Technology Research Papers Post 2001

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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.

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.

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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.

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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Berry J. and Sharp J., 2002, The Use of Technology in Developing Mathematical Modelling Skills
Proceedings of the International Teachers Teaching with Technology Conference, Calgary, Canada, 2002
Much has been said about the importance of mathematics in the school and college curriculum and much has been written about what should be included in the curriculum for the beginning of the twenty first century. Often there is a tension between the school curriculum and the perceived needs of college and university mathematicians. Too often the mathematics curriculum at all levels is seen as a ‘body of knowledge’ which needs to be delivered in order to provide an ‘acceptable graduate in mathematics’. In this era of powerful software on hand-held and computer technologies we need to review the procedures and rules that have been the central focus of the mathematics curriculum for over one hundred years. That is not to say that we do not need some of the traditional skills so that students can make effective use of the technology. However there are important generic skills that mathematics provides, and to the employer of our graduates these skills are often more important than the actual mathematics that they have learnt. In this paper we explore the use of graphic calculators in developing mathematical modelling skills.

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Berry J. and Sharp J., 2001, Using Graphic Calculators to Develop the Feel for Mathematics Proceedings of the International Teachers Teaching with Technology Conference, Columbus, Ohio
During the past five years the authors have run many training courses on the use of hand-held technology in the teaching and learning of Mathematics. These courses have been offered throughout the United Kingdom and internationally from The Hague in Europe to Phuket in Thailand. One of the common views expressed by many of the teachers that we have met is that because of the use of technology today's students will not 'see and feel mathematics' in the same way as we do. Our argument is that with technology students may not need the same feel for mathematics and that this does not matter. What we need to do as mathematics educationalists is to explore the different needs of mathematicians in the twenty-first century. In this paper we provide examples of approaches to learning and doing mathematics that will enhance a good conceptual understanding and that this will help to develop a 'feel for mathematics' that is as useful as a traditional one that is based on a purely algebraic approach.

Sharp J and Berry J., 2001, These have worked for us! Proceedings of the International Teachers Teaching with Technology Conference, Columbus, Ohio
The Centre for Teaching Mathematics at the University of Plymouth run a Mathematics Enrichment Programme which provides extra curricular activities for students from year 4 through to 12. Often we work with the able students who need the extra activities to maintain their interest in the subject - the bright ones will soon get turned off if they are not being challenged. However we also work with students who are not particularly motivated about mathematics. This means that the tasks and activities we provide need to cover both a wide age range and mixed ability. We often use hand-held technology in the workshops and activities and this session presents some of the activities that have worked both for us and for the students.

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

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.

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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.

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'.

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Berry J., 2002, Developing mathematical modelling skills: The role of CAS , ZDM, 34 No 5
Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters. This is an important part of validating a mathematical model formulated to describe a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.

Berry J., 2002, Do You Know What Your Students are doing on their Graphic Calculators? Proceedings of the International Teachers Teaching with Technology Conference, Calgary, Canada, 2002, CD Rom

The aims of this session are to explore the following issues:

  • what are your expectations of the working styles of students when they are doing mathematics with graphic calculators?
  • how can we find out?

We introduce a piece of Applications software for the Texas Instruments TI-83 Plus which enables the researcher to capture exactly the students use of a graphic calculator. The software, which has been developed in co-operation with Texas Instruments records the key presses a student makes as they use the calculator and saves them within the calculators’ internal memory. The paper also reports on the working styles of a group of teachers attempting to draw the graphs of three functions.

Smith A. and Berry J., 2002, Observing Student working styles when using graphic calculators In: Eds. Borovcnik, M. & Kautschitsch, H. Technology in Mathematics Teaching. Proceedings of ICTMT5, Klagenfurt, Austria, 2001. Schriftenreihe Didaktik der Mathematik, 25, Oestererreichischer Bundesverlag and Hölder-Pichler-Tempsky, Vienna,
In this paper we introduce a piece of Applications software which enables the researcher to capture exactly the students use of a graphic calculator. The software, which, has been developed in co-operation with Texas Instruments records the key presses a student makes as they use the calculator and saves them within the calculators’ internal memory. The paper also reports a small-scale study on graphing, which we used to test the software. One of our aims at this conference is to generate a discussion and use the experience that many of you have in using graphic calculators to come up with innovative and novel ways that the software may be used.

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Berry J. and Fentem R., 2002 Investigation into Student Attitudes to using Calculators with CAS in learning Mathematics In: Eds. Borovcnik, M. & Kautschitsch, H. Technology in Mathematics Teaching. Proceedings of ICTMT5, Klagenfurt, Austria, 2001. Schriftenreihe Didaktik der Mathematik, 25, Oestererreichischer Bundesverlag and Hölder-Pichler-Tempsky, Vienna,
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 (CAS) and the other two groups had full access to graphing calculators without CAS. 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 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. 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.

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