David A. Smith

The Active/Interactive Classroom

 

1. CREATING A CLASSROOM ENVIRONMENT:
HARDWARE AND SOFTWARE

For the past year, I have had the pleasure of teaching in a new Interactive Computer Classroom (ICC), a ‘studio’ classroom designed for hands-on student work with computers. The students refer to this room as a ‘lab’, but it has several characteristics that distinguish it from a more traditional computer laboratory:

For more details about the classroom, including pictures of students at work, see ICC Task Force (2000).

In addition to the physical facility, our environment is enhanced by the Blackboard CourseInfo software for Web delivery of course materials. This product enables an instructor who knows little about Web pages to construct and maintain a substantial course Web site. You may visit any of the CourseInfo course sites (on any campus) as a guest by logging in as guest with password also guest. You will be able to see everything on the site except pages that identify students or that enable the instructor to control the content. In particular, please visit my most recent course sites at Smith (1999), (2000a), and (2000b).

The principal categories on every CourseInfo home page are

This structure enables an instructor to easily manage a multifaceted course with multiple classroom strategies, multiple assessment tools, multiple due-date structures, and a mix of individual and group work. It also enables students to stay in touch with each other and with the instructor, which is very valuable in building a sense of community. Students quickly learn that they need to visit the course page every day, where the current Announcements are always displayed on the home page. The instructor can use this page for announcements made or forgotten in class, reminders of due dates or test dates, corrections of mistakes in the text or lectures, pointers to new content on other pages, and many other things. The examples that follow are taken from the home page for my differential equations course. The other course pages are not identical, but they use CourseInfo in similar ways.

My Course Information page contains what I used to provide on many pieces of paper at the beginning of each course: textbook and other materials, course goals, syllabus, requirements, and policies. Students generally lost my paper handouts and could never remember what I had told them about the course. Now I rarely get questions about course matters, and when I do, I (or another student) can reply "Look on the Web page." Most of this information is static, but it resides on live Web pages, so it is easy, for example, to make a change in the syllabus if that becomes necessary. It is also easy to include external links in these documents. For example, where the syllabus specifies an on-line computer module for a lab exercise, there is a link that will bring up the module. My goals for engineering courses are usually quoted from the ABET 2000 criteria, and there is a link to the standards page of the Accreditation Board for Engineering and Technology (1999).

Requirements and policies are organized in folders to enable easy retrieval during the semester without scrolling through lengthy documents. The Requirements folder contains documents titled Reading, Writing, Homework, Computer Assignments, Test Information, Electronic Journal, and others. I quote selections from these to indicate the structure of a typical course.

Writing. You will be expected to ... write clear, direct English prose about your thought processes and your work on various sorts of problems. You will find it useful to read A Guide to Writing in Mathematics Classes  by Annalisa Crannell (Franklin and Marshall College) ....

Homework. ... In general, no solution will be given full credit unless you have written an explanation of why you know  it is correct. For example, an acceptable explanation for a solution of an equation is that you have substituted the proposed solution into the original equations and found that it satisfies the equation identically -- and you have to show your work. Three examples of unacceptable  explanations:

Tests. All tests will be open-book and take-home. ...

There are, of course, some policies imbedded in the Requirements documents. Others are in the Policies folder -- again, I quote selectively:

Teamwork. Computer work will be done by teams of two students, and some class work will be done by teams of four, usually by pairing two computer teams. The instructor will make team assignments and will change them about once a month. ... You will find useful information about working in groups in Ten Guidelines for Students Doing Group Work in Mathematics  by Anne E. Brown, Indiana University South Bend. ...

Cooperation. ... Learning is a cooperative activity -- and the ‘real world’ demands workers who have learned to cooperate in order to solve more substantial problems than individuals can solve working alone. Therefore, some tasks in this course will be assigned to designated groups, and such tasks will receive a group grade.

Open- vs. Closed-Books. The ‘real world’ is ... a very large collection of ‘open books’. Success does not require memorizing the contents of those ‘books’ but rather understanding how to use available resources in an intelligent way. Closed-book tests do little to measure such understanding, so all activities in this course -- including tests -- will be open-book.

