For the technologically challenged: four free online tools to liven up a mathematics classroom.
Northcote, Maria
Do you struggle to find meaningful ways to integrate technology
into your mathematics lessons? Maria Northcote provides an insightful
discussion on the purposeful use of technological tools and gives
examples of four specific tools with ideas for integrating their use
with the use of concrete manipulatives.
Introduction
Mathematical ideas have evolved across all cultures over thousands
of years, and are constantly developing. Digital technologies are
facilitating this expansion of ideas and providing access to new tools
for continuing mathematical exploration and invention. (Australian
Curriculum Assessment and Reporting Authority [ACARA] 2013b)
Use of technology in the mathematics classroom has the potential to
advance children's learning of mathematics and enhance their
attitudes about mathematics. When used in conjunction with purposeful
planning, teachers can use technological tools to reinforce their
pedagogical intentions and to facilitate relevant learning activities
for students. Students can increase their understanding of mathematical
concepts by manipulating on-screen representations (Goodwin, 2008) and
by engaging in the use of digital resources which are interactive and
mathematically appropriate (Haldane, 2007; Higgins, Beauchamp &
Miller, 2007; Kennewell & Beauchamp, 2007). As Attard and Curry
explain: "One of the most significant influences on student
engagement is the teacher's pedagogical practices, including the
incorporation of technology into the teaching and learning of
mathematics" (2012, p. 75). Bouta, Retalis and Paraskev (2011)
reinforce the significance of a teacher's pedagogical skills in
integrating the use of virtual tools into effective learning sessions:
"careful design is necessary in order to use their full
potential" (p. 501).
Digital resources
Online and digital resources have the capacity to "provide a
motivating and engaging learning experience through which thinking
capabilities and relationship skills can be enhanced" (Falloon,
2010, p. 626). Such resources can also form part of a collection of
resources, experiences and activities (practical and virtual) by
presenting multiple perspectives of a mathematical concept (Sfard,
2008). The benefits of using online resources in the mathematics
classroom have been further explained by Kissane (2009) who identified
five uses that internet resources offer mathematics teachers and
learners, including: 1) interactive opportunity 2) reading interesting
materials 3) reference materials 4) communication and 5) problem
solving. However, a teacher's pedagogy should drive the technology,
rather than technology being used for technology's sake.
In the context of this article, digital or technological tools
(specifically, freely available online tools) are not intended to be
used alone but in conjunction with purposefully designed learning
activities and hands-on materials. The use of virtual manipulatives
should be purposeful, systematic and carefully organised (Falloon, 2010;
Swan & Marshall, 2010). Indeed, the overuse or segregated use of
virtual tools may even cause conceptual problems in the way children
understand complex mathematical concepts, if they are not meaningfully
integrated with the use of practical applications and hands-on
resources. Furthermore, while the tools described in this article can be
used by teachers to facilitate opportunities for students to enhance
their numeracy understanding and skills, there may be ways in which
these tools can support children's development of literacy skills.
Examples of how each tool can be used in practical classroom
situations have been interspersed throughout this article. While these
tools are presented as examples of how to introduce technology into
mathematics lessons and activities, they are not presented in a 'be
all and end all' manner: instead, they are intended to be used in
conjunction with a teacher's pedagogical intentions, and integrated
into associated mathematical programs, lessons and activities. The tools
presented here are not put forward as being better or worse than, or
pedagogically superior or inferior to, concrete, hands-on or hand-held
technological devices, but as alternatives or supplements that are
convenient to access and duplicate within classroom groups of learners.
It is intended that these tools may be used in circumstances when sets
of classroom hands-on materials are not available, or when it is not
possible to access some of the more traditional forms of these tools
easily or quickly. In many cases, they offer additional interactive
affordances that are not always possible in physical versions of these
tools. In some cases, the use of online tools may also compensate for
potential mechanical problems associated with the use of hand-held tools
(such as the continual renewal of batteries or mechanical parts). Access
to a substitution tool--such as an online stopwatch in place of a
hand-held version of the tool--is presented in this article as one way
to encourage experimentation and use of new technology in mathematics
classrooms.
