"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-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 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 Is not totally complete in responding to all aspects of the problem.
o Shows some deficiencies in understanding aspects of the problem.
o Exhibits a moderate amount of reasoning but reasoning is incomplete.
2 Assign to those responses falling between 3 and 1.
1 An inadequate response:
o Attempts, but fails to answer or complete the question.
o Shows very limited or no understanding of the problem.
o Contains words, examples, or diagrams that do not reflect the problem.
0 No response:
o The question was left blank.
o
The written information made no attempt to respond to
the problem.
o The written information was insufficient to allow any judgment [9].
As for the 15 multiple-choice questions of the student survey, they were scored as a raw percentage of those students who selected a particular response choice. The results of the additional 5 free-response questions were analyzed for patterns of response along with student sentiment towards the project in general. I was particularly interested in obtaining additional insight into the students’ experiences with the modules and their own perceptions thereof. It was believed that the ability for students to express themselves in an open format would broaden the range and quality of their reflective responses.
Results,
Data Analysis & Discussion of Free-Response Element
The student subjects for the project were 16 beginning calculus students who had a wide variety of incoming mathematical experiences. They all participated in both the traditional and interactive computer activities and upon completion received the two assessment instruments: free-response exam and student survey. Because this project did not involve a control verses treatment group scenario, the data collected from this project and its quantitative and qualitative analysis represents a baseline upon which future classes can be compared to. The free-response test items were carefully graded based on the aforementioned 5-point rubric and their numerical results were examined keeping in mind Bloom’s Taxonomy as the benchmark qualifier. The arithmetic mean, standard deviation, range, and median were all examined and carried weight in the formulation of the conclusions and general comments that accompany each individual test item.
Free-Response Results
The analysis of the free-response test items is broken down into Bloom’s individual taxonomy levels followed by a combined summary of results. Each level includes a graph displaying the arithmetic mean for each test item and is accompanied by a brief commentary.
Knowledge
Most students scored quite high at the knowledge level with little variation within the 5 test items. Questions 1 and 4 had the greatest variability (standard deviations of 1.41 & 1.65) while the other 3 items showed very little deviation from their means. At this level of understanding the 5-point rubric was less effective at delineating the quality of student responses due to the lack of sophistication inherent to the questions. I expected high marks at this level but did notice that several students responded with an expression when an equation was explicitly called for.
Example:
Write down the equation for the FTC1.
Erroneous response:
Teaching Implications: I will emphasize the distinction between equation and expression to a greater extent.
Comprehension
The comprehension level of understanding provided more varied results. All of the questions except for number 3 had a range of 0 – 5. Questions 1 and 4 required students to summarize and interpret in written form the meanings of particular symbolic relationships and mathematical phrases. Students struggled to write clear and comprehensive responses. This could reflect an absence of meaningful written tasks in their previous mathematics instruction. Question 3 had students describe the affect a change in a diagram would undergo given a particular variable change. Needless to say, the group scored extremely high on this “visual based” question which may have been a direct result of their interaction with the computer modules.
Teaching Implications: I see a need to infuse more writing activities throughout the semester if I expect to get better results on questions that require such responses. One can argue that perhaps the students did understand what an adequate response should have been but were unable to express it in the form asked for (i.e. written in paragraph form) and, hence, the structure of the question invalidated any reliable measure of understanding. I reject that notion due to the actual content of their responses. Since I consider it highly advantageous for students to be able to communicate a mathematical concept through a variety of means I will seek to minimize my students’ shortcomings in this area.
Application
At the application level of understanding we see more consistent results again. All the questions except for question 4 had very high means. The questions themselves progressed from simple “evaluate/solve” types to elaborate word problem scenarios. It bears notice that as the problems became more demanding in reading and interpretation the scores dropped. Question 4’s results were significantly lower due in part to the complexity of its structure. However, once I included a visual component (question 5) the students performed much better. This again could point to the visual dynamics of the computer modules as contributing to greater understanding of problems reflective of the modules’ own characteristics. It was quite interesting to note how many current textbooks categorize problems as “analysis and synthesis” even though they are merely application level problems that are elaborately written, multi-task, or simply algebraically complex to solve.
Teaching Implications: I certainly do not want to fall into the “because it is harder to solve, it must be a higher level of understanding” trap. Texts often try to appeal to instructors by claiming they are ripe with real-world examples and problems for students to solve. Naturally, we want students to become excellent problem solvers on a variety of application problems but let’s not use this as our understanding threshold level. It takes time to construct meaningful questions and activities in the upper three cognitive levels as posited by Bloom but it is undoubtedly a worthwhile pursuit.
Analysis
Progressing past
the application level it became increasingly challenging to write quality
representative questions. In order for students to have adequate response time
this level consisted of 4 questions rather than the previous 5. Although the
mean scores are lower overall, the consistency in their values is striking. If
one examines the individual data one sees that the range of answers was 0 – 5
while the modes were 5. Most students did very well while others chose to
simply leave many of the questions blank. Questions 2 & 3 of this level asked
for some reflective prose and seemed to be the most difficult while the
dissection of a particular theorem into its components (question 1) was not as
challenging. The last question required students to extract meaning from a
given graph. The solution sought was not arrived at trivially and even though
the question was given in graphical form students exhibited a sub-par
performance on it (mean = 2.82).
Teaching Implications: As stated earlier, quality questions at this higher level were much more difficult to compose. I am confident that I can create better questions in the future and look forward to doing so. I will continue to accentuate the connections and components of particular theorems and the students’ ability to elaborate on them.
Synthesis
The synthesis level was very rewarding to construct and to grade. Students were required to take what they had learned about the FTC and extrapolate new meanings. The graph above shows the great disparity in the response means from problem to problem. All 4 items were unique in their solution emphasis with problem 4 being the most demanding (develop the mean value theorem for integrals). Students did best with question 3 that asked them to propose a method of solution to determine the area between two functions when presented graphically. The high level of student achievement on another visually aided problem seems to be consistent with the previous levels’ results. Question 2 asked students to construct a geometric argument demonstrating that the integral of an odd function over an interval from [-a, a] is zero. Only one student claimed general symmetry for odd functions while all others who had responses appealed to specific functions (i.e. sin(x), x3, etc.). This lack of “generalizing” seems consistent with past introductory calculus students I have had. The students did, however, create very nice example drawings of odd functions and did not seem to be intimidated by the directive as a whole.
Teaching Implications: Students indicated to me after the test that they found the problems at this level very interesting and uniquely different than what they had experienced in the past. I will certainly make a conscience effort to interject such activities in my future lesson plans.
Evaluation
The two tasks I proposed at this level were:
“Assess
the strengths and weaknesses of the Fundamental Theorem of Calculus.”
“Assess
the theoretical importance and utility of the Fundamental Theorem of Calculus.”
The quality of the responses varied considerably (ranges: 0 – 5, and 1 – 5). Many students failed to stress the practicality of the FTC when compared to Riemann Sum Approximation Methods. I concede that the tasks for this level may need greater clarification in order to elicit the responses I seek. Perhaps if they were phrased differently the scope and depth of the responses would have been better. Students who left the items blank or nearly blank may have done so due to confusion of intent. It would be interesting to reverse the question order of the entire test to see if fatigue and waning enthusiasm could have been contributing factors.
Teaching Implications: I was not pleased with the quality of the answers I received and will put greater emphasis in this level. There was a consistent thread throughout the responses: those who wrote proficiently continued to do so. For those who provided inadequate responses a lack of writing skills may have been a significant contributing factor. Since the questions on this level did not ask for any mathematical symbolism or graphical representations, I might consider including such components in the future to better reflect the nature of the utilized materials.
Comparative Summary
The overall results are consistent with educational research and my own past experiences in which students perform best at the lower levels of understanding. Since I did not have a control & treatment group to run a comparison study the results above will be used to establish a baseline upon which to do future analysis. The results have, in my opinion, satisfactorily met the project’s goals. Modifications and improvements in the overall structure and delivery are certainly in order. In the student survey participants identified several key areas for improvement. It is precisely this type of input for which the student survey was designed.
Results,
Data Analysis & Discussion of Student Survey Element
There were 17 students who provided responses to the student survey assessment component. It provided students the vehicle by which to candidly express their participatory experiences in an unfettered format. I wanted the computer modules to be user-friendly, of high-quality construction, interesting, and beneficial in developing mature concept images of the Fundamental Theorem of Calculus. Additionally, I wanted students to have a positive mathematical experience that they could look back upon fondly. When one peruses the response data given below of the 15 multiple-choice questions and the 5 open-ended questions it is evident that in the eyes of the students the labs accomplished what I had desired.
Student Survey
Directions:
Please allot some time to
answer the following questions for they are designed for me to better assess
how the computer labs worked and how I can make them better for future
students. I am very interested in your personal experiences and attitudes
towards them. I wish to provide the best calculus experience for all students
and this survey will help me in doing so. Thank you for your time.
Circle the response that
fits best.
#1. How often have you used a computer lab
(not calculators) in your past math classes?
a. Very
Often 0%
b. Occasionally 0%
c.
