Technology Research Papers

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Watson, B. and Graham, E. (2009), Rural Schools Mathematics Project, Mathematics in School, Vol.38, No. 3, 17-20.

This article describes an ongoing initiative which uses computer desktop conferencing to link remote rural schools to the Centre for Teaching Mathematics (CTM) at the University of Plymouth, with the aim of enhancing and extending the mathematical experience of able pupils. Funded by the South West Gate, a pilot project was initiated in Devon in 2005. Since then, rural primary schools in Cornwall, Somerset and Wiltshire have also welcomed participation in projects. Typically, between 5 and 8 schools at a time are involved, each school putting forward two pupils from year 5 or year 6.

Graham, E., Headlam, C., Sharp, J. and Watson, B. (2008), An investigation into whether student use of graphics Calculators Matches their Teacher’s Expectations, International Journal of Mathematical Education in Science and Technology, Vol. 39, 179-196.

This research examines students’ use of graphics calculators and investigates the extent to which the students’ use meets their teachers aim when using graphics calculators in the classroom. The teacher’s use of her graphics calculator was analysed over a week using Key Record software. The teacher was questioned about her aims and expectations for the students when using a graphics calculator. As a result an interview schedule for students was constructed in order to determine whether the teacher’s aims had been met. It was found that in general all of the teacher’s aims were met to some extent by most of the students.

Berry J, Lapp D and Nyman M (2008) Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra Journal of Teaching Mathematics and its Applications, 27 No 2, 102 - 111
This paper discusses student difficulties in grasping concepts from Linear Algebra. Using an example from an interview with a student, we propose changes that might positively impact student understanding of concepts within a problem-solving context. In particular, we illustrate barriers to student understanding and suggest technological interventions to address these barriers.

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Berry, J., Graham, T., Honey, S. and Headlam, C. (2007) A Case Study of the Issues Arising When Teachers Adopt the Use of a New Form of Technology in their Teaching for the First Time.
International Journal for Technology in Mathematics Education 14 (3) 150–160

Introducing any new initiative into teaching involves professional development and training. This paper investigates the reactions of three teachers to the introduction of graphics calculators into their department. Each teacher was followed through one academic year. They were interviewed formally on two occasions and also met informally with the researchers to discuss how they were using the calculators. The interviews were used to develop profiles of the three teachers. From these profiles a set of recommendations was developed that could guide other schools introducing graphics calculators for the first time. These recommendations were (i) that the department should have an action plan which describes where and why the calculators are to be used (ii) both initial and on-going training is necessary (iii) appropriate support in the form of both teaching resources and hardware should be readily available.

Graham, E., Headlam, C., Sharp, J. and Watson, B. (2007) An investigation into whether student use of graphics calculators matches their teacher's expectations International Journal of Mathematical Education in Science and Technology, 39 (2) 179 – 196

This research examines students’ use of graphics calculators and investigates the extent to which the students’ use meets their teachers aim when using graphics calculators in the classroom. The teacher’s use of her graphics calculator was analysed over a week using Key Record software. The teacher was questioned about her aims and expectations for the students when using a graphics calculator. As a result an interview schedule for students was constructed in order to determine whether the teacher’s aims had been met. It was found that in general all of the teachers’ aims were met to some extent by most of the students.

Berry, J., Graham, E. and Smith, A. (2006), Observing Student Working Styles when Using Graphics Calculators to Solve Mathematics Problems, International Journal of Mathematical Education in Science and Technology, Vol. 37, 291-308.
Some research studies, many of which used quantitative methods, have suggested that graphics calculators can be used to effectively enhance the learning of mathematics. More recently research studies have started to explore students’ styles of working as they solve problems with technology. This paper describes the use of a software application that records the keystrokes made by students as they use calculators, in order to enable researchers to gain better insights into students’ working styles. The recordings obtained from this software can be replayed to observe how students have actually used their calculator in tackling a problem. The paper describes three pilot studies from quite different contexts, in which the software reveals how the calculators have been used by the students. In all of these studies the software provides insights into the working that would have been very difficult to obtain without the record of the keystrokes provided by the software.

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Berry J., Graham E. and Smith A., 2005, Classifying Student’s Calculator Strategies. The International Journal for Technology in Mathematics Education, 12 No 1, 15-31
When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on a case study that follows the progress of twelve first year University students as they solve twelve mathematical problems associated with the investigation of functions and their graphs. We use software that works in the background of the graphic calculator capturing the students’ keystrokes as they work with the calculator. This software runs in the background of the graphics calculator, discreetly capturing the students’ keystrokes as they make use of the calculator. The aim of the research study described in this paper is to provide insights into the problem solving strategies of these students. Through a detailed analysis of their graphic calculator keystrokes and associated written solutions, we have proposed a classification of different working styles that identifies the effectiveness of the solution strategies and the efficiency of the use of the technology. This classification has important implications for the teaching of functions and their graphs with such technology.

Berry J., and Graham E., 2005, On high-school students’ use of graphic calculators in mathematics. ZDM, 37 No 3, 140-148
When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on two case studies that follow the progress of students as they solve mathematical problems. We use software that works in the background of the graphic calculator capturing the students’ keystrokes as they use the calculator. The aim of the research studies described in this paper was to provide insights into the working styles of these students. Through a detailed analysis of their graphic calculator keystrokes, interviews and associated written solutions we will discuss the effectiveness of their solution strategies and the efficiency of their use of the technology and identify some barriers to the use of graphic calculators in mathematical problem solving.

Picker, S.H. (2005). Mindful Calculator Use: Dealing With the Critics and Naysayers. Proceedings of the Texas Instruments International Conference, Washington, D.C.

