Winter 2006

The ICT agenda

Graphing to learn

How can technology help students master the big ideas in mathematics? Jeremy Roschelle and Corinne Singleton review the research behind handheld graphing calculators and discuss new technological capabilities that are further enhancing mathematics education.

Research indicates that students who use calculators score higher on tests, explore more maths concepts, develop flexible strategies for problem-solving and develop deeper appreciation of mathematical meaning. Schools can effectively integrate graphing calculators into maths classrooms because they are relatively inexpensive and their functionality is aligned with curriculums, instructional practices, and assessments. Graphing calculators can provide teachers with a platform for improving their mathematics teaching.

The research base

Researchers in different settings have investigated the effectiveness of graphing calculators in relation to students, teachers, and schools with diverse characteristics. One such study examined the effectiveness of graphing calculators in algebra classrooms in New Zealand (Graham and Thomas, 2000). The study compared pre-test and post-test scores for students in treatment and control group classes in two schools. In all classes, the regular classroom teacher taught the ‘Tapping into Algebra’ curriculum module.

In treatment groups, each of the students used a graphing calculator throughout the module; students in control groups did not use graphing calculators. All students had similar background characteristics and maths abilities. Graham and Thomas found that students in the treatment groups performed significantly better on the post-test examination than students in the control groups.

Effective use in the United States

In the United States, graphing products are integrated with national and state standards and supported in some curriculums. Using graphing products, teachers can enhance their classrooms by:

• increasing the attention to conceptual understanding and problem-solving strategies by off-loading laborious computations
• examining the related meanings of a concept through display of multiple representations, such as exploring rate of change in a graph (i.e. slope) and a table
• engaging students with interactive explorations, real-world data collection and more authentic data sets
• giving students more responsibility for checking their work and justifying their solutions
• introducing topics that were previously too difficult for many students (e.g. modelling).

Research shows that daily use of graphing calculators is generally more effective than infrequent use (Heller, 2005). Similarly, unrestricted use of calculators, including the opportunity to use calculators on tests, appears to be more beneficial than restrictive calculator usage for maths learning.

Why have calculators been so successful?

A number of key features contribute to the success of graphing calculators in bolstering learning in maths. Graphing calculators are relatively simple, robust and cheap; they are also remarkably free of much of the complexity that accompanies full-featured computers. More importantly, there is a deep scientific link between the capabilities of the technology and the way people learn. Students learn best with increased learning time, scaffolding, formative assessment, and opportunities for reflection and revision—qualities that can be achieved with graphing technology.

Two less obvious factors also contribute to the success of graphing calculators. Firstly, the adoption of the technology has been led by teachers who function as the key champions and influencers in a professional community. Secondly, efforts to integrate graphing calculators did not begin with the expectation of a rapidly transformed classroom, but rather provided a context to support a long, steady trajectory of continuous improvement. In this way, teachers can begin with one or two relatively simple applications of the technology and gradually increase the depth and breadth of their calculator integration as they grow more comfortable with the technology. At each stage, graphing calculators can provide concrete enhancements for teaching and learning maths.

Tackling big ideas

Until recently, graphing calculators were used primarily as a general tool. Increasingly, researchers are developing more targeted uses of the technology in order to address particularly important yet difficult ideas. For example, the ‘SimCalc Project’ is investigating new applications for graphing technology that will enable more students to develop a conceptual understanding of key concepts in advanced maths. In particular, SimCalc builds on the strengths of graphing calculators by incorporating new capabilities for dynamic representations and classroom networking into the curriculum.

Networking capabilities and dynamic representations allow teachers to deepen maths content and increase student participation in their classrooms.

A glimpse at an 8th grade Algebra lesson designed by Hegedus and Kaput shows why these investigators are enthusiastic. In class, the teacher asks students to count off and then poses a mathematical challenge that varies according to their count-off number. So, each student is working on an individual challenge. In this case, the challenge is to make a function whose graph starts at your number and goes through the point (6, 12).

Using a calculator, the students each specify mathematical functions. Then, using the classroom network, the teacher rapidly ‘harvests’ every student’s unique solution for display on an overhead panel. The students now see their work on a shared screen, leading to passionate talk about the mathematics they created. In addition, the teacher can guide them to investigate new structure that appears in the aggregated set of lines—e.g. how do their slopes vary? Furthermore, the graphed functions can also control a motion animation on both the student units and the classroom display. Each student’s function thus becomes part of a mathematical model of a race, dance or parade. (See www.simcalc.umassd.edu for more information.)

Teachers can take advantage of networked capabilities by assigning both individual and group tasks, where the individual tasks contribute to the larger group solutions. This set-up allows students to collaborate with peers and to see the interaction between different elements of a larger mathematical concept. As students complete the task, they can submit their solutions electronically to the teacher’s device. The teacher’s device can instantly aggregate responses and display simple graphs of student response patterns, etc. This rapid accumulation of student work allows teachers to assess individual and overall student understanding immediately and precisely.

Teachers can adjust instruction, provide feedback, and require students to revise their work as needed.

Teachers can display student work to strategically direct attention to certain concepts and underlying mathematical structures. Classroom networks facilitate a blend of public anonymity and individual accountability that can reduce academic anxiety, while still encouraging all students to work hard.

By facilitating motion, connectivity and collaboration, networked handheld devices are creating a transformation in maths classrooms. In transformed classrooms, technology is not merely a medium for individual practice with maths content, but rather it is a pervasive medium in which teaching and learning take place. Thus, as more work happens through collaborative interaction, learning increasingly occurs in the social space (Stroup et al, 2002).

Collaborative learning augments the learning that occurs through individual interaction with technology devices. With careful pedagogical guidance by teachers, students can progress through a trajectory of understanding in which their focus advances ‘from static, inert representations, to dynamic personally indexed constructions in the SimCalc environment on their own device, to parametrically defined aggregations of functions, organised and displayed for discussion in the public workspace’ (Hegedus and Kaput, 2004).

Enabling more students to grasp the big ideas of mathematics is vital for preparing youth for success in the 21st century. Research shows that graphing calculators can help students to understand important mathematical concepts. Further, new technological capabilities, including motion and connectivity, promise to transform teaching and learning in mathematics, making critical maths concepts accessible to all students.

References

Graham, A & Thomas, M (2000). ‘Building a Versatile Understanding of Algebraic Variables with a Graphic Calculator’, Educational Studies in Mathematics, 41(3), pp 265–282.

Hegedus, S & Kaput, J (2004). ‘An Introduction to the Profound Potential of Connected Algebra Activities: Issues of Representation, Engagement, and Pedagogy’, Proceedings of the 28th Conference of theInternational Group for the Psychology ofMathematics Education, 3, pp 129–136.

Heller, J et al. (2005). ‘Impact of Handheld Graphing Calculator Use on Student Achievement in Algebra 1’, Heller Research Associates.

Roschelle, J et al. (2004). ‘The Networked Classroom’, Improving Achievement in Mathand Science. 61(5), pp 50–54.

Stroup, W et al. (2002). ‘The Nature and Future of Classroom Connectivity: The dialectics of mathematics in the social space’, paper presented at the Psychology and Mathematics Education North America conference, Athens, Georgia.

Jeremy Roschelle is director of the Center for Technology in Learning at SRI International in California.

Corinne Singleton is research social scientist at the Center for Technology in Learning at SRI International in California.

The author owns the copyright in this article. For information related to the reuse of this work in any form please contact the publisher denise.quinn@curriculum.edu.au