Seldom 3%
d. Never 82%
Indicates
that most students have never used a computer in their mathematics courses.
#2. How would you describe your overall experience with the 3
labs we did?
a. Very
Positive 47%
b. Somewhat
positive 53%
c.
Somewhat
negative 0%
d. Negative 0%
e. Very
Negative 0%
Displays
an overall positive experience for all students.
#3. Which of the 3 labs did you like the most?
a. Riemann
Sums 24%
b. Accumulation
Function 65%
c.
Fundamental
Theorem of Calculus 12%
Curiously,
the most inquisitive based lab was the one they preferred. Students needed to
determine the Accumulation Function from the animations. This could imply an
important element that all labs should contain and reflects the call for
inquiry-based investigations.
#4. Which of the 3 labs did you like the least?
a. Riemann
Sums 41%
b. Accumulation
Function 12%
c.
Fundamental
Theorem of Calculus 47%
Reflects
the results of #3.
#5. How difficult were the labs to run?
a. Not
difficult – they were easy 100%
b. Somewhat
difficult 0%
c.
Difficult 0%
d. Very
Difficult - too hard 0%
Validates
“user-friendly” goal. Students today are more adept with computers so their
utilization should not be dismissed based on complexity grounds. Having the
students work in pairs may have also contributed to the results.
#6. How would you rate the quality of the labs?
a. Excellent 12%
b. Good 82%
c.
Average 6%
d. Below
Average 0%
e. Poor 0%
Supports
the student view that the labs were of good quality or above. I would liked to
have seen more “excellent” responses and will seek greater clarification from
the students regarding what they consider excellent.
#7. How helpful were the labs in understanding the concepts?
a. Very
helpful – should always be a part of the class 65%
b. Somewhat
helpful 35%
c.
Not
very helpful 0%
d. Not
helpful at all – waste of time 0%
The
responses here support the project’s educational aim and learner objectives.
The students unanimously felt that the labs were of some conceptual help.
#8.
Do you think the animations helped
you visualize the concepts better than just the textbook?
a. Yes 88%
b. No 6%
c.
Undecided 6%
Supports
the usage of animations.
#9. Do you think the lecture time in class
combined with the labs is a good approach to learning the Fundamental Theorem
of Calculus?
a. Yes 100%
b. No 0%
c.
Undecided 0%
These
results overwhelmingly support the teaching strategy/methods chosen and
advocate a multi-pronged approach.
#10. Were the examples and student problem sets interesting?
a. Yes,
very interesting 18%
b. Somewhat
interesting 71%
c.
Not
very interesting 12%
d. Boring 0%
This
could be an area of improvement for the future. Choosing worthwhile activities
may not always be akin to student preferences but a concerted effort needs to
be made to obtain as best a fit as possible. I will ask my future students
which activities they liked best and why.
#11. Do you think you enjoyed calculus more because of the labs?
a. Yes 76%
b. No 18%
c.
Undecided 6%
It
is nice to see that many students felt that the labs increased their enjoyment
for the subject.
#12. Did you like working with a partner next to you?
a. Yes 71%
b. No
6%
c.
No
opinion 24%
Experience
shows that some students are ambivalent when it comes to partner or group work
and the results here bear that out. I will continue to follow this set-up until
I see more negative responses.
#13. Would you recommend that calculus classes
have some computer lab activities that use animations in them?
a. Yes. 100%
b. No 0%
c.
Undecided 0%
Students
strongly advocate the usage of animations in a lab setting for learning
calculus.
#14. Do you plan on taking another math class?
a. Yes 76%
b. No 12%
c.
Undecided 12%
Most
of the students will move on.
#15. If you were to take another math class
would you like it to contain some computer animations for the concepts?
a. Yes 94%
b. No 0%
c.
No
opinion 6%
The
results indicate that students strongly desire an opportunity to have computer
animations in their future math classes.
The
results of the 15 multiple-choice questions indicate that the major goals of
the lab construction and intent were accomplished. The additional 5
free-response items allowed the students to elaborate on varying aspects of the
labs and provided me further evaluative information. For each question I have
copiously rewritten the student responses so that the reader may get an
unbiased and panoramic view of the results. Concluding each question are some
of my own personal notes regarding my impressions of the student responses. It
is hoped that readers may gain some added insight into the student viewpoints
expressed.
#16. Did using the animations on the computer help you focus and learn the concepts better than just using the textbook and listening to a lecture?
“Yes,
using the animations helped me better understanding the concepts than just
using the textbook..(unreadable) ..or that then I could visualize how the area
were increasing and in what form.”
“Yes,
the animations helped.”
“I
have always relied on the internet to provide things such as in depth
discussion, or visualization of difficult concepts. Long before we were using
the labs, I was utilizing these sources + the disk provided with the book.
Early in the semester I learned the usefulness of the animations, so the labs
were a continuation of this.”
“I
liked the animations more because I could step to a point. The fact that I
could stop the animations where I wanted to was nice. The animations for the
Rieman Sums where very nice. It gave a good feel for how close the rectangles
get to the curve.”
“Yes,
it changed things up, when it seems we are starting to get pounded with
material, the labs lighten the mood and helps me be more relaxed while I
learn.”
“As
a visual learner I found it extremely helpful.”
“Yes
the labwork that I did was helpful.”
“They
may have helped some but I don’t think they were necessary.”
“Yes.
I understood things, better when there is visual aid. I feel that computers are
just a supplement to the lectures.”
“Yes,
the animations on the computer made the visualization process much more real. I
found that I was able to understand the concepts much better than just doing
the assignment from the book. I learned so much more by being able to play and
work with the programs.”
“Yes,
not only did the animations help, but you could study the concepts as long as
you wanted. It’s not like a class where you take notes and if you miss
something you’re lost; with the computers you could go over the same
problem/concept as many times as you wanted to get a good grasp on the topic.”
“I
don’t think they helped me understand the concepts better but they were a good
break from the norm.”
“Yes,
by using the animations you could get a visual idea of what was happening.”
“The
animations were equivalent to seeing multiple drawings on the whiteboard during
lecture. The nice thing was you could back up at your own pace and visualize
the concepts we were learning.”
“Yes.
The animations helped provide a visual context to the math and better
illustrated the theorem then just using the text and trying to grasp the
concept.”
“Textbooks
have never been much help to me. Even when they do example problems, they jump from
step to step. Maybe its to save space, but the connection from one step to
another isn’t always clear. Also, the examples don’t always cover the several
different types of problems that can occur in a chapter. The problems in the
prob. Sets have so many situations, that the examples in the book aren’t much
help. On the harder concepts this year, just watching a couple of problems
solved from beginning to end in front of us helped me see the bigger picture
and then I could grasp the concept for myself and understand the laws and rules
of calculus. This works better than reading a laws definition in the text. A
combination of visual animations and real life problems solved on the board
teaches concepts better.”
“Yes,
it is much better to see an animation rather than reading “image the number of
rectangles over the interval growing”. Many times descriptions written by text
book authors are limited and only make sense to one who already knows the
concept, and not one who’s trying to learn it. Lectures are better because
they’re interactive, but still lack the simplicity of seeing the movement in an
animation.”
The prevailing sentiment is that the students believe the
labs helped them understand the concepts. They pointed out that such labs
allowed for greater visualization, an easing of time constraints, flexibility
in usage, and clearer explanations. This is a positive response as it validates
some of the project’s working assumptions that were based on cited educational
research.
#17. Did the labs make the classroom lecture and
textbook easier to understand? Would you do the labs first or the lectures?
“I would do the labs first, and the reason is the labs start very simple and move slow, with each example, a little more in depth. Once the concepts are visualized. Then it would help to see those concepts applied in solving problems, by a live person in real time, not only examples in the book.”
“Yes.
Lecture first, then lab, then a wrap up/ summary lecture. Without the lecture,
the lab was more difficult to understand. The lecture provided a framework for
the lab to slide into, and the lab took the lecture information and made it
easier to understand the concept.”
“Hands
on learning always help. Lecture first then labs. Unless you make the labs more
like a science experiment. Then during lecture you can discuss what you saw or
found.”
“I
would do the lectures first w/textbook because after using the labs – I felt so
confident in the FTC stuff I really avoided the text, because the text is
always difficult to understand. It would be better to get everything you could
out of the textbook, then make it seem like less of – foreign language w/ the
computer labs.”
“The
labs were much better than the textbook but I think I learned more and better
from the lectures.”
“Yes
the labs make it easier to understand and lab should be first.”
“Yes,
the labs made it easier to understand because if you don’t really know what
your teacher and the book are saying how can you visualize the concept. The
computers allowed you to see the animation to help you understand. I think a
particle introduction into the lab would be good then the lab and then the
lecture. This way the students are not lost completely when they go to do the
lab. But I think that the labs did a good job of giving information about the
concepts that the lecture in the end sums everything up.”
“Yes,
It cleared up some confusing points for me. I would do the lab first because
they were more of a discovery type lab.”
“Somewhat
easier to understand. Labs first.”
“If
I had completed more labs than I would have been able to understand more. I
think lectures are necessary to begin with and the labs are helpful as
additional learning activities.”
