In the last years, several studies have investigated the role of technology in teaching and learning mathematics. However, the specific role of computer algebra systems (CAS) in early algebra in contrast to graphic calculators (GC) is still unclear. The CAYEN project is researching this field by comparing 13-year-old pupils—one GC class and two CAS classes have been observed while acquiring elementary algebraic competences with nearly the same teaching sequence. The field of algebraic competences is split into syntactic abilities and symbol sense. The results of this explorative case study show that the development of symbol sense is influenced by the adoption of CAS in the learning process. Especially when transitioning from arithmetic to algebra, the pupils’ views of algebra as well as their conceptions of algebraic objects seem to be affected by the availability of CAS.

To improve mathematics in low-performing schools, educators should address a broad range of factors systemically, including an intensification strategy, coherent curriculum, effective pedagogy, deeper teacher mathematics knowledge, positive social factors and supportive organizational structures.

The Phase 2 ALN research yielded the following key findings:
• TI-Nspire Navigator is an important component of the ALN resource suite that complements other TI resources (i.e. ALN and TI-Nspire)
• The research found a marginally significant increase in learning associated with TI-Nspire Navigator, above and beyond TI-Nspire and ALN. (This result may be an underestimate of the full impact, due to implementation variation and other factors inherent to the research.)
• The ALN resource suite influenced teacher pedagogy to emphasize deeper learning for students
• Teachers believe that the ALN resource suite contributed to increased student engagement and supported deeper student learning
• In comparison with their less experienced peers, teachers with more experience with TI-Nspire and TI-Nspire Navigator:
▫ Were more likely to shift their pedagogy to include more high-level instructional activities
▫ Reported that their students were more engaged and learned more
▫ Had students who learned more math
• Teachers who used the technology more frequently also reported that their students were more engaged and learned more math

In this systemic improvement program, district-wide proficiency on the standardized state test climbed steadily during three years from 39% to 62% in both 7th and 8th grades.

Summary Slides for Case Study #15.  This is a preliminary report on this study.

a 5% increase in learning occurred directly due to the addition of technology in the classroom….The class that used technology was able to have a maximum score of 100% on the assessment, whereas the non-technology group only scored a maximum of 95%

TI-Nspire technology made possible a whole new way of teaching with major improvements in performance on the new NY State Regents exam.

The class was mostly paperless, and had no textbook.  Everything was distributed electronically via the TI-Nspire Handhelds and class web site.

“Calculators belong in the math classroom…The students are excited about them and enjoy any activities presented that allow them not only to use the calculators but to learn more about the possibilities surrounding them

Dr. Lapp gives an example of how he posed a problem to his students and they used multiple representations to solve the problem, building their own deeper understanding of the behavior of functions.

Using TI-Nspire, at midyear Ms. Hoyt had 2 to 4 times as many students in the “Basic” or “Proficient” level, compared to other teachers’ classes not using TI-Nspire

Results on Learning of Algebra from the CAYEN project
1. In lessons using the black-box-approach (in which the equation is unknown), CAS pupils who were taught with an emphasis on math principles mastered new challenges in algebra well and were able to work independently. 
2. Use of CAS makes it possible to get an overview of a topic at the beginning, simply by trying out new commands. Thus, with CAS it is easily possible to learn many aspects of a mathematical topic in parallel.
3. In our analysis of pupils’ written comparisons of graphic, numeric and symbolic representations, in the CAS-group we noticed many positive comments about advantages of algebra. An effect was that they were more motivated to use algebra and inserted it more often in open tasks.
4. CAS pupils master the transition from arithmetic to algebra more easily.
CAS students accept the output of the calculator as a common means of expression and realize the relevance of algebra. Furthermore, early in the curriculum they perceive the versatility of algebraic work in contrast to arithmetic approaches. By using CAS the pupils learned many commands and algebraic transformations; it did not matter that they could not do them all in a technology-free way. By contrast, GC-pupils sometimes had difficulties in accepting that the same underlying rules are valid in algebra and arithmetic. They argued that their calculators should be able to handle expressions with variables, if the same rules would be valid.
5. CAS-pupils’ argumentation concerning algebra included more mathematical arguments and was more objective than the argumentation of the GC-pupils.
6. We observed that the thoughts of pupils using CAS were on a high algebraic level and included reference to many concepts.

