During student efforts to attack and solve complex, technology-based problems there is rich opportunity for assessment. The teacher can assess student initiative, creativity, and discovery; flexibility and tolerance; communication, team, and group self-assessment skills; mathematical knowledge; implementation of established and newly discovered mathematical concepts; and translation from physical descriptions to mathematical models.
Background and Purpose
Complex, technology-based problems worked in groups during class over a period of several days form a suitable environment for assessment of student growth and understanding in mathematics. The teacher can make assessment of student initiative, creativity, and discovery; flexibility and tolerance; communication, team, and group self-assessment skills; mathematical knowledge; implementation of established and newly discovered mathematical concepts; and translation from physical descriptions to mathematical models. Viewing students' progress, digression, discovery, frustration, formulation, learning, and self-assessing is valuable and instant feedback for the teacher.
During student efforts to attack and solve complex, technology-based problems there is rich opportunity for assessment. Active and collaborative learning settings provide the best opportunity for assessment as students interact with fellow students and faculty. The teacher sees what students know and use and the students receive immediate feedback offered by the coaching faculty.
Student initiative is very easy to measure, for the students voice their opinions on how they might proceed. Creativity, discovery, and critiquing group member's offerings come right out at the teacher as students blurt out ideas, build upon good ones, and react to ideas offered by others. Listening to groups process individual contributions will show the teacher how flexible and tolerant members are of new ideas. The write-up of a journal recording process and the final report permits assessment qualities such as follow-through, rigor, and communication.
I illustrate success using this assessment strategy with one problem and direct the reader to other problems. I used the problem described below at Rose-Hulman Institute of Technology several times  as part of a team of faculty teaching in an Integrated First-Year Curriculum in Science, Engineering, and Mathematics (IFYCSEM). In IFYCSEM the technical courses (calculus, mechanics, electricity and magnetism, chemistry, statics, computer programming, graphics, and design) are all put together in three 12 credit quarter courses . This particular problem links programming, visualization, and mathematics.
This activity fits into a course in which we try to discover as much as possible and reinforce by use in context ideas covered elsewhere in this course or the students' backgrounds. Further, it gives the teacher opportunities to see the students in action, to "head off at the pass" some bad habits, to bring to the attention of the class good ideas of a given group through impromptu group presentations of their ideas, and to have students "doing" mathematics rather than "listening" to professorial exposition.
The activity is graded and given a respectable portion of the course grade (say, something equal to or more than an hourly exam) and sometimes there is a related project quiz given in class after the project has been completed.
In IFYCSEM we used Mathematica early in the course as a tool for mathematics and as the introductory programming environment. Thus we were able to build on this expertise in a number of projects and problems throughout the course. I have also used such projects in "stand-alone" calculus courses at West Point and they work just as well.
A suitable class size for "coaching" or "studio modeling" such a problem is about 24 students eight groups of three. It is possible to visit each group at least once during an hour, to engage the class in feedback on what is going on overall, and to have several groups report in-progress results. I have used this problem with two class periods devoted to it at the start, one class period a week later to revisit and share progress reports, and one more week for groups to meet and work up their results.
One has to keep on the move from group to group and one has to be a fast read into where the students' mindsets are, to try to understand what they are doing, to help them build upon their ideas or to provide an alternative they can put to use, but not to destroy them with a comment like, "Yes, that is all well and good, but have you looked at it this way?" (Translated into adolescent jargon "You are wrong again, bag your way, do it my way!")
The problem is to describe what you can see on one mountain while sitting on an adjacent mountain. See Figure 1. In a separate paper  I discuss how students, working in groups, attack the problem and the issues surrounding the solution strategies. This problem develops visualization skills, verbalization of concepts, possibly programming, and the mathematical topics of gradients, projection, optimization, integration, and surface area.
The problem stated
For the function
suppose your eye is precisely on the surface z = f(x, y) (see Figure 1) at the point (2.8, 0.5, f(2.8, 0.5)). You look to the left, i.e., in the direction (roughly) (-1, 0, 0). You see a mountain before you.
(a) Determine the point on the mountain which you can see which is nearest to you.
(b) Describe as best you can the points on the mountain which you can see from the point (2.8, 0.5, f(2.8, 0.5)).
(c) Determine the amount of surface area on the mountain which you can see from the point (2.8, .5, f(2.8, 0.5)).
I introduce the above problem in a room without computers so the students will visualize without turning to computers to "crunch." I give the students the statement of the problem with the figure (see Figure 1) and say, "Go to it!" Working in groups, they turn to their neighbor and start buzzing. Circulating around the class, I listen in on group discussion.
Typically students offer conjectures which can be questioned or supported by peers, e.g., the highest point visible on the opposite mountain can be found as the intersection of a vertical plane and the opposite mountain. Students with hiking experience will say you cannot see the top of the mountain over the intervening ridges. A recurring notion is that when placing a "stick" at the point of the viewing eye and letting it fall on the opposite mountain this stick is tangent to the opposite mountain. Other students can build upon this vision, some even inventing the gradient as a vector which will necessarily be perpendicular to the stick at the tangent point on the opposite mountain. Others can use this latter tangency notion to refute the vertical plane notion of the first conjecture.
I also assign process reporting (summative evaluation) at the end of the project in which students describe and reflect upon their process. This is part of their grade. They keep notes on out of class meetings, report the technical advances and wrong alleys, and reflect on the workings of the group. Assigned roles (Convenor, Recorder, and Reactor) are changed at each meeting.
Use of Findings
The formative assessment takes place in the interaction among students and between students and teacher. Basically, the students "expose" their unshaped ideas and strategies, get feedback from classmates on their ideas, hone their articulation, and reject false notions. In so doing they clarify and move to a higher level of development. Observing and interacting with students who are going through this problem-solving process is an excellent way for the teacher to assess what students really understand.
Reading the journal accounts of progress gives the teacher insight into the learning process as well as group dynamics. The former helps in understanding the nature of student difficulty. The latter can help the teacher see why some groups are smoother than others and thus be aware of potential problems in the next group activity assigned.
The only real difficulty in this entire approach is how to address the many conjectures and convictions the students bring forth in the discussion. One has to be thinking and assessing all the time in order to give meaningful, constructive feedback. The teacher has to watch that one idea (even a good one) does not dominate from the start and that all ideas are given fair consideration. The entire process never fails if the problem is interesting enough.
A Mathematica notebook, ASCII, and HTML version of this problem, with solution and comments is available (under title "OverView") at web site http://www.rose-hulman.edu/Class/CalculusProbs. This is a part of a larger National Science Foundation project effort, "Development Site for Complex, Technology-Based Problems in Calculus," NSF Grant DUE-9352849.
 Winkel, B.J. "In Plane View: An Exercise in Visualization," International Journal of Mathematical Education in Science and Technology, 28(4), 1997, pp. 599-607.
 Winkel, B.J. and Rogers, G. "Integrated First-Year Curriculum in Science, Engineering, and Mathematics at Rose-Hulman Institute of Technology: Nature, Evolution, and Evaluation," Proceedings of the 1993 ASEE Conference, June 1993, pp. 186-191.
*This work was done at the author's previous institution: Rose-Hulman Institute of Technology, Terre Haute, IN 47803 USA.