Implementation of an integrated, technology based, discovery mode assessment item involving an incubation period to enhance learning outcomes for engineering mathematics students by Neil Kelson
and
Anand Tularam
School of Mathematical Sciences, Queensland University of Technology
Introduction
For the teaching and learning of mathematics, there is a growing consensus among educators that the role of routine procedural skills should diminish to allow more emphasis to be placed on learners gaining conceptual insights and analytical skills that appear essential in real-life mathematical problem solving. Central to this view is the modern constructivist's theory of how learning occurs, an appreciation of which does not appear to be widespread in the tertiary education arena, despite its growing acceptance in the secondary education sphere as part of effective pedagogical theory and practice. On the other hand, the notion that "what gets assessed is what gets taught" (O'Day & Smith, 1993) is widely appreciated by educators and students alike. It follows that if we want our students to tackle realistic problem solving tasks, engage their higher order thinking skills, or actively construct their own knowledge generally, then we need to devise assessment instruments which move away from the convergent style of questioning in unseen written exams which has been prevalent in tertiary mathematics in past era.
A discussion of the constructivist theory, and the important implications in terms of assessment, is presented in a companion paper to this one (Tularam & Kelson, 1998). The scope of the present work, however, is more specific. Here we address the question of how can a constructivist assessment item be introduced into the assessment profile of a target group of about 100 second level engineering mathematics students. In addition, we also focus on the question of whether an assessment task can be used to promote higher order thinking, rather than the routine testing of domain specific knowledge and skills.
A third question of interest to us is one which relates to the management of students' learning and the close relationship between assessment and instruction. We are interested here in creating an assessment item which encourages students to build in-roads into potentially difficult subject matter that they will be exposed to latter in their course. This should be contrasted with one of the main concerns about unseen written exams, namely, that they can force students into surface learning, and into rapidly clearing their minds of previous knowledge when preparing for the next exam (Race & Brown, 1998).
The fact that assessment instruments may actually encourage students to actively clear knowledge stored for successive exams provides an interesting challange for educators who want to cultivate more integrated approaches to learning in their students. In this work, we investigated one such approach which addresses this issue. Here, we introduce an assessment task which is designed to be intimately linked to the overall course content, in that it aims to introduce important concepts and skills (via individual student exploration) which will be revisited somewhat later in the pedagogically more restrictive formal lecture component of the course. Furthermore, a time delay, or "incubation period", is incorporated to allow students the opportunity to undergo potentially important internal re-adjustments in their understanding, before being exposed to lectures where related course content is invariably delivered in a more condensed, formal and mathematically rigourous manner.
As we shall describe in more detail shortly, we employed a novel computer based method (CBM) of teaching Fourier series to second year engineering mathematics students. In addition to producing technologically literate engineers, the CBM has a number of potential advantages which may lead to better student learning outcomes, as we have discussed elsewhere (Kelson & Tularam, 1998). Here we have exploited the algebraic and visualisation capabilities of the symbolic algebra package Maple to provide students with a pre-lecture laboratory experience. The use of a package such as Maple has the additional advantage that certain mathematical concepts and skills can be investigated "on-line" using visual and intuitive reasoning, rather than a formal or rigourous approach. With regard to the choice of subject matter, Fourier series is an important topic which has a number of applications in engineering mathematics. However, anecdotal evidence suggests that students find the complexity the Fourier series difficult to deal with, and it would seem that any suggestions for improvements in the teaching of this topic would be appreciated by many tertiary mathematics educators.
In the following, we elaborate further on the design and rationale of the present approach, followed by a discussion of student learning outcomes we have observed in our trial.
An overview of how the present assessment instrument fits in with the overall assessment profile for our target group is available from the authors on request.
Laboratory Experience: "Visual Introduction to Fourier Series"
In this work, we trialed an assessable pre-lecture laboratory experience on Fourier series. The lab was intended to replace, in part, the usual tutorial format, and aimed to introduce students to some basic ideas in the theory of Fourier series before lectures on this topic were given.
In this laboratory, we aimed to create opportunities for students to engage in higher order thinking via "on-line" exploratory or discovery mode tasks. Also, we tried to answer the question. "How can we give students an assessment task which has student learning outcomes (as opposed to measuring/ranking/testing) as its primary focus, asks them to engage their higher order thinking skills, is exploratory or discovery in design, reflective, non-routine, allows them to tap into powerful visual representations of theory, permits intuitive reasoning to sense patterns and form tentative conclusions, and most importantly, has the potential to aid their understanding of difficult subject matter in the course (as, for example, an effective tutorial session should do).