Grading. There is no intellectually respectable justification for pretending that understanding of mathematics ... can be measured by numerical point scales. ... [E]ach graded activity will ... be assigned a letter grade relative to the instructor’s expectations .... Such grading is sometimes called ‘subjective.’ The alternative to subjective grading is not ‘objective’ -- it’s arbitrary.

Your letter grades from the different components of the course will be weighted approximately in the following way. ...

Team work

 

Individual work

 

Computer reports
(9 to 10)

25-30%

Homework

15%

Class participation
and projects

10-15%

Electronic journal

5%

   

Take-home tests (2)

20%

   

Final Exam

20%

Total

40%

Total

60%


The next important segment of CourseInfo is Course Documents. The instructor can post any type of document that Microsoft environments recognize -- text, Word, HTML, PowerPoint, jpeg, and so on. With a little care in opening and saving, this facility can also be used to provide unrecognized file types, such as Maple. I find it useful to construct my lectures (yes, there are some) in Maple so that I can vary examples on the fly, with instant access to precise and dynamic graphics, and little possibility of an arithmetic or algebraic error. The ‘lecture notes’ are immediately available to students, and they can manipulate the Maple code to enhance their understanding, further modify the examples, or adapt code to address their homework exercises. Because these notes are available in the classroom, I can build follow-up exercises and worksheets that will be used in the same class period. The most important folder in Course Documents is the Lecture Support file.

The Course Documents page also contains assignments of students to teams (often more than once per semester), a summary of the class demographics and individual goals, explanations of troublesome concepts, pictures taken in the classroom, slides from lectures given elsewhere that relate to the course, and any other documents that may be useful.

There is much more in CourseInfo, but I will add just a few words about my Assignments pages. The major folders here are Weekly Plans, Worksheets (for planned group classroom activities), and Review Problems (plus solutions that are posted after students have worked on the problems in the classroom with their teammates). The Weekly Plans are the organizing heart of the course. Each plan announces the major concepts for the week, makes connections to work already completed and to the overarching themes of the course, sets the daily homework assignments, maps out what will happen in the classroom meetings, and sets and reviews due dates for homework, in-class work, computer modules, and any other assignments.

2. CREATING A CLASSROOM ENVIRONMENT: PEOPLE

Only so much can be done with hard and soft course structures. The most important characteristic of a classroom in which everyone can succeed is trust.  And trust is possible only if the environment is cooperative and collegial, not competitive. If the highest grades -- the ‘coin of the realm’ for students -- are available only to a small percentage of the class, then trust and cooperation are not likely. If the instructor is seen as an adversary -- always ‘taking points off’ rather than supporting student growth -- trust and cooperation are simply impossible.

Students often go to great lengths to ‘psych out’ a course and its instructor, optimizing strategies to achieve high grades with (what they think is) minimal effort. If they can postpone study until the week of an exam, cram for one or two nights, and get a good grade, that is what they will do, no matter what the instructor says to do. Our elite colleges, such as my own, tend to be populated with students who have succeeded with such strategies -- and who are reluctant to give them up. However, they will  give them up if they can be convinced that

  1. that won’t work in this class,
  2. there are quite different strategies that will lead to good grades, and
  3. real learning can be a lot more fun than periodic cramming and regurgitation -- just as real dining is a lot more fun than bingeing and regurgitation.

The policies announced on my CourseInfo pages make point a) -- there are no available ‘points’ for memorization. I reinforce that frequently in class by reminding students that the only rule for my tests is that they will see problems they have never seen before. Furthermore, the tests account for only 40% of the course grade. I have to teach to points b) and c). I address b) with a variety of assessed activities, most of them with grades that ‘count’, but some not, most of them likely to lead to success and to a sense of growth and preparation for that still-distant exam. Point c) is addressed mainly by careful selection of quality materials and interesting problems, plus frequent interaction with working groups to keep them moving forward and to help them learn group skills.

The most important fact about trust is that someone has to go first. Since I’m the one who really wants that atmosphere of trust, that would be me. I give out my home phone number on day 1, and I don’t mind if they use it. I generally respond to e-mail in a reasonable time, and I pay attention to the electronic journal entries. I make it very clear to my students that I trust them totally, that I simply don’t believe it is possible that they would cheat, because there is essentially nothing that can be gained that way, and it would be destructive to oneself and one’s teammates.

