Volume 6, No. 1, Winter 2002

Handhelds Track Student Progress

Instant Feedback Through Beaming Identifies Student Misconceptions

by Carolyn Staudt

When the Apple Newton was released in the early '90s we recognized that small portable computers had the power to change education.1 We chose the Newton, a portable computer a student could hold in one hand, in 1996 for our Science Learning in Context project in order to test the impact on student learning of portability and simple, easy access to a computer.

While the first handheld computers lacked educational support, the new generation of handhelds now emerging are smaller, more powerful, and have educational applications in all subject areas.2 Their potential to improve education is so significant, we call them "equity computers": low-cost computers that can open the door for all students, regardless of circumstances, to quality education.

One of the most exciting advances in handheld computers is the incorporation of wireless communications. Using an infrared beam built into each handheld, students can share their work through wireless communication with each other and the teacher. "Beaming" has opened up new opportunities in the classroom for collaboration and student assessment.

Figure 1: Students can write equations and test the results immediately, even displaying two equations at the same time.

Figure 2: Definitions for slope and y-intercept were entered in their handhelds using the Notes function.

Figure 3: Saved notes and graphs were shared with other students using the infrared "beaming" capabilities of the handheld.

Figure 4: Students took longer to reporduce a line with a zero slope, but seeing the result of their trial and error efforts helped.

Figure 5: By reproducing a y-intercept at -3 for both a positive and a negative line, the students demonstrated that they understood the concepts of slope and y-intercept.

Figure 6: The ImagiWorks Sonar Ranger attaches to a Palm handheld computer. It allows a student to experiment with the concept that motion can be measured and represented on a graph.

How It Works

In order to test the utility of personal computers and beaming, we distributed handhelds to a middle school mathematics class working on linear equations. Each student was given a Palm computer with Imagigraph equation graphing software and infrared beaming capabilities. Having a personal computer allowed each student to independently test theories, collaborate and share work with other students, and document their process. At the end of class students could send their results to the teacher's computer using a serial port in the handheld.

The class objective was to understand linear equations through explorations of positive and negative slopes and y-intercepts. The handhelds allowed students to write equations that were translated into visual graphs on the screen.

Students were given the following equation:

y = mx + b

The slope is "m" and the y-intercept is "b." On the blackboard the teacher drew an intersecting x-axis and y-axis and a line with a positive slope and asked the students to write an equation that reproduced the line on their handheld computers. Using the graphing software and working in groups of four, students were able to formulate the correct linear equation, which was shown as a positive sloping line on their screens. They even went a step further and displayed two equations simultaneously (see Figure 1). It is important to note the eagerness with which they worked on the problem.

Promoting Individual Student Thinking

While this type of investigation is nothing new in classrooms that use graphing calculators, the teacher with the handhelds could now go a step further and test student understanding.

Many handhelds contain a writing/drawing tool for writing notes. The teacher asked the students to explain the terms "slope" and "y-intercept" and to identify each value on their individual graphs by writing in the Notes section of their handhelds (see Figure 2). (The Palm has a digital alpha-numeric keypad and handwriting recognition software that allows students to enter information with a stylus.) This information was then saved individually in a way that allowed identification of the author. The descriptions and graphs were then beamed to students in other groups who were able to test and manipulate the results, learning from each other's work (see Figure 3).

The teacher then asked each group to come up with a common definition for slope and y-intercept. These group definitions were then beamed to the teacher who shared them with the entire class. The ensuing discussion unveiled new terms such as "coefficient" and "vertical axis" based on the students' own operational definitions, but these terms were introduced only when needed to help students understand their graphs.3

The importance of immediate feedback to the teacher was enormous. It quickly revealed misperceptions that could be corrected and also identified concepts that the students understood.

Guiding Student Interpretation and Reasoning

To further test student collaborative definitions of mathematical terms, the teacher drew another x- and y-axis on the board, but this time a line with a negative slope. Students were again asked to formulate an equation on their handheld computers that would reproduce the teacher's line.

Students were more perplexed this time and tried formulas using lesser or greater positive slopes and y-intercepts. Many students placed negative signs randomly in their equation. Through trial and error, and being able to immediately view the results of their equations, at least one member within each group managed to create an equation that displayed the teacher's line.