Hands-on and virtual tools
In their discussion of the value of hands-on and virtual
manipulatives, Swan and Marshall (2010) reviewed Perry and Howard's
research on mathematics manipulative materials (Perry & Howard,
1997). Combined, their work extends our understanding of how hands-on
mathematics manipulative materials can incorporate the use of virtual
manipulatives. For example, Attard and Curry (2012), in their
exploratory study of engaging students in learning about mathematics
concepts using iPads, found that these technological tools appeared to
"have had a positive impact on the teaching and learning of
mathematics for the participants involved ... and increased student
engagement by providing a resource that promoted interactivity,
immediate feedback, challenge and fun" (p. 81). Some learning
benefits of using tools in mathematics which present graphical displays
in virtual contexts, have been found as far back as 2000 (Cannon, Heal,
& Wellman) and 2002 (Moyer, Bolyard, & Spikell).
In more recent times, the potential offered by the use of virtual
tools in mathematics classrooms to encourage collaboration and
discussion among students has also been reported (Bouta et al., 2011).
In their work on using virtual manipulatives in mathematics with
children with special needs, Bouck and Flanagan (2010) explain the link
between motivation and learning as follows:
Students' interest in computers and the
accompanying motivation can be captured
with virtual manipulatives, and teachers can
take advantage of their students' increasing
ability to use this technology (p. 186).
Together, these researchers and educators provide a contextual
backdrop of the issues impacting on mathematics classrooms. As a result,
these issues have informed the creation of an evaluative guide that can
be used to determine the appropriateness of an online tool for
mathematics learning in primary schools.
Four tools
This article focuses specifically on the use of a handful of free
online tools that can be used to liven up and support mathematics
learning in lessons and activities that take place inside (that is,
mathematics activities that take place within the walls of a classroom)
or outside (that is, mathematics activities that take place beyond the
walls of a classroom such as in a playground). The examples provided
have been specifically drawn from the Foundation to Year 6 levels of the
Australian Curriculum. Teachers are invited to evaluate each of the four
tools outlined in this article according to the Technological Tool
Evaluation Guide, presented later in this article. However, it is
recognised that the value of the tools can only genuinely be realised by
considering the practical context in which the teaching and learning
occurs.
It is recognised that there is a growing collection of online tools
that are clearly and specifically designed for use within mathematics
classrooms, such as Geogebra (http://www.geogebra.org/), some apps
available to use on hand-held tablets (for example, Number Pieces,
Geometry Pad and Geoboard) and the tools offered through the National
Library of Virtual Manipulatives by the Utah State University the US.
Nevertheless, not all of the tools presented in this article have been
specifically designed for mathematics teaching purposes. While tools
such as the interactive number board, and virtual chance tools (such as
virtual dice and virtual spinners) are examples of online tools that may
have been purposefully designed for use in mathematics learning
activities, the online stopwatch has not necessarily been created for
such purposes. Even so, it can be used when students are learning about
many aspects of the Australian Curriculum. Furthermore, a language-based
tool such as the online game-creation tool, Quizlet, has not necessarily
been created with mathematics in mind but may be used in mathematical
teaching and learning contexts.
As well as recognising the functional uses of the four online tools
presented in this article, teachers are also encouraged to evaluate them
in terms of their capacity to engage and motivate students in primary
mathematics classrooms. While this article primarily focuses on the
value of using these tools as a way of introducing some purposeful
technology into mathematics teaching and learning, there is also the
possibility that the use of technology may enhance students'
engagement with mathematics, an idea already put forward by a number of
researchers and educators (Attard & Curry, 2012; Bouck &
Flanagan, 2010; Bouta et al., 2011; Reimer & Moyer, 2005). To
further contextualise engagement in a mathematics teaching environment,
Attard's framework for engagement with mathematics in the primary
classroom reminds us of the powerful role of motivation as well as the
value of acknowledging students' views about mathematics (Attard,
2012a, 2012b). Specifically, Attard (2012b) explains the importance of
integrating technology into student learning opportunities:
"technology is embedded and used to enhance mathematical
understanding through a student-centred approach to learning" (p.
