"Assessing Student Understanding of the Fundamental
Theorem of Calculus on all Levels of Bloom's Taxonomy"
By: David Schultz
Montana State
University
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-naďve 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 University of Plymouth write, “Our experience is that the vast majority of students in introductory calculus courses do not develop an appreciation of the theoretical concepts or an intuitive ‘feel’ for the ideas. Integration is seen as the opposite of differentiation and techniques in integration are little more than a ‘bag of tricks’.” Orton [28] revealed that students are able to apply some of the basic techniques of integration but that they generally possess fundamental misunderstandings about underlying concepts and that their view of central calculus concepts (i.e. The Fundamental Theorem of Calculus) are exceptionally primitive. These findings are representative of my own experiences and bolstered my justification to pursue this project’s goals.
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 developed materials and classroom experience and do not attempt to ascertain or address any mathematical gaps in the their prior mathematical experiences. Admittedly, this omission is a shortcoming of the project for such omissions most certainly exhibited themselves in the students’ assessment responses. Even so, I believe that the project’s overall design and results offered valuable quantitative and qualitative in-sight into my students’ mathematical thinking and understanding as they progressed through the project elements. Such measurements may be regarded as baseline standards upon which to make future comparisons.
The Formulation of an Educational Aim
The structure and writing of learner objectives commands a large place in educational literature. There is a wealth of manuals, textbooks, in-service programs and the like which are all geared to writing worthwhile objectives as judged by specific criteria [10], [15], [39]. In the development of learner objectives I considered two specific tasks:
1. Identifying the objectives to be demonstrated.
2. Elucidating the objectives in written form.
In thinking through the first task I came to the conclusion that there must be an educational aim which acts as the umbrella for a collection of underlying behavioral objectives which are composed of both learning outcomes and classroom processes. Davidson and McKeen [15] define an educational aim as a legitimate and essential part of the objective defining process which is characterized by the utilization of such vague descriptors as foster, understand, appreciate, and enhance. The selection of an educational aim must be regarded as an enduring idea germane to the subject and supported by national, state, and district standards. The educational aim of increasing student understanding of the Fundamental Theorem of Calculus meets both criteria and qualified it as the canopy for my project. The educational aim of this project is stated as follows:
“The
student will increase his/her understanding of the Fundamental Theorem of
Calculus.”
The reader may note the absence of any specific performance task within the educational aim itself. Such elements are contained in the behavioral objectives that acted as the foundational glue of the assessment design and teaching strategy selection process.
Defining
Characteristics & Construction of the Behavioral Objectives
The selection of carefully chosen behavioral objectives acted as the structural support for my educational aim. In order to provide specificity and clarity in their formulation I chose to use the framework by Cook and Wahlbesser [10]. Their framework identifies 4 criteria that should be considered when writing behavioral objectives:
1. Does
the objective identify who is to exhibit the performance?
2. Does
the objective describe an expected observable outcome?
3. Does
the objective identify any materials or directions needed for the learner?
4. Does
the objective identify what constitutes an acceptable response?
Each of the 25 student behavioral objectives formulated for the project exhibits these four criteria. Additionally, the collection of objectives was specifically designed to be representative of all six levels of Bloom’s Taxonomy of Understanding. Bloom’s Taxonomy provided the hierarchy of understanding upon which to align the behavioral objectives to best achieve the stated educational aim. This process of identification and alignment of objectives in accordance to Bloom’s Taxonomy revealed to me that through the years I had often succumbed to emphasizing one particular level too heavily while shortchanging or ignoring others. The specific listing of the behavioral objectives deemed as being worthy of student understanding and uncoverage is found in Appendix A.
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.