For the most part, the experiment could not discern an impact as a result of providing the equipment and training for TI-Navigator. As shown in the figure below, we found a modest effect for Geometry achievement using the NWEA End of Course Geometry test. This figure shows the outcome measure in standardized units. However, this impact was not reflected in CST Geometry scores. In Algebra, while we found no overall difference, there was some evidence of a small negative impact for students scoring “below basic” on the CST and, holding pretest score constant, for English proficient students. The results of the NWEA End of Course Algebra I test did not reflect those same results. Implementation must be considered in interpreting these findings. Our surveys and observations make clear that this implementation was not a fair test of the difference TI-Navigator might make if used more extensively. Of the 19 teachers originally assigned to the treatment group, about half did not use the system at all for instruction. Of the remaining nine teachers, only three could be considered “Comprehensive-Implementers.” Of those three, only one used TI-Navigator daily. Technical glitches deterred many from using the system after previous failed attempts. Overall use may have been constrained by the fact that California prohibits calculator use on state tests. Impact on NWEA Geometry Achievement Our results also must be qualified by the fact that, while finding differences on one test, we did not find differences on the other test. The  significant amount of attrition, both at the teacher and student levels, although not believed to be associated with the program being tested, raises issues about generalizability. For example, it is clear that in both experimental conditions, lower scoring students were significantly more likely to not have posttests, indicating that our findings are not applicable to the lowest scoring students in these districts. Overall, we found that the TI-Navigator affected the average number of minutes the technology was used. The teachers with TI-Navigator reported using the technology about 15 minutes more per week per class period than teachers without. Future exploratory analyses may prove useful in suggesting whether extent of usage can account for student outcomes. In particular, since TINavigator resulted in greater technology use, examining the correlation between technology use and achievement may suggest a mechanism by which TI- Navigator could be effective. Future studies of TI-Navigator will benefit from greater support for implementation. We also recommend continuing to include Geometry in the topics to which TI-Navigator is applied, since the positive result found in this experiment should be replicated.

The purpose of this action research study was to determine whether a more frequent integration of the TI-NspireTM into the pedagogy for my Advanced Algebra class would enhance the students’ achievement and increase their comfort with, usage of, and knowledge of graphing calculators in general and the TI-NspireTM in particular. I also wanted to determine students’ perceptions about the use of TI-NspireTM graphing calculators and technology in the teaching of Advanced Algebra. Pre- and post-surveys of the students were used to measure the students’ comfort with, usage of, and knowledge of graphing calculators in general and specifically the TI-NspireTM graphing calculator. Achievement was measured by students’ ability to reach scores of 89% on quizzes and tests. Students’ perceptions about the use of TI-NspireTM graphing calculators and technology in the teaching of Advanced Algebra were measured using a questionnaire at the end of the study. End of project interviews with participants and teacher journaling were also part of the study. Integration of TI-NspireTM graphing calculators into my pedagogy did have a positive effect on enhancing my students’ achievement on exams while also increasing their comfort with, usage of, and knowledge of graphing calculators in general and specifically the TI-NspireTM graphing calculator. I was also able to determine that my students had positive perceptions regarding the TI-NspireTM and the use of technology in the teaching of Advanced Algebra.

This study investigated the relationship between instructional use of handheld graphing calculators and student achievement in Algebra 1. Three end-of-course test forms were administered (without calculators) using matrix sampling to 458 high-school students in two suburban school districts in Oregon and Kansas. Questions on two forms were drawn from Texas and Massachusetts publicly released standardized test items, and the third form was custom-designed to emphasize conceptual understanding and math applications. All classes used Key Curriculum Press’s Discovering Algebra textbook. Results showed that the more access students had to graphing calculators, and the more instructional time in which graphing calculators were used, the higher the test scores. In addition, scores were significantly higher where teachers reported receiving professional development on how to use a graphing calculator in math instruction.