13). As Attard has done in her work with Curry (Attard & Curry,
2012), this article uses Fredricks, Blumenfeld and Paris' (2004)
approach to defining the multi-faceted concept of engagement as
including:
1) cognitive engagement, incorporating "thoughtfulness and
willingness to exert the effort necessary to comprehend complex ideas
and master difficult skills" (p. 60);
2) behavioural engagement, involving active participation in
academic and social activities; and
3) emotional engagement, which is seen as involving attitudes,
reactions and a willingness to engage.
Consequently, in this article, the term engagement is intended to
encompass the actions, thoughts and feelings that children experience
through their involvement in mathematics activities.
What is a good tool in a mathematics classroom?
There are many ways to define a tool or resource in the mathematics
classroom. While some learning contexts suit the use of hands-on tools,
others are more suited to virtual use of online resources, and other
contexts may be suited to the parallel use of both hands-on and virtual
tools. While some researchers have used the term "mathematics
tool" to describe linguistic devices (Collins-Browning, 2009) and
communication journals (Amaral, 2010) used in mathematics, others
consider mathematics tools to be technological devices (Attard &
Northcote, 2012) or specifically designed tools for the purposes of
teaching and learning mathematics (Arnold, 2012). Goodwin (2008)
categorises technological tools according to whether or not the tool has
the capacity to be instructive, manipulable or constructive.
Furthermore, these types of tools could be further categorised into
teacher-made, student-made and commercially available tools, or tools
that have been tailored for a mathematics classroom and those that have
not. In light of this vast range of tools available for mathematics
teachers and students, this article recognises the value of using
virtual manipulatives alongside or to support the use of hands-on
manipulatives.
Whether or not a virtual or online tool is a 'real tool'
is debatable. Whereas a hand-held tool such as a stopwatch may be
considered more 'real' to a digital immigrant, a digital
native (Prensky, 2001) child in a current Australian classroom may view
an online stopwatch to be more 'real' than a hand-held
stopwatch stored in a school cupboard. Pedagogically, whether or not
these online tools are labelled as 'real' or not when compared
to traditionally hand-held mechanical tools, is possibly not as
important as how they are used, how accessible they are, how functional
they are, and how useful they are for teachers and learners in practical
mathematical contexts.
Consequently, a 'good' technological tool in the
mathematics classroom can be defined according to the learning or
teaching purpose. The use of technology for technology's sake will
not necessarily enhance students' mathematical learning.
Nevertheless, with the expansion of freely available, interactive online
tools and resources that has occurred over the last few decades, there
are now multiple opportunities for teachers and students of mathematics
to use these tools in purposeful, targeted ways. Focused use of these
tools has the capacity to engage students in the mathematics classroom
and to enhance enjoyment, motivation and relevance (Falloon, 2010;
Kennewell & Beauchamp, 2007). They should be used in pedagogically
sound ways to enhance learning through the use of modern digital
technology that is familiar to most primary-school-aged children (Reimer
& Moyer, 2005), the digital natives of this century (Prensky, 2001).
Evaluation of tools for use in mathematics learning
For the purposes of this article, the criteria for an effective and
motivating technological tool that is suitable for use in a mathematics
learning context in a primary school, has been informed by Attard's
Framework of Engagement for Mathematics in the Primary Classroom
(Attard, 2012a, 2012b) and Swan and Marshall's discussion of both
hands-on and virtual manipulatives (2010). Furthermore, the emphases in
the recently published Australian Curriculum (ACARA, 2013b) on the
general capabilities of literacy and Information and Communication
Technology (ICT) have provided additional input to the development of
the Technological Tool Evaluation Guide used to evaluate the four online
technological tools described in this article in relation to:
* mathematics and English (the content);
* learners and teachers (the people);
* possible learning environments (the context); and
* the tools themselves (the technology).