In order to address some of the issues described above, we decided to trial a design which we derived from a theory due to Rubinstein (1975) with regard to expert problem solving and creativity. Rubinstein suggested that creativity involves the linking and 'connectiveness' of knowledge, and argued that such a process usually occurs after an 'incubation period'.
It seemed to us that applying this incubation period concept to the teaching of difficult higher level concepts of mathematics may be a useful teaching method. For example, the learners can be given the opportunity to learn concretely, visually, and symbolically first and then after an incubation period re-examine the content and/or procedures in a more formal manner. In this way, the learners can reflect on their work and examine the content more critically. According to constructivist educational theorists such a process may facilitate the connectiveness of knowledge through the linking of concepts and procedures into a logical/coherent cognitive structure (Clements & Battista, 1990; English, 1997; Neiss,1993; Sternberg, 1988).
The basic design is shown schematically as follows.
Pre-Lab
workLaboratory
Exercise?
Delay time or
'incubation period'?
Formal Lectures
on Fourier seriesAs part of the design, a collection of pre-laboratory homework problems were set. The pre-laboratory work primarily set exercises which involved demonstrating a working knowledge of relevant theory and skills introduced in earlier courses. In particular, concepts and properties relating to odd, even and periodic functions, as well as simple examples of integrals involving products of trigonometric functions were the focus at this stage. However, a small individual research component aimed at extending and synthesising the student's existing knowledge base on piecewise defined functions, limits and discontinuities was also included as a way of leading them away from more familiar "closed" questions and convergent thinking. Furthermore, students were required to submit their completed pre-laboratory homework just before commencing the laboratory exercise, in order to assist their recall of essential theory and skills needed for the exploratory tasks. By incorporating the pre-laboratory homework into the overall design in this way, we envisaged that students would establish a reasonable prior knowledge base which would allow them to engage in the investigative laboratory work with some degree of confidence.
Regarding design of the pre-lecture laboratory on Fourier series, a set of instructions and accompanying text were prepared in the worksheet environment of the symbolic algebra package Maple. The main concepts explored in the laboratory program included visual "proofs" of orthogonality properties of trigonometric functions, examination of piecewise defined functions and finite jump discontinuities, Fourier series representations and partial sums, the Gibbs phenomenon, half range series, and continuity of Fourier series. However, the laboratory was not simply an on-line visual based tutorial but rather a combination of hands-on, visual and written/research work. In this manner we aimed to examine attitudes, beliefs and patterns in thinking to decipher the nature of thinking undergone during the tutorial (Schoenfeld, 1985; Tularam, 1997).
The Fourier series program was designed to examine whether, given the opportunity, students self-engage their higher mental faculties of metacognitive and critical thinking while working on-line. We envisaged that the questions posed in the CBM tutorial may have created some tension and conflict in students' minds, and to resolve such conflicts, students would need to move away from the computers and engage in written, reflective and/or critical thinking. Indeed, this "movement away" from the computer to written or other reflective research work was an important aspect of the self-learning laboratory program. In such actions the students' thinking can be explored to determine whether they were indeed engaging their higher faculties (Flavell, 1987).
Regarding assessment, we wanted to attach importance to the pre-lecture laboratory experience by making it an assessable activity. In this regard, the existing institutional arrangement of having to allocate marks to every assessment item proved restrictive. With an assessment instrument as described above, we were conscious of asking students to take intellectual risks, and it was imperative that a relatively "safe" environment is created for them to do so without excessive fear of failure.
With the preceding comments in mind, we opted for a two part scheme where students were awarded marks equally on the basis of both their pre-lab and laboratory work. Thus, for half the total marks assigned to this assessment instrument, students were awarded marks by comparing their pre-lab work against fully worked solutions with a detailed marking scheme. On the remaining marks to be allocated, our view was that marks awarded for the laboratory work should be based mainly on the extent to which higher order thinking is attempted or demonstrated, rather than the amount of CBM tasks completed by the student in a given time frame. However, as a practical measure, we decided instead to simply instruct the students that clear evidence of "engaging the task" (as judged by their written responses) would result in high marks being awarded for their laboratory work.
Results
Initially, students were allocated to 3 groups which undertook the computing laboratory exercises within a fixed one hour time frame over 3 tutorial sessions, each one week apart. Both in and after each session, students' attempts were evaluated, and minor modifications in the form of corrections and clarifications to the program were made. On our initial examination of the total body of work, we felt that the one hour time frame was too short, with many students spending a significant portion of the designated time in familiarising themselves with the essential functions of the Maple software.
In light of our preliminary observations, we offered an optional laboratory, where students were to complete the Fourier series exercises without time constrants. About half of our target group chose to complete the optional laboratory, and we report the results of their efforts here.
