LEARNING IN A COMPUTER ALGEBRA SYSTEM (CAS) ENVIRONMENT Michael Meagher 1
The Ohio State University [email protected]
Introduction
This study is a qualitative case study focusing on the question “What are the processes of learning in a Computer Algebra System (CAS) environment for college students learning calculus?” The study is designed to research the impact on student learning of a new software available for mathematics education. The research aims to provide insight into the nature of learning in a technology-rich environment.
Motivation for the Study
Calculating technology in mathematics has evolved from four-function calculators to scientific calculators to graphing calculators and now to calculators (or computers) with Computer Algebra System (CAS) software. The advent of CAS software, which can do a great deal of the problems in a standard algebra or calculus text book at the push of a few buttons, truly represents a quantum leap in technology. The community of mathematics educators is in the throes of a great debate as to whether this is one of the most exciting or most frightening developments in the history of mathematics education (Kutzler, 2000; Waits & Demana, 1999) as mathematics educators struggle with the implications of having software in the classroom which can, for example, expand and factorise algebraic expressions, solve equations, differentiate functions, and find anti-derivatives. For centuries now the emphasis in algebra and calculus curricula has been on the development of technical skills through practicing algorithms: making of students efficient machines for performing the algorithms now automatised by CAS. Machines with CAS capability have, therefore, brought into focus fundamental questions about the purpose of mathematics in school, and the nature and content of the mathematics curriculum. The use of CAS in education is still relatively rare but the growing body of research and the interest of such organisations as the National Council of Teachers of Mathematics (Cuoco et al., 2003) suggests that its extended use is imminent. It is important, therefore, that there be a firm research base for the implementation of CAS capable technology in schools, colleges, and universities which has addressed the processes of learning and the conceptual development of students learning using CAS as well as recognising obstacles to learning with CAS.
Since the wide availability of CAS in the late 1980s, research on its impact on mathematics education has developed in two main strands. The first has concentrated on showing the effectiveness of technology in supporting the learning of specific topics most of which are part of a traditional curriculum (Judson, 1990; Mayes, 1995; Palmiter, 1991). The second strand examines the question of what a technology-enhanced curriculum should consist of, suggesting new topics (cryptography, chaos theory, etc.), suggesting changes to the order in which topics are introduced and suggesting assessments which emphasise problem solving ability over technical skill (Heid, 1988; Herget et al, 2000; Kokol-Voljc 1999).
There is, however, a gap in the research in the area of investigations into the process of student learning and their development of concepts while using CAS. The importance of investigating these processes lies in understanding the impact on learning so that an appropriate
1 Proceedings of the international meeting of Psychology of Mathematics Education North American Chapter XXIV. 2004. Toronto, Canada, pp. 1457-1464.
balance can be found in the integration of technology into teaching ensuring that the merits of traditional methods are not lost and that the merits and demerits of CAS use are fully understood.
The aim of this research is to address this gap by investigating, as a case study, a group of students taking a college-level mathematics course in which a CAS (in this case called Mathematica) is at the core of the course design. A case study approach will be taken so that the research can focus intensely on students’ work and discussions in class to capture the processes of their learning and their conceptual development in calculus. The research question is:
What are the processes of learning in a Computer Algebra System (CAS) environment for college students learning calculus?
Theoretical Framework
This research employs two theoretical frameworks through which to approach student learning while using CAS. The Rotman Model of Mathematical Reasoning (1993) is used as a macro-framework for the place of technology in the learning of mathematics. This framework is useful for addressing the question of the effect of technology on learning by positioning technology in the activity of mathematical reasoning. The framework also provides a lens for interpreting the role of experimentation in the learning of mathematics and the relationship between technology and learners. The Pirie-Kieren Model of Mathematical Understanding is used as a micro-framework and as a lens through which to interpret and analyse specific learning episodes as they take place in the classroom. This frame for the analysis of learning episodes helps in the interpretation of the process of learning in a CAS environment as well as answering questions about formalisation of mathematics in a CAS environment and questions of students’
conceptions of mathematical objects and mathematics as a subject. The two frameworks together provide a vehicle for understanding learners’ mathematical activity, reasoning and development in a CAS environment across a period of time.
