# Teachers’ views on dynamically linked multiple representations, pedagogical practices and students’ understanding of mathematics using TI-Nspire in Scottish secondary schools

## Abstract

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons and what evidence do they provide to support their findings? Throughout session 2008–2009, this empirical research project involved six Scottish secondary schools, two mathematics teachers from each school and students from different ages and stages. Teachers used TI-Nspire PC software and students the TI-Nspire handheld technology. This technology is specifically designed to allow dynamically linked multiple representations of mathematical concepts such that pupils can observe links between cause and effect in different representations such as dynamic geometry, graphs, lists and spreadsheets. The teachers were convinced that the use of multiple representations of mathematical concepts enhanced their students’ relational understanding of these concepts, provided evidence to support their argument and described changes in their classroom pedagogy.

## 1 Introduction and background

This paper considers the use of representations of mathematical concepts, relationships between these representations and also pedagogical approaches which highlight and emphasise links between representations. The underlying mathematical concepts may be represented in different ways and a deep understanding of the concept includes the ability to fully appreciate each representation, its connection to the concept itself and also the links that exist between the representations. The representational theory underpinning this paper is based on the work of Duval (2006), who believes that semiotic systems of representation may be used to designate, communicate, work on or work with the mathematical objects being represented. He describes the different semiotic representations (number systems, geometric figures, algebraic symbols, graphs and natural language) as ‘registers’ and argues that a characteristic feature of mathematical activity is the simultaneous use of at least two registers of representation, or possibly the changing from one register to another. He uses the term ‘conversion’ to describe the transformation of representation from one register to another without changing the object being denoted, for example passing from an algebraic equation to its graphical representation, and considers that any conversion in register requires recognition of the same represented object between the two registers whose contents may appear quite different. Crucially, he argues that what he terms ‘standard teaching’ never focuses on this recognition of the mathematical object across registers and sees this as a possible cause of incomprehension among learners. He goes on to state that “changing representation register is the threshold of mathematical comprehension for learners at each stage of the curriculum” and that comprehension depends on coordination of several representation registers. He also postulates that the direction of any conversion between registers is crucial, one direction being more challenging than another, but this is based on his observations of ‘standard teaching’. It may be that the continuous observation by students of the impact of a change in one register on another dynamically linked register, possibly simultaneously visible, in either direction, may reduce or even eliminate this directional issue.

In the field of secondary school mathematics in Scotland many topics are taught in a disjointed way with little emphasis on the connections or links across representations. This may have resulted from the cyclic nature of the national mathematics curriculum, where topics are initially introduced and then revisited and augmented after a period of time during which other topics are considered, or from the use of school textbooks where topics, and their associated concepts and the representations of these concepts, are separated out into chapters using a cyclical approach. This fragmented practice fails to place sufficient emphasis on the links between the various representations and may result in a lack of relational understanding of the topics and their related concepts. The review of literature below would indicate that this problem is not unique to Scotland.

The topic of quadratics provides a useful illustration. It is not untypical to have a situation in which students’ first experience of quadratics is of multiplying out brackets such as (*x* + 3)(*x* − 2), a purely algebraic task. Some weeks or even months later the reverse procedure of factorising a quadratic expression is introduced, again as a purely algebraic exercise. After another gap in time, students may be taught how to solve quadratic equations algebraically, first by factorising and then later using the “quadratic formula”. Students may then be introduced to the graph of a quadratic function and eventually to the graphical solution of quadratic equations. Hence, in common with many other mathematics topics, the quadratic function has several representations, namely algebraic (symbolic), numeric, including a table of values, and graph (pictorial), each involving its associated verbal terminology, and the fragmented teaching approach fails to emphasise the links between these representations.

Graph drawing software and the graphing calculator provide one way of demonstrating and emphasising the links that exist between representations, for example between the factors and the roots or between the numerical values and the graphs. The use of several dynamically linked representations of a mathematical concept and a teaching emphasis on the links that exist between them may help to deepen students’ understanding of the concept. This study examines whether this may be true when using technology specifically designed for the purpose and also considers associated pedagogical practices.