Once it is settled that I have refused the role of adversary, I see the bonds of trust gradually strengthen. Students often sign up for my section (so they tell me) because they have heard that I give good grades. A few back off when they realize how much work I expect, but most stick around because they believe the good grades are achievable, even if not the way they expected. Almost all of my students are in class almost all of the time, and almost all earn a B or better -- even if they have never done that before in a maths class. And the small number of students who do not achieve at that level generally concede that it was their own fault, not mine.

3. SUPPORT IN THE RESEARCH LITERATURE

There is a vast and growing literature supporting the use in college classrooms of the practices outlined above: active learning, group work, continuous and mixed assessment, reading and writing to learn, intelligent use of technology, and above all, community. A useful resource for pointers to this literature is The National Teaching And Learning Forum; some of the articles I have found helpful are those by McLeod (1996), Rhem (1995) and Zull (1998). A concise summary of research findings prior to the reform efforts of the 1990s, applied specifically to mathematics education, is Everybody Counts (NRC, 1989). A more recent and more general survey, which includes results of both cognitive psychology and neurobiology in the last decade, is in Bransford, Brown and Cocking (1999) and its companion volume Donovan, Bransford and Pellegrino (1999). For the reader unfamiliar with the learning implications of recent neurobiological research, Hannaford (1995) and Sylwester (1995) are also helpful.

I have written at some length elsewhere (see Smith, 1998, 2000c) about some aspects of this literature and their relevance to college mathematics classrooms and curricula. I will summarize here some key points that support my current classroom practices.

In the late 1980s, Chickering and Gamson (1991) surveyed "50 years of research on the way teachers teach and students learn" and distilled their findings into Seven Principles of Good Practice in Undergraduate Education:

Good practice:

  1. encourages student-faculty contact;
  2. encourages cooperation among students;
  3. encourages active learning;
  4. gives prompt feedback;
  5. emphasizes time on task;
  6. communicates high expectations;
  7. respects diverse talents and ways of learning.

They summarize, "While each practice can stand on its own, when they are all present, their effects multiply. Together, they employ six powerful forces in education: Activity, Cooperation, Diversity, Expectations, Interaction, and Responsibility."

Kolb’s research in experiential learning (see Wolfe and Kolb, 1984, pp. 128-133) led to his description of a learning cycle as a two-dimensional way to think about students’ preferred learning styles and about ways to build better learning environments (see Figure 1). The four stages of this cycle, interpreted in terms of brain research, are

The ideal learner cycles through these stages in each significant learning experience. The AE stage represents testing in new situations the implications of concepts formed at the AC stage. Depending on the results of that testing, the cycle starts over with a new learning experience or with a revision of the current one. The ideal learning environment is designed to lead the learner through these stages and not allow ‘settling’ in a preferred stage.

Figure 1. Kolb Learning Cycle

In fact, there are very few ideal learners. Most students have preferred learning activities and styles, and they are not all alike. This is one reason why learning experiences work better for everyone in a diverse, cooperative, interactive group.

Learning takes place by construction of neural networks. External challenges (sensory inputs) select certain neural connections to become active, and this is a random selection among many possible connections that could occur, not something that happens by deterministic design. The sensory input can trigger either memory, if it is not new, or learning, if it is new. The cognitive psychology term for this process is constructivism:  The learner builds his or her own knowledge on what is already known, but only in response to a challenge or ‘disequilibration.’ In particular, knowledge is not a commodity that can be transferred from knower to learner.

Deep learning, learning based on understanding, is whole brain  activity -- effective teaching must involve stimulation of all aspects of the learning cycle (Rhem, 1995). Research by Entwistle, Marton, Ramsden, Schmeck, and others (see especially Chapters 1, 2, 3, 7, and 12 in Schmeck, 1988), has shown that a student’s learning approach  or strategy  may be more important than learning style -- a student may exhibit different approaches in different courses at the same time, depending on perceptions of what is expected for success. Furthermore, some course characteristics have been found to foster surface approaches and some to foster deep approaches:

Course characteristics that encourage surface  approaches:

Course characteristics that encourage deep  approaches:

The first of these lists describes the way I used to teach, more than a decade ago. The second describes what I am trying to do now. When we free ourselves from the tyranny of the packed syllabus, we can give our students many freedoms, with or without computers. There is much we don’t know yet about the World Wide Web as a learning tool, but having an on-line studio classroom allows us to conduct and observe some of the necessary experiments.

