Mathematics, Technology Interventions and Pedagogy

Seeing the Wood from the Trees

Aibhin Bray and Brendan Tangney

Centre for Research in IT in Education (CRITE), School of Education and School of Computer Science & Statistics,

Trinity College Dublin, Dublin, Ireland

Keywords: Mathematics Education, Technology, Classification, Guidelines.

Abstract: This research explores recent technological interventions in mathematics education and examines to what

extent these make use of the educational opportunities offered by the technology and the appropriate

pedagogical approaches to facilitate learning. In an attempt to address this, a systematic literature review has

been carried out, and a classification is presented that categorises the types of technology as well as the

pedagogical foundations of the interventions in which those technologies are used. The potential of

technology to fundamentally alter how mathematics is experienced is further investigated through the lens

of the SAMR hierarchy, which identifies four levels of technology adoption: substitution, augmentation,

modification and redefinition. Classification of the interventions in this paper thus ranges from enhancing

traditional practice, to transforming teaching and learning through redefinition of how tasks and activities

are planned and carried out. The results of the research will be beneficial for guiding teaching, increasing

our understanding of learning in a technology rich environment, and improving mathematics education.

1 INTRODUCTION

When mathematics teachers first consider the

integration of technology into their daily class

activities, they can be faced with an overwhelming

array of devices, software, and instructional

approaches, with no clear guide as to best practice.

Hoyles and Noss (2003) highlight that the focus of

research on digital technologies for mathematics

education tends to concentrate on identification of

the potential of a particular technology and the

pitfalls or obstacles to its integration, and then

discusses its mediation through activities, and the

role of the teacher. The aim of this research is to

gain some clarity regarding pedagogical approaches

to technology interventions in post-primary

mathematics education, as documented in recent

literature, with the goal of generating an overview of

the properties of interventions that are deemed to be

successful. The main objectives are to increase the

understanding of the kinds of teaching and learning

of mathematics that technology has the potential to

enhance and the generation of a set of guidelines for

the implementation of such activities. A long-term

goal is to create, and test, a pragmatic and

comprehensive 21st Century model of classroom

practice for mathematics education.

In order to address these issues, this paper first

conducts a review of research that discusses issues

and approaches to technology interventions in

mathematics education. Following from this, the

need for a system of classification is investigated,

along with some existing models. A systematic

literature review on a selection of twenty five papers

that discuss specific interventions is carried out in

accordance with the methodology discussed in

section 4. The resulting data are analysed through

the lenses provided by the emergent classification. A

set of guiding principles for the appropriate use of

technology in mathematics education is presented in

the final section of the paper.

2 BACKGROUND

2.1 Pedagogic Approach

Many of the empirical studies examined for this

paper are limited in that they concentrate on

implementations of specific technology and do not

focus on the more pragmatic issues around

technology interventions that teachers require.

Bray A. and Tangney B..

Mathematics, Technology Interventions and Pedagogy - Seeing the Wood from the Trees.

DOI: 10.5220/0004349100570063

In Proceedings of the 5th International Conference on Computer Supported Education (CSEDU-2013), pages 57-63

ISBN: 978-989-8565-53-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

However, there is a clear trend towards a socially

constructivist approach, indicating that technology

interventions in mathematics education may be

suited to an active and collaborative environment.

This pedagogical theory has its foundations in the

work of Kolb, Vygotsky and Bruner, and its positive

effects have been borne out through the results of the

longitudinal SPRinG study in the UK (Blatchford et

al., 2003).

The emphasis on sense-making and problem

solving, in particular in a social context, that

becomes evident through the classification of the

literature, informs the development of some of the

guiding principles presented in this paper.

2.2 Further Areas of Consideration

Prior to the initiation of any intervention, it is of

utmost importance to look at the circumstances

under which learning can be enhanced by

technology (Means, 2010). Oldknow (2009)

suggests that the transformative potential of ICT is

not restricted to new, or purpose built technology,

but also lies in the innovative uses of everyday

equipment.

Oates (2011) and Geiger, V., Farragher, R., and

Goos, M. (2010) provide evidence that the

outsourcing of computation through the use of

technologies such as Computer Algebra Systems

(CAS) has the potential to do more than just improve

speed and accuracy. It can also provide increased

opportunity for the development of investigative

skills and problem solving.