## 2 Review of relevant literature

### 2.1 Research on multiple representations

The National Council of Teachers of Mathematics (NCTM, undated) places considerable emphasis on the use of representations in its Principles and Standards for School Mathematics. Members of the council recommend that students should be able to create and use representations to organise, record, and communicate mathematical ideas; select, apply and translate among mathematical representations to solve problems and use representations to model and interpret physical, social and mathematical phenomena. They argue that representations are necessary for students’ understanding of mathematical concepts and relationships, and that they allow students to be aware of connections among related concepts. They see representations as a means of facilitating students’ learning of mathematics and their communication with others about mathematical ideas. We need to ask on what evidence are their decisions and recommendations based?

Brenner et al. (1997) point to the crucial role of representation in mathematical problem solving which appears in a considerable body of research in cognitive psychology, cognitive science and mathematics education throughout the 1980s and early 1990s. Their study focuses on the representation of functions and relationships with particular emphasis on the use of tables (ordered pairs of values), graphs (pictorial representation) and equations (algebraic notation). They conclude that students could be successfully taught both to represent function problems in multiple representations and to translate between these representations. Kieran (1993) places emphasis on the integration of the various representations of functions such as graphical, algebraic and tabular, and Williams (1993) highlights the importance of being able to move comfortably between and among these three different representations of function.

Adu-Gyamfi (2002) provides a useful review of the extensive literature describing research on multiple representations and its use in mathematics education. The purpose was to examine information from available studies to assess whether evidence obtained supported or refuted the assertion that utilising multiple representations in mathematics teaching enables students to develop deeper understanding of mathematical concepts, relationships and problem solving. None of the studies examined reported any negative impact on students’ learning and he concluded that “students experiencing multiple representations type instruction demonstrated deeper understanding of mathematical concepts and demonstrated at par or superior performances during problem solving situations” (p. 45). He went on to recommend that more studies be done to find out the impact of using multiple representations in mathematics teaching and to inform curriculum decisions and hence bring its potential to the forefront of the mathematics education and mathematics teaching community.

A word of caution is provided by Even (1998) who looked at factors involved in linking representations of functions and found that subjects who participated in the study had difficulties in working with different representations and went on to stress the importance of understanding how these subjects think when they work with different representations of functions. Amit & Fried (2005) support this viewpoint and argue that we may have to challenge a multiple representations approach as a framework to begin with in teaching and think of it as a distant goal that may not be achieved until the learner has had considerable experience in the kinds of thinking that potentially link representations.

### 2.2 Research on multiple representations and the use of ICT

Much of the literature pertaining to the use of ICT in the teaching of mathematics relates to the benefits to be gained from the use of graphing calculators. There are now a number of reviews on this graphing calculator literature (Hembree & Dessart, 1986, Burrill et al., 2002, Ellington, 2003, Roschelle & Gallagher, 2005). A surprisingly small proportion of this literature relates specifically to the use of multiple representations but a number of studies are of interest.

Research indicates that the way teachers use technology in their classrooms is related to their beliefs about mathematics, in general, and that teachers who emphasise conceptual understanding, making sense of mathematical ideas and drawing conclusions based on mathematical grounds will reflect their beliefs in their use of the technology. “Teachers who emphasise connections among representations and sense making in working with both the mathematics and the tool see the results in the performance of their students.” (Burrill et al., 2002, p. iv). Burrill et al. also argue that students with access to handheld graphing technology are more flexible in their solution strategies, make conjectures and move more comfortably among algebraic, numeric and graphical approaches (Ruthven, 1990, Hollar & Norwood, 1999).

“it also builds connections between principles and concepts which are often taught in isolation (sometimes years apart, in conventional curricula). The ability to move easily across a connected network of knowledge, and to change representation systems, is critical to high-level problem solving when doing “real” mathematics and science.” (Hegedus & Kaput, 2007, p. 3).

As a number of the above studies refer specifically to teachers’ classroom practices it is worth considering related research findings.

### 2.3 Research findings on teachers’ classroom practice

Clearly, the way teachers use the technology and the associated teacher behaviour and teaching methodology will have an impact on the success or otherwise of its use. Roschelle (1992) suggests that given the support of teacher-guided and collaborative conversations about multiple representations, students may better understand the meaning of mathematical expressions as well as other mathematical concepts.

Burrill et al. (2002) are acutely aware that the use of handheld technology is not a single variable which can be easily isolated but is part of a very complex teaching and learning environment where a great number of factors are intertwined and inter-related. They suggest several areas for further study, one of which is the role of technology in providing access to particular mathematics curricular content earlier than would traditionally have been done.

