Durmus, S., & Karakirik, E. (2006). “Virtual Manipulatives in Mathematics Education: A Theoretical Framework”

From MathWiki

From The Turkish Online Journal of Educational Technology – TOJET January 2006 ISSN: 1303-6521 volume 5 Issue 1 Article 12.

Available through York’s eResources at http://www.eric.ed.gov.ezproxy.library.yorku.ca/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/28/06/d8.pdf

Table of contents

Focus of the article

Durmus & Karakirik (2006) seek to outline some relevant theoretical elements when considering the role of virtual manipulatives in mathematics education. Before they turn their attention to virtual manipulatives, they consider mathematical modelling, abstraction and the use of physical manipulatives in the teaching and learning of mathematics. The authors give links to some specific virtual manipulatives available online, and provide (from the literature) some criteria for evaluating the design and use of virtual manipulatives.

Ideas on virtual manipulatives & Initial thoughts/reations

A common definition

“A virtual manipulative is defined as "an interactive, web-based visual representation of a
dynamic object that presents opportunities for constructing mathematical knowledge" (Moyer et
al., 2002, p. 373)” (Durmus & Karakirik, 2006, p.5)

This is a definition that I have come across several times in the more recent literature on this subject. When we speak of virtual manipulatives in our discussion group, we have agreed upon a more inclusive definition – one which includes computer software packages like The Geometer’s Sketchpad, which is not web-based. Indeed, when the authors of this paper consider items that would fall under our broader definition, they are sometimes referred to as “computer manipulatives”.

Virtual and physical manipulatives in action

“Visual representations of concepts and relations help learners to gain insight in
mathematics. Virtual manipulatives enable as much engagement as physical manipulatives do
since they are actual models of physical manipulatives mentioned above including Tangram and
Geoboard (Dorwand & Heal, 1999)” (Durmus & Karakirik, 2006, p.5, emphasis added).

While I believe that virtual manipulatives can serve as useful tools for students to develop and deepen their understanding of mathematical concepts, I am unsure about the validity of equating virtual and physical manipulatives. The authors suggest that virtual manipulatives provide as much mathematical engagement for students as their physical counterparts. I feel (and this was the sense in our group as well) that physical manipulatives should not be abandoned for any software set. This feeling is supported by some examples that highlight, for example, the kinaesthetic element of learning with manipulatives (e.g. Gutiérrez, 1996, with some more to follow).

However, if we consider the recent research into mirror neurons and mental rotation, wouldn’t it seem like students could achieve the benefits of physical manipulatives through simply visual stimulus? (Please correct me here: I am under the impression that when I watch someone performing a task, the same neurons fire as if I were performing that same task myself.) This line of reasoning might support the use of virtual manipulatives as support for student learning, but not to the point where we should stop encouraging the use of physical manipulatives. The statement, “Virtual manipulatives enable as much engagement as physical manipulatives” is bold: they provide an opportunity for different kinds of engagement (this is a good thing!), but not to the point where one latter should abandoned in teaching/learning practices.

“Although virtual manipulatives might simulate manipulatives in flesh, they are much more
abstract since they do not allow hands-on activities. However, it is suggested that virtual
manipulatives could be employed interchangeably with physical manipulatives in mathematics
since manipulatives are not expected to make mathematical concepts “touchable” but to
highlight the salient features of the concept to be covered. Hence, it is necessary to design
specific math manipulatives focusing at different mathematical concepts. Virtual manipulatives
might also provide further advantages over physical manipulatives by eliminating some of the
constraints they impose on the task. Some computer manipulatives may be more beneficial than
any physical manipulative” (Durmus & Karakirik, 2006, p.5).

This passage raises two important points. The first is that, by their nature, virtual manipulatives are an abstraction from their physical counterparts. Therefore, it may be difficult for students to move directly to virtual manipulatives. Nevertheless, the authors suggest that virtual manipulatives should be used “interchangeably” with their physical counterparts. This equality of the virtual and the physical is again problematic for me, since it detracts from the value of handling physical objects. There are a few examples in the literature of instances where students have found it difficult to move directly into the use of a virtual manipulative, without any prior experience in handling physical models (reference/s to follow). I agree that the idea of ‘interchangeable use’ in the sense of weaving back and forth from the physical to the virtual – but some more thought (and research) needs to be given to the importance of which comes first.

The second important point here is that virtual manipulatives can provide certain advantages over their physical counterparts. In order to make a stronger case for the use of technology and virtual manipulatives, we should be able to demonstrate some of the advantages of virtual manipulatives that are not already associated with existing physical models. This can also be a driving force to improve software packages: what would we like this software to do, which would otherwise be impossible (or extremely difficult) with physical manipulatives or models?


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