In brief, we examined in detail sixteen students' written work. These scripts revealed a number of interesting features. For example, in some papers we observed evidence of conflict in students' written responses. "How can a Fourier series be continuous but the underlying function be discontinuous?" was one recurring theme. There was also evidence of connective thinking, particularly with reference to subject matter studied in previous courses relating to properties and convergence of series. In one instance, a student wrote of an admission of lack of understanding, but later indicated that problems were clarified and resolved as work progressed on the assigned laboratory tasks. Here, the student has used an off-line written exercise to document or journal their thoughts as they progressed through the on-line tasks in a highly reflective way. In some scripts, students speculated on the link between the rate of convergence of the Fourier series representation and the continuity of the underlying function it is trying to represent, while in other written responses, students attempted to present intuitive and geometric arguments which related functions with infinite slopes (i.e. discontinuities) to the slow convergence properties of the corresponding Fourier series.
The noteworthy and exciting point to be made here is that some of these "leaps of faith" described above are very close to the mark in terms of the formal theory that was presented to them later in the course.
In only one of the written responses examined in detail, the student conveyed a reluctance to attempt discovery mode tasks. Nevertheless, a positive constructivist outcome was observed, in that the student actually moved off-line to do independent research, saying "...once looked up the relevant material and understood process a little made a lot more sense." In terms of Kolb's group identifications of how people prefer to process information, namely assimilators, accomodators, convergers and divergers (for a discussion of these see Tularam, 1998), this observation suggests that this student processes information as mainly an assimilator and converger. An observation such as this is invaluable, as it clearly highlights the need for a variety of assessment instruments which cultivate a wider range of thinking and information processing skills, in order for students to be adequately equiped to solve real life problems in their future professional practice.
Finally, with regard to student feedback, at the present time we have available only informal feedback obtained from students while the laboratory sessions were in progress.
During these laboratory sessions, students have described an intellectual "tension", as evidenced by student (S) comment such as:
S1: "What the hell's going on here?"
"Do I actually have enough knowledge to do this work?"In this case, the student is actually being highly metacognitive, in that they are demonstrating an awareness and evaluation of their knowledge base. Of particular concern is our informal observations that many students, particularly the male students, seem to be locked into a convergent style of thinking. Student comments and exchanges between the student and the instructor (I) along the lines of the following were often repeated.
S2: "What do you want for the answer to this? It could be anything!"and
S3: "So what's the answer you are after...?" I: "There is no correct answer. Just tell me what you think is going on." S3: "What..!?"From these observations, it seems that students are less familiar or comfortable with problem solving tasks which involve more divergent thought processes. As educators, it appears that we need to monitor our students' progress in a variety of ways, to ensure that they are developing appropriate critical thinking and problem solving skills that they will need in their future professional activities.
Concluding Remarks
In this paper, we have discussed a novel on-line teaching and learning strategy based on the incubation period concept of Rubinstein (1975), for a target group of 2nd level engineering mathematics students. At the present stage, we have preliminary evidence that students engage in higher order thinking when engaged in this approach, but nonetheless, there are a number of areas that need further investigation. For example, to what extent do the students who experience technology based teaching actually gain metacognitive, reflective or critical thinking skills while working on computers in an "on-line" manner? While we have evidence to suggest that students do gain the necessary higher-order thinking skills there needs to be more work done in this area. Also, it remains to demonstate the extent of benefit of the present approach in terms of student learning outcomes. At the present time, for example, we have no indication as to the extent to which a pre-lecture laboratory experience assists the student in making inroads to theory presented later in their course. These and other questions remain the subject of future work.
References
Note: a full reference list is available from the authors on request.
Kelson, N. & Tularam, A. (1998). Tutoring with higher mathematics and the use of technology. Proc. 2nd Conf. on Tutoring. The University of Queensland.
O'Day, J.A. and M. Smith (1993). Systemic school reform and educational opportunity. In S, Fuheman (Ed.), Designing coherent educational policy: Improving the system. San Francisco: Jossey-Bass, pages 250-311.
Race, P. & Brown, S. (1998). The lecturer's toolkit. Kogan Page Ltd.
Rubinstein, M.F. (1975). Patterns of problem solving. Prentice Hall. California
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Tularam. G. A. (1997). The role of higher-order thinking, algebraic knowledge and affective factors in novel algebraic problem solving. Unpublished PhD Thesis. Queensland University of Technology, Brisbane.
Tularam, A. (1998). Individual differences in learning styles. Proc. 2nd Conf. on Tutoring. The University of Queensland.
Tularam, A. and Kelson, N. (1998). Assessment in tertiary mathematics. Proc. 5th Conf. on Effective Assessment. The University of Queensland.
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