This guide offers classroom teachers a structured approach to
evaluating the capacity of online tools according to how they would
enhance the content being taught, the people involved in teaching and
learning, the context of the teaching and learning, and the technology
available in their mathematics classrooms. As well as proposing this
guide as an instrument to evaluate the tools presented in this article,
it may also be used to evaluate a range of other online, virtual or
technological tools that have the capacity to enrich mathematics
teaching and learning situations.
The tools
The four technological tools described below have been especially
selected for their efficient and effortless nature (Reys et al., 2012)
to ensure their use is both incidental but pedagogically purposeful
within a mathematics learning context. After considering their
descriptions and some suggested uses, teachers are encouraged to
evaluate each of the tools, using the Technological Tool Evaluation
Guide, according to the tool's capability to engage teachers and
students with relevant content within the context of primary school
mathematics.
Tool no. 1: Online stopwatch
Most students will have access at some stage of their young lives
to mobile stopwatch devices, some of which may be hand-held mechanical
devices or digital tools, including wrist watches, tablets (such as
iPads), and mobile phones. These tools are typically visible and
accessible by individuals in everyday life.
Alternatively, access to an online stopwatch on an interactive
whiteboard or a digital projector provides full class access to such a
device which, when required, can also be used as a timer for full class
activities. Use of interactive whiteboards to manipulate online tools
has been associated with learning engagement and resource management in
primary mathematics classrooms (Mildenhall, Swan, Northcote, &
Marshall, 2008). While the availability, distribution and management of
hand-held stopwatch devices may present resource problems in the context
of a busy classroom, the duplicability offered by online versions of
tools such as stopwatches can enhance student access and enable multiple
users of the same device. Availability of online tools from a full class
or individual learner perspective also has pedagogical value. For
example, the use of an online stopwatch on a large screen can illustrate
how a technological timing device can be used incidentally to check time
estimates, to track the time taken to play mathematics games and races,
and to time the period between events (for example, the time between
recess and lunch time). Classes of children who all have individual
access to such online tools on internet-connected tablets have the
freedom to manipulate these devices according to their own learning
needs and styles.
[FIGURE 1 OMITTED]
From Year 1 onwards, according to the Australian Curriculum,
children are expected to "explain time durations" (Year 1
Achievement Standard), "investigate the relationship between units
of time" (Year 3, ACMMG062) "convert between units of
time" (Year 4, ACMMG085) and "solve problems involving time
duration" (Year 4 Achievement Standard). The use of an online
stopwatch, which is easily accessible and used by students of all ages,
can put students in control of their own learning and engage them in
meaningful and incidental use of time devices during purposeful
activities.
Furthermore, online stopwatches can be used to enhance
collaborative learning opportunities. If accessed via mobile devices
(e.g., iPads), online stopwatches can facilitate activities that focus
on developing concepts such as the difference between seconds and
minutes as students "investigate the relationship between units of
time" (Year 3, ACMMG062; ACARA, 2013a). Groups of students may
investigate problems such as: "How can we time our movements around
the classroom? How are activities that take longer than 60 seconds timed
and how are these measurements recorded? How do we time our recess and
lunch breaks?" Students could be placed in groups of 3-4, each with
a tablet device on which they can access a stopwatch and a recording
application (such as Notepad). When such an activity begins with a set
of inquiry-focused questions, students are presented with numerous
opportunities to investigate, question and engage with practical
activities that support their conceptual development of different units
of time. As Attard and Curry (2012) found: "The iPads allowed the
teacher to introduce a wider range of teaching strategies that included
group work and a rotation of tasks within each lesson as opposed to
whole-class, worksheet based lessons" (p. 81). While manual,
hand-held stopwatches could also be used in such activities, and records
of time-based experiments could be documented on either paper-based or
digital resources, the use of an online stopwatch on a mobile
internet-connected device allows students to use a single device to
access a measuring tool and record their measuring results.