“I
think it was easier to have some understanding of the concept before jumping
into the lab.”
“Yes,
because I could visually see what the text book is talking about.”
“I
liked having the lab first. It made the lecture easier since I already had an
idea what we would be learning.”
“I
would briefly discuss the concepts first. The accumulation lab was the least
easy to understand, although if I had read the book before class…J.
The labs are great tools for understanding the concepts, I am glad you have
left them available on the internet. Google searches don’t always point to
quality.”
“The
labs do make the lecture material more clear, I think the labs should be after
the lecture.”
“Yes
the lab made easier everything for my understanding I would do the lectures
first and then the labs. Because the lectures only gives theory so with the
labs I can (illegible) and play around to see what is going on.”
The use of computers and lectures as tools for understanding
was relatively unanimous. The order in which they should proceed was quite
interesting. The students were about split on which should be first with the
lecture component having a slight edge. It seems that the students believed
both elements were beneficial. My sense is that I should carefully examine how
I can fuse them more advantageously in the future.
#18. Do you think the labs motivated you to
learn the concepts more?
“I’m always motivated to learn new concepts. I think the labs facilitated learning the concepts by making them easier to understand. I feel the labs were encouraging, which in turn is motivating, and more could be easily understood in a shorter amount of time.”
“Yes
I thought that everything that you can visualize is better for a good
understanding you can (illegible) more information if you have the opportunity
to see something.”
“Yes,
I think the labs helped me a lot. They gave me the motivation and knowledge to
comprehend the concepts.”
“Yes,
the labs provided a quality means to understand a complex concept. Anytime
learning is made easy, it is a motivator. Computers can help provide the needed
“push” over the learning curve. The labs are clearly laid out w/ clear
instructions which make it easy to follow & complete.”
“I
don’t think so. I wanted to learn these tings anyways. I would work just as
hard if they had come from a textbook.”
“Not
necessarily, because I know I need to learn them no matter what, the labs gave
another way of doing things as opposed to the same old lecture everyday.”
“I
think that just by doing the labs I learned the concepts better.”
“Yes.”
“No.”
“I
didn’t experience any added motivation from the labs.”
“Yes,
they were fun and it was easier to understand so by doing the lab we were more
motivated. The labs gave more practice than just the class examples or
textbook.”
“Ya,
they did a little.”
“No,
but the didn’t discourage me either.”
“Yes.
I can’t say why, but I definitely was more interested now than before with
class.”
“Sure,
it took away from the same routine day in and day out of coming in and just
listening and writing. It got us involved.”
“Motivation
by the end of the semester is non existent. There is only a drive to survive
the last month of classes. The labs enabled me to get a much better grasp
(quicker) on the concepts then working through and rereading the Schaum’s calc
outline and the textbook usually takes.”
“Yes,
it is nice to have more than one resource. Than one being the text. I feel
texts don’t always communicate to everyone. They explain things in one rigid
way. The labs are a nice change in approach to a stale way of just reading and
taking a stab at applying.”
The responses indicate that most students prefer variations in their learning and many felt the labs had some motivational benefit. Several students indicated that they were not additionally motivated because they needed to learn the material anyway. What I took away from reading these comments is that more needs to be done throughout the year to make students feel as active participants.
#19. In your
opinion, what were the strengths and weaknesses of the 3 labs?
“The
one weakness is that the lab on the web is not interactive, I don’t know enough
about computers to know if it is possible, but it would be nice to have that
convienience of doing it @ home. The strengths are the visualization of the
concepts actually happening.”
“Somewhat
repitious, and hard to finish in one class period. Some of the wording is a bit
ambiguous on the second lab. Good summarization of the topics, visual
representation, and organization to lead students to grasping the concepts in
an understandable manner.”
“Weakness
– Wasn’t able to replicate them at home, a little vague on what the goal was
until we got into the middle of the lab. Strengths – Could be repeated over and
over. Graphics and animations.”
“Refer
to #16, 17, 18 for strengths. The weakness was the inaccessibility outside of
class. I know this will be resolved in the future, but it was really a drawback
this year. Also, Maple 6 isn’t user friendly as a media player for the
animations – it req’d time to explain usage while a more user friendly program
would be easier to accustom myself to.”
“The
labs provide another way for people to learn. Different people learn in
different ways. The greatest weakness would have to be that we didn’t really
have any discussion involving the whole class when it comes to the labs.”
“The
strengths were that you could use the animations and run them and also you got
a feel for the section and how it worked. There were some part that wjere a
little confusing and didn’t quite understand.”
“The
strengths were that it helped tie everything together and let us visualize
problems that would occur in everyday life. The weaknesses were that there were
some problems that were much harder to understand and we never went over any
questions that we had about the labs in class.”
“Weak:the
questions in the lab didn’t stretch me, the student. Some questions hard to
comprehend. Strengths: visual teaching students work at own pace.”
“Weaknesses:
code was visible. Strengths: all the information in one place.”
“The
greatest weakness was that I didn’t do all three of them.”
“One
major weakness was the accessibility. Getting to the lab to complete it was
difficult and not being able to complete it all from my computer at home was
annoying (changing the code). As for strengths, again the visual aspect was
very helpful.”
“Being
able to visualize the concepts is a big strength. I really didn’t see a
weakness except we lacked the time to go through the labs sufficiently and get
all accomplished.”
“The
labs were good at showing many things at once. I also liked that they focused
on one example for so long. Textbook problems tend to jump around. Didn’t like how hard it was to work on the
labs at home. The first lab required so many changes it was almost impossible
over the web.”
“The
strengths include: incorporation of computer technologies, clear instructions,
ease of completion. Weakness include: Not enough class room time to complete. I
couldn’t see the Foxtrot cartoon to finish the project.”
“The
strengths were that they were very animated, a weakness was the had to be done
in-class or in the lab.”
“The
strengths were the graphics from Maple. They were very good (illegible)… The
weaknesses were the concepts, I couldn’t find more examples an some formulas.”
“I think the Riemann Sums were lacking in
instruction and guidance. We had not been lectured on midpoint sums and the lab
did not have an explanation, yet it asked for them to be calculated. I feel the
strengths of the labs were a more interactive and visual approach to learning concepts
which allows students to go at their own pace . I think the weaknesses were a
lack of them being completely polished, the took away valuable lecture time,
and they were not completely or easily accessible besides class time.”
A significant number of responses indicate that the labs
were not portable enough for home usage. Although there were modifiable
downloaded versions in our computer lab accessible for nearly the entire day
and early evening, most of our students are on campus for a short duration and
have work and family obligations which make it difficult to utilize many of our
services consistently. The web versions of the labs are view-only because our
District did not have a “student-user” agreement with Waterloo Maple so the
cost to students for the needed software was far too expensive. This problem
has just been resolved for SY 2004/2005. Students may now purchase Maple 9 for
a very low price. There is talk in the district of making this fee mandatory
for the calculus classes which would go a long way toward alleviating the
students’ justifiable access complaint. One very encouraging aspect to the
students’ responses is that they were logging onto my website and viewing the
running versions. This indicates a willingness to do their work when time
permitted. In general, the “strengths” responses echoed earlier comments while
the “weaknesses” responses offered some additional insight into the flow and
construction of the labs themselves. Since portability, access, and quality are
desirable components to the entire lab experience, I appreciated the students’
candor.
#20. Please share anything else you would like
to about your overall experience this past week. (You may attach additional
paper if needed)
“I find concepts are much easier to learn from a “top down approach”. For instance , you gave six levels of learning (?) or comprehension (?) I find if I learn starting from the sixth stage, and working my way back, that I learn a concept more quickly, easily, and thoroughly. I find knowing how an application is to be used, how it might be extended upon or altered, etc. makes for the mechanics to make sense. If I must learn the mechanics first before being filled in on the “secret” of what they’re good for I lack encouragement and comprehension, and what I do learn is only temporary and to finish an assignment. A top down approach works much better and I am confused as to why it is ignored in mathematics. The labs allowed me to take this approach, which made the topics more enjoyable. Contact me if you have any questions or would like additional feedback. (Student Name)”
“It
was a very good experience, because it was very interesting see how the (?) of
every figure were moving. I think the Math lab MC 114 should be opened the
whole day so you can provide and put more (?) in your assignments.”
“Keep up the good work.”
“It seemed to
make the class more relaxed & fun. It even mayed the teacher seem more
relaxed.”
“I really
liked these labs overall. It seems to me that I learned the material better.”
“I think the
labs were very helpful in helping us learn by being able to visualize. I think
students in the future should continue to do these labs. But I think that more
help on lab three should be provided. So that students can ask questions that
they may have.”
“I thought it
has been a great experience in general. The only thing I think could have been
improved didn’t deal with the labs. I thought some of the homework assignments
were excessive. I often felt burnt out of math after trying to complete the
homework assignments.”
“Can I have
that add’l sheet of paper?No, just kidding, but finally I felt like I was
really getting calculus (until I took the test), but really, I felt more
confident with the concepts of Ch. 5 with the labs.”