Design of the Study and Methods
The research is a case study of three students in a Calculus & Mathematica (C&M) class as individuals and in a group. This number of students was chosen because it is the natural group size in the class. Since much of the work in the class is group work so it is appropriate to study an entire group although the analysis will consider the individuals within the group as well as the group as a whole. The primary data is audio tape and video capture of computer screens of the group’s discussions and collaborations in the C&M classroom. To provide layers of context for the case study and for the purposes of triangulation, the video and audio data is supplemented by interviews with the students and their written work for the class. Also, a grounded survey was administered to all students in the C&M class. An approach grounded in the traditions of qualitative research provides the best opportunity of describing, analysing, and understanding the specific impact of learning in a CAS environment on student conceptual development. As noted above, analyses which are largely quantitative in nature, focussing on increase in student achievement have been done in similar contexts in the past (Judson, 1990; Mayes, 1995;
Palmiter, 1991) and do not provide sufficient insight into students’ specific conceptual development or the particular impact of CAS on the learning process. This research focuses, not on the outcomes of student learning in a final exam, but rather on precisely these processes of learning and student conceptual development.
Analysis and interpretation of the audio and video data, through the Rotman Model of Mathematical Reasoning (1993) and Pirie-Kieren Model of Mathematical Understanding (1994), is the primary source of answers the research question. The Rotman Model is used as a macro- framework for understanding the place of technology in the learning of mathematics. The Pirie- Kieren Model is used as a micro-framework in which to make a detailed analysis of the work and
discussions of the students as they take place in the classroom. The two frameworks together provide a vehicle for understanding learners’ mathematical activity, reasoning and development in a CAS environment across a period of time.
Data Presentation and Analysis
This section consists of a detailed analysis of a learning episode using the Pirie-Kieren model and some more general conclusions framed by both the Rotman and the Pirie-Kieren models.
An Example
Find the highest point on the graph f[x] = e^(-x^2) (2 + Cos [x] + (Sin[x])/2). Is there a lowest point on the graph?
Perhaps prompted by the instruction to find the highest point on the graph the students begin their analysis by plotting the curve:
A: OK. Is there a lowest point on the graph?
C: I don’t know. Let’s plot it and find out.
They decide, as an afterthought, to attempt a formal analytic solution:
C: Let’s solve for the maxs and mins here as well.
But find nothing satisfactory on either front:
(silence)
A: Did you put the “Solve” function in wrong or is something that’s too complex for it?
Thus the analysis begins on a formal level but the students find that Mathematica is unable to help them find an algebraic solution. They descend to the Image Making level in order to get a picture of what the solution might be. The students try to make a better image by changing the lower and upper limits of the graph from –10 to 10 to –100 to 100.
This doesn’t help in finding a maximum point but rather than making more images the students switch tactics to trying to get Mathematica to find a solution again. This time they try to get a numerical answer from the input NSolve [f’[x] == 0, x]
They give up on the “Solve” function in any of its forms at this point although they will return to a numerical approach at a later stage. For the moment the students return to a visual approach changing the limits of the graph to –1 to 1.
It should be clear to the students now, at least what an approximate solution to the problem is since they now have an excellent representation of the function. However, they fail to recognise this and they then get progressively worse images, with limits of –5 to 5; –25 to 25.
The Image Making approach has again failed but the successful representation of the function inspires a move upwards helped by the Property Noticing by C of the values of the gradient function:
C: Try “f’(0)” and see if that works.
[Output: 1/2]
T: Go to the right and see if it’s still increasing.
C: f’(1). Wow.
T: Put a 1.
[Output –2.38879]
C: Yeah. So it’s decreasing at 1.
T: So it’s between 0 and 1.
C: Well, it’s between 1/2 and 1. You’re right 0 and 1. Closer (as A tries other values).