 TI-Nspire technology, a new generation of graphing calculators, was integrated into high school integrated algebra curriculum. Four teacher-participants were supported through a yearlong professional development emphasizing the use of technology through an inquiry based approach. The data included the teachers’ perceptions about TI-Nspire technology, teachers’ proficiency with TI-Nspire technology, quality of instruction determined through classroom observations, and the frequency of technology use in the classroom based on questionnaire completed by the students. Data analysis indicates that there is a significant positive correlation between quality of instructional practice, quality of use of technology, and teachers’ level of TPACK. In general, teachers with better perceptions used technology in the classroom more frequently, were more proficient with the technology, had higher quality of instructions, and higher level of TPACK. Implications on years of experience and pre-service training are discussed

The data represent a reduced achievement gap between Integrated Algebra and College Prep Algebra. In Year 1, the gap declined from a 14.02% difference to a 8.18% difference in average number of proficient students. In Year 2, the gap declined from a 14.91% difference to 7.54% difference in the average number of proficient students.

Classroom Connectivity Technology (CCT) can serve as a tool for creating contexts in which students engage in mathematical thinking leading to understanding. We theorize four principles of effective mathematics instruction incorporating CCT based on examination of teachers‟ use of CCT within their Algebra I classrooms across four years. Effective implementation of CCT is dependent upon (1) the creation and implementation of mathematical tasks that support examination of patterns leading to generalizations and conceptual development; (2) classroom interactions that focus mathematical thinking within students and the collective class; (3) formative assessment leading to teachers‟ and students‟ increased knowledge of students‟ present understandings; and (4) sustained engagement in mathematical thinking. Each of these principles is discussed in term of its implications for teacher knowledge.

This paper reports preliminary analyses comparing results on the state-administered 8th Grade and 9th Grade algebra Texas Assessment of Knowledge and Skills (TAKS) for a treatment and a control group. The treatment group consisted of 127 students from algebra classes at a highly diverse school in central Texas taught by two relatively new teachers using a network-supported function-based algebra (NFBA) approach as integrated with the ongoing use of an existing school-wide algebra curriculum. The control group was comprised of 99 students taught by two more-senior teachers in the same school using only the school-wide algebra curriculum. The intervention consisted of implementing 20-25 class days worth of NFBA materials over an eleven-week period in the spring of 2005. Because the students were not randomly assigned to the classes, the study is a quasi-experimental design. Using a two sample paired t-Test for means, statistically significant results for the treatment group (p-value one tail = 0.000335  > alpha = 0.05) were obtained.  We can conclude the NFBA intervention was effective in improving outcomes related to learning the algebra objectives assessed on the 9th Grade TAKS.

There are many arguments for and against the use of CAS.  This book sets out to provide an argumentative case for the use of CAS.  The first section outlines of the key issues surrounding the use of CAS and the related research, which supports these findings.  The next section details the ideas of teachers from various countries who already use CAS in teaching, learning and assessment.  It sets out their experiences in terms of lesson ideas, teaching sessions, and the adaptations which have to be made to question types and examination papers, to assess suitably.  The final section of the book outlines the arguments for and against the use of CAS ("the Advocatus Diaboli"), providing what the authors believe is a convincing case for the use of CAS in the teaching and learning of Mathematics.

This paper details the motivation, background, and analysis for studying the effect that an Increasing Achievement on Algebra Assessment (IAAA) workshop for a group of Florida high school teachers had on student pe/jormance in statewide testing. The main focus ofthe workshop was to provide participating teachers with both instruction and activities related to problem-solving techniques using the TI-83 Plus handheld graphing calculator that the teachers could then use in an effort to better prepare their students for state-wide testing. Overall, students (n = 328) who were taught by IAAA-trained teachers between consecutive annual statewide assessments had a greater gain in test scores than students (n = 202) who were taught by non-trained teachers. In particular, the data indicates a significant increase in scores for students taught by IAAA-trained teachers in a general mathematics course, as compared to no significant increase in scores for students taking that same course taught by non-trained teachers.