Means (2010) and Oates (2011) highlight that if

technology is to be truly integrated into teaching and

learning then the assessment potential that it offers

needs to be utilised where possible. Assessment can

be administered through computer based testing,

intelligent tutoring systems, use of collaborative

documents or knowledge fora (Lazakidou and

Retalis, 2010), or student devices networked to the

teacher console (Noss et al., 2012).

It is also noted (Means, 2010) that teachers who

actively facilitate and scaffold their students

interactions with the technology are in a position to

use their insights to refine the activities and inform

instruction. In essence, the students’ interactions

with the technology can contribute to their formative

assessment.

Innovation with regard to the working

environment and class routine are seen as necessary

in order to fully exploit the potential of technology

in the teaching and learning of mathematics. Means

(2010) points out that, contrary to popular belief,

higher learning gains are evident when there is not a

one-to-one relationship between the student and the

technology, thereby encouraging collaboration and

team-work.

Issues such as the professional development of

teachers also emerge as essential for the successful

integration of technology in educational settings, but

these concerns are beyond the scope of this research

3 CLASSIFICATIONS

Classifications are not definitive descriptions, but

should reflect a theory about the current situation;

they should be dynamic and able to keep pace with

the changes to the status quo. They should permit

generalisation, and provide a basis for explanation of

the emerging argument. In this case, the

classification system is being developed to shed light

on the current trends in technology usage in

mathematics education, with a view to informing a

set of guidelines for future interventions in the field.

Prior to the development of the classification of

the literature presented in this paper, some existing

systems were identified and considered. Four areas

emerged as being of interest: technology, levels of

adoption, learning theory and instructional approach.

These will now be discussed in more depth.

3.1 Existing Classifications

of Technology

The classification systems of Clarebout and Elen

(2006) and Passey (2012) were considered, but were

deemed unsuitable due to issues around relevance to

mathematics and levels of complexity. Two

classifications of technology for mathematics

education by Hoyles and Noss however, are

influential in this research. They are specific to

mathematics education and, while being concise,

provide an appropriate level of detail.

The first, (Hoyles and Noss, 2003) distinguishes

between programming tools and expressive tools.

Programming tools, such as microworlds, are

defined as lending themselves to individual

expression and collaboration. Expressive tools on

the other hand, provide easy access to the results of

algorithms and procedures, without the user being

required to understand the intricacies of their

calculation. The category of expressive tools is

further broken down into pedagogic tools, designed

specifically for the exploration of a mathematical

domain, and calculational instruments, which are

frequently adapted to, rather than designed for,

pedagogic purposes. Dynamic Graphical

CSEDU2013-5thInternationalConferenceonComputerSupportedEducation

Environments, such as GeoGebra, are examples of

pedagogic tools, and spreadsheet programs would

fall into the category of calculational instruments.

In their later research, Hoyles and Noss (2009)

classify tools according to how their usage shapes

mathematical meanings. They refine and extend

their previous framework differentiating between

dynamic and graphical tools such as Cabri and

Geometers Sketchpad; tools that outsource the

processing power, of which computer algebra

systems are an example; new semiotic tools, which

may have the potential to influence how

mathematics is represented; and tools that increase

connectivity, such knowledge fora.

3.2 Emerging Classification

of Technology

In this study, the classifications by Hoyles and Noss

are further refined and amalgamated to provide the

foundation for the technological component of the

emerging classification. There is no evidence in the

papers reviewed of semiotic tools that change the

representational infrastructure of mathematics, and it

has thus been removed from the presentation of the

findings. Through the review of the papers it

emerged however, that an extension of the Hoyles

and Noss classification was required. The category

of toolkit is therefore added as a distinct class.

Integral to the definition of this new category is the

design of technologies in accordance with a specific

pedagogical approach, along with the provision of

support for the student and the teacher through tasks

and lesson plans, and feedback for assessment, all

founded in the relevant didactic theory. Examples

include Noss et al. (2009) and Tangney et al. (2010).