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Berry J., Fentem R., Partanen A-M., and Tiihala S., 2005, An Investigation into Students’ Attitudes about Using Advanced Calculators in Learning Mathematics. Hiroshima Journal of Mathematics Education, 11, 1-20
Using technology in the teaching and learning of mathematics requires, among other things, positive attitudes from students and teachers. In this study, two teachers helped design and then taught two courses of mathematics to four groups of upper secondary school students. Two of the student groups had full access to graphic calculators with computer algebra systems (the TI-92 calculator) and the other two groups had full access to graphing calculators without computer algebra (the TI-85 calculator). In this paper we report on the student attitudes and beliefs about using technology in their course. The outcomes suggest that integrated technology teaching impacts in a complex fashion on student attitudes. The results of this study suggest that the nature of the assessment system, which forms part of the curriculum, influences the attitude towards rich usage of the technology. We also found that mathematical maturity leads to changes in attitude towards using the technology in the study. Several issues for future study are also identified.

Berry J., Fentem R., Partanen A-M., and Tiihala S., 2004, The Use of Symbolic Algebra in Learning Mathematics: the barrier from formal examination schemes. Nordic Studies in Mathematics Education, 9 No 4, 49 - 64
Using technology in the teaching and learning of mathematics requires, among other things, positive attitudes from students and teachers. In this study, two teachers helped design and then taught two courses of mathematics to four groups of upper secondary school students. Two of the student groups had full access to graphic calculators with computer algebra systems and the other two groups had full access to graphing calculators without computer algebra. In this paper we report on the student attitudes and beliefs about using technology in their course within a curriculum that restricts its use in a ‘final matriculation examination. The outcomes suggest that integrated technology teaching impacts in a complex fashion on student attitude. The results of this study suggest that the nature of the assessment system, which forms part of the curriculum, influences the attitude towards rich usage of the technology.

E Graham and P Smith (2004), An Investigation into the use of Graphics Calculators with Pupils in Key Stage 2, International Journal of Mathematical Education in Science and Technology, Vol. 35, 227-237.
This paper outlines an experiment in which pupils in Key Stage 2 were encouraged to use graphics calculators, in particular two simple programs, which helped them develop recall of their tables and allowed them to practice multiplication. The pupils responded very well to the calculators and seemed to have been motivated by them. The pupils did not find them difficult to operate and experienced very few technical problems. The authors concluded that the graphics calculator has considerable potential to enhance the mathematical experience and learning of pupils at this level, and that although the extent of the investigation was fairly limited the results were encouraging enough to justify further work in this area.

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Berry J and Nyman, M., 2003, Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behaviour, 22, 481-497
Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were audio tape recorded and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data we were able to construct a picture of the students' graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to 'walk' an associated displacement time graph we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a 'physical feel'.

S Honey and E Graham (2003), To Use or not to Use Graphics Calculators on Teaching Practice: A Case Study of Three Trainee Teachers' Beliefs and Attitudes, International Journal of Computer Algebra in Mathematics Education, Vol. 10, No. 2, 81-101.
This paper reports on a pilot study involving three PGCE (Post-Graduate Certificate in Education) students’ initial beliefs and attitudes towards graphic calculators, and their subsequent classroom practice whilst on their school-based training. This case study investigates whether the trainee-teachers modify their behaviour to meet the ‘ideal’ expected of them by their university tutors whilst on teaching practice, or do they revert to teaching the way they were taught? Are their beliefs and attitudes about mathematics and mathematics education evident from their classroom practice? These questions are considered with respect to the use of graphics calculators in mathematics classrooms. The initial questionnaire suggested three differing viewpoints, but the lesson observations and interviews suggest that the student with the positive attitude was almost as reticent to use graphics calculators as the one with the negative attitude.

Berry J., Graham E. and Smith A., 2003, Student Difficulties in Graphing Effectively with a Graphics Calculator The International Journal of Computer Algebra in Mathematics Education, 10 No 4, 251-264
The key-recorder is a software application that has been developed to capture the key strokes made by a graphics calculator user. This paper looks at data that has been gathered, using this method, for students who are working on graph sketching problems. The analysis has led to the identification of both technical and conceptual areas of difficulty. The interplay of these two different types of difficulty is discussed as examples of students’ work are considered. The paper notes that to avoid the types of difficulty that have been observed, students need to have both technical competence with the calculator and the ability to develop an enquiring approach when using graphics calculators to explore or sketch the graphs of functions.

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Graham T, Headlam C, Honey S, Sharp J and Smith A., 2003, The Use of graphics calculators by students in an examination: what do they really do? International Journal of Mathematical Education in Science and Technology, Vol. 34 No 3, pp 319 - 334
In many British schools, A-Level Mathematics students are advised to purchase Graphics Calculators at the start of their Mathematics course, and there has been considerable research into the effectiveness of the use of Graphics Calculators in developing mathematical understanding. Recent U K examination regulations have prohibited the use of Graphics Calculators in certain module examinations but allowed them to be used in others. This study set out to investigate how a small group of students actually used their graphics calculators under examination conditions.

The students sat an externally set practice examination paper for a statistics module. The examination paper was analysed by the research team in order to identify the potential use that the students could have made of the graphics calculators in each question. When they took the examination the students were provided with specially adapted calculators; these calculators were virtually identical to the students’ own calculators but contained specially written software which enabled the students’ keystrokes to be captured and saved. After the examination the keystrokes were replayed and studied by the research team with reference to the students' examination scripts. Each of the students was interviewed. The interviews were based on the students' use of calculators on the examination paper and more generally in their study of mathematics.

The research found that very little use was made of the graphics calculator in the examination, with most of the students using a scientific calculator in preference to their graphics calculator, unless a graph was specifically requested in the question.

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