*Effecting working processes and improving production,*notably by increasing the speed and efficiency of such processes, and improving the accuracy and presentation of results, so contributing to the pace and productivity of lessons;*Supporting processes of checking, trialling and refinement*, notably with respect to checking and correcting elements of work, and testing and improving problem strategies and solutions;…*Enhancing the variety and appeal of classroom activity*, notably by varying the format of lessons and altering their ambience by introducing elements of play, fun and excitement and reducing the laboriousness of tasks;*Fostering pupil independence and peer exchange*, notably by providing opportunities for pupils to exercise greater autonomy and responsibility, and to share expertise and provide mutual support. (Ruthven et al., 2009, p. 280).

Ruthven et al. (2009) used this ‘practitioner model’ to analyse how teachers adapt their classroom practice and develop their ‘craft knowledge’ in order to effectively introduce the use of software into their lessons. A particularly noteworthy aspect of the study is the weight and credence given to teachers’ comments in relation to their decision making and practice.

One more relevant issue cited in Ruthven et al. (2009) is the work of Farrell (1996) who studied the classroom practice of teachers involved in a development project in which the use of graphing technology was integral. The study concluded, with some caution, that there was a tendency for the technology use to help teachers to shift their classroom activity such that there was less teacher exposition and more student investigation and group work, allowing for both teacher and students to adopt the roles of explainer, consultant and co-investigator.

These issues are discussed below in the section describing teachers’ awareness of changing classroom practice and ways of teaching topics.

## 3 Instrumental and relational understanding

The terms instrumental and relational understanding are associated with the work of Skemp (1976, 1987). Instrumental understanding, on the one hand, is characterised by the ability to recall an appropriate rule or algorithm for particular circumstances and to execute it correctly. Relational understanding, on the other hand, involves knowing why the algorithm applies as well as why it works. It also involves a sound understanding of the links both within and between mathematical ideas and concepts. An example might contrast the instrumental understanding needed to appropriately choose and apply Pythagoras’ Theorem and/or the Cosine Rule with the relational understanding with which a student would be able to explain the link between the two theorems. Both types of understanding are necessary in that even relational mathematicians will apply instrumental methods. For example, it is not always necessary to differentiate from first principles, applying the rule is easier and more efficient. It may be argued that instrumental understanding is contained within the relational understanding that permits the relational mathematician to make the informed choice. Thus, instrumental understanding may be seen as a necessary prerequisite for the development of deeper relational understanding.

[The term ‘instrumental’ here should not be confused with other terminology such as ‘instrumental genesis’, ‘instrumentation’ and ‘instrumentalisation’ (Guin & Trouche, 1999, Drijvers & Trouche, 2008) related to the instrumental approach used elsewhere in this ZDM issue. In the Scottish study (Duncan, 2010), the matter of gaining mastery of the instrument, TI-Nspire, was found to be less of an issue for students as for teachers, who desired rapid development of expertise. Also, instrumentation issues applied to the students whereas instrumentalisation applied more to the teachers, who wanted the software to meet their particular pedagogical needs.]

Weber (2002) extends these types of understanding to include advanced mathematical concepts and compares relational understanding to Tall & Vinner’s ‘concept image’ (Tall & Vinner, 1981).

## 4 The research questions and setting

The present study involves an investigation into the use of software and handheld technology, which allows dynamically linked multiple representations of mathematical concepts within a single document, in the teaching of mathematics in secondary schools in Scotland. In particular, teachers are asked for their views and comments on how the use of this technology might help students to understand and make connections between the representations such as algebraic, geometric, graphic and numeric, of any mathematical topic?

The research considers one main question and other subsidiary questions. The main research question is:

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons or not, and what evidence do they provide to support their findings?

In what ways is the learning and teaching of mathematics changing as a result of using the software plus handhelds?

When using the technology, are teachers conscious of changing the way they teach particular topics?

When using the technology, are teachers conscious of changing the way they teach in general? If so, what are these changes and how are they justified?

When using the technology, what is the impact on students’ motivation and engagement?

A group of teachers were supplied with TI-Nspire software and handhelds, on a long-term loan basis, for use with students for an initial period of one academic session (2008–2009). The schools were chosen to represent a range of types from a range of geographical locations and to be representative of most Scottish secondary schools, each being fully comprehensive (all state schools in Scotland are fully comprehensive), ranging from rural to city and covering a large range of socio-economic backgrounds. The school rolls ranged from ca. 500 to ca. 1,200 students. Two mathematics teachers from each school were chosen by their Principal Teacher to be involved in the study on the basis that they could benefit from the experience and were willing to participate. Three of the teachers were the Principal Teachers of their departments. The teachers had a range of background experience with the number of years of teaching prior to the start of the project ranging from 1 to 35, with a mean of 17.5. There was also a wide range of experience with ICT, in general, and with mathematics software, especially in its use in classrooms (see Sect. 6).