New Jersey’s urban students traditionally don’t do well on the high stakes NJ High School Proficiency Assessment. Most current remedial mathematics curricula provide students with a plethora of problems like those traditionally found on the state test. This approach is not working. Finding better ways to teach our urban students may help close this achievement gap. This study examined whether a problem/project-based data analysis unit incorporating the document features of the TI-Nspire would help students master data analysis concepts. The study used a quasi-experimental pre/Post-test design enhanced by a qualitative component. A four-week problem/project based data analysis unit served as the curriculum for the intervention treatment. Students were assigned either the TI-84 or the TI-Nspire calculator. Twelve sections of ninth grade students were divided into four basic study groups: (Intervention (TI-84), Traditional (TI-84), Intervention (TI-Nspire), and Traditional (TI-Nspire)).
The quantitative component of the study analyzed differences between students’ pre/post- Total, Multiple-choice, Open-ended mean scores and quantified attitudinal responses. The analysis showed students in the TI-Nspire groups improved more on the Total test and Multiple-choice questions while students in the TI-84 group performed better on Open-ended questions. The Intervention Curriculum was more effective for Multiple-choice questions, Traditional Curriculum for Open-ended questions and Total scores. Student interviews revealed they didn’t like taking notes and answering questions on the TI-Nspire. Some students liked referring to the information in the calculator while others felt that accessing information was too time consuming. The merits of the TI-Nspire document feature needs further exploration. Analysis of the quantified attitudinal survey showed an increase in the positive attitudes of students using the TI-Nspire.
Both qualitative and quantitative evidence showed the Traditional TI-84 group had fewer changes in attitude and content knowledge than everyone else combined, suggesting the need to change how we teach data analysis. Problem/project-based learning, if introduced gradually, may prove to be an effective teaching/learning educational practice.
Further exploration needs to match students’ technological and data analysis proficiencies when determining readiness for student-centered learning that expects students to be calculator proficient and comfortable with basic quantitative procedures such as finding measures of central tendency and variation.

A qualitative study in one algebra geometry III classroom of students using TI-Nspire CAS showed TI-Nspire CAS  had a positive effect on students’ understanding of solving equations, using parentheses, and understanding equivalent operations.

This exploratory study focused on what students learned in algebra, how it was different for students with differential access to graphing calculators, the use of the technology on tasks of different cognitive demand, and whether the teachers' background and experience with graphing calculators might be related to student outcomes. The study considered two conditions: high quality professional development and high frequency calculator use on the part of the students and involved three different populations: 1) teachers who seldom or never used graphing calculators in their classrooms; 2) teachers who used graphing calculators in their classrooms but without a high degree of support and ongoing professional development; and 3) teachers with a high degree of support and ongoing professional development in the use of graphing calculators for instruction. Statistically significant results (p,<004 or better) indicate that access to and use of graphing calculators seems to increase achievement, achievement decreases for both users and nonusers of calculators as the cognitive demand of the tasks increases, and while the background and experience of the teachers seems to make a difference for the top 75 percent of the students, some students perform at very low levels with or without the technology.