Tool no. 2: Virtual chance tools
The Australian Curriculum: Mathematics outlines a range of Content
Descriptions in the content strand of Statistics and Probability, that
describes what primary school students should learn about chance and
probability in their everyday lives. The topic of Chance is explored
from Year 1 onwards, as described in the Australian Curriculum. For
example, in Year 3, students are provided with opportunities to
"conduct chance experiments, identify and describe possible
outcomes and recognise variation in results" (Year 3, ACMSP067) and
Year 6 students "conduct chance experiments with both small and
large numbers of trials using appropriate digital technologies"
(Year 6, ACMSP145; ACARA, 2013c). While the use of hands-on devices
(such as dice and spinners) are ideal for individual and small group use
in mathematics games and activities--to facilitate turn-taking and
prediction activities --enlarged virtual chance tools can also be used
to illustrate the processes and outcomes associated with probability of
events to a whole class. Once children have an understanding of how to
use physical chance tools (such as plastic die and gameboard spinners),
their understanding of such tools may be extended to the use of virtual
tools which simulate the actions of the physical tools.
Virtual online chance tools can be used to initiate conversations
about the possible number of combinations of outcomes that may occur for
particular events. For example, virtual dice can be used to
"conduct repeated trials of chance experiments" (ACMSP067;
ACARA, 2013c). Virtual spinners can assist with common everyday events
such as allocating students to groups for team activities. Whereas the
physical versions of chance tools are difficult to modify, these
free-of-charge and freely accessible online devices can be easily
modified to increase or decrease the complexity of probability, or to
enable students to experiment with the different parameters associated
with chance activities including the number of sides on a die, the
number of segments on a spinner and the labels on dice sides. The
flexible size and appearance of online dice and spinners can further
engage students in mathematics activities as they can modify the tools
to suit their own aesthetic preferences and their own learning levels,
thus providing them with greater control over their own learning
contexts.
The use of virtual chance tools, due to their customisability and
flexibility, can provide students with opportunities to devise their own
probability experiments that can be initiated by inquiry questions such
as "How many times will this virtual die land on 5 when I throw it
20 times compared to when I throw this plastic die 20 times?"
Use of multiple virtual and physical tools can stimulate students
with a range of possibilities for creating problems to be solved by
other children or by groups of children. Such mathematics activities,
while assisting in children's conceptual development of
understanding how to "conduct chance experiments, identify and
describe possible outcomes and recognise variation in results"
(Year 3, ACMSP067; ACARA, 2013c), can also facilitate the development of
communication skills as they explain their problem to others and record
their results. Possibilities presented by the use of varied mathematical
tools can enhance collaboration and engagement (cognitive, affective and
behavioural) as reported by Bouta, Retalis and Paraskev (2011) in their
investigation of student engagement in a virtual environment: "the
3D virtual environment actively engages the students' interest and
leads to richer interaction between them. This in turn results in a
higher level of student engagement in the collaborative learning
process" (p. 501).
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
When engaging in learning about probability and chance in Year 6
with larger numbers of chance trials, for example, children can easily
use virtual chance tools to "conduct chance experiments with both
small and large numbers of trials using appropriate digital
technologies" (Year 6, ACMSP145; ACARA, 2013c). If children choose
to extend their investigations after school hours, during which time
their access to physical chance tools may be restricted, easy access to
online tools such as virtual spinners and virtual die can facilitate
their extra-curricula work. As such, the use of virtual mathematics
chance tools may present opportunities to engage students cognitively
and socially, and to increase their interest in investigating
mathematical concepts and sharing them with others.
Tool no. 3: Interactive number board
One of the major content strands of the Australian Curriculum:
Mathematics is Number and Algebra. This strand incorporates wide-ranging
and comprehensive topics such as place value, whole number, number
patterns, addition, subtraction, multiplication and division (ACARA,
2013c). While this article does not claim that an online version of a
number board is preferable to a physical version of this device (for
example, in wooden or plastic form), a virtual version of a number board
offers different learning opportunities for children when they are
learning about number concepts.