“I wish every
math concept had a lab that went with it. They are very helpful in learning the
material because you can go over the material again and again. It would be
nice, if they could be used on the concepts that individual students are having
issues with and ignored on issues that come easily to one. Put them on CDs so
we can run the Maple applications at home. In my opinion if you did the whole
class as a lab with lecture coming from you and getting rid of the text book it
would have been much easier to learn everything in a shorter amount of time.”
“The labs
provided a focus for the week.”
“With calc.
Most things I’ve learn are concepts and ideas that are not tangible. For me, I
can read and understand a lower concept, get it into my mind at the time, but
in a couple of days. I have to re explain and get it straight in my head again.
It seems that there is no retention power in self explanation. The power of
retention that visual learning has is not typically used in math classes. I
believe that is why trig identities, rules, laws always have to be reviewed and
relearned by myself. If you can see something rather than just learn it I think
it will stay with you indefinitely longer.”
The last question of the survey was an opportunity for students to add any further comments. I anticipated some interesting responses to this open-ended question and was not disappointed. Several students wrote very elaborate paragraphs. The information contained in the responses offers illumination into the students’ perceptions on how the project meshed with their perceived learning styles. This information directly impacts this and future instructional attempts towards concept deliverance and acquisition. I would be wise to review the student responses often when considering modifications to this project and the construction of others.
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:
§ The Fundamental Theorem of Calculus
Teaching
Strategies Consistent with the Nature of Calculus
Calculus is a
rich topic whose concepts are at their very core dynamic in nature. As Anton [1] succinctly states in his introductory paragraph
from Calculus: Early Transcendentals:
” Calculus, sometimes called the mathematics of change, is the branch of mathematics concerned with describing the precise way in which changes in one variable relate to changes in another.”
In nearly every calculus text one can find a similar statement about the nature of calculus. Most calculus texts go to great lengths trying to emphasize the relationships between changing quantities and the limiting process in general. They do so by presenting various sequential static diagrams and discrete tables of values. Additionally, phraseology is introduced in an attempt to relay the continuous interplay of quantities and often includes an appeal to the students’ ability to formulate mental images of such actions on their own. Here are some such phrases taken from various current and popular texts:
1. “ …over ever smaller intervals…”(Anton) [1]
2. “… as x approaches a.” (Stewart) [30]
3. “…by seeing what happens to the sum as n grows without bound.”(Strauss, Smith & Bradley) [31]
4. “…one can form a mental picture of the functional values f(x) increasing without bound whenever x is close to a” (Zill) [40]
The previous representative samples provide a glimpse of the attempts at providing students with the dynamic nature of calculus through approaches that are actually inconsistent with that nature. As an instructor of many years, I, too, fumbled my way trying to convey the dynamic nature of calculus through chalkboard scribbles and oral arguments. It is my belief that such approaches have been providing an inadequate conceptual representation of calculus to our students denying them ample opportunity to create “elaborate concept images”. Since the Fundamental Theorem of Calculus is grounded in the interplay of changing quantities and the limiting process, it seems reasonable to examine what the research says about students’ abilities and shortcomings in these areas. By reviewing such literature teachers can gain valuable insight into helping their students reach higher levels of conceptual understanding and imagery of crucial concepts. Important research has been done by Cottrill et al [11], Monk & Nemirovsky [24], and Thompson [35], that consistently shows that students at the college level have a weak understanding and difficult time modeling the rate of change in one quantity which depends on a continuous change in another. These works offer great insight into the difficulties students have understanding the Accumulation Function (Fundamental Theorem of Calculus Part 1) and the antiderivative approach to evaluating the definite integral (Fundamental Theorem of Calculus Part II) which are presented below respectively.
Fundamental Theorem of Calculus Part 1 – Accumulation
Function
If
f is continuous on an interval [a, b], then the function A(x)
defined by
is
continuous on [a, b] and differentiable on (a, b), and A’(x) = f(x).
Fundamental Theorem of Calculus Part II – Antiderivative
Evaluation
If f is continuous on [a, b], then
where F is any antiderivative of f , that
is, F’ = f.
Patrick Thompson [35] observes that, “…students’ difficulties with the Fundamental Theorem of Calculus can be traced to impoverished images of rate.” His work emphasizes the need for students to develop “mature” images focused upon the dynamics of accrual and accumulation. The recent efforts of Carlson [9] have supported such conclusions and Cottrill [11] recommends that the limit concept should begin with the informal dynamic notion of “values of a function approaching a limiting value as the values in the domain approach some quantity.”
The previously cited research articles solidified for me the need to incorporate teaching strategies/methods that exemplified the dynamic nature of calculus while fostering a mature concept image of the Fundamental Theorem of Calculus.
Teaching
Strategies/Methods Selected
In selecting appropriate teaching strategies/methods I identified several key criteria.
· Soundness of Pedagogy: Are the methods I am going to use based on sound educational pedagogy and consistent with both research and relevant standards?
· Prior Evidence: Is there prior evidence that the strategies/methods I selected offer a reasonable expectation of success?
· Ability to Implement: Can I implement my choice of strategies/methods effectively given time, facility, and student constraints?
· Measurability: Do the strategies/methods selected contain an appropriate element of measurability in order to gauge their effectiveness?
· Quality, Portability & Utility: Do the materials to be implemented exhibit quality of design? Do they offer benefits to my students, the mathematics department, and the District in general?
In light of the
above criteria and my overall desire to improve my students’ understanding of
the Fundamental Theorem of Calculus, I settled on the following teaching
strategy: Utilize 3 student-centered interactive computer modules which use
animations to aid in concept image formation reinforced by traditional
classroom instruction. The following paragraphs offer an elaboration on the
justifications and rational for this selection.
Soundness
of Pedagogy
Using
3 Student-Centered Animated Computer Modules:
The National Council of Teachers
of Mathematics (NCTM)
[42], the American
Mathematical Association of Two-Year Colleges (AMATYC)
[43],
and the Mathematical Association of America’s Undergraduate Program in
Mathematics (CUPM)
[44]
and its subcommittee on
Calculus Reform And the First Two Years (CRAFTY) have all provided
guidelines and various Principles/Standards for the mathematics community which
strongly advocate an increased emphasis on the proper utilization of
mathematics software and technologies.
Explicitly stated within these documents is the need for the appropriate
usage of various technologies in order to enhance student understanding of
mathematical concepts. Additionally, it is recommended that students actively
interact with these technologies and that teachers routinely incorporate their
usage in the daily lesson planning process. The inclusion of technology in the
classroom is recognized as an integral part of the instructional delivery
method for mathematics, as a viable means of increasing student understanding
of mathematical concepts, and as a necessary element in preparing individuals
to interact successfully in a modern society
[42].
The various Standards/Guidelines clearly state that a teacher can take advantage of technology’s power to efficiently and effectively aid in a student’s ability to visualize concepts. It is precisely this benefit that convinced me to select the inclusion of student-centered animated computer modules as a component of my teaching strategy. It is well documented that one major type of learning style is the visual learner. In an article by Keith Devlin [47] who writes Devlin’s Angle for the MAA, he references the work done on learning styles by Suzanne Miller at Diablo Community College in which she found through survey methods that, “By far the most powerful method of learning among all age groups (at the community college) is visual nonverbal.” One can take the test at
http://silcon.com?~scmiller/lsweb/dvclearn.htm.
I recently had my students from a past semester’s calculus I class take the survey and found that the majority of students, indeed, had a propensity to learn best through a visual medium. With such high numbers of visual learners in our classrooms we would be irresponsible if we did not provide activities that address that learning style. Knowing the import of this I was faced with discerning which types of visual medium were best used with calculus concepts: Static diagrams/tables/pictures which current texts are laden with or dynamic diagrams/tables/and pictures? The dynamic nature of calculus discussed earlier in conjunction with research to be discussed later led me to select a dynamic/interactive method of display. Based on those two elements it was my contention that the inclusion of dynamic representations (animations) in the computer modules offered students a distinct advantage both from a pedagogical and motivational standpoint. I further posited that students who participated with such modules would exemplify higher levels of understanding as measured by Bloom’s Taxonomy.
The 3 models are entitled “Riemann Sums”, “The Accumulation Function”, and “The Fundamental Theorem of Calculus”. The series of three mesh together to present a comprehensive approach to the Fundamental Theorem of Calculus beginning with an intuitive and less rigorous investigation into its plausibility and foundational underpinnings while culminating with the theorem itself and its implications. Although the modules can be run independently of each other, they are most effective when viewed as a whole and implemented in close succession. Each of the three modules was written in MAPLE 6 and is comprised of several components that closely follow stated objectives. Elements within the three modules were carefully written to ensure conceptual flow and are intertwined to reflect this aspiration.
Module
Components:
1. Student Module Component: An interactive student-centered computer activity which enables students to change the MAPLE code parameters in order to see the covariational aspect of variables, run adjustable animations repetitively in order to better analyze the dynamic nature of individual concepts, reflect upon their experiences through leading and probing questions, review the correct mathematical symbolism, and interface in a technologically sophisticated and stimulating environment. Students may work through the stated objectives individually, in pairs, or in larger groups. The component is suited for a lab setting, mathematics resource center, library, or at home. The student may modify the MAPLE code independently and is often required to do so in some of the component tasks. MAPLE 6 or higher is required to run this component.