Selection of Assessment Items and their
Implementation
The choice of assessment types and their constructions was paramount in determining if the project’s goals were achieved. Jody O’Neal [27] of North Georgia College & State University points to the choice of assessment instruments as critical in capturing the clearest possible picture of what students know and are able to do. With that in mind, I selected two varieties of assessment instruments: a 25 question free-response exam and a 20-question student survey. The free-response exam was broken into 2 separate parts. Students had 50 minutes to complete each part and took the two parts on consecutive days immediately following the last of the three instructional modules outlined in Phase III. The student survey was taken during the class period following the second part of the free-response exam. Both assessment types were chosen based on their ability to reflect and measure the behavioral objectives while providing the necessary latitude needed for implementation given time and facility constraints. Informal assessment was garnered through observations of discourse between students working with the computer modules and during the allotted direct instructional time.
Construction
of the Assessment Instruments
The formal free-response assessment instrument was constructed to accurately reflect all six levels of Bloom’s Taxonomy for the Fundamental Theorem of Calculus. Creating assessment items according to models termed levels of cognition has been recognized as a valuable tool in the assessment construction process for creating an effective means of measuring mathematical understanding. Bloom’s model has been noted to be especially successful in the mathematical sciences. In A Handbook for Mathematics Teaching Assistants, by Tom Rishel [46], of Cornell University, he specifically encourages teaching assistants to incorporate Bloom’s Levels in their teaching activities and assessment instruments. Other examples heeding similar advice for assessment item construction for the calculus curriculum include the works of Neil Davidson & Ronald McKeen [15]. As mentioned previously, the free-response assessment instrument was divided into two parts. This was done in order to ensure that students had ample time to demonstrate their level of understanding for each question. The first part of the exam consisted of 15 questions that were representative of Bloom’s first three levels of understanding: Knowledge, Comprehension, and Application. The second part of the exam asked students to respond to 10 questions which were representative of Bloom’s last three levels of understanding: Analysis, Synthesis, and Evaluation. All 25 questions were carefully aligned to the specified learner objectives and paralleled the students’ experiences in the development of the key concepts during the computer modules and classroom instruction. The careful alignment of the free-response assessment test items to the stated objectives and student activities provided the authenticity and reliability upon which to draw meaningful measurements and conclusions. This close alignment between instruction and assessment benefits both teacher and student. As Glaser and Silver[17] note:
“Closer
ties between assessment and instruction imply that the nature of the
performances to be assessed and the criteria for judging those performances
will become more apparent to students and teachers…”
In fashioning the student survey I first reviewed several examples of previously administered student surveys [4], [12], [25]. After cautiously reviewing the questions contained for their appropriateness to this project’s goals, I settled on a 20-question survey consisting of 15 multiple-choice questions and 5 short-answer questions where most were of my own design. The overriding purpose of the survey was to elicit the students’ own perceptions on the effectiveness of the project and to provide me with insight as to where the project could be modified in the future. The 15 multiple-choice questions allowed me to quantify some of their responses while the 5 short-answer questions provided me with additional elaboration on pertinent aspects of the project in general. The free-response test can be found in Appendix B while the student survey can be found in Appendix C.
Rubric
Selection and Scoring
Any rubric selected for the scoring of the free-response test needed to accurately gauge the depth of student understanding on each test item. Constructing rubrics that focus on “fleshing out” the students’ depth of understanding, as opposed to simply measuring the progression of skill development, is strongly advocated by Wiggins & McTighe [38]. Additionally, the rubric chosen needed to assign a meaningful quantity to each student response in order to provide reliable evidence upon which to draw valid comparisons between individuals/groups. The rubric chosen for the free-response test instrument of this project was based on a collaboration of efforts from several faculty members at Arizona State University and is a hybrid of the rubrics used in the state mathematics exams from Kansas and California. The general rubric is as follows:
Score
5 Superior response:
o Complete in responding to all aspects of the question.
o Shows complete mathematical understanding of the problem’s ideas and requirements.
o Includes only minor computational errors, if any.
4 Assign to those responses falling between 5 and 3.
3 Adequate response:
o Demonstrates understanding of the main idea of the problem.
o