The technologies in the literature reviewed in

this paper are thereby classified as follows:

 Outsourcing of Processing power

 Dynamic Graphical Environments (DGE)

 Purposefully Collaborative

 Simulations/Programming

 Toolkit

3.3 Classifications of Technology

Adoption

Two perspectives on the adoption of technology in

mathematics interventions were considered: the

FUIRE model (Hooper and Rieber, 1995) and the

SAMR hierarchy (Puentedura, 2006). While the

FUIRE model provides information on an

individual’s use of the technology and their level of

adoption of it in the classroom, the SAMR model is

better fitted to describing the level of adoption

present in a given intervention and as such, is the

model selected for this classification of the papers.

The SAMR hierarchy (Figure 1) is broken down

into the two broad categories of Enhancement and

Transformation, each of which has two further

subsections. The lowest level of Enhancement is

classed as Substitution. This describes situations in

which the technology is used as a direct substitute

for the traditional method, without functional change

as exemplified by the reading of classic texts online.

The second level is that of Augmentation, in which

the technology is used as a substitute for an existing

tool, but with some functional improvement, e.g. if

the text being read contains links to online study

guides.

Figure 1: The SAMR Hierarchy (Puentedura, 2006).

The Transformation space on the SAMR hierarchy

describes interventions that either significantly

redesign the tasks provided through the use of the

technology (modification), or that have used the

affordances of the technology to design new tasks

that would previously have been inconceivable

(redefinition).

3.4 Additional Dimensions

of the Classification

In addition to grouping the papers reviewed in this

study by technology and level of adoption, this

research also classifies them according to learning

theory and instructional approach. These additional

dimensions will now be expanded.

3.4.1 Learning Theories

The learning theories considered fall into two main

Mathematics,TechnologyInterventionsandPedagogy-SeeingtheWoodfromtheTrees

camps: Behaviourism (Skinner, 1938) and

Cognitivism (Bruner, 1977). Some cognitive

learning activities can be further classified as

Constructivist (Kolb, 1984), and within this, as

Constructionist (Papert, 1980) or Social

Constructivist (Vygotsky, 1978).

3.4.2 Instructional Approaches

The Instructional Approaches taken into account are:

 Drill and practice.

 Task Based.

 Individual work.

 Contextual.

 Inquiry.

 Plenary or whole class discussion.

 Realistic.

 Sense making.

 Active learning.

 Problem solving.

 Collaborative.

Most of the interventions examined adopted more

than one instructional approach, with up to five

distinguishable in some cases.

4 METHODOLOGY

The electronic databases searched in the review of

recent literature were chosen for their relevance to

education, information technology and

mathematics:::ERIC (Education Resources

Information Center), Science Direct, and Academic

Search Complete. The general search terms used

were:

math* AND (technolog* OR tool*) AND education

These were used in an initial pass over the

databases, and the results were then refined by

limiters such as “secondary education”. In order to

further restrict results, only peer-reviewed, journal

articles, issued between 2009 and 2012, were

considered. A preliminary set of thirty four papers

were selected for initial analysis, and of these,

twenty five make up the final data set. The

remaining nine papers were not included as they did

not discuss specific interventions. However, a

number of them did compare interventions in

general and were useful in informing the set of

guidelines that aim to describe a method of

successful integration of technology in mathematics

education.

5 ANALYSIS OF THE DATA

The data emerging from the literature review were

coded and stored in a spreadsheet pivot table. This

allowed the information to be arranged, related and

visualised in diverse and meaningful ways. A

summary of the classified papers is presented in

scss.tcd.ie/~braya/csedu/The%20Papers.pdf.

Through this process, a number of interesting

patterns became available. Figure 2 illustrates the

clear socially collaborative trend in the literature, as

well as the concentration on Outsourcing of

Processing, and Dynamic Graphical Modelling

Environments as the technologies of choice.

Figure 2.

CSEDU2013-5thInternationalConferenceonComputerSupportedEducation

A correlation between instructional approach and

learning theory also emerged from the data, as

illustrated by Figure 3.

The interventions were also coded according to

where they fell on the SAMR scale. It became clear

that none of the papers considered dealt with

technology as a direct substitute for traditional

methods, without functional change (Figure 4).

Roughly 30% fell into the sphere of augmentation,

an example of which is the paper by Kay and

Kletskin (2012), who evaluate the use of video

podcasts in mathematics education.

