An Evaluation of the CAS Pilot Project
This research was jointly funded by the Ministry of Education and the New Zealand Qualifications Authority. We particularly thank Geoff Gibbs and Steve Bargh for their interest, encouragement, and for including us in the professional development sessions.
Author(s): Alex Neill and Teresa Maguire, New Zealand Council for Educational Research, Report prepared for the Ministry of Education.
Date Published: 2006
This report is available as a download (please refer to the 'Downloads' inset box). For links to related publications/ information that may be of interest please refer to the 'Where to Find Out More' inset box.
If one word were to be used to sum up the CAS Pilot Project, it would be the word "pedagogy".
Using CAS (Computer Algebraic Systems) is all about promoting a well-founded pedagogy of teaching and learning that can be supported and enhanced with technology. The teachers in the pilot were supported in achieving this by quality professional development, and classroom resources. They were striving to perform assessments of learning in ways that support and value the pedagogy of exploration, discovery, and understanding.
The dominant aspect of this project was changing pedagogy to a more exploratory, discovery-based approach that enhances understanding, rather than a rules and algorithmic approach. An exploratory approach is the intent of the current mathematics curriculum, but has not always been the focus in the classroom. This pedagogy is not dependent upon technology, but good technology use in the pilot was enhancing it.
The teachers in the CAS pilot were all firmly in support of this pedagogical approach. They were either believers before the pilot began, or were won over by their participation in the project. The teachers and students reported changes to a more student-led, interactive, exploratory, collaborative, discussion-based style of teaching and learning. Teachers and students agreed that the focus was constructivist, with an emphasis on understanding rather than having a focus on rules and procedures. The students largely saw these lessons as different from other mathematics lessons.
Teachers were implementing the pedagogy of exploration in a variety of ways including: group work with a focus on peer learning and teaching; students working at their own pace; interactive discussions between the teacher and the class; or short cycles of teacher instruction followed by student work.
Teachers found that the pilot project was time consuming for them. Because this type of pedagogy was less familiar they needed to attend to a number of organisational and planning issues. They needed to spend time familiarising themselves with the CAS, refining resources, and making new resources or assessment instruments.
There was a consensus amongst the teachers that the understanding of students of all ability levels had increased, without a negative impact upon their more traditional manipulative skills. This is very much in line with discussions in the research literature. While the evidence from the pilot was largely subjective, teachers who had performed common assessments in their school did not see the CAS students doing less well in algorithmic questions than non-CAS students. One teacher taught several lessons using the traditional pencil-and-paper approach part way through his algebra unit. He found the students covered this work in a far shorter time than in previous years, when his students had not been exposed to the exploratory style of teaching. The pace of lessons for high ability students may need to be faster.
Students had mixed views on whether their mathematics understanding was higher. Many thought it was. A significant minority believed that their understanding was worse. Often this was because the lessons did not emphasise traditional skills. These students were often the more able ones, who had experienced success in a predominantly algorithmic approach to mathematics. Students generally enjoyed using the CAS and were confident about using it.
Professional development (PD) was characterised by the providers modelling a pedagogical approach that teachers could use in their classrooms—that of using exploration and discovery to enhance understanding. Teachers experienced the PD in much the same way as their students experience mathematics lessons. Much of the guidance about how to use the technology was done on a "just in time" basis, so that teachers would not be swamped with technological details, but could see and explore the mathematical issues involved. A major benefit of the PD sessions was the interaction and sharing between the teachers in the pilot.
The initial training familiarised the teachers with the technology. Most of them were satisfied with this, but some felt the PD could become more focused by using only examples from junior high school mathematics, and having written guides for verbal instructions so teachers could replicate the learning sequences that were demonstrated.
The PD also provided teachers with frameworks for teaching units. These resources were intended to include sound pedagogy, a coherent and linked sequence of teaching, and complete instructions both on the mathematical ideas and instructions needed to operate the CAS in the prescribed manner. Largely the provided resources met these criteria, but part of the ongoing project is to refine the resources to better meet teacher needs and expectations. Teachers often saw a need to refine or adapt the resources to their own classrooms.The PD provided to teachers in the pilot was of a very high standard. PD for the wider mathematics teaching fraternity needs to adopt an excellent, sustainable model. Facilitation of PD is a professional skill that differs from classroom teaching and requires trained facilitators with time to perform this task. Schools also need leadership commitment to the pedagogy. Using the PD expertise and values of the Numeracy Project should be considered.
Assessment, both formative and summative, needs to reflect the values of exploration and understanding that define the pedagogical thrust of the CAS pilot. This poses challenges for traditional assessment practices.
Teachers were performing high levels of formative assessment in all their classrooms, but they did not always recognise this, usually because they were not employing many formal assessments.
Schools were still debating and experimenting with a variety of school-based summative assessments. New forms of summative assessment that reflect the nature of, and the values behind, the teaching in the CAS pilot are needed. Some of these may need to be "CAS-resistant", which means that students who do not have a CAS in the assessment are not disadvantaged. Teachers need models of assessment styles that will be used in the high-stakes NCEA assessments in time to prepare their students for them. The role of CAS in assessments of other subjects needs to be clarified. More time and resources will be needed to develop appropriate assessment material.
The teachers and the PD providers saw the technology as one tool for learning, not the driver of learning. The teachers were largely positive about the technology. They could see that it allowed more authentic contexts to be used, and a more problem-solving approach to be taken. Students generally enjoyed the technology, but most of them encountered specific problems concerning its use. Some of them could not always see the mathematical reasons for what they were doing on the CAS.
Care will need to be taken that the technology is used in an illuminating "white box" way (aiding in the construction of meaning for mathematical concepts), rather than in a "black box" way (press these buttons to get the answer). An appropriate pedagogical approach and assessment consistent with exploratory learning to enhance understanding will be needed to support this.
The price of the technology could be a barrier for many.
For the project to become sustainable, both parents/caregivers and mathematics teachers need to come on board with, and understand the values of, this pedagogical approach. The schools were employing a range of effective strategies to communicate with parents or caregivers, and this will need to continue. Communication within the mathematical education community is also needed.
School leadership needs to be supportive of the values and practice of the CAS pedagogical approach within their school or department. System-wide leadership is needed as well, in policy, communication, and assessment issues.
More classroom resources across all the areas of the curriculum are needed.
Where to find out more
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