2. Teacher Module Component: An abbreviated version of the student component that is designed to enhance and augment the traditional classroom discourse by allowing for the easy importation of pertinent concept visualizations. The teacher may modify the MAPLE code to better suite his/her personal style and student needs. Delivery may take place through a fixed computer or a laptop. MAPLE 6 or higher is required to run this component.
3. Stand-Alone Module Gifs: These are “graphics interchange format” visualizations designed to allow virtually any instructor an easy way to interject animations into their lessons without being dependent on any expensive software or coding knowledge. Having this option provides the technology-adverse instructor a non-threatening portal in which to include innovative teaching methods. For those instructors who are comfortable with technology they provide a “no frills” way of quickly and efficiently displaying difficult concepts to students to better develop mental constructs. No MAPLE software is required to run the Gifs but they do require a player such as Quicktimeã.
Using
Traditional Classroom Instruction and Dialogue:
To have simply relied on the computer modules to be the sole source of concept transference would have failed to target the auditory leaner adequately. Although many students are predominantly visual-nonverbal and kinesthetic learners, there is still a significant number who are auditory learners. These students often learn best through the lecture format [45]. Since I seek to be as inclusive and effective as possible it was prudent to take the CUPM 2004 Curriculum Guide’s [44] advise and “present key ideas and concepts from a variety of perspectives and instructional formats.” The computer modules were only part of the concept construction process and were certainly not all encompassing. There are aspects of the Fundamental Theorem of Calculus that are simply too complex to be entirely exemplified in the modules. The use of traditional classroom instruction provided me with active dialogue to informally assess student understandings and misconceptions while reinforcing certain conceptual aspects. Active teaching seeks informal student assessment and welcomes student input. In the traditional group setting I would be able to address the entire group on important matters, elicit valuable informal assessment, and receive student commentary on their interactive experiences in near real-time. Furthermore, the usage of the Teacher and Gif components of the Modules foreshadowed and reinforced many of the activities the students would be engaged in, thus developing stronger conceptual bonds.
Prior
Evidence & Support
Including a technology component as an augmenting teaching strategy for a particular concept in no way guarantees success. The words “appropriate” and “judicious” often occur in the calculus reform literature and the calls by professional mathematics organizations. In considering the creation of the 3 modules I wanted to ensure that there was ample evidence that such modules and their individual component designs had the potential to actually cause the constructive change I sought in my students’ understanding of the Fundamental Theorem of Calculus. The evidence I found varied from formal, to anecdotal, to merely speculation. The most intriguing aspect of my hypothesis is that the addition of animations in the modules will strongly influence the ability for students to make elaborate concept images. It is the inclusion of animations for calculus concept acquisition that provides an element of innovation in the computer modules that is seldom seen and will be the focus of the following paragraphs.
What evidence suggests that
the modules I have written should place an emphasis on animations?
There are literally thousands of articles that discuss the usage of technology as a teaching aid, but the number dwindles rapidly when we discuss using dynamic interactive visualizations in sophisticated software for the acquisition of calculus concepts. There are several benchmark studies that indirectly address this concern by revealing an immature or absent concept image of across-time graphical representations by students. Monk [23] and Swan [32] observed that students have serious difficulties with understanding an across-time graphical representation which is invaluable to understanding such calculus concepts as the Fundamental Theorem of Calculus. Both of these individuals are not alone in their conclusions. Orton’s [28] paper entitled, “Students’ Understanding of Integration”, notes that the procedure of breaking up an area or volume, making use of a limit process, and providing the reasons why such a method works were not part of the students’ understanding of the integral. Orton suggests that there is “a need for activities” which focus on letting students explore limiting processes and that technology can be extremely beneficial in that aim. Additionally, both Tall [33] and Cunningham [14] have written papers about the advantages of dynamic visualizations over static visualizations in the success rate of their own students. Tall [33] comments:
“I soon realized that graphics alone were unsatisfactory,
and saw the need for versatile movement between representations.”
Cunningham [14] elaborates further and states:
“Most educational visualization now uses
postprocessing, since it focuses on presenting finished concepts to the
students…However, the author’s (Cunningham) informal studies across the
sciences indicate that students respond much more strongly to dynamic images
than to static ones.”
In addition to
having a real-time element of changing quantities in the modules, I recognized that
a student-centered activity includes actually interfacing with the modules in a
constructive way. I constructed the computer visualizations to be modifiable so
that students could have control over speed, direction, and quantity rather
than merely being passive observers. Such an active design approach is
indirectly supported by current work done at Arizona State University by
Carlson, Jacobs, Coe, Larsen, and Hsu
[9]. They noted during their work on functions
with undergraduates that, “The use of physical enactment appeared to provide a
powerful representational tool that assisted these students in reasoning about
the change in one variable while concurrently attending to the change in the
other variable.” The call for interactive, inquiry-based learning, has been
championed through the years by individuals such as Bitter [4], Birkhoff
[3],
Wahlberg [37], and Berry & Nyman [2].
Are there others
implementing a strategy closely paralleling mine in their classrooms?
There are many college and university calculus courses that include technology in some form or another. Several examples that specifically use student-centered computer extensions that contain elements of animation are listed below and are representative of a cross-section of programs.
Example 1.
Raouf Boules and Mike O’Leary [6], of Towson University, Towson, Maryland, have been implementing a computer laboratory element in their classrooms for the past few years. They use the interactive modules to supplement the traditional classroom lecture and focus on illustrating concept development through the usage of animations and discovery learning. Although they use MathematicaÔ as their software platform, it is quite evident from previewing their labs that their philosophy closely parallels mine.
Example 2.
Between 1991 and 1995 every Calculus I student (about 1000) at Rensselaer Polytechnic Institute, had taken a computer-intensive calculus course with MAPLE as the software platform of choice. According to professors William Boyce and Joe Ecker [7] the inclusion of MAPLE’s graphics capabilities had a substantial impact on conceptual understanding, modeling, and higher-level problem solving. They specifically noted that the visualization utilities of MAPLE were extremely helpful for students to see the relations between variables and instrumental in clarifying the conceptual basis of calculus. The authors also point to a spike in student enthusiasm for calculus based on positive survey responses up from 65% to 80%.
Example 3.
The Illinois Institute of Technology first-year Calculus courses (Math 151, 152, 161, and 162) are all 5 credit hours, of which 4 hours are devoted to lectures and 1 hour is spent in a computer laboratory where students get a chance to solidify their basic understanding of the principles learned in class. In these labs animations play an important role by providing powerful visualizations of concepts (http://www.iit.edu/computing/).
Example 4.
The Department of Technology at Telemark College in Porsgrunn, Norway, has been using MAPLE constructed computer animations to provide engineering students with powerful visualizations. As Dr. Harald Pleym [29] puts it, “ The MAPLE system enables us to develop both simple and complex mathematical models in classrooms and labs, run them, analyze their output, modify them and rerun them easily. This makes mathematics more relevant and motivating for engineering students and provides valuable insight into the underlying dynamics, which helps the learner of mathematics to gain a better feel for what is going on than has hitherto been possible.”
Earlier
Examples:
Other notable past collaborative efforts to include computer-extended instruction include the University of Denver’s two-year National Science Foundation funded Computing and Mathematics Curriculum Project (CMCP) and the construction of a one-semester experimental beginning calculus course at State University of New York (SUNY) which utilized a mini-computer as an augmentation to the lecture format [4], [26]. Both efforts point to the benefits of including the visualization component to enhance concept acquisition. Forty-one percent of the students involved in the CMCP stated that computer assignments were helpful in understanding the integration concept. The results at SUNY indicate that the simultaneous graphing and computing of the lower and upper sums (Riemann Sums) aids the student in developing a strong intuitive understanding of the definition of the definite integral. These conclusions were arrived at using a control group that received only traditional lecture compared to a treatment group that received a combination of lecture and computer related activities.
Anecdotal/Tangential
Evidence:
Other non-formal evidence exists in the form of anecdotal and tangential forms. Firstly, there has been a tremendous increase of animations added to textbook ancillaries in the form of CD-ROMs, supplemental books, and web support. While chairing our calculus textbook adoption committee in the spring of 2003, it was quite apparent that nearly all of the current market offerings included some element of “active” graphical concept development. Large publishing companies often respond to the consumer’s demand for the inclusion of particular pedagogical elements or aids. Textbooks that had been staunchly categorized as “traditional” heeded the calls for reform and have now included a technology component in order to stay competitive.
Secondly, the explosion of other individuals across a variety of higher education landscapes writing computer extended activities for their own courses or for published print has also taken place. One merely needs to do a web search to find plenty of examples of worksheets and course syllabi containing design elements which utilize Maple, Mathematica, MatLabÔ, or some other powerful software package to offer illumination of concepts from Calculus I, II, or III.