The purpose of this study was to determine the effect of TI-Nspire graphing calculator use on student achievement and on teacher behavior variables of planning, teaching, and assessing. This study investigated the teaching of functions by teachers using the TI-Nspire graphing calculator versus teachers using a non-graphing scientific calculator.
A review of the literature found that the emergence of calculators and computers has changed the way mathematics is both done and used (Ellington, 2006; Thorpe, 1989; & Kieran, 1992). Research also showed that students can effectively use a graphing calculator as an instructional tool to make and understand different types of representations (Choi-Koh, 2003; Colgan, 1993; and Drijvers & Doorman, 1996). Other studies have shown how graphing calculator use has engaged students in higher level thinking skills (Dessart, DeRidder, Charleen, & Ellington, 1999; Ellington, 2006; Graham & Thomas, 1998; Keller & Hirsch, 1998; Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; & Ronau et al., 2008). Since it is a relatively new tool, there is a limited amount of research on the classroom use of the TI-Nspire. The TI-Nspire is designed to link together multiple-representations within a single problem, so the concept of functions is an ideal context within which to study the impact of the TI-Nspire.
This was a quasi-experimental study. The researcher gathered and analyzed pre-test, post-test, and post post-test data on student performance on function concepts. The study included a 90 minute classroom observation of each class as well as document analysis of weekly questionnaires, daily lesson plans, and daily assessments. Vignettes employed classroom observations, document analysis, and thick description to triangulate the results of the qualitative analysis.
During the summer prior to this study, all teachers attended 12 hours of training over the course of two days with a National Texas Instruments Instructor in which they were trained to use the TI-Nspire graphing calculator. Teachers were then given a TI-Nspire, TI-Nspire emulator and access to online Atomic learning video training (Atomic Learning, 2011), to continue their exploration of the TI-Nspire. The week prior to the study, the teachers attended another day of professional development activity taught by a Texas Instruments Trained Cadre member. This “Function Focused Session” was six hours long and provided review on the TI-Nspire, specific training about teaching the function concept with the TI-Nspire, and time to create lesson plans and activities for this study. During the two weeks of treatment and two weeks of follow up, teachers met once a week for “Weekly Touchdown Sessions,” a 90 minute meeting held after school to complete a weekly questionnaire, turn in lesson plans, assessments, and receive further professional development on the TI-Nspire. Providing a trained Texas Instruments Instructor on a weekly basis to answer questions, assist in providing direction for the following week, and meeting weekly with the teachers to complete questionnaires were vital strategies necessary to support teachers with this new technology tool and to assure their fidelity in treatment implementation and control maintenance. All professional development sessions were taught by Texas Instruments trained Instructors.
The results from four teachers, each with one treatment class using the TI-Nspire and one control class using a non-graphing scientific calculator, were significant on the pre-test with the control group having a higher mean score than the treatment group and statistical significance on the post post-test with the treatment group having a higher mean score than the control group. While there was a statistically significant effect of Teacher Zeta on the post-post test in comparisons with the other teachers, most of the teacher effect was controlled for within the design of the study. To control for teacher effect, all teachers taught both a treatment and a control class. For each teacher, one of their two algebra classes was randomly assigned to treatment and the other was then assigned to control. There was not enough power in the data to properly analyze the effect of socioeconomic status and special education.
This study supports the use of TI-Nspire graphing calculators in Algebra classrooms while studying the concept of functions. This study shows that, while using the TI-Nspire graphing calculator, the use of multiple representations and higher Depth of Knowledge activities can be used to improve student achievement, and impact classroom teaching, and lesson planning. While this study shows the impact of the TI-Nspire graphing calculator for the concept of functions, further research is needed to continue evaluating the impact of the TI-Nspire across additional mathematics topics.

A UK review of 14 studies of multiple representations found that gains in understanding and visualization can result from multiple representations, when:
• Software is well designed
• Teachers are skilled
• Students know how to use the device critically, including use of visualization to solve problems in graphing. 
Lower attaining students preferred working with tables, and had trouble moving between symbolic, tabular and graphical forms, depending on the nature of the task assigned.

Students showed improvement in the areas of: conceptual understanding, classroom interactions, quantity and quality of responses, time on task and time to start tasks

 The Math Nspired series of curriculum supplements currently include Algebra Nspired and Geometry Nspired. The supplements grew from research on the “tough to
teach/tough to learn” topics, which our item analysis of state tests showed to be common points of difficulty for many students. To determine the underlying reasons for
the difficulty, we consulted the research on the reasons why students struggle with key concepts in Algebra and Geometry.