• by Paula Johnson, M.A., Veronica Betancourt, M.A., and Hector Bojorquez • IDRA Newsletter • September 2011 •

Paula JohnsonVeronica BetancourtHector Bojorquez

Though children are exposed to and begin development of algebraic models at a very young age, there is no true relevance in the early stages of their education. The idea of a natural mathematics progression in subsequent grade level knowledge is lost on the students themselves. For example, the formulae for rate and slope are relatively intangible concepts. We simply ask young scholars to take our word for it that if an unknown variable continually increases by the same value in subsequent steps in a pattern, it has a constant rate. And if this rate were to be graphed, it would indeed have a slope consistent with that rate, thus creating a linear function… What?

We can find similar scenarios in science as well. For instance, imagine graphing data collected during an experimental investigation that centers on the enzymatic effect on reaction rates of chemical combinations. It is difficult enough to understand the periodic table and how elements combine to form compounds, but now ask students to create a visual representation of data that demonstrates how enzymes affect the rate of reaction when two compounds are mixed together. Again, what?

Technological applications enable students to visualize scientific and mathematical concepts they may not be able to otherwise imagine. As children progress cognitively through their educational career, they are asked to give a variety of relational descriptions that are truly abstract notions. Asking a student to take such great leaps of faith is optimistic at best. In a generation dedicated to technology, it is in our best interest to take advantage of the engagement built into the devices at our disposal.

Options for Dynamic Learning

Technology is evolving and becoming more accessible to us instantaneously. It is certainly true that the availability of resources provided to individual teachers can vary dramatically and so does their level of technology efficacy and implementation. Dynamic learning can start off simply by including interactive websites that demonstrate conceptual targets, such as functions in math or chemical reactions in science. It also can embrace whatever technology is available at that moment, such as graphing calculators.

Mathematical Ownership

It often is questioned whether or not calculator-based ranger (CBR) activities actually improve student graphing aptitude and understanding. This device presents data in real time on calculators which may help students better visualize physical problems and check their intuitions. Kwon (2002) compared the mathematics achievement of students (in grades seven and eight) who used CBRs against those who did not (11th graders with no experience with CBR or graphing calculators). The findings suggest that use of CBR activities enhance students’ graphing abilities. These results speak to the endless possibilities that exist even with what seem to be the most archaic technologies.

The point here is that we must seize the opportunities to embrace the technologies available. Imagine the feeling of accomplishment as you witness learners overcoming their initial preconceptions or misunderstandings about graphs (including their previous interpretations). In a discovery-based environment, students are less apprehensive and can actually enjoy themselves while making these connections. Moreover, an experience with CBR technology leads students and teachers to recognize the value of communication (oral and written) in learning mathematics (Stylianou, et al., 2005) and can set the foundation for further exploration of web-based technological applications as co-learners in the classroom.

Scientific E-Learning

In science, graphing calculators also can be used to find pH, temperature changes, acidity levels, change in rate and dissolved oxygen readings. The possibilities are numerous. However, we must find the courage to venture into the realm of the Internet and explore the possibilities of interactive technologies available for use in math and science.


Paper and pencil tasks offer a limited and narrow opportunity for understanding concepts. A slope is not simply a shape on a graph, it can represent the movement of concrete real-world applications. Yet, on paper it is static. CBRs, online applets and any form of technology that enables students to view in real time the effects of changing variables brings mathematical and scientific concepts to a deeper level of understanding.

What makes more sense to us: that the slope is a shape on cartesian coordinates or that the slope of a line can be representative of steepness of real-world objects (mountains, ladders, buildings) or exponential functions (temperatures, weight)? Technology offers us the opportunity to bring these concepts from abstractions to concrete examples by simulating the effects of numbers.


Kwon, O.N. “The Effect of Calculator-based Ranger Activities on Students’ Graphing Ability,” School Science and Mathematics (February 2002).

National Council of Teachers of Mathematics. Principles and Standards for School Mathematics (Reston, Va.: NCTM, 2000).

Stylianou, D.A., Smith, B., & Kaput, J.J. “Math in Motion: Using CBRs to Enact Functions (calculator-based-rangers),” Journal of Computers in Mathematics and Science Teaching (September 2005).

Texas Instruments. “Move My Way – A CBR Analysis of Rates of Change,” Explorations (Texas Instruments, 2004).

Paula Johnson, M.A., is an education associate in IDRA Field Services. Veronica Betancourt, M.A. is an education associate in IDRA Field Services. Hector Bojorquez is an education associate in IDRA Support Services. Comments and questions may be directed to them via e-mail at feedback@idra.org.

[©2011, IDRA. This article originally appeared in the September 2011 IDRA Newsletter by the Intercultural Development Research Association. Permission to reproduce this article is granted provided the article is reprinted in its entirety and proper credit is given to IDRA and the author.]