The teachers decided upon the classes with which to use the handhelds. Some decided to use them as and when they thought appropriate with any of their classes, while others chose to use the technology with very similar classes so that they could work in a collaborative and mutually supportive way. As a consequence of these decisions the handhelds could not be issued on a full time basis to any students. Also, students were not permitted to borrow them for use outside school.

The handhelds (and software) were non-CAS [this decision was based on the fact that CAS calculators are not permitted in Scottish Qualification Authority (SQA) national examinations].

Each school was supplied with 30 handhelds and the teachers used TI-Nspire Teacher Edition software. All the teachers had some facility for projecting an image of their work with the software.

## 5 Methods

The method involves ‘mixed methods’ empirical research whose central premise is that the use of quantitative and qualitative approaches in combination provides a better understanding of issues under investigation than either approach alone (Cresswell & Plano Clark, 2007; Tashakkori & Teddlie, 2003; Johnson & Onwuegbuzie, 2004). Qualitative information was collected by a variety of means; however, there is also quantitative data arising from the teacher lesson evaluations questionnaires as described below.

The study describes the findings and experiences of 12 teachers from 6 schools and no attempt is being made to generalise to a wider population. In this sense it is phenomenological in essence.

In one school, one of the teachers had two very similar classes and she used the handhelds with one class but not with the other. It was hoped that a small scale control/experimental comparison could be made for this situation but in practice it was impossible to eliminate other contaminating influences such as time spent on a topic or use of graphing calculators with the non-experimental class for a topic where a very similar approach was used with both classes.

It would have been possible to use a pre-test/post-test model for the project as a whole but this model was rejected as it would have required the use of a test to gauge students’ understanding of the mathematics syllabus at the beginning of the project, the teaching of completely new topics for a whole academic year followed by a post-test covering the same material as the pre-test and not on what is taught during the study. Even with control and experimental classes, it is extremely difficult to eliminate other possible causes of any recognised change in performance. In this study it would also have required the construction of different tests for the various age groups involved, depending on their prior knowledge and the timing and sequencing of the teaching of the concepts being tested. The pre-test/post-test model would also require the construction of tests which specifically test students’ relational understanding of mathematics topics as opposed to memorisation of facts, formulae, routines and other measures of instrumental understanding. It was considered more worthwhile to elicit the views of a highly professional group of teachers on what they consider to be evidence of relational understanding.

It is interesting to note that in schools where an experimental/control model might have been possible, the teachers opted to both run ‘experimental’ classes (experimental in the sense that they were using new technology) in order to provide each other with mutual support in the development of resources for use with the software and handhelds. Most often this collaboration simply involved one teacher creating a TI-Nspire file and using it with their class and the other either using it unaltered or amending in some way based on the first teacher’s experience. The classes did not follow the same timetable so there was time for discussion and alteration before the second teacher used the file. It would appear to be the case that this mutual support and saving of time is an important issue for teachers especially when initially gaining familiarity with the technology. With evidence from a pilot study in England, Clark-Wilson noted that

“The time spent in collaborative professional dialogue was most valued by the teachers in the project. Their discussions focused upon: the management of the technology in the classroom; evaluation of the learning outcomes; refining TI-Nspire files; overcoming usability issues; developing ideas for future lessons…” (Clark-Wilson, 2008, p. 87).

The research method involves the use of a teacher ‘lesson evaluation’ questionnaire proforma which requested details of how the lessons and topics were taught, which mathematical representations were used and posed further questions which appear in Sect. 7. The questions were each followed by blank spaces for teachers’ open written responses. Each of the 12 teachers undertook to complete six lesson evaluations relating to lessons specifically designed to make use of at least two representations and using TI-Nspire technology, and they were asked to provide negative as well as positive feedback. This was a considerable undertaking and the teachers displayed a very high level of commitment and professionalism both in the content and number of their responses. It should be noted that the teaching workforce in Scotland is generally well qualified and committed. Only those with a degree containing a high proportion of mathematics and also a teaching qualification specifically in mathematics are permitted to teach mathematics in secondary schools.