As students beamed their results back and forth, they realized that a descending line had a negative slope. The teacher asked the students to revise their definitions for slope within Notes and encouraged them to cite examples. For the first time, they understood the significance of sign, both for slope and y-intercept values.

The students were then given yet another challenge. The teacher drew an x- and y-axis on the board, but this time she drew a flat line that intercepted the y-axis at the number 3. The students were again asked to reproduce the line in their Palms by writing a function. After several minutes, one student yelled, "I've got it!" When he beamed his solution, y=3, to his group, they looked puzzled (see Figure 4). His formula did not fit the normal equation format. The teacher encouraged them to think about how slope and y-intercept related to this particular line, and revise their definitions. Remarkably, each group prepared a written explanation that included a definition of zero slope and the conclusion that "any number multiplied by zero is zero."

Testing Student Understanding

At this point the teacher was confident that a workable understanding of slope had been achieved, but could students master the concept of y-intercept? The teacher drew yet another set of axes on the board and two sloping lines, one negative and one positive, that intersected the y-axis at –3.

Initially, several students in each group mistakenly placed the intercept at positive 3, but they quickly realized their error when they saw the line graphed by the software. It only took a short time for every student to reproduce the teacher's graph accurately. Students then spontaneously produced more equations for others in their group, showing that they understood the concept of y-intercept (see Figure 5).

Relating to Real World Phenomena

One more activity was presented, and its outcome demonstrated the unique importance of handhelds to mathematics. Each group was provided with an Imagiworks Sonar Ranger (release date March 2002), a device that looks like a small egg and attaches by a phone cord to the interface box that fits onto the bottom of the Palm. Holding the Palm in one hand and the Sonar Ranger in the other, it generates a graphical display of motion (see Figure 6). The Sonar Ranger transmits a sound pulse and receives its reflective beam, much like a radar gun. While experimenting with the Sonar Ranger by walking up to and away from a wall, students watched a graphical representation of their movements displayed on the handheld.

After these preliminary investigations, the teacher drew a set of axes on the board, but this time labeled the y-axis "position" and the x-axis "time." She then drew a positive sloping line in the first quadrant. Students were asked to reproduce the line by walking in front of the Sonar Ranger. During this investigation, they described their movements in Notes and tested their ability to mathematically describe their physical movements using the graphing application. By relating the motion of their own bodies with a graphical representation, they were meeting one of the National Science Standards that is typically hard to achieve: the concept that motion can be measured and represented on a graph.

Teachers Monitor Individual Student Understanding

How do you assess student understanding when using handhelds? We are addressing this concern by embedding assessment capabilities within learning materials that reside on the handheld computer. Our assessment tool allows students to not only select answers but provide justifications for their choices.4 These justifications can be beamed between group members or directly to the teacher, who uses them to decide how to proceed with the lesson. The reasons students select their answers are often as meaningful to the teacher as the answers. Our research has shown that this type of embedded assessment can also prompt students to discuss differences among themselves, which serves to further test their beliefs and theories.5

Handhelds with beaming capabilities enable the teacher to make a rapid assessment of each student's comprehension of the concepts. It allows the teacher to build on the current state of student understanding in order to provide on-going challenges that enhance each student's conceptual model. And it frees the teacher to analyze student weaknesses and strengths, misconceptions, and process skills -- all during the lesson.

Carolyn Staudt (carolyn@concord.org) is a curriculum and professional development specialist for the Concord Consortium and vice president of KidSolve.

Notes

Tinker, R. and Krajcik, J., eds. (2001). Portable Technologies: Science learning in context. New York: Kluwer Academic/Plenum Publishers
See the Concord Consortium's Palm Applications in Education database reviewed by teachers.
Alerting the teacher to commonly held student ideas is one of the criteria in the AAAS Project 2061 "Middle Grades Science Textbooks Evaluation: Criteria for Evaluating the Quality of Instructional Support."
For more information, see our Technology Enhanced Elementary and Middle School Science (TEEMSS) Project.
For more information, see our Data and Models project.

The projects described in this newsletter are supported by grants from the National Science Foundation, the U.S. Department of Education, the Noyce Foundation and others. All opinions, findings, and recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agencies. Mention of trade names, commercial products or organizations does not imply endorsement.

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