"Increasing Student Understanding of the Fundamental
Theorem of Calculus Through Interactive Computer Modules"
By: David Schultz
Summer, 2004
Introduction
Over the past 15 years I have been continually forced to reexamine just what my role is as a teacher and how it affects my students. Influencing this reflective journey has been national and local mandates, educational research into best practices, and my own personal experiences of observing student success and failure. Although the maturation process still continues to this day, I have settled on a simple tenet that I’ll identify as the core essence of teaching. Teaching is getting an individual or group of individuals to understand what I already know. I can design and disseminate what I believe is an excellent lecture or develop a seemingly engaging and informative activity, but, have the participants constructed an enduring level of understanding of the concepts or have they merely been an audience in another mathematical show? The goal of having students become an integral factor in the construction and ownership of non-naïve concept images is both lofty and elusive. In order to achieve such a goal demands that the teacher seriously consider what is worthy of being taught, what evidence validates that learning occurred, and what is/are the most effective pedagogical approach(s). This Capstone Project represents an in-depth attempt at developing a richness of understanding within my students with regards to the Fundamental Theorem of Calculus through the combination of animated computer modules and traditional lecture constructed in accordance to Bloom’s Levels of Taxonomy [5]. I have never truly been satisfied with the depth of understanding that my students have exhibited over the years in this area and wanted to help rectify this situation through this endeavor. The construction and flow of the project follows the backward design process succinctly presented in Understanding by Design, by Wiggins and McTighe [38]. Relevant research and example cases are peppered throughout the manuscript in order to provide a sound foundational basis for the selection and usage of various methodologies and criteria. It is my desire that upon reading this project one will walk away with new insights into helping students construct a highly developed concept image of the Fundamental Theorem of Calculus, an appreciation for the judicious usage of technology as a tool for understanding mathematical ideas, and a potential template for future concept design considerations in one’s own teaching strategy.
Capstone Project Focus and Navigation Pages
The working hypothesis for this project is stated in the Focus that follows below. The Focus was based on my desire to increase my students’ understanding of the Fundamental Theorem of Calculus in light of relevant concept acquisition research and educational best practices trends.
The Focus:
After receiving direct classroom instruction and interacting
with 3 student-centered calculus concept modules students will be able to
demonstrate a high degree of understanding of the Fundamental Theorem of
Calculus by exhibiting competency at all levels of Bloom’s Taxonomy.
In fashioning the above Focus the three previously mentioned questions were intently considered.
1. What is worthy of being taught?
2. What evidence validates that learning occurred?
3. What is/are the most effective pedagogical approach(s)?
The answering of these three questions led to the division of the project into three distinct phases followed by summative results and personal reflections. The project’s subsections are presented in a linear fashion in order to facilitate the reader’s ability to navigate efficiently throughout the manuscript. In the web-based version each heading is a live link that can be selected for a smooth transition to the section of interest. It is hoped that regardless of viewing format the reader will appreciate the reflective and narrative writing style and my attempt to provide a seamless integration of rational thought throughout.
Phase I – What Should Students Know About
The Fundamental Theorem of Calculus?
·
Why Focus on the Fundamental Theorem of
Calculus?
·
What About The Fundamental Theorem of
Calculus is Worthy of Knowing?
§
Formulation of an Educational Aim.
§
Defining Characteristics & Construction
of the Behavioral Objectives.
§ The Specific Listing of the Behavioral Objectives – Appendix A.
Phase II – What Evidence Validates That
Learning Has Occurred?
·
Selection of Assessment Items and their
Implementation.
·
Construction of the Assessment Instruments.
·
Actual Assessment Instruments.
§
Free-Response Test –
Appendix B.
§
Student Survey – Appendix C.
·
Rubric Selection and Scoring.
·
Assessment Summary & Conclusions:
§ Results, Data Analysis & Discussion of Free-Response Element.
§ Results, Data Analysis & Discussion of Student Survey Element.
Phase III – What Is/Are the Most Effective
Pedagogical Approaches?
·
Which Teaching Strategies/Methods is Best
Suited for my Educational Aim and Learner Objectives?
§
Teaching Strategies Consistent with the
Nature of Calculus.
§
Selection of Teaching Strategies/Methods.
§
Prior Evidence Supporting the Selection
Choices.
§ Implementation Framework.
·
Computer Modules:
§ Riemann Sums - Student Lab:
§ Accumulation Function - Student Lab:
§ The Fundamental Theorem of Calculus - Student Lab:
§
Teacher components available by request from
the author.
What Do I Want My Students To Know About The Fundamental Theorem of Calculus?
·
Why Focus on the Fundamental Theorem of
Calculus?
·
What About The Fundamental Theorem of
Calculus is Worthy of Knowing?
§
Formulation of an Educational Aim.
§
Defining Characteristics & Construction
of the Behavioral Objectives.
§ The Specific Listing of the Behavioral Objectives – Appendix A.
Why Focus on the Fundamental Theorem of Calculus?
When one considers the calculus curriculum as a whole there are central questions that provide the impetus for entire thematic units. One of these central questions is often referred to simply as “the area question”. The question is often posed as follows:
“Given
a function, f(x), which is non-negative over an interval [a, b], what is the
area beneath the function over the given interval?”
The pursuit of answering that question occupies the introductory calculus teacher for a significant part of the semester. The mathematics underlying the answer is profound in its beauty and critical in providing the building blocks for a full understanding of integration and future topics in the calculus curriculum. It is traditionally in this environment that students are introduced to the linchpin concept of the Fundamental Theorem of Calculus whose unification of differentiation and antidifferentiation is considered one of the most important theorems in mathematics. Thompson [35] notes that in the classic text Differential and Integral Calculus, by R. Courant (1937), the Fundamental Theorem of Calculus is referred to as “the root idea of the whole of differential and integral calculus”. A more recent excerpt from Thomas’ [34] Calculus reads:
“The discovery of the Fundamental Theorem of Calculus brought differential and integral calculus together to become the single most powerful tool mathematicians ever acquired for understanding the universe.”
The implication for teachers is
obvious: The Fundamental Theorem of Calculus is a central calculus concept of
which students must have a sophisticated level of understanding. The theorem
lies at the very core of the calculus curriculum and instructors must make a
concerted effort in its teaching. The lack of student understanding of this
theorem and its role in the calculus curriculum is duly noted in the research
and in classroom experiences. John Berry and Melvin Nyman [2] from the Center for Teaching
Mathematics at the
Mathematical theorems like the Fundamental Theorem of Calculus are multifaceted in their compositions and must be viewed both as a whole and in parts. In order to increase the likelihood of promoting a deeper understanding within my students regarding this theorem I had to first identify just what exactly I wanted my students to know about this most ‘fundamental’ theorem. To pursue that end I needed to carefully dissect both parts of the theorem and reflect upon the various characteristics, connections, and implications each element possessed. Furthermore, the theorem and its components needed to be considered in light of previous mathematical concepts that establish its foundation. I quickly came to the realization that in identifying what I wanted my students to gain from this project I needed to surrender to the fact that they may be harboring a whole host of mathematical deficiencies in their previous mathematical concept images and that to try and identify those deficiencies was simply beyond the scope of the project. Research indicates that when a student constructs a concept image they will often hold onto it vigorously even if it is incorrect [16]. Thus, the educational aim and behavioral objectives that I developed for this project are based on what I wanted my students to gain from their interaction with the dev