Finally, there has been an avid interest on the part of members from the mathematical community to attend conferences, mini-classes, workshops, and lectures on how to incorporate such technologies into their teaching repertoire. When I began this project many fellow mathematicians both on-site and off-site came to me with topics that they were having a hard time getting students to “see”. Many asked if I could develop or consult on animations of specific concepts. Although most were not basing the inclusion of animations (or technology for that matter) on educational research, they had an “intuitive” sense that such an inclusion may help propel their students to higher levels of success.
Ability to
Implement
The ability to implement the modules was, of course, a major concern. Facilities and access to them could have been a debilitating factor to the project but was not a factor here at Mesa Community College. Fortunately, I had 28 computer terminals all loaded with MAPLE 6 software and were readily accessible to my students. This ability permitted a seamless implementation of all 3 Modules effectively
Measurability
The teaching strategies/methods selected were done so to correspond to the desired learner outcomes and assessment items developed. The backward design process focuses on the identification and delineation of desired student outcomes first (Phase I), followed by the selection and creation of appropriate evaluative instruments for those stated outcomes (Phase II), and the discernment and creation of teaching tactics/materials that bring coherence to the entire process in hopes of providing the optimum opportunity to evoke the desired understandings (Phase III). The strategy of coupling student-centered computer modules characterized by interactive and reflective activities with direct classroom instruction was done so in accordance to the backward design process and, hence, it was reasonable to hold a high degree of confidence in the agreement between the results of the evaluative instruments and the instructional materials and processes. Walter Zimmermann [41] sums up this entire design process with regards to visualization quite nicely:
“In the
absence of a set of guiding principles to relate visualization to the content
and learning objectives of the course, any use of computer-based or non
computer-based graphics is likely to be ineffective”.
Quality, Portability & Utility
Quality:
There was a conscience effort on my part to make 3 high quality computer modules as least restrictive as possible so that they would be student friendly, teacher friendly, and community friendly. I thoughtfully took into consideration the various participatory audiences and what elements should be contained in “quality” technology materials. Identifying such criteria is an important element of the design method of any educational component that incorporates technology due to the variety of user experiences, abilities, and facility access issues. When one peruses the multitude of MAPLE worksheets, tutorials, and student handouts currently found on the world-wide web or in materials developed by textbook writers and other publishing entities, one quickly notices that the quality of these items vary dramatically. The bulk of the offerings are simply static displays coupled with a few lower level questions providing little to no student exploration of the concept(s) intended. In my opinion, the quality of most materials currently available for the calculus curriculum do not fully meet the intentions outlined in the guidelines and standards of the major mathematical associations. The majority of materials lack the interactive discovery-learning element whereby students can construct meaningful and lasting concept images. Besides the overall lack of depth in the offerings there is a conspicuous absence of materials that utilize the powerful animation capabilities of these software utilities. Granted, there is a higher level of complexity when one introduces the element of animations (clearly demonstrated by the number of inquiries to the web-based Maple Users Group), but the benefits to students far outweigh the complexity hurdles. The following is a list of 10 items that I considered “quality” markers necessary for inclusion in each of the three modules.
1. The objectives should be clearly stated and the user should be able to identify and navigate through the module’s sections easily.
2. There should be a reference to the approximate amount of time needed to complete each component.
3. Directions for running the module’s components should be explicit and lucid.
4. The delivery of concepts should logically coincide with classroom instruction, yet, be flexible enough to allow the user to engage in meaningful investigative activity in order to take ownership of them.
5. All composition within the module should be consistently accurate in mathematical notation, inquiry based, and easily distinguishable.
6. The animations should be clearly designated and students should be directed to “explore” variable relationships by adjusting time, direction, and speed.
7. The student exercises and interactive graphs should be reflective of Bloom’s Level of Taxonomy.
8. There should be the meaningful connections to future activities and/or stimulating queries.
9. Each module should contain engaging examples, visually stimulating graphics, and an element of “fun”.
10. Each module should
conclude with a summary paragraph of the concepts presented.
Portability:
Since many of
the students at Mesa Community College transfer within district or to the
State’s three major universities it made sense to me that the mathematical
software platform for the calculus modules had to be transferable and flexible.
The Maricopa Community College District has a district-wide site-license
agreement with Waterloo MAPLE and this same mathematics software platform is
currently being utilized at the State’s universities. This uniformity in
software and program syntax offered the portability and transference that would
be most beneficial to both students and the overall mathematics community.
The modules were specifically constructed in 3 parts: Student Component, Teacher Component, and Stand-Alone Gifs. These can be implemented as a trio, as pairs, or individually depending on the instructor’s desires. The 3 modules have been copied onto all of the lab computers for ease of in-class usage, burned onto CDs so students may use them at our open-lab facility, and presented in print form so that students and teachers alike may use them as readily available hard copies. Additionally, students may check out a CD version to take home or simply use the web-version at:
www.mc.maricopa.edu/~dschultz/
Utility
Having high quality computer extended activities is of prime importance for the students in our District. The Maricopa Community College District services a student body of more than 277,000 students year-round and being the major contributor to the State’s three major universities (54% of Arizona State University’s baccalaureate degree recipients transferred credits from one of the Maricopa Community Colleges), there is added incentive to produce quality calculus materials/activities that increase student success. Our District is recognized as an important conduit for students to pass from into the State’s major universities. Those same universities have expressed their desire for the community colleges to provide richer and more abundant experiences with technology. This directive has trickled down to each of the 10 individual Maricopa Community College campuses. It is my belief that this project provides a small step towards meeting that directive.
This project additionally provides the District with a technology-based calculus package of pedagogically sound resources that meshes with the desires of the universities while providing continuity of curriculum within the individual and satellite community college campuses themselves. Furthermore, the project’s on-site implementation increased the dialogue within my own department concerning the proper usage of technology as a tool for new and innovative approaches to concept delivery and student exchange.
Implementation Framework
The implementation framework was as follows:
1. 1 class period (50 minutes) for each of the three computer modules. Each student had access to a computer and worked in pairs through the modules. They were responsible for completing all tasks required by the module and to turn in a hard copy of the student problem sets.
2. 2 class periods of direct instruction for Riemann Sums, 2 class periods for the Accumulation Function, and 1 class period for the Fundamental Theorem of Calculus. They also received selected homework problems from Stewart’s Calculus: Early Transcendentals, 5th Edition, sections 5.1, 5.2, and 5.3.
The framework equated to nearly 7 total hours of instructional time relatively split between the computer modules and the classroom lectures. Both of those elements were accompanied by a variety of student homework tasks. The order of the concepts presented was: Riemann Sums, The Accumulation Function, and The Fundamental Theorem of Calculus. In addition to the formal instructional process, students were encouraged to investigate the modules during open lab time, by taking home a disk copy, or by visiting the created web version. They had 1 week for this type of independent review before the assessment process.
The design and implementation of this Capstone Project reflected a concerted effort on my part to affect the levels of understanding that my students attained with regards to the Fundamental Theorem of Calculus. Current research and educational trend-lines throughout the mathematical community strongly advocate the responsible use of technology in the development of lasting mature concept images. The inherently dynamic nature of calculus bespeaks for activities that accurately reflect the notion of changing quantities over time. Such a notion can be accurately portrayed through the animation utilities found in today’s mathematical software packages. The belief that the utilization of 3 student-centered interactive computer modules coupled with traditional lecture aligned to the 6 levels of Bloom’s Taxonomy would increase student understanding provided the motivational impetus and theoretical foundation upon which this Capstone Project was built. Upon analysis and reflection of the 2 assessment items designed to measure student attitudes and the attainment of the behavioral objectives as generalized in the Capstone Project’s focus, several conclusions can be confidently drawn:
1. The participatory students demonstrated average to above average mathematical ability throughout all six levels of Bloom’s Taxonomy.
2. The numerical data collected from the free-response test can be reliably used as a baseline benchmark for future comparisons.
3. The project’s choice of teaching methods/strategies, structural design, and overall tone is consistent with the students’ perceived learning realities.
4. The computer modules exhibited the desired characteristics of pedagogical soundness, ease of implementation and usage, measurability, structural quality, and utility. The only substantial shortcoming noted was that of portability which was a factor of District constraints.
5. Students indicated that continued use and even expansion of the project’s approach to concept acquisition should be a part of the mathematical experience.
6. Student responses indicated that animations offer a significant advantage to the visualization process of mathematical ideas and should be threaded throughout calculus.
7. Most students considered their experience during the entire project process as positive to highly positive.
In addition to the direct conclusions mentioned above, the following auxiliary claims may be stated:
1. Materials have now been designed which offer greater articulation between the mathematical experiences desired by the State’s universities and the mathematics preparation received by students attending Mesa Community College.
2. Materials which reflect a serious effort at addressing student understanding of the Fundamental Theorem of Calculus are now web-accessible to all interested parties.
3. The project’s on-site implementation has stimulated dialogue among colleagues concerning the appropriate and advantageous use of technology.
In a paper entitled, “Perspectives on Instructional Technology”, presented by the Teaching Academy University of Wisconsin-Madison, May 1997, the following statement occurs:
“Visualizing complex processes
is often difficult for students to achieve with standard textbook
illustrations, and is challenging for blackboard and overhead
presentation...Animations enhance those topics that involve movement.