The research results mentioned here all reinforce each other, and together they constitute a clear and consistent message about good design and execution of curricula and pedagogy.

4. A SAMPLE LEARNING ACTIVITY

I will illustrate implementation of the research findings in the preceding section by way of a learning activity from a differential equations course. In the sixth week of the course we were studying undriven harmonic oscillators and their representation by second-order linear, homogeneous, constant-coefficient differential equations. (See Section 1 for instructions for viewing the syllabus and the Week 6 Plan on the course Web page.)

The activity begins with students gathering position data on a spring-mass system with a Calculator-Based Laboratory (CBL) and motion detector. At Interactive Computer Classroom Pictures 2000, the last six pictures show this experiment in progress. As students observed the spring motion under various damped and undamped conditions, the data were displayed in real time by way of a Texas Instruments TI-89 calculator with a view screen on an overhead projector. The calculator is quite capable of completing the analysis of the data, but we transferred the data sets to a computer in order to make them available in CourseInfo documents for easy copying into a computer worksheet. In this example, the physical experiment is the principal part of the CE phase of the Kolb cycle, but well-designed computer simulations can play this role as well.

Next the students worked in teams of two on a Maple-based Connected Curriculum Project module (Moore and Smith, 2000). We devoted a 50-minute class period to this part of the activity, and then students had several days to finish the module on their own time.

The module begins with student entry of the undamped data and an attempt to fit the data with a cosine function, which they already know from assigned reading and classroom discussion should be an appropriate model. The following figures and text illustrate work in progress. Student inputs to the Maple worksheet and their comments are in boldface, and Maple outputs are centered.

> y0:=(0.835273-.6787664300);
>
Y:=t->y0*cos(sqrt(K/m)*t + Pi/10);

[Maple Math]

[Maple Math]

We add the factor of amplitude of y0 to our model, taking the highest peak from our data and subtracting the mean from it. We do this, instead of merely taking the value at t=0 because at t=0, we see that the spring is not at its peak. Hence, we adjust for the actual amplitude by taking the highest peak of the data. We then had to adjust for the period (knowing that the data graph does not start at its peak at t=0). Thus, we adjusted for this by adding a delay factor of Pi/10, (which seems to be sufficient).

Next we graph the data and the model function together.

> graph2:=plot(Y(t), t=0..7,color=red):
>
display({graph1,graph2});

[Maple Plot]

Our model seems to fit the data pretty well. With our adjustments for amplitude and our delay (because the data did not exactly start at the peak for t=0), we have a model that seems to show an undamped spring motion. The only difference comes where as t grows larger, the amplitude of the data graph decreases. This is because the data was not ideally undamped (it had a CD attached to the weight), and so air resistance was damping the spring as time went on.

This student sample needs a few words of explanation.

  1. Some details that we don’t want students to waste time on are already provided in the worksheet. For example, they didn’t have to do anything with the graphing commands except execute them. But the plot command here does not do anything until the model function Y(t) has been defined, and that’s the key thing we want students to think about at this stage. We (the authors of the module) chose to display the data as discrete boxes and the model function as a continuous curve so the distinction would be instantly clear.
  2. Our intention was that students would start their data gathering at the instant the extended spring was let go, so y0 would be one of the minimum points on the plot, eliminating the need to think at this stage about ‘delay factor.’ Since the experiment did not go according to plan, students had to think about something  that would account for not starting at a peak or valley. Some students discarded the first eight data points, and some (like the team quoted here) more or less invented phase shift. The first quoted paragraph is their justification of this procedure. Note the revelation of ‘prior knowledge’: They felt compelled to express that horizontal shift as a multiple of pi.
  3. In the second quoted paragraph, the students are tentatively taking credit for a good fit. They are also noticing the obvious fact that some damping is present and attributing that to the cause they could see: a CD attached to the weight. (Don’t throw away all your ‘Free AOL’ CD-ROMs. They make excellent lightweight targets for a motion detector.) In fact, there is just as much frictional damping without the CD, but ‘seeing is believing.’ Their conclusion here may also be influenced by a later stage of the experiment, in which we used a one-foot square of cardboard (in place of the CD) to induce significant air damping.