An interactive online number board can be used as a natural
extension to using a wooden or plastic number board that children and
teachers can modify to explore and demonstrate number patterns,
sequencing and place value. Because of their size and expense,
mathematics classrooms typically tend to be equipped with one or two of
the hands-on variety of number boards. However, once children are
familiar with the three-dimensional, hands-on version of a number board,
the interactive number board can provide individual children with
multiple, alternative and extended opportunities to engage in number and
pattern activities--colour coding even and odd numbers, exploring
multiples and factors, identifying a fractional part of a collection,
and practising skip counting. Because the use of online interactive
number boards enables more than one small group of children to use the
tools at any one time, individual children's progress can be
tracked for assessment purposes, especially when work samples are
captured.
As children develop abilities to "describe number patterns
formed by skip counting and patterns with objects" (ACMNA018;
ACARA, 2013c), the use of a virtual number board can be seen as the next
step in their conceptual understanding as they progress through to the
use of concrete to representational to abstract thought in association
with number sequences; a progression also noted by Moyer and Bolyard
(2002) who recommend using virtual representations of mathematics
processes to facilitate development of mathematical thinking from visual
to abstract. While a hands-on version of a number board may also
facilitate such thinking, the virtual number board enables customisation
of skip counting sequences, record-keeping by children and teachers
(saving and printing), and the capacity to allocate varied colours to
depict different number sequences. The use of a virtual number board may
provide a supplemental or alternative view of how to sequence numbers
for children who experience difficulties understanding such a concept.
In fact, Bouck and Flanagan (2010) consider the use of virtual
manipulatives with students with disabilities to be "best
practice", recommending that "special educators need to
consider manipulatives as a means of helping their students learn
mathematics and should be open to the use of virtual manipulatives"
(p. 186).
[FIGURE 4 OMITTED]
Tool no. 4: Quizlet
With the online game-creation tool, Quizlet (http://quizlet.com/),
teachers and students can quickly create content-specific games on any
topic, or access games made by other Quizlet users. Of course, as with
any mathematical games, the game itself does not teach children, but the
way it is integrated into purposeful and pedagogically sound learning
activities increases its capacity to assist learning. When designed
well, mathematical games give children the chance to engage in
"sustained attention, high-level thinking and collective as well as
individual effort" (Booker, Bond, Sparrow, & Swan, 2009, p.
28). This is especially the case if mathematical games are presented in
classrooms for both individual and group use. The online tool Quizlet
provides teachers especially with a tool that can be used to produce
customised, timely resources for use in their mathematics classrooms.
Typical games that can be made using Quizlet include virtual flash
cards, multiple choice questions, study games and drag-and-drop matching
games. By providing the game user with direct control of the game pieces
on the screen, students can quickly engage with the activity presented
to them. On-screen instructions are typically short and succinct to
ensure direct communication of what is expected of the user, thus
facilitating self-directed learning. Although primarily used by students
and teachers of arts and language-related courses, Quizlet can also be
used to create mathematics activities and games. While children's
engagement in Quizlet games would not necessarily be encouraged at the
beginning of a learning sequence (for example, when Year 3 children are
grappling with the concept of odd and even numbers--ACMNA051), these
type of online games are useful to reinforce concepts taught, to
emphasise links between concepts, and to revise the appropriate use of
mathematical language.
By using Quizlet, teachers can create a set of 'matches'
in which questions are matched with answers, or definitions are matched
with words or digits (see Table 2: Ten matches made using the online
game, Quizlet). For example, number words (five, seventeen) can be
matched with numerals (5, 17).
In the game in Table 2, Numbers 1 to 20, a set of ten matches were
made.Voice-overs can also be incorporated into the sets of matches.
The matches created can then be integrated into a variety of games
including flash cards and scatter games (see Figure 5).
After the matches were created (see Table 2), the
'Scatter' game type was selected (see Figure 6) which triggers
the creation of the game.
[FIGURE 5 OMITTED]
Selection of this game type automatically generates a URL (website
address) that teachers can share with students. Game users are then
faced with the challenge of making all of the numbers disappear from the
screen by dragging and dropping matching halves. When learners drag a
numeral to a word match (for example, when the numeral 5 is dragged to
the word five), both instances of the number disappear. Conversely,
learners can drag words to number matches (for example, when the word
twenty is dragged to the numeral 20). Learners can play the game
multiple times in an attempt to beat their personal best score. Feedback
is instant and the game enables individual engagement, without the
competitive pressure of team games. If students are currently learning
another language, Quizlet provides game creators with over 50 languages
to choose from.