Over 40% of the papers came under the heading

of Modification, these include Ruthven, Deaney et

al., (2009) analysis of the use of graphing software

to teach about algebraic forms, and Lazakidou and

Retalis’ (2010) work on the use of technology

supported collaborative learning strategies for

problem solving in mathematics education.

The technology in each of the interventions

classified in this way has facilitated significant

redesign of the tasks and the learning experience.

Articles within the category of Redefinition make

up the remaining 30% of the papers. These describe

tasks and activities that would not have been

possible without the use of the technology in

question (e.g. Noss et al., 2012; Tangney et al.,

2010). All of the studies that fell into the technology

classifications of programming (e.g., Noss et al.,

2012) or toolkit (e.g., Noss et al., 2009; Tangney et

Figure 4: Technology according to SAMR hierarchy.

Figure 3.

Mathematics,TechnologyInterventionsandPedagogy-SeeingtheWoodfromtheTrees

al., 2010), are also classified under the remit of

redefinition. Figure 4 illustrates the breakdown of

the technologies according to the SAMR hierarchy,

and Figure 5 indicates how the learning theories

observed in the interventions are grouped in terms of

the hierarchy.

6 DISCUSSION

Having looked at the general literature on uses of

technology in mathematics education, this study

took a structured approach, facilitated by a

classification system, to analyse specific

interventions.

It is interesting to note that all of the

constructionist interventions use programming tools

or toolkits (Figure 2), and are classed in the sphere

of redefinition (Figure 5). The only other incidences

of redefinition are seen within socially constructivist

settings. Another interesting result is the lack of

papers in the literature review that are classified at

the level of Substitution on the SAMR hierarchy.

There are a variety of possible reasons for this: one

may be that while it is occurring in everyday usage

of technology, it simply may not be reported in the

literature. Alternatively, it is possible that the

selection process of the papers under review was too

narrow, an issue that will be addressed in future

work.

Based on the general literature review and the

classification of selected papers, a set of guidelines

is now proposed outlining an approach to the design

of learning experiences that fully employ the

educational potential of technology and appropriate

pedagogical approaches to facilitate learning.

Interventions designed in accordance with these

guidelines should significantly modify or redefine

the learning experience through the affordances of

the technology. That is, interventions of this type are

likely to be classified within the Transformation

space on the SAMR hierarchy.

6.1 Guidelines

An appropriate and innovative technology

intervention in mathematics education should thus:

1. Be collaborative and team-based in accordance

with a socially constructivist approach to

learning.

2. Exploit the transformative as well as the

computational capabilities of the technology.

3. Involve problem solving, investigation and

sense-making, moving from concrete to abstract

concepts.

4. Make the learning experience interesting and

immersive/real wherever possible, adapting the

environment and class routine as appropriate.

5. Use a variety of technologies (digital and

traditional) suited to the task, in particular, non-

specialist technology that students have to hand

such as mobile phones and digital cameras.

6. Utilise the formative and/or summative

assessment potential of the technology

intervention.

7 CONCLUSIONS AND FUTURE

WORK

Through the literature review and the development

of a classification system, pedagogic approaches

have been identified which are appropriate for use in

technology enhanced teaching and learning. These

are based in socially constructivist/constructionist

learning theory and emphasise problem solving,

investigative, and realistic instructional approaches.

Use of assessment potential provided by the

Figure 5: Learning Theory according to SAMR hierarchy.

CSEDU2013-5thInternationalConferenceonComputerSupportedEducation

technology and flexibility with regard environment

and class routine also emerged as important aspects

of a successful intervention. From this data, a set of

guiding principles were extracted that have the

potential to form the basis of a 21

Century model

for the integration of technology into mathematics

education.

In order to gauge the efficacy of the guidelines,

initial exploratory interventions are being developed.

As part of this, a number of pilot activities have

already been implemented, with very encouraging

results.

Further studies which implement the guidelines

will be used to build up a strong evidence base for

the potential of such activities to enhance

mathematics education. Such activities will require

execution in traditional school settings as well as

purpose designed environments, in order to

investigate their potential to scale.

The literature review will continue to be

expanded in order to confirm the results and keep

the system of classification up to date. This will be

an iterative process and will, along with the results

of the studies, continue to inform and refine the

guidelines.

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Mathematics,TechnologyInterventionsandPedagogy-SeeingtheWoodfromtheTrees