At this early stage of the module, students have already completed the Kolb cycle at least once. They have reflected on the concrete experience and visualized the motion as something familiar, namely, a sinusoid. They have connected this with their prior experience (in particular, reading) to abstract the motion into a formula, and they have tested and adjusted their formula until they are confident it is right. We don’t actually see this last phase in their report -- that’s where the instructor comes in. I know  that formula was not copied from a book or from another student team because I watched the students working in class and interacted with them as they did so. Collectively, the class had as many different, correct  formulas as there were teams, and the learned concept was the same for all of them.

The second Kolb cycle in this module uses the damped data and proceeds in a similar manner, but success is more difficult to achieve, because there are now four parameters in the model, only one of which can be estimated directly from the data. We provide some direction about the order in which students should attempt to find the other three. Here are the results from the same student team -- note their judgment about quality of the data:

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Plot]

Our approximation is pretty good. It fits the period and the same damping and the same amplitude. One difference is in the first valley, where we see our model goes down farther than the data. However, looking at the graph of the data, we see that this seems to be a data error, since the graph of the data is not very smooth here. The device did not seem to collect enough data here to make the graph smooth.

[Click here to see the entire student worksheet down to this point, and here to see the instructions for the module.]

To reinforce abstract conceptualization about the harmonic oscillator, the module has a third part on critical damping -- which cannot be tested by the physical apparatus at hand, but for which computer testing of the model can still be carried out. The outcome of this exercise is that most students have a clear understanding of the significance of the constant, exponential, and trigonometric factors in the solution formula and of the parameters as well.

Each CCP module ends with a Summary section in which students respond with paragraphs to integrative questions that force them to think through internal and external connections among the various experiences and their prior understandings. In addition to the explicit connections with the Kolb cycle, it should be clear that this learning activity resonates with the characteristics that foster deep learning and with Chickering and Gamson’s Seven Principles of Good Practice in Undergraduate Education.

The key findings of the National Research Council study How People Learn  (Bransford, Brown and Cocking, 1999) are summarized in Donovan et al., (1999, pp. 10-15):

  1. Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.
  2. To develop competence in an area of inquiry, students must:
    1. have a deep foundation of factual knowledge,
    2. understand facts and ideas in the context of a conceptual framework, and
    3. organize knowledge in ways that facilitate retrieval and application.
  3. A ‘metacognitive’ approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.

A lot of implications are packed into those three statements, and one purpose of the NRC study is to unpack them. Suffice it to say that the CCP learning environments and my other classroom activities are consciously designed to align with these findings. The statement about metacognition brings us to some final observations.

5. PROGRESS TOWARD GOALS

On homework and tests, I refuse to give full credit even for correct answers unless accompanied by a correctness argument. I also use frequent and varied assessments, formal and informal, to assist students in monitoring their progress. The CCP materials referred to in the preceding section contain many self-monitoring steps.

Of course, one cannot monitor progress toward goals if there are no goals. Early in each of my courses, I engage the students in a goal-setting exercise. Not all of the expected learning gains are necessarily identified among students’ own goals, but their goals usually have a substantial overlap with my own -- which I post on the course Web page before the semester starts. Here are the goals for the differential equations course -- as noted in section 1, these goals are stated in the language of the ABET 2000 criteria.

Students who complete this course should be able to:

Of course, we don’t have the opportunity to observe whether students actually do engage in life-long learning, but progress toward the other six goals could be observed and measured during the course. I set high expectations for my students, and almost all of them respond with a high level of performance.

References

Accreditation Board for Engineering and Technology (1999). ABET Engineering Criteria 2000. URL: http://www.abet.org/eac/EAC_99-00_Criteria.htm#EC2000. [2000, June 21].