A mobile app version of Quizlet is also available for use on phones
and tablets.
[FIGURE 6 OMITTED]
While the example of Quizlet utilises a mixture of both literacy
and numeracy skills, the game options within Quizlet lend themselves to
a single focus on topics related to mathematics or literacy, or an
integration of the two. Further possibilities for incorporating Quizlet
into mathematics lessons may include:
* matching definitions of shapes with numbers of shape sides;
* learning basic number facts;
* number sequencing;
* sorting even and odd numbers;
* listening to recognise the names of large numbers; and
* skip counting.
Special note regarding the use of Quizlet by teachers: the free
version of Quizlet was used in the preparation of the examples provided
in this article. When registering for a Quizlet account, teachers may be
offered to upgrade to Teacher Quizlet at the cost of $25 per year.
Unless the teacher wishes to use Quizlet extensively, the author
recommends clicking on the "Maybe later. Continue using free
Quizlet" link at the base of the registration screen which will
allow free use of the tool.
Using the Technological Tool Evaluation Guide
Each of the four technological tools outlined in this article can
be evaluated using the Technological Tool Evaluation Guide (see Table 1
earlier). Teachers may decide to use this Guide on a purely qualitative
basis; that is, use the descriptor criteria to examine various aspects
and potential uses of the technological tools. On the other hand,
teachers may decide to evaluate individual technological or online tools
by giving each of the dimensions (content, people, context and
technology) a star rating such as a scale from one star (not useful for
my context) to five stars (highly useful for my context). A grid such as
the following may be a useful way for teachers to record their
evaluative comments or ratings about particular online tools. Overall,
however, despite the suggested criteria outlined in the Technological
Tool Evaluation Guide (see Table 1), it is the pedagogical context in
which the tool will be used that should drive the way in which it is
evaluated.
While the advantages of such tools can be explained in terms of
their easy access (Goodwin, 2008), nil cost, flexibility,
customisability, motivational value and even "unlimited
supply" (Kissane, 2009, p. 137), these attributes may also point to
some of their limitations. Just as Swan and Marshall remind us:
"simply placing one's hands on the manipulative materials will
not magically impart mathematical understanding" (2010). So it is
with online tools. The use of online tools will not necessarily result
in mathematics learning without purposeful planning and the facilitation
of relevant learning activities. Concern has been expressed by some
educators about the over-use of virtual manipulatives (Swan &
Marshall, 2010) and the over-use of online resources that focus on the
right-wrong nature of mathematics to the detriment of the richer side of
mathematics. The highly engaging nature of some of these resources may
prove distracting for some children and, because they are so easy to
access, teachers and students alike may favour them in a busy classroom
context over their valuable, hands-on counterparts. Furthermore, because
some of these tools can be customised in a variety of ways (colour,
size, font, style, etc.), some users may become sidetracked by the
endless opportunities offered by tweaking the resources to suit their
personal preferences.
Whereas the convenience, flexibility and capacity to motivate
engagement are clear advantages of these online tools (depending on how
they are used in the context of practical classroom teaching) they can
offer students the chance to manipulate mathematical ideas across a
range of contexts, and provide opportunities to integrate literacy and
numeracy skills. Whereas some tools lend themselves more to individual
use to encourage personal understanding and reflection, others are more
appropriate for use in small or large groups of learners to encourage
collaborative learning and dialogue.