Interactive computer display systems are optimum tools for teaching about
dynamic processes because they allow both the teacher and the student to be
fully engaged with what is being shown. Interaction with data or system of
equations allows both students and instructor to become the directors of their
own animations. Students can change the variables being viewed to best
illustrate the phenomena of interest.”
The enhancement
of student concept acquisition through dynamic displays reflected in the above
quote is echoed and championed by many throughout the education community. Such
a belief of utility was also a significant element woven throughout this
Capstone Project. However, this Capstone Project was not solely about using
animations as an educational device in a learning setting, but rather identifying
a global desirable educational aim and then utilizing the backward design
process to obtain the best series of actions to reach it. The inclusion of
animations within computer modules, specific assessment formats and items, and
the alignment of materials to Bloom’s Taxonomy was an outgrowth of the
design process and reflected a best practices approach to teaching the
Fundamental Theorem of Calculus as cited in the educational literature. As the
pipeline of new technologies expands teachers are faced with more instructional
options for their students to engage in. Starting with a clear understanding of
the nature of a particular concept and the students’ relationship to it acts as
the beacon from which an informed choice of a specific technology and teaching
strategy can be made. It was in this spirit that I chose to attack the linchpin
concept of the Fundamental Theorem of Calculus with the desired outcome of
increasing my students’ success in its understanding. The nature of such an
important undertaking does not lend itself to conclusion but must be viewed as
an ongoing process of refinement. I encourage the reader to continue the
investigation of creating a mature concept image in the mind of students with
regards to this most “fundamental” theorem.
There are many people who helped contribute to this paper and no contribution was too small. Hopefully, the following acknowledgements will not resemble an album sleeve too closely. I would like to thank the entire Mathematics Education Department at Montana State University with a special “thanks” to Dr. Janet Sharp for her guidance in the project’s development. I additionally want to thank my 7:00 am Calculus I class for allowing me to photograph them and attending class faithfully each morning to do integrals with their morning coffee. Finally, the biggest thanks go to my wife April and three boys Alex, Kai, and Zachary. Their love and patience during this “ordeal” was immeasurable.
References
1. Anton, H., Bivens, I., and Davis, S. Calculus: Early Transcendentals, 7th Ed., John Wiley & Sons, New York, New York, 2002, 394 – 426.
2. Berry, J. & Nyman M. (2003). Promoting students’ graphical understanding of the
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4. Bitter,
G. (1970). Calculus and the Computer: An Evaluation by Participants. The
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12. Crowell, R. & Prosser, R (1991). Computers with Calculus at Dartmouth, Primus, 12, 149 – 158.
13. Curjel, C. & Sylvester, J. (1993). A Computer Lab for Multivariable Calculus, The College Mathematics Journal, Vol. 24, No. 2, 175 – 177.
14. Cunningham, S., (1991). The Visualization Environment for Mathematics Education, MAA Notes and Reports Series: Visualization in Teaching and Learning Mathematics, the Mathematical Association of America, 1991.
15. Davidson, N. & McKeen, R. (1979). An Expanded Domain for Objectives in Mathematics Education, The American Mathematical Monthly, Vol. 86, No. 10, 858 – 862.
16. Ferrini-Mundy, J & Graham K. (1991). An Overview of the Curriculum Reform Effort: Issues for Learning, teaching, and Curriculum Development, The American Mathematical Monthly, Vol. 98, No. 7, 627 – 635.
17. Glaser, R., & Silver, E. Assessment, Testing, and Instruction: Retrospect and Prospect. Review of Research in Education, Vol. 20, edited by Linda Darling-Hammond, pp. 393 – 419. Washington, D.C.: American Educational Research Association, 1994.
18. Harel, G. (1998). Two Dual Assertions: The First on Learning and the Second on
Teaching (or Vice Versa). The American Mathematical Monthly, Vol. 105, No. 6, 497 – 507.
20. Hill, D., Mabrouk, S., and Roberts, L.(2002) Demos And Projects That Make A
Difference. 13th Annual ICTCM Proceedings, Addison Wesley, Boston, Mass., 170 – 174.
21. Kaput, J. (1997). Rethinking Calculus:
Learning and Thinking, The American
Mathematical Monthly, Vol. 104, No. 8, 731 – 737.
Monthly, Vol. 104, No. 8, 724-727.
27. O’Neal, J. (2002) Instruction, Assessment, And Technology: Teacher Preparation Revisited. 13th Annual ICTCM Proceedings, Addison Wesley, Boston, Mass., 287 - 291.
28. Orton, A. (1983). Students’ understanding of differentiation, Educational Studies in Mathematics, No. 15, 235 – 250.
29. Pleym, H. ( 2004 ), Samples of Animations from Teaching Engineering Mathematics with Maple V R5. (contact: herald.pleym@hit.no).
30. Stewart, J., Calculus: Early Transcendentals, 5th Ed., Thomson – Brooks/Cole, United States, 2003, 369 – 404.
31. Strauss, M., Bradley, G., and Smith, K. Calculus, 3rd Ed., Prentice-Hall, New Jersey, 2002, 271 – 308.
32. Swan, M., (1988). On Reading Graphs, Talk at ICME-6, Budapest, Hungary.
33. Tall, D. (1992). Students’ Difficulties in Calculus, Proceedings of Working Group 3 on Students’ Difficulties in Calculus, ICME-7, Quebec, Canada, pp. 13 – 28.
34. Thomas, G., Finney, R., Weir, M. Calculus: Early Transcendentals, 10th Ed., Addison – Wesley, Boston, Mass., 2001, 363 – 386.
Fundamental Theorem Of Calculus, Educational Studies in Mathematics, Vol. 26, 229 – 274.
37. Wahlberg, M. (1997). Lecturing to the “Bored”. The American Mathematical Monthly,Vol. 104, No. 6, 551-556.
38. Wiggins, G. & McTighe, J. Understanding by Design. Prentice Hall, 1998.
39. Wood, R. (1968). Constructing teacher objectives. Educational Research, No. 10, 83 - 98.
40. Zill, D. Calculus, 3rd Ed., PWS Publishing Company, Boston, Mass., 1993, 264 – 296.
41. Zimmermann, W. (1991) Visual Thinking in Calculus. MAA Notes and Reports Series: Visualization in Teaching and Learning Mathematics, the Mathematical Association of America, 1991.
E - References
45. Devlin, K. (2003). Johnny might not be math-challenged; his problem could just be that he’s an auditory learner. Devlin’s Angle available from The Mathematical
Association of America ‘s Online Resources,
http://www.maa.org/devlin/devlin_1_00.html.
47. Rishel, T. A Handbook for Mathematics Teaching Assistants, Preliminary edition
available from The Mathematical Association of America ‘s Online Resources, 21- 25.
http://www.maa.org/pubs/books/teaching_rishel.html
Appendix A
A
Listing of the Behavioral Objectives
Knowledge:
1. The student will be able to properly write both parts of the FTC.
2. The student will be able to state in writing all the conditions needed for the FTC.
3. The student will be able to accurately graph an example of the accumulation function on graph paper.
4. The student will be able to properly name the accumulation function.
5. The student will be able to correctly match the FTC1 and FTC2 to the words function or constant.
Comprehension:
6. The student will accurately summarize the FTC1 in written form when given the representative equation.
7. The student will accurately summarize the FTC2 in written form when given the representative equation.
8. When given a graph of a function, the student will correctly interpret how a change in the interval length affects the accumulation function.
9. The student will correctly interpret the phrase “differentiation is the inverse process of integration.”
10. The student will provide an appropriate example where the FTC fails for a definite integral.
Application:
11. The student will correctly apply the FTC2 to solve a definite integral.
12. The student will correctly take the derivative of the accumulation function when written in integral form.
13. The student will correctly apply the FTC2 to a specific real-world application.
14. The student will utilize the FTC1 on a graph to determine the total accumulation.
15. The student will demonstrate the proper usage of both the FTC1 and FTC2 in the solving of a multi-layered non-routine problem.
Analysis:
16. The student will break down the FTC2 into its various component parts accurately explaining the meaning of each.
17. The student will explain the derivative/integration connection inferred by the FTC1 equation.
18. The student will explain the derivative/integration connection inferred by the FTC2 equation.
19. When given a rate graph, the student will correctly deduce when the accumulation is greatest.
Synthesis:
20. The student will construct a new formula from old through algebraic means.
21. The student will create a geometric argument for the generalization of an integrals value.
22. The student will propose a general method of solution to determine the area between to curves.
23. The student will develop the Mean Value Theorem for Integrals given a graph of the Mean Value Theorem for Derivatives.
Evaluation:
24. The student will thoroughly assess the strengths and weaknesses of the FTC.
25. The student will judge the theoretical importance and utility of the FTC.
Appendix B
LEVEL 1: (Knowledge)
Define, list, name, label, identify, match, state,
identify…
#1. State in writing both equations associated with the Fundamental Theorem of Calculus.