Bransford, J. D., Brown, A. L. and Cocking, R. R. (Eds.). (1999). How People Learn: Brain, Mind, Experience, and School. National Research Council. Washington, DC: National Academy Press.

Chickering, A. W. and Gamson, Z. F. (Eds.). (1991). Applying the Seven Principles of Good Practice in Undergraduate Education. New Directions for Teaching and Learning No. 47. San Francisco: Jossey-Bass Publishers.

Donovan, M. S., Bransford, J. D. and Pellegrino, J. W. (Eds.). (1999). How People Learn: Bridging Research and Practice. National Research Council. Washington, DC: National Academy Press.

Hannaford, C. (1995). Smart Moves: Why Learning is Not All in Your Head. Arlington, VA: Great Ocean Publishers.

Interactive Cluster Classroom Task Force(2000). URL: http://www.aas.duke.edu/admin/icc/. [2000 June 21].

Interactive Computer Classroom Pictures (2000). URL: http://www.math.duke.edu/education/icc/iccpix.html. [2000, June 21].

Materials for Engineering Mathematics (2000) [Online]. Connected Curriculum Project. URL: http://www.math.duke.edu/education/ccp/materials/engin/. [2000, June 21].

McLeod, A. (1996). Discovering and Facilitating Deep Learning States. National Teaching And Learning Forum, 5 (6), 1-7.

Moore, L. C. and Smith, D. A. (2000). Spring Motion [Online]. Connected Curriculum Project. URL: http://www.math.duke.edu/education/ccp/materials/diffeq/spring/. [2000, June 21].

National Research Council (1989). Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, DC: National Academy Press.

Rhem, J. (1995). Deep/Surface Approaches to Learning. National Teaching And Learning Forum, 5 (1), 1-5.

Schmeck, R. R. (Ed.). (1988). Learning Strategies and Learning Styles. New York: Plenum.

Smith, D. A. (1998). Renewal in Collegiate Mathematics Education, Documenta Mathematica  Extra Volume ICM 1998 III, 777-786. URL: http://www.math.duke.edu/~das/essays/renewal/. [2000, June 21].

Smith, D. A. (1999). Applied Mathematical Analysis II  URL: http://cinfo.aas.duke.edu/courses/MTH114.01/. [2000, June 21]. (Note:  The published URL was changed by Duke University in January, 2002, to https://courses.duke.edu/bin/common/course.pl?course_id=_1226_1&frame=top. The link has been corrected.)

Smith, D. A. (2000a). Elementary Differential Equations. URL: http://cinfo.aas.duke.edu/courses/MTH131.03/. [2000, June 21]. (Note:  The published URL was changed by Duke University in January, 2002, to https://courses.duke.edu/bin/common/course.pl?course_id=_1229_1&frame=top. The link has been corrected.)

Smith, D. A. (2000b). Introduction to Linear Programming and Game Theory. URL: http://cinfo.aas.duke.edu/courses/MTH126.01/. [2000, June 21]. (Note:  The published URL was changed by Duke University in January, 2002, to https://courses.duke.edu/bin/common/course.pl?course_id=_1228_1&frame=top. The link has been corrected.)

Smith, D. A. (2000c). Renewal in Collegiate Mathematics Education: Learning from Research. Chapter 3 in S. L. Ganter (Ed.), Calculus Renewal: Issues for Undergraduate Mathematics Education in the Next Decade, pp. 23-40. New York: Kluwer/Plenum.

Sylwester, R. (1995). A Celebration of Neurons: An Educator’s Guide to the Human Brain. Alexandria, VA: Association for Supervision and Curriculum Development.

Wolfe, D.M. and Kolb, D.A. (1984). Career Development, Personal Growth, and Experiential Learning. In D. A. Kolb, I. M. Rubin and J. M. McIntyre (Eds.), Organizational Psychology: Readings on Human Behavior in Organizations (4th ed.), pp. 128-133. Englewood Cliffs, NJ: Prentice-Hall.

Zull, J. E. (1998). The Brain, The Body, Learning, and Teaching. National Teaching And Learning Forum, 7 (3), 1-5.


David A. Smith

Duke University, North Carolina, USA

das@math.duke.edu