Nine of the teachers completed all six lesson evaluation questionnaires, two completed five and one completed only two before withdrawing from the project for personal reasons. Thus, 66 out of a possible 72 (92%) lesson evaluation forms were submitted, a very high rate of return. All 66 lesson evaluation questionnaire responses were analysed and the results appear in Sect. 7. Written responses were analysed, categorised with regard to content, labelled using a summarising statement for each category and given an abbreviation code for ease of reference. For example, teacher responses relating to the issue of multiple representations and relational understanding (MRRU) were categorised and coded as MRRU1 to MRRU9 (see Table 1).

The qualitative data (teachers’ statements) were supported with quantitative data obtained by noting the number of comments falling into each category. The raw frequencies were then converted to percentages of the total number of response comments for each question in order to provide a possible indication of level of importance in the minds of the teachers.

The teacher evaluation form included a student feedback sheet which was issued separately by teachers to each student in their class for each evaluated lesson. These were completed by the students and then forwarded to the researcher along with the associated teacher lesson evaluation form. The third part of a triangulation method for scrutiny of findings within this investigation involved lesson observations by the researcher. Both student comments and researcher lesson observations support teachers’ observations and statements; however, space does not allow for an analysis of these within this article which focuses on the teachers’ comments. Further detail is available in the full project report (Duncan, 2010).

## 6 Ongoing continuing professional development (CPD) days for teachers

A total of six training days were provided for the teachers and they were released from class contact in order to attend. During the first 2 days, before the project got underway in schools, all the teachers met together and were taught how to operate the software and handhelds and were given the opportunity to discuss possible lessons and teaching approaches. The teaching approaches and exemplar files placed particular emphasis on the use of multiple representations and stressed the need for relational understanding of the underlying mathematics. The teachers were also provided with an outline of the research and an introduction to and discussion relating to the terminology being used. The terms instrumental and relational understanding were explained in detail as in Sect. 3 and discussed in terms of a variety of mathematical topics. In order to deepen a shared understanding of the terminology, the teachers were asked what they would consider to be reliable evidence of, for example, relational understanding in their students. Initially, individual responses were gathered then shared and discussed. Seven of the teachers referred to students making connections/links between topics or a single concept from different perspectives; six of them suggested students correctly explaining topics/lessons to others verbally and four mentioned students asking/answering questions and wanting to know ‘why’ rather than just ‘how’.

Some months later, the second two CPD days were provided locally in three areas. The first of these days was used for classroom lesson observation and the second for focused feedback and discussion of local issues. Only the teachers from the particular area were present at these training days.

Six months into the project, the final two CPD days were again delivered in one single location bringing all the teachers together. Once again some specific training on use of the software and handhelds was provided along with further opportunities for raising issues and sharing experiences.

The trainers for the CPD days were practising teachers and members of T^{3} (Teachers Teaching with Technology, a network of teachers providing professional development in the use of TI technology, funded by Texas Instruments) and were undeniably enthusiastic and capable users of the technology. Assuming that a degree of facility with ICT, in general, would be a factor contributing to facility in the use of TI-Nspire, a short questionnaire was designed to investigate the use made by the teachers of particular software for teaching mathematics. It was also necessary to test the conjecture that the project involved a biased sample of teachers who were proficient and experienced users of technology in teaching mathematics. This was certainly not the case. On the contrary, there is even a suspicion that some of the teachers were selected by their Principal Teachers as a way of initiating their involvement with technology. Indeed, 10 of the 12 teachers had never used either spreadsheets or dynamic geometry before the start of the project and most had made very little use of graphing calculators.

As mentioned above, time was allocated during each of these CPD days for the purposes of the research. The days were used for explaining the purpose and nature of the research, information gathering and data collection, classroom observation and sharing of findings to date. Teachers in effect took on the role of action researchers and displayed a generally high level of commitment. Questionnaire returns were conscientiously completed and contain rich and detailed information.

## 7 Findings

Teachers’ responses were both detailed and extensive and for this reason it was decided that crucial questions would be analysed in turn. What follows is a summary of these findings.

### 7.1 Multiple representations and relational understanding

“In your view, did the use of multiple representations with TI-Nspire enhance students’ relational understanding of the mathematics involved in this lesson or not?”

An overwhelming 80% of the 66 returns had a positive (Yes) response to this question in comparison to only 3% saying No with 12% being categorised as ‘undecided’. Teachers provided explanations for their decisions. The remaining 5% were either blank or related to lessons which did not involve multiple representations.