#2. State the conditions needed for the Fundamental Theorem of Calculus to hold.
The function f is continuous
on [a, b] and the function A(x) needs to be an antiderivative for f over
the entire interval [a, b]..
#3. Draw and label an example graph of an accumulation function, A(x).
#4. Name the function, A(x), defined as.
The accumulation function.
#5. Match the two expressions below to the words “ function” or “value”.
The first expression is a function while the second expression is a value.
LEVEL 2: (Comprehension)
Summarize,
interpret, contrast, distinguish, describe in your own words, group...
#6. Summarize in your own words the first part of the Fundamental Theorem of Calculus (FTC 1) shown below.
If in a definite integral the upper limit of integration is a variable, then the integral is a continuous function that accumulates a net change.
#7. Summarize in your own words the second part of the Fundamental Theorem of Calculus (FTC 2) shown below.
A definite integral’s value is equal to the difference of the antiderivative of the integrand evaluated at its endpoints.
#8. Describe from the graph what happens as x increases.
As x increases the graph accumulates more signed area.
#9. Interpret the phrase “differentiation is the inverse process of integration.”
If we differentiate the function defined as
then we obtain the function f(x). This is written symbolically as:
#10. Give an example where The Fundamental Theorem of Calculus fails to apply.
Let f(x) = 1/x2 on [-1,1]. Then the Fundamental Theorem of Calculus fails to apply due to the infinite discontinuity at zero.
LEVEL 3: (Application)
Solve, calculate, demonstrate, apply, use of facts/rules,
modify, construct…
#11. Solve.
#12. Evaluate.
#13. The total cost of purchasing a piece of equipment and maintaining it for x years is given by:
Find C(10). (Source: Calculus: Early Transcendental Functions, Larson & Hostetler, 1995)
#14. It is 10:00 a.m. and five ants have already entered Kendra and Kelly’s picnic basket. Ants are notorious followers, so ants from all over the vicinity follow their brethren into the basket. The culinary treat awaiting them is unsurpassed elsewhere, so once an ant enters the basket he does not leave. If the rate at which the ants are climbing into the basket is well modeled by Ants(t) = 100e-0.2t ants per hour, where t = 0 is the benchmark hour of 10:00 A.M.
Write an integral expression to find the number of ants there will be in the basket x hours after 10:00 a.m.
If the girl’s dig into the basket at 1:00 p.m., how many ants will be inside? Give an exact answer.
(The approximate answer, which I did not ask for is about 231 ants.)
#15. #2. The graph below displays the velocity of a car on a straight road for a 24-hour period.
- Approximate the total distance traveled.
About 650 miles.
- Express the car’s total distance traveled as an integral with an unknown velocity function v(t).
- During what intervals of time was the car accelerating?
That would
be positive slope on the velocity graph:
About the
times (0, 5) hours and (14, 22) hours
- Sketch an approximate graph for distance traveled verses time using 3-hour increments as the independent variable.
LEVEL 4: (Analysis)
Seeing patterns, recognition of hidden meanings, explain the connection, identification of components, separate, break down…
#16. Break down the FTC 2 shown below into its parts and explain each part’s meaning while classifying it as function, variable, scalar, or notational device?
“a” a scalar which is called the lower limit of integration and provides us with our starting domain value.
“b” a scalar
which is called the upper limit of integration
and provides us with an ending domain value.
a notational device
called the integral sign devised by Leibniz to signify an infinite sum.
“f(x)” a function termed the integrand upon which we are seeking the net change .
“dx” a notational device that identifies the independent variable
“A(b)” a scalar obtained by evaluating an antiderivative to f(x) at the ending interval
value.
“A(a)” a scalar obtained by evaluating an antiderivative to f(x) at the ending interval
value.
“A’(x)”a function which is equivalent to the integrand f(x).
#17. Explain in your own words the derivative/integration connection obtained from the FTC 1 shown below.
This equation says that if we take a function, say f, integrate it and then differentiate it, then we arrive back at the original function f which is equivalent to the derivative of f’s accumulation function.
#18. Explain in your own words the derivative/integration connection obtained from the FTC 2 shown below.
This equation says that if we take the derivative of a function’s accumulation function and then integrate it, we obtain the original function’s net change over the interval.
#19. Analyze the graph
below and deduce when the line is the greatest if the checkers can process 20
people per minute. (Source: Calculus: Instructor’s Manual with Sample Exams Hughes-Hallett,
Gleason, et al, 1994)
The line will be growing in length
as long as the arrival rate exceeds the checkers’ processing rate. This
continues until about 7:17 am.
LEVEL 5: (Synthesis)
Relate
knowledge from several areas, predict, generalize, reconstruct, summarize…
#20. If f is continuous and g and h are differentiable functions, construct a formula for.
#21. Design a geometric argument to demonstrate that the integral of any odd function over the interval [-a, a] is 0.
If f(x) is odd it has symmetry about the origin. On any closed interval [-a, a] the net accumulation will be zero. See example graphs above.
#22. Propose a formula to determine the area between the two given functions shown in the graph below.
23. Earlier we learned the Mean Value Theorem
that stated if a function was continuous and differentiable on a closed
interval then there exists a point in that interval such that the slope of the
tangent line at that point is equal to the secant slope (see example picture).
Develop a Mean Value Theorem for integrals.
LEVEL 6: (Evaluation)
Assess value of
theories, make choices based on reasoned argument, recognize subjectivity.
Rank, recommend, judge, compare summarize, justify, critique…
#24. Assess the strengths and weaknesses of the FTC.
The FTC 2 allows us to compute a wide variety of integrals both definite and indefinite. It is both elegant in its simplicity of notation yet powerful enough to be utilized in nearly all fields of science. Its relatively ease of application to so many branches of mathematics makes it a crowning achievement in the field of mathematics. As Newton and others quickly noticed is that its biggest drawback is that one must be able to determine an antiderivative of the integrand function. For simple functions this is usually not that difficult but there are many important functions out there whose antiderivatives are difficult to determine by hand or may not exist in a closed form.
#25. Assess the theoretical importance and utility of the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus was a crowning achievement in mathematics because it opened the door to compute integrals exactly if the antiderivative could be found. It draws the connection between differential and integral calculus and provides the basis for the determination of the net accumulation of any rate function. Its utility is found in nearly every branch of science.
Appendix C
Student Survey
Directions:
Please allot some time to
answer the following questions for they are designed for me to better assess
how the computer labs worked and how I can make them better for future
students. I am very interested in your personal experiences and attitudes
towards them. I wish to provide the best calculus experience for all students
and this survey will help me in doing so. Thank you for your time.
Circle the response that fits best.
#1. How often have you used a computer lab
(not calculators) in your past math classes?
a. Very
Often
b. Occasionally
c. Seldom
d. Never
#2. How would you describe your overall experience with the 3
labs we did?
a. Very Positive
b. Somewhat
positive
c. Somewhat
negative
d. Negative
e. Very Negative
#3. Which of the 3 labs did you like the most?
a. Riemann
Sums
b. Accumulation
Function
c. Fundamental Theorem of Calculus
#4. Which of the 3 labs did you like the least?
a. Riemann
Sums
b. Accumulation
Function
c. Fundamental Theorem of Calculus
#5. How difficult were the labs to run?
a. Not
difficult – they were easy
b. Somewhat difficult
c. Difficult
d. Very
Difficult - too hard
#6. How would you rate the quality of the labs?
a. Excellent
b. Good
c. Average
d. Below Average
e. Poor
#7. How helpful were the labs in understanding the concepts?
a. Very
helpful – should always be a part of the class
b. Somewhat helpful
c. Not very helpful
d. Not
helpful at all – waste of time
#8.
Do you think the animations helped
you visualize the concepts better than just the textbook?
a. Yes
b. No
c. Undecided
#9. Do you think the lecture time in class
combined with the labs is a good approach to learning the Fundamental Theorem
of Calculus?
a. Yes
b. No
c. Undecided
#10. Were the examples and student problem sets interesting?
a. Yes,
very interesting
b. Somewhat interesting
c. Not
very interesting
d. Boring
#11. Do you think you enjoyed calculus more because of the labs?
a. Yes
b. No
c. Undecided
#12. Did you like working with a partner next to you?
a. Yes
b. No
c. No opinion
#13. Would you recommend that calculus classes
have some computer lab activities that use animations in them?
a. Yes.
b. No
c. Undecided
#14. Do you plan on taking another math class?
a. Yes
b. No
c. Undecided
#15. If you were to take another math class,
would you like it to contain some computer animations for the concepts?
a. Yes
b. No
c. No opinion
Free-response. Please take your time and
write clearly.
#16. Did using the animations on the computer help you focus and learn the concepts better than just using the textbook and listening to a lecture?
#17. Did the labs make the classroom lecture and
textbook easier to understand? Would you do the labs first or the lectures?
#18. Do you think the labs motivated you to
learn the concepts more?
#19. In your
opinion, what were the strengths and weaknesses of the 3 labs?
#20. Please share anything else you would like
to about your overall experience this past week. (You may attach additional
paper if needed)