Using Technology to Develop Mathematical Thinkers

— Marcella McConnell

Marcella McConnell

Marcella McConnell

Below I outline ways that teachers of mathematics (and those who teach the teachers, like me) should approach the uses of technology in the mathematics classroom, with a special emphasis on how technology can enhance mathematical thinking and the development of technologically-astute mathematical thinkers. However, many of the issues that, at first, seem specific to my discipline and my student teachers may well apply to all our disciplines and to teaching at both the university and secondary levels.

I prepare students to teach mathematics in a world of evolving technologies. I want students to have an in-depth understanding of mathematics, both its teaching and learning, but also have an understanding of how to incorporate technology appropriately and effectively. Technology is an essential tool for learning mathematics.

Effective teachers maximize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. The National Council of Teachers of Mathematics (NCTM) Standards and various state standards call for the use of graphing calculators and/or computers in teaching mathematics (Chamblee, Slough, & Wunsch, 2008; NCTM, 2009). Since technology is a tool, a means rather than an end for the mathematics teacher, merely adding technology will not improve mathematical instruction: “simply using technology is not enough” (Campe, 2011, p. 620). Teachers need to know the mathematics behind the technology and how to implement it in a meaningful way.

When working with student teachers, I begin activities by looking at the ‘hows’ of the technology. Before they can implement the technology, they must know how to use it themselves. Most technology is not as straightforward as a mere calculator. Teachers need to go through the activities themselves and experience both mishaps and successes so that they can help and guide their students. Additionally, the student teachers learn how to write directions, so that they do not have to repeat themselves over and over, and so their students can move at their own pace.

Teachers need to consider when to use technology and when not to use it. Careful decision making about technology use is essential, especially when integrating technology into the already established curriculum (Ball & Stacey, 2005). When requiring my student teachers to include technology in their unit plans, I emphasize the difference between improving the lessons with technology versus just adding fluff. Technology should be included when it helps the students discover, visualize, and investigate the mathematics and helps them do mathematics with higher cognitively demanding tasks; it shouldn’t be added merely because it is required. Both Battista and Van Auken Borrow (1998) and Scher (2005) stress that the main purpose for using computer software is to open up discussions with students and to help students to think about the mathematics in greater depth.

Technology gives students the opportunities to see multiple representations and numerical examples. “Anyone who has seen trained teachers use calculators knows that they can be used to teach ‘thinking’” (Martin, 2008, p. 22).  In Using Spreadsheets, Battista and Van Auken Borrow (1998) discuss how the implementation of technology in the classroom might help students’ thinking progress from numerical procedures to algebraic reasoning.  If a main goal in mathematics instruction is to create innovative mathematical thinkers, technology gives many students plenty of numerical data to investigate.This approach can help students develop a deeper understanding rather than only mimicking procedures.

Technology has improved many of my previously traditionally based lessons by adding more connections, understanding, and inquiry. I can remember, when teaching high school, holding up a piece of paper and spinning it around an imaginary axis to simulate determining the volume of a function revolved around the x-axis. With the program Autograph, the revolution comes to life for students. They can see the function, the area under the function, and how revolving the area creates a three-dimensional figure. Now, when students ask about how the Simpson’s Rule works, technology easily shows parabolas graphically and can be connected to solving systems. These technologies provide higher-level connections than would not occur if one merely gave students the formula. Technology, especially the graphing calculator, has made factoring a foundational concept for teaching zeros, continuity, and other higher-level mathematics concepts.

The role of the teacher is different when teaching with technology. When students struggle, teachers must avoid pushing the right buttons for them, but, instead, advance their students’ understanding with prompting questions. When their students are trying to make conjectures, they need to be careful not to give answers away too soon. Teachers might even need to pretend they don’t know so their students will continue to investigate.

IMG_0783

Marcy and her student teachers.

Another important aspect of teaching with technology is teaching the students when to use the technology versus when to use pencil and paper or even mental strategies. I go to numerous school districts to observe my student teachers out in the field. With this experience, I think the two main issues that need to be addressed with technology are how and how much should it be implemented to advanced mathematical instruction. Many of the in-service teachers I have worked with either have students using calculators all the time or none of the time. The message that needs to be communicated is that students should have a balance of experiences with technology and without it.

To make these approaches effective, teachers need to have access to training and professional development to learn the hows of various technologies on a continual basis because technology is changing constantly. Technology is not something that holds students back from learning basic facts or pencil and paper skills. A thoughtful teacher with the right tools and training holds the key to identify and implement technology that will be a tool for higher-level, cognitively demanding thinking and more meaningful learning.

How do the issues of technology in teaching mathematics compare with your experiences in your disciplines? For those of you preparing students to teach in our schools, what might you emphasize as key problems with technology in the classroom? Key possibilities?

References

Ball, L., & Stacey, K. (2005). Teaching strategies for developing judicious technology use. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments: Sixty-seventh yearbook (pp. 3-15). Reston, VA: National Council of Teachers of Mathematics.

Battista, M. T., & Van Auken Borrow, C. (1998). Using spreadsheets to promote algebraic   thinking. Teaching Children Mathematics, 4, 470-478.

Campe, K. D. (2011). Do it right: Strategies for implementing technology. Mathematics      Teacher, 104(8), 620-625.

Chamblee, G., Slough, S., & Wunsch, G. (2008). Measuring high school mathematics          teachers’ concerns about graphing calculators and change: A yearlong study. Journal of Computers in Mathematics and Science Teaching, 27(2), 183-194.

Martin, A. (2008). Ideas in practice: Graphing calculators in beginning algebra. Journal      of Developmental Education, 31(3), 20-37.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making in algebra. Reston, VA: Author.

Scher, D. (2005). Square or not? Assessing constructions in an interactive geometry software environment. In W. Masalski & P. Elliot (Eds.), Technology-supported mathematics learning environments (pp. 113-124). Reston, VA: National Council of Teachers of Mathematics.


Marcella McConnell recently graduated from Kent State University with a PhD in Curriculum & Instruction concentrated in Mathematics Education.  She has been working for Clarion University in the Chemistry, Mathematics & Physics and Education departments for the past five years.  Her scholarly interests at Kent State University and the University of Pittsburgh have involved investigations into low achieving students’ academic achievement and success in mathematics.

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