Of the eight ‘undecided’ votes, six were shared equally among three very thoughtful teachers who had to be completely convinced before voting ‘yes’. Other possible reasons for enhanced relational understanding were carefully considered. Also, in one lesson which involved ‘walking a graph’ using TI-Nspire and a motion detector, the teacher was not convinced that the lesson involved multiple representations because she assumed that all representations must be on the technology itself and had not considered the physical action of ‘walking’ (faster, slower, different directions and standing still), to be a representation. She was also unsure of exactly how much prior knowledge the students brought to this lesson. Such evidence of careful consideration by teachers provides further support to their conclusion that the use of multiple representations of mathematical concepts enhances their students’ relational understanding of these concepts.

Categorisation of teachers’ comments on multiple representations and relational understanding

Comments ( | % |
---|---|

MRRU1: Evidence detailing specific use of multiple representations | 33 |

MRRU2: Evidence detailing verbal or written responses from pupils | 13 |

MRRU3: Evidence of improved discussion | 12 |

MRRU4: Evidence of ‘aha’ moments—‘seeing’ pupils’ understanding | 12 |

MRRU5: Evidence of improved retention | 10 |

MRRU6: Evidence believed by teacher to be inconclusive | 8 |

MRRU7: Evidence detailing increased motivation, engagement/encouragement | 7 |

MRRU8: Evidence to support a ‘NO’ response | 3 |

MRRU9: Evidence from formal assessment | 2 |

Most teachers made specific mention of the multiple representations used along with evidence of students’ understanding either overheard or observed (MRRU1, 2 and 4). One rich lesson involved finding the maximum area of a rectangle with constant perimeter. Students constructed the rectangle on the geometry screen, dragged one vertex to see the rectangle change from long and thin to tall and narrow and noticed that the area increases then decreases between these extremes. Data capture was used to collect values of the area and length for a spreadsheet page and a graph was also created and appeared along with the spreadsheet on a split screen. The classes involved were two-second year second top sets and a third year second bottom set. The teacher comments related to this lesson are interesting. The first stated that “pupils were looking at a spreadsheet, seeing the range of their answers on the graph opposite and talking about an imagined rectangle that wasn’t there to see” and the second that;

“It was very telling that the pupils deduced and concluded on the rectangle with equal dimensions yet failed, across 3 classes, to identify that this was a square. I think it says something about how pupils learn concepts (today?)—compartmentalised and not seeing links. Many went on to argue afterwards that a square isn’t a rectangle—yet they had just ‘built’ one from a rectangle. I think it’s also worth noting that few seemed perturbed by a parabola—they saw it drawn point by point and knew the area would wax and wane so the graph seems to have been of no great surprise.”

Such evidence also relates to improved discussion (MRRU4) which helped to resolve the cognitive conflict faced by students who would not previously have been asked to argue that a square is a special rectangle.

It is noteworthy that there were very few negative statements and even those categorised as inconclusive or undecided (MRRU6) had comments such as “While the use of multiple representations would have certainly enhanced the students’ relational understanding of this topic, the benefits of this may not be obvious for a few years until the pupils reach the stage of transformation of graphs.”

Unfortunately only two statements are related to formal assessment evidence, but it is worth quoting one which was made by a very experienced teacher. She commented that “The standard of answers to similar triangles in the block test was significantly better than expected and the setting down of working was also better than expected from a group of this ability.” (This teacher was also teaching a very similar class of students but without TI-Nspire. The comment relates to the class using the technology but no direct comparison is made.)

Overall then, the teachers involved in this study, no matter what their background, length of experience as a teacher or extent of experience with ICT were convinced that the use of multiple representations of mathematical concepts generally enhances their students’ relational understanding of these concepts and were willing to provide evidence to support their argument.

### 7.2 Ways of teaching a topic (WTT)

“Were you conscious of changing the way you teach this topic?”

Categorisation of teachers’ comments on ways of teaching the topic

Comments ( | % |
---|---|

WTT1: Changing the way I teach the topic | 42 |

WTT2: Evidence of more active involvement from pupils | 20 |

WTT3: Using TI-Nspire to support my normal teaching methods | 16 |

WTT4: Evidence of links across maths topics | 10 |

WTT5: More opportunity for more open questioning and discussion | 6 |

WTT6: Teaching topics earlier than normal | 5 |

WTT7: Use of more mathematical language | 1 |

“A very clear ‘yes’ to this question. Whenever I’ve taught areas of circles previously, it has always been based around a factual ‘introduce the formula with follow up examples’ format. Use of the handhelds allowed a much more investigative, stimulating and meaningful way of teaching this topic and I am sure the understanding obtained by pupils was correspondingly ‘deeper’.”

