SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

This chapter presents conclusions drawn from this project, recommendations for mathematics education, and implications for further study. This research project involved technology intensive instruction to facilitate students' understanding of mathematical concepts (rate and reading, interpreting graphs) and help improve their attitudes toward mathematics. Technology intensive instruction through the use of tools like those used in this research can align conceptual understanding of mathematics with positive attitudes toward mathematics. This study suggests that during the instructional period some low performing and high performing were engaged in learning mathematical ideas related to rate, and reading and interpreting information from graphs.

Low performing and high performing students alike responded favorably to the instructional techniques. Students in both of the classes demonstrated increased knowledge about the concepts explored as suggested by the results of the pretest and posttest. In addition to an increase in knowledge gained from the instructional period, students in each class demonstrated positive attitudes using technology to learn mathematics and more positive attitudes toward their mathematics experiences as suggested by the surveys and classroom observations.

Some students began to identify more with their mathematics as suggested by classroom observations that showed increased ability to engage in experimentation and communication of ideas. The students in each class also demonstrated that they are able to identify with the mathematics and transfer some concepts to the world outside their classroom.

The results of this study are important to analyze the impact of technology-intensive instruction on more classes. Questions need to be asked about the content delivered to the traditionally lower-tracked classes, and the instructional methods with low performing students. Several students who had not participated in class very much prior to the instructional period became very involved in the instruction during the research time with the technology. Several of these students actively described events outside the classroom in relation to concepts explored, requesting more instruction of this sort, or participating in ways the classroom instructor had not seen previously. Results and observations like these imply the need for new methods of instruction to engage newer and more students in mathematics. Steele describes students choosing to disidentify while this project recognized several students who actively chose to identify with the concepts that were explored during the instructional period.

Many students in these classes demonstrated abilities to communicate, reason, connect, and explore problems during the instructional period as suggested by the classroom observations.

The conclusions regarding this study and the experiences of the students support mathematics education reform movements' recommendations for the infusion of technology with mathematics education programs. The technology was not reserved for the students enrolled in the advanced mathematics class but was available for all the students in the study. This arrangement gave low performing and high performing students opportunities to explore complex mathematical ideas that they had not had an opportunity to before this instruction.

__Limitations__

There are several limitations to this research project. This study was administered to small samples of specific students (2 classes of nearly 20 students). More conclusive evidence of the effects of technology on these students could be gained through the involvement of larger classes and more students at different grade levels. This study focused on two middle school classes. However, research on graphing calculators in the mathematics classroom (Hembree & Dessart, 1986) has shown there is an increase in positive attitudes toward mathematics and an increase in self-concept in mathematics with students using calculators across all grades and ability levels.

This research took place over a short time period (10 class periods) of instruction. More time spent using the technology is necessary to gauge how the students use the technology every day in more areas of mathematics. Technology should be infused with the curriculum so that it is used regularly during instruction. This type of arrangement will encourage students to continually consider experimentation and exploration as reasonable approaches to learning mathematics. The novelty of the technology used in this study may have attributed to students' positive attitudes. They had not experienced instruction like this before and they may have responded out of excitement for the novelty. This research was also limited in the amount of topics covered. There are many more topics that can be explored in greater depth with graphing calculators and data collection devices such as those used in this research.

This research focused on the experiences of low performing and high performing middle school mathematics students. However in concluding this study serious questions arise for areas associated with the future of these students and the future for mathematics education. Some future areas of research deal with extended issues within mathematics education. Some areas that can be explored are:

- continuation of positive attitudes toward mathematics in future mathematics classes;
- transfer of rate concepts to other mathematics classes or other fields;
- preservice mathematics teacher education with this technology; and
- continued professional development of current teachers with this technology

__Continuation of Positive Attitudes toward Mathematics__

The students in this research project demonstrated gains in achievement and conceptual development of mathematics topics. They also gained more positive views of mathematics and school mathematics indicating identification with mathematics, at least immediately following the instructional period. It remains to be seen if the students at both levels can retain this positive attitude in current classes or if more work with technology would be beneficial. More research toward these ends would be worth pursuing for the mathematics education community.

__Transfer of Rate Concepts__

Many students transferred concepts from the classroom to the "real-world" they encountered daily. An extension of this transfer would be the study of how the students transfer graphing concepts, and rate concepts to more advanced mathematics classes or even to science classes. Data collection devices of this type are frequently used in science classes yet there is rarely a relationship drawn between the mathematical concepts and the science concepts.

__Preservice Mathematics Teacher Education__

The activities in this research project were piloted with an undergraduate mathematics education course of preservice teachers. The preservice teachers learned the concept of rate in addition to the inclusion of technology in the mathematics classroom. Some of the students had not realized that some of the most fundamental ideas of calculus are explored in activities like these. They remembered or relearned the vertical line test for a function while they explored these experiments and they discovered new ways of working with real-world graphs.

They enjoyed the experience and participated just as enthusiastically in the activities as the middle school students. More research could be designed to see how much of the concepts the students have retained in relation to their teaching of mathematics. How comfortable do preservice mathematics teachers feel exploring mathematics concepts with technology, especially concepts that can now be explored at the middle school that were previously reserved for later grades? How do preservice teacher feel about introducing higher concept topics to classes that have traditionally been designated as low performing students?

__Continual Professional Development__

Another interesting extension to this research occurred when an abbreviated version of this unit was presented at a workshop for teachers currently instructing middle school mathematics. Some of the practicing teachers encountered the same misconceptions that middle school students did. Including but not limited to describing the graph as a map, attempting to create a graph while walking parallel to the sensor and not perpendicular. Some of them also revisited fundamental calculus concepts; a vertical line slope is undefined, while a horizontal line has no slope, and there is a very important difference.

These practicing teachers enjoyed using the equipment and were excited about the possibilities of this equipment and areas where they can include the equipment in their classes. This workshop took place over a four-hour time period. It included instruction with an abbreviated version of the materials intended for the two-week instructional period in the middle school. This is interesting and important as more technology especially calculator and hand-held technology enters mathematics education as a viable tool for exploration. How comfortable and experienced are current mathematics teachers with the technology that is being introduced?

This study and other reports of studies using graphing technology indicate the potential for technology to dramatically influence the way students learn and the way mathematics taught in the mathematics classroom. The potential for all students to explore powerful mathematical ideas is exciting. Recommendations from this study include letting more low performing students experiment with complex mathematical ideas at an early age in dynamic classroom environments instead of continually instructing in a traditional row-and-column format that focuses on drill and procedural skills. More teachers should be given support to learn about this equipment to find interesting and comprehensive methods for exploring mathematics. The mathematics curriculum as it is right now does not serve all students and from this research technology has the potential to introduce some of the students who are currently not being served by the curriculum to powerful mathematical ideas. As such then, schools and teachers should consider recommendations made by groups promoting the infusion of the mathematics curriculum with technology. These groups are the NCTM, PCAST, and several other researchers who have documented the effectiveness of technology-intensive instruction.