Conclusion
While there is no doubt as to the value of young children using
hands-on, concrete tools, there is also a place for technological tools
or virtual manipulatives in the modern primary mathematics classroom
(Goodwin, 2008; Kissane, 2009; Reys et al., 2012; Swan & Marshall,
2010). This is especially the case when technological online tools are
used to support the use of hands-on mathematics manipulatives within
carefully planned mathematics programs, lessons and activities. The four
online tools presented in this article provide students with a variety
of learning opportunities to engage them in mathematically-focused
activities and games, which have the potential to extend students'
understanding of mathematical concepts in a virtual environment. Many of
these tools can be manipulated by students as well as teachers and offer
potential integration with literacy. However, it is always timely to
consider Cotton's (2006) reminder that a learning resource cannot
teach children: good teachers do that. No matter how full of potential
these tools appear to be, it is their application within a well-planned,
pedagogically purposeful mathematics activity that will determine their
genuine value or otherwise.
Notwithstanding some of the limitations of online resources, the
use of such tools can liven up a mathematics classroom by introducing
purposeful, engaging tools that enhance teaching and learning processes.
The tools offered for consideration in this article represent a small
selection of such tools and they have been presented as flexible,
convenient supplements to good classroom practice. The affordance that
online tools offer in terms of their use on large and small screens
alike, ensure they are suitable for use by the whole class, small groups
and individual learners. They have the potential to further highlight
for students the relevance of mathematics in their lives and the
ever-present links between literacy, numeracy and technology.
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Maria Northcote
Avondale College of Higher Education
<maria.northcote@avondale.edu.au>
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Table 1: Technological tool evaluation guide
Dimension Evaluation criteria Descriptors of criteria
Use of the tool
Content ... in mathematics * clearly suitable to
mathematics learning
and teaching
* promotes active
engagement with
mathematics hands-on
resources and
manipulation of ideas
and resources
* helps obtain exact
results, where
appropriate (e.g.
stopwatch)
* does not only focus
on mathematics
problems or questions
that require
polarised (right or
wrong) responses
* clearly relates to
Content Descriptions
and strands in the
Australian Curriculum
... in English * offers opportunities
to integrate numeracy
and literacy
* use of mathematics
vocabulary is
facilitated
* use of oral language
is facilitated
* provides opportunities
to use problem
solving language
... to develop numeracy * realistic use of
and literacy skills numeracy and numeracy
skills is promoted
* assists with the
solving of authentic
numeracy and
literacy problems
People ... by the student * tool use can be
initiated by students
* can be used by either
primary school aged
students as well as
their teachers
* caters for diverse
learners
* encourages students
to develop their
own views
... with other students * tool can be used by
pairs or groups
of students
* mathematics
conversations may
occur while tool
is being used
* tool can facilitate
collaborative problem
solving
... by the teacher * motivating for the
teacher to use
* easy access,
distribution and
management
* can be used in
mathematics and
other subjects
Context ... in the classroom * suitable for use in
context (within the a mathematics
walls of a classroom) learning context by
primary school aged
children
* can be used inside
the classroom
... outside the * can be used outside
classroom (beyond the the classroom
walls of * convenient access
a classroom) via mobile devices
* incidental use
is encouraged
... with other materials * can be initiated by
the use of hands-on
manipulatives
* can be supplemented
by the use of hands-on
manipulatives
* does not disregard
the value of hands-on
manipulatives
* can extend understanding
gained by hands-on
manipulatives
Technology ... as a learning * promotes active
technology engagement with
mathematics hands-on
resources
* encourages active
participation by
students with the tool
* technology use is
embedded in
mathematics activities
* an element of choice
is incorporated into
the tool
... with other ICT * uses desktop or mobile
technology
* can be used on large
(e.g., interactive
whiteboards), medium
sized (e.g., computers)
or small screens
(e.g., mobile devices)
* accessed online
... to facilitate * free of charge
realistic and * offers repeated use
convenient use of * registration or
technology membership not
required
* no need for downloading
or installing software
Table 2: Ten 'matches' made using the
online game, Quizlet
two 2
five 5
seven 7
nine 9
ten 10
eleven 11
sixteen 16
seventeen 17
eighteen 18
twenty 20
Table 3: Suggested evaluation grid
Evaluation criteria
Tool Content People Context Technology OVERALL
Online stopwatch
Online dice and
spinner
Number board
Quizlet game maker