This exemplifies the repeatedly occurring situation where previous teaching practice had involved straightforward exposition even to the extent of telling students a formula, showing an example followed by students doing an exercise from a textbook for consolidation and practice. In contrast, the teachers were now finding more practical and investigative approaches to topics where they had not done so in the past. A good example is shown in Fig. 1. The teacher created a TI-Nspire file using the software, transferred it to the students’ handhelds and they initially followed the teacher’s steps. Using the dynamic geometry screen he had created a circle and measured the circumference, diameter, area and radius. He then dragged the circle to produce different sizes and captured measurement data which were transferred to the lists and spreadsheet application. He had previously graphed circumference against diameter but in this lesson, which was observed by the researcher, he graphed area against radius. The students had learned about using linear regression to get the equation for circumference versus diameter and used the same strategy (unsuccessfully) to try to obtain the equation connecting area and radius. The teacher then suggested creating another column in the spreadsheet for the square of the radius and then linear regression produced the equation plus something which meant almost zero. (An introduction to standard form was postponed to a later date.) The equation was then translated to Area = *πr*^{2}. With time to spare, the teacher then suggested that if they returned to the first graph then possibly another form of regression might work. A number of hands were instantly raised showing that these first year secondary school students (age 12) had independently taken the initiative to investigate the various other options appearing in the regression menu and both quadratic and power regression were confirmed as being appropriate.

It appears that the move to more investigative teaching approaches consequently led to more active involvement by the students. Almost two-thirds of all the comments related to this combination of changing practice and/or more active involvement of students (WTT1 and 2). These findings support those of Ruthven et al. (2009) and Farrell (1996).

In one lesson the students discovered a way of determining the axis of symmetry of a quadratic graph which was quite different from either the teacher’s method or the method appearing in the majority of textbooks. The ‘normal’ method involves ‘completing the square’, a mathematical manipulation which was possibly beyond the ability of the students in the class and which they had not yet been taught. The students discovered that you can simply ignore the constant term “because it just moves the graph up or down and that does not affect the line of symmetry”, factorise the first two terms and then take half way between these values. For example, for the quadratic expression *x*^{2} + 6*x* + 7, the students factorised *x*^{2} + 6*x* to get *x*(*x* + 6) then found the roots of the equation *x*(*x* + 6) = 0 namely 0 and −6. They finally found half way between these and deduced that the line of symmetry must have equation *x* = −3.

I mention this situation not solely because it demonstrates the successful results of an investigative approach and active involvement of students but also highlights the links being made between their algebraic and their graphical understanding (WTT4). It is also relevant as a representative example of a situation in which the teacher can possibly feel threatened. It may be the case that if a teacher’s subject knowledge or pedagogical content knowledge (Shulman, 1986) is weak they could be unreceptive to such suggestions from students and in the worst case scenario could actually reject students’ ideas just because they do not conform to their normal practice (see following section).

### 7.3 Ways of teaching in general (WTG)

**“**When using the technology, are you conscious of changing the way you teach in general? If so, what are these changes and how are they justified?”

Categorisation of teachers’ comments on ways of teaching in general (total of 99% caused by rounding error)

Comments ( | % |
---|---|

WTG1: Allows students more freedom to investigate possibilities | 28 |

WTG2: Conscious of changing classroom dynamics | 13 |

WTG3: Allowing/encouraging more discussion with and among students | 13 |

WTG4: Consciously making an effort to link topics together | 11 |

WTG5: Consciously thinking about how to utilise the facilities/benefits of the technology | 11 |

WTG6: Consciously aiming to improve/deepen students’ understanding | 9 |

WTG7: Less teacher exposition or direction from the front | 4 |

WTG8: Students’ own discoveries pose a challenge to teachers’ subject knowledge | 2 |

WTG9: Encouraging students to be more responsible for their own learning | 2 |

WTG10: N-Spire has the potential to be a distracter | 2 |

WTG11: Teaching topics earlier than normal | 2 |

WTG12: Changed practice with less able pupils | 2 |

Comments appear to indicate a change in general classroom teaching pedagogy with a move towards more investigative work (WTG1) with more discussion among students (WTG3) and a consequent reduction of teacher exposition and direction from the front (WTG7). These findings also support those of Ruthven et al. (2009) and also of Farrell (1996). The results would further suggest that by being involved in this project for a whole session the teachers are now more conscious of trying to make good effective use of the technology (WTG5) in order to highlight the links that exist across mathematical topics (WTG4) and help improve and deepen students’ mathematical understanding (WTG6).

