# MET:Graphing Calculators

This page originally authored by Sarah Rowe (2011).

## Definition

A graphing calculator is a hand-held device that combines the calculation abilities of a traditional calculator with other more complex computing functions. Depending on the type of device, graphing calculators are able to provide the user with the ability to graph and solve equations, create spreadsheets, and display dynamic images and text.

## Types of Graphing Calculators

### Standard Graphing Calculators

A standard graphing calculator can perform arithmetic calculations, solve equations, and create graphs based on user input. Graphing calculators are also referred to as programmable calculators, as their enhanced memory allows the user to store information and program the various steps of graphing equations. Manufacturers of graphing calculators provide pre-programmed calculator options, which can include financial, data analysis, and other equation solving applications.

### Computer Algebra Systems

Computer Algebra Systems (CAS) for graphing calculators allow for an expansion of the abilities of a standard graphing calculator. CAS calculators have additional applications beyond simple equation solvers. In addition to calculations and graphs, CAS calculators can create and analyse spreadsheets, create diagrams for geometry or trigonometry, and append diagrams with text and images. Worksheets in CAS systems have the ability to link to other in-calculator applications, and can be saved to the calculator’s memory. Software and expansion kits allow collaborative calculator-computer interfacing, publication to the Internet, and linking together of calculators via a wired or wireless connection.

There are several manufacturers of graphing calculators, including Casio, Hewlett-Packard, and Texas Instruments. Calculators range in price, mainly due to memory size and availability of CAS applications.

#### CAS Applications

- Calculator: enter, save, and evaluate mathematical equations and expressions
- Graph: create and explore graphs of functions based on their equations or other parameters
- 3D Graphs/Conics: create and explore 3-dimensional graphs and graphs of conics
- Geometry: create and solve problems involving geometry and trigonometry
- Lists/Spreadsheets: use lists and spreadsheets to store and work with data
- Data/Statistics: apply statistical analysis to data (includes financial calculator)

## Theoretical Frameworks

Graphing calculators provide students with the ability to represent mathematical thinking in a number of different forms that stray from the traditional “pencil-and-paper” approach to teaching and learning mathematics (Doerr & Zangor, 2000). With the CAS' ability to link students’ calculators with one another and with a central teacher system, graphing calculators support social learning and allow students to construct meaningful learning of mathematics (Doerr & Zangor, 2000; Cobb & Yankel, 1996). The extremely customizable nature of graphing calculators and the abilities of calculator systems to interact with others and the Internet promotes a social constructivist theory of learning. Further, the use of graphing calculators can promote deeper understanding of mathematical concepts rather than mathematical procedures, preventing the common problem of rote memorization (Hollar & Norwood, 1999; Smith & Shotsberger, 1997).

## Learning Affordances

### Exploration

Graphing calculators allow students to represent their thinking in multiple ways, such as linking between forms of a function (table of values, graph, algebraic equation, words) (Brown, 2004). The different data formats, ability to link to the Internet and download questions, diagrams, and graphs afford exploration of mathematics.

A study by Duda (2010) introduced a group of students to a type of function they had not yet learned in a classroom setting and used an open-ended research problem to encourage students to explore the graph of the function and answer questions. Duda found that the majority of students were able to take this challenge and successfully apply and extend previous knowledge, using graphing calculators to support their exploration and confirm their thinking (Duda, 2010).

### Collaboration

Current instances of CAS graphing calculators, such as the TI-Nspire, allow calculator-to-calculator linking, as well as linking to central teacher hubs. Students can collaborate with each other, sharing data and calculations and send their information to the teacher’s central hub. Teacher software allows the teacher to monitor student progress, synthesize class information, and display step-by-step methods for solving problems. This could be used to lead a group investigation, gather information for a problem-based learning assignment, or to demonstrate for students in a synchronous or asynchronous distributed learning environment.

### Personalization

CAS graphing calculators can link to digital data-collection devices, such as Vernier LabPro tools, which allow students to collect real-time lab data. This allows students to engage in interactive, authentic data sets with which they have a personal connection and can be amended with personalized text and images (Roschelle, 2006). New Texas Instruments (TI) graphing calculators afford for personalization of learning by also allowing students to create PublishView documents, which incorporate CAS calculator data, images, and other calculations into a web-compatible document that can be viewed by anyone. Students and teachers can publish these files to a website or blog.

