as competition, and a few were just trying to get by on a mediocre level. I however saw this class, which presented the pinnacle of modern Analysis (Calculus), as a chance to mature into an able mathematician. When the course was over, it felt like I have gone through a training period harsher than ever. Yet by overcoming independently the challenges I have faced, I have reached a new level of mathematical awareness and understanding, inaccessible via any other learning process. Then and there, I became a mathematician.

How was any professor able to evoke all these feelings alone? Analyzing my memories, I believe the answer is that he did not do it alone, I was an equally necessary ingredient. In fact, being the experienced educator he was, I believe he tailored his classes to specifically the kind of student that I was. This was quite irresponsible, since the rest of the class was left behind. I have observed however, that many professors grow tired over the years of having to deal with other - perhaps less motivated - students, and hold their classes for the most worthy. I must however voice my criticism, being a great believer in student potential. While I may have felt good in this particular classroom experience, it was unfair to the other students. Therefore I once again arrive at the conclusion, that a good educator creates classes which run on multiple levels, as an orchestral concert involving winds, strings, and percussion, for various musical sounds which may appeal in varying degrees to individuals in the audience. For instance, there are lovers of rhythm (percussion) and those who are more receptive to tunes (winds or strings), whether brief or overarching. A great symphony like a good lecture, weaves elements in a magical fashion, sparking the minds of as many listeners as possible, ie. optimizing the learning experience for the entire class.

In conclusion, I deduce that the ideal learning experience for me is one where I am granted freedom to explore, and the material is shown to me to be a toolset for practical application in the gradual exploratory process. Even though I have always thought that the personality of the instructor has no effect on me as a learner, I have now realized that it is indeed a major factor towards connecting with the material. A demandingly harsh yet nurturing environment, can be better than the nowadays common easy-going attitude of many professors, on the lookout for high student ratings. A professor can still remain accessible for a personal connection, in many ways. In regards to presentation of material, I have reasoned for a multilevel orchestral manner of information propagation, for optimal exploitation of student potential. Though certain manners of instruction I may see selfishly as ideal to me personally, as a teacher I feel I must become a more complete individual, growing beyond my own natural ways of learning.