This exchange represents the mathematical breakthrough of the episode. C has noticed the property of the relationship of the value of the gradient function to the maximum value of the function. The noticing of this property is constrained by, or informed by, the formal problem of finding the maximum value of a function. The students continue using better approximations culminating in a formal numerical solution correct to 15 decimal places. They spend quite a lot of time on finding better approximations and don’t seem to have a good notion of how accurate an acceptable answer is. B speculates on whether the answer they are converging on is the square root of a “nice” decimal like 0.5 without offering any algebraic/analytic suggestion as to why this should be so.
A Pirie-Kieren map of this learning episode is as follows:
We see in this episode that in Week 7 of the 10 week course the students are able to implement graphical, numerical, and algebraic solution strategies but that there is a lack of sophistication in how they use Mathematica. We see this in the creation of less successful graphical representations after they have made an excellent representation, as well in the lack of a good sense of the acceptable level of accuracy of answers and in the speculation on “coincidence” in the decimal representation they achieve.
Further Themes
Framing of Technology
The introduction of Mathematica to the students as well as the logistics of using Mathematica had a considerable influence on the students’ progress and behaviour during the time of the study.
The students were very slow to make a transition to simply regarding Mathematica as a calculating tool/calculator which happened to sit on a computer. This is evidenced by their reliance on calculators or by hand calculations in preference to Mathematica for drawing certain graphs or performing certain symbolic manipulations. The reasons for this seem to be two-fold:
the instructor/facilitator of the class, and the set up of Mathematica in notebooks.
The instructor/facilitator was not very familiar with Mathematica and had not used it in a pedagogical context before. At the beginning of the quarter he did not introduce the students to
the capabilities of Mathematica as a calculating tool either by relating it to calculating experience they had (arithmetical calculations, graphs of functions) or exploring new possibilities such as simplifying expressions or solving equations. A consequence of this deficit was that the students took some time to understand what exactly Mathematica is and how it can be used. The other issue holding the students back was that the content for the course was presented as an, in effect, an interactive electronic text. The set up was that in any given section the students would have some questions with some sample code which the students ran and then composed their own code to answer the questions. This restrictive format meant that students were, generally, reluctant to use Mathematica in the same way as they would use a blank piece of paper any more than they would do calculations by hand on the pages of a textbook. They tended to wait until they had what they considered to be a correct strategy before they would employ Mathematica. This problem lessened somewhat as the quarter passed as can be seen in the episode analysed above which took place in Week 7 of a ten-week quarter.
The overall consequence of the framing of the technology was that the students’ use of the Computer Algebra System was clearly hampered by the way the technology was presented to them at the beginning of the quarter and how it was set up for them to use throughout the quarter.
Experimentation
Waits and Demana (1999) and Kutzler (2000) have argued for the importance in CAS as a tool for reintroducing experimentation into the learning of mathematics. As we can see, for example, in the episode above there is a clear difference between awareness of strategies to use in experimenting to find an answer and sophisticated implementation of those strategies. Students were able to employ analytic/algebraic, graphical, and numerical strategies in attempting to solve the given equation. However, the lack of sophistication shown in the students’ lack of recognition of when they had a good graph is noteworthy as is their “blind” application of the numerical strategy to get a numerical approximation to 15 decimal places without giving any consideration to what might be a reasonable answer. This lack of sophistication is all the more noteworthy since, as noted above, it took place in Week 7 of a ten-week quarter.
During the quarter it was also the case that the students’ willingness to experiment was extremely limited. There were no instances in the ten weeks of the study of students failing to solve a problem and using the power of Mathematica to solve a related but simpler version of the problem to gain some insight into the nature of the problem. This constrained experimentation is in many ways related to the framing the technology discussed above since the students never made the move to considering Mathematica as a calculating tool independent of the set of problems they were asked to solve in the Mathematica notebooks which constituted the content of the course.
Conclusions
This study aims to present close readings of learning episodes to provide, so to say, existence proofs of student learning in a CAS environment. As well as providing examples of this learning the broader conclusions of the study are (i) that the framing and introduction of technology at the beginning of an instruction period impacts crucially on student behaviour and use of technology throughout that period and (ii) that while students will naturally experiment in a CAS environment intervention is probably required fro them to develop sophistication in their experimental behaviour and strategies.
References
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