This situation is well explained by one particularly thoughtful teacher who wrote:

“My normal classroom teaching style is relatively didactic (i.e. initial class explanation and/or demonstration of new topic/theory, etc.), but with regular use of both open and targeted pupil questioning to try and get the pupils to come up with at least some of the new conceptual understanding where and when possible. Apart from use of graphic calculators, the amount of group and/or investigative work carried out in my classes and–for that matter—the whole Department is relatively small.

Using the TI-Nspires this year has then increased the number of practical lessons I teach and the general response of pupils to this has been very favourable. Lessons in which I have used the handhelds have normally been chosen as suitable for a more investigative type of approach, and there is no doubt in my mind that the calculators (handhelds) do lend themselves very well to this style. When planning such an investigative type lesson, I have normally found that opportunities for examining/discussing/encouraging some of the deeper relational understanding that maths teachers often yearn for in pupils, become more obvious and certainly easier to build into the lesson structure. Most, if not all, of my TI-Nspire sessions have attempted to provide the opportunity for pupils to extend their understanding beyond ‘rote learning’ level and I think many have been successful with this aim.”

### 7.4 Pupils’ motivation and engagement (PME)

**“**Describe the impact of the software and handhelds on both your motivation and the pupils’ motivation and engagement in this lesson.”

More than half the comments related to the positive impact on pupils’ motivation and engagement (PME1).

Categorisation of teachers’ comments on pupils’ motivation and engagement (total of 101% caused by rounding error)

Comment ( | % |
---|---|

PME1: Positive impact on the pupils’ motivation and engagement | 56 |

PME2: Positive contribution to pace and amount of learning | 14 |

PME3: Negative comment | 8 |

PME4: Positive contribution of linked multiple representations | 8 |

PME5: Comment related to work possible with other software | 6 |

PME6: Improved discussion | 5 |

PME7: Positive comment about individual pupil | 2 |

PME8: Positive teacher experience | 2 |

The negative comments (PME3) all related to the operation of the handhelds and the occasional frustration of forgetting how to do something in particular, but most were tempered by some relatively positive interpretation.

Other comments related to the advantages gained using multiple representations, improved discussion in class, improved performance and engagement of particular individual students, usually students who were normally much less interested, and also to positive experiences for the teacher.

The final category is for comments which are specifically related neither to multiple representations nor to TI-Nspire but which could equally be applied to other graph drawing software or graphing calculators. These comments pointed out how much more work (graphs) could be covered using the technology than would be possible when drawing by hand! This observation coincides with Ruthven’s first theme as described earlier, effecting working processes and improving production.

It appears then that, in general, teachers involved in this project found the technology and its use led to positive motivation and engagement among their students.

## 8 Conclusions

This paper considers the views and experiences of 12 mathematics teachers who were given the opportunity to use TI-Nspire PC software and handheld technology in their classrooms. The study is essentially phenomenological and no attempt is being made to generalise the findings beyond this group of teachers. On the other hand, the teachers were not exceptional in any respect and their observations of their experiences are of value for mathematics education literature. All of the teachers concluded that teaching using dynamically linked multiple representations of mathematics concepts enhances their students’ relational understanding of these concepts and they provided evidence to support their argument. The findings support Duval’s argument that a characteristic feature of mathematical activity is the simultaneous use of at least two registers of representation and the changing from one register to another (Duval, 2006).

The pedagogical situation is complex. It appears that by being asked to teach using multiple representations and highlighting connections among them using handheld technology, teachers not only changed the way they introduce and develop particular topics but also the way they teach in general, allowing more active involvement from students and tending to move away from teacher exposition to more investigative approaches involving more discussion between teacher and students and among the students themselves. These findings lend support to those of Ruthven et al. (2009) and Farrell (1996). Teachers also commented positively on the relationship between using the technology and students’ motivation and engagement. All the findings were sustained throughout a whole academic session and were as strong at the completion of the project as they had been at the outset. They are supported by comments in student questionnaires and by lessons observed by the researcher.

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