## Classroom Applications

### Student Success

Studies have shown that graphing calculators positively impact students’ understanding of mathematics concepts (Milou, 2010). Despite these findings, additional studies have concluded that there are limitations to the successful integration of graphing calculators in the classroom, mainly due to teacher concerns about students' abilities to use the devices and reluctance to incorporate technology into the curriculum. Texas Instruments funds research of how graphing calculators are used and their degree of effectiveness in improving student success in mathematics and the sciences. They have determined that the use of graphing calculators can yield average learning gains of 14-50%.

### Appropriateness of Course Material

Students’ abilities and course topics lead teachers to be reluctant to apply graphing calculators in their mathematics courses (Milou, 2010). In Algebra I classes, teachers reported that the use of a graphing calculator for day-to-day work would be inappropriate, while teachers of Algebra II and Pre-Calculus found the material more conducive to graphing calculator use (Milou, 2010; Dion, Harvey, Jackson, Klag, Liu, & Wright, 2010).

### Teacher Experiences

Well-integrated use of graphing calculators in the classroom may largely depend on the individual teaching styles of teachers (Doerr & Zangor, 2000). Teachers who favour non-traditional styles of teaching, such as constructivist collaboration and group work often use graphing calculators more frequently in the classroom, and also ensure that classroom activities are structured to best take advantage of graphing calculator technology (Tharp, Fitzsimmons, & Ayers, 1997).

### Use in Distributed Learning

Graphing calculators can be successfully incorporated into distributed or distance learning programs due to their powerful computing capabilities. Providers of graphing calculators offer pre-programmed activities online, allowing students to use the calculator for advanced calculations without the need for face-to-face instructor support (Kuhn, 2003). Graphing calculators’ portability and moderate costs make them effective tools for distributed learning programs (Moore, 1997).

### Use in Standardized Tests

The use of graphing calculators is wide-spread in several national standardized tests in North America (primarily, in the United States of America). Many standardized tests require students to use a graphing calculator, although some simply recommend their use. For instance, the Advanced Placement exam for Statistics requires the use of a graphing calculator, while graphing calculators are recommended for other exam such as those for Physics or Chemistry.

### Graphing Calculator Stop Motion Movie

Created by Raymond Kline https://youtu.be/yfgzpIxaXB8

## Resources

- Casio Calculator Teacher Guides
- Texas Instruments Activities Exchange
- Texas Instruments Instructional Guidebooks
- TI-Nspire Tutorials
- The Case For CAS

## References

- Brown, J. (2004). The affordances of technology for student understanding of functions.
*Proceedings of The Mathematical Association of Victoria*. Retrieved from http://www.mav.vic.edu.au/files/conferences/2004/Brown-formatted.pdf - Cobb, P. & Yankel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research.
*Educational Psychologist, 31*(3/4), 175-190. - Dion, G., Harvey, A., Jackson, C., Klag, P., Liu, J., & Wright, C. (2010). A survey of calculator usage in high school.
*School Science and Mathematics, 101*(8), 427-438. - Doerr, H. M. & Zangor, R. (2000). Creating meaning for and with the graphing calculator.
*Educational Studies in Mathematics, 41*(2), 143-163. - Duda, J. (2010). Mathematical creativity and the graphic calculator.
*International Journal for Technology in Mathematics Education, 8*(1), 3-14. - Hollar, J. C. & Norwood, K. (1999). The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function.
*Journal for Research in Mathematics Education, 30*(2), 220-226. - Kuhn, J.R. (2003). Graphing calculator programs for instructional data diagnostics and statistical inference.
*Journal of Statistics Education,11*(2). Retrieved from http://www.amstat.org/publications/jse/v11n2/kuhn.html - Milou, E. (2010). The graphing calculator: a survey of classroom usage.
*School Science and Mathematics, 99*(3), 133-140. - Moore, D.S. (1997). New pedagogy and new content: the case of statistics.
*International Statistical Review, 65*, 123-165. - Roschelle, J. (2006). Effective integration of dynamic representations and collaboration to enhance mathematics and science learning.
*Proceedings of The Curriculum Corporation 13th National Conference*. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.7560&rep=rep1&type=pdf - Smith, K.B. & Shotsberger, P.G. (1997). Assessing the use of graphing calculators in college algebra: Reflecting on dimensions of teaching and learning.
*School Science and Mathematics, 97*(7), 368-376. - Tharp, M.L., Fitzsimmons, J.A., & Ayers, R. L. (1997). Negotiating a technological shift: Teacher perception of the implementation of graphing calculators.
*Journal of Computers in Mathematics and Science Teaching, 16*(4), 551-575.