#### << PART II: The development of conceptual understanding

### The research and its limitations

Proponents of virtual manipulatives suggest they hold significant advantages over their physical counterparts. Many of these are easy to observe and measure while others are more difficult to establish. Virtual manipulatives are still new and as such, the research is relatively limited and can be contradictory (Hunt, Nipper & Nash, 2011). It can also be tempting to generalise based on studies involving relatively few students or specific mathematical concepts (McNeil & Jarvin, 2009).

However, it is possible to make some reasonable, initial observations about their potential. These will be measured against the criteria established earlier. For each criterion, the research will be presented, exploring the strengths and limitations offered by the virtual space.

**Criteria:**

### 1. Creates links between multiple physical and symbolic representations

*Strengths*

Virtual manipulatives can represent and link multiple versions of a concept simultaneously and the links are immediate, direct and changeable (Clements, 1999). In the interactive below, a student creates an amount using virtual coins. They can view its symbolic representation (the numerical amounts), and see this change as they add or remove coins.

**HINT: **Where can I find this interactive? *Coin Count – Mathletics eBook section Year 4*

The student might also change a symbol and watch how this transforms the objects. Thompson (1990, as cited in Clements 2009) suggests that working on the symbols rather than the objects allows students to see connections and establish mathematical rules more easily. For example, when constructing a square a student must make decisions about the size and number of sides and angles and the relationships between them; a far richer learning experience than clicking on or picking up a shape.

This ability to link representations appears to be a key advantage of the virtual space. Physical manipulatives must be manually linked to other representations, which students may fail to do, or they may make faulty connections that go unrecognised by the student or teacher (Moyer-Packenham & Westenkow, 2013).

*Limitations*

Whilst multiple models do appear to build conceptual understanding and encourage flexibility of thinking (Moyer et al, 2008), Mix notes that many Asian nations prefer to use just one physical model at a time, as it is felt too many models can be confusing. She also observes that exposing students to multiple models may not give students enough time to establish stable patterns with one model before they move onto the next (2009).

It seems reasonable to assume that this may also be the case with virtual manipulatives, particularly if the student is not yet able to make connections with some of the information on the screen. Some manipulatives have hide functions, which can be used to limit what appears, though there is then the danger that the learning will be limited as well.

Ainsworth (1999) suggests that it is important to consider the purpose of multiple representations. They can:

- complement each other, providing different but similar information and processes
- constrain the interpretation; a known representation supports the understanding of a new and different one
- help construct a deeper understanding of the concept; students use multiple representations to explore, link and explain their personal understandings

Considering these functions in the light of learning objectives may help teachers decide which representations to use and when.

### 2. Consistency with cognitive and mathematical structures and processes

Many physical manipulatives are carefully designed to represent their mathematical properties; a hundreds ‘flat’ is the same size and shape as 100 unit blocks and a brown Cuisenaire rod (8 cm) is twice the length of a crimson rod (4 cm). Good virtual manipulatives are also true to these same properties. Recent research has also highlighted the link between movement and cognition; movement provides a steady stream of information that we use to help construct knowledge (Mix, 2009). Manipulatives provide us with models to act upon.

*Strengths*

Virtual manipulatives appear to have an added advantage in that they can be pre-programmed to represent the actions we want learners to internalise as mental processes (Samara & Clements, 2009). A base-10 block can be broken apart into smaller units with a click or tap (useful for addition and subtraction with exchanging), objects can be composed and decomposed easily to see how they relate to each other and shapes can be rotated, resized and re-coloured while still maintaining their geometric properties.

Such processes are generally far harder to replicate with physical manipulatives, although this does depend on the fine motor skills of the user. Manches, O’Malley & Bedford note that young learners in particular may find physical manipulatives easier when they move groups of objects at a time (2009); selecting multiple objects on a screen can be a challenge. Tablet versions of applets may be easier for young learners as they can use gestures that are both easier to perform and a more intuitive match to the mathematical process.

Importantly, virtual manipulatives often provide the user with instant feedback on their actions. This feedback is immediate and non-judgmental and actions can be repeated or altered as users tests out their thinking (Petit, 2013). It also makes it harder for users to overlook the power of their actions. Samara & Clements argue that this is a major weakness of physical manipulatives; students can act upon them in ways that are personally but not necessarily mathematically meaningful (2009). Manches et al see that because virtual manipulatives are easier to change, students are more likely to persevere with actions, which in turn, creates the likelihood of connections and learning (2009).

**HINT: **Where can I find this interactive? *What Triangle? – Mathletics eBook section Year 5*

*Limitations*

Boedemer, Ploetzner, Feurelein & Spada found that students must interact with virtual manipulatives in a goal-oriented way for learning to be enhanced (2004). Manipulatives that constrain the users to perform certain actions or ones that require students to solve specific problems may be useful in this regard.

Research also shows that the students must perform the actions themselves (McNeil & Uttal, 2009). Often, the teacher controls the manipulative, be it physical or virtual, and assumes the students are making the same connections they are. To construct their own understandings, students must be directing the actions (McNeil & Jarvin, 2007).

### 3. Scaffolds and supports the learner as they construct new understandings

The use of manipulatives to scaffold learning is not new; they have long been used to distribute the cognitive load, direct focus, and provide reliability as a basis for experimentation (Mix, 2009).

When students can rely on models as stable representations of a concept, they can attend more closely to discovering and connecting mathematical relationships and rules.

*Strengths*

Recently it has been proposed that manipulatives may do too much of the work for students and that realistic models may actually hinder learning as they draw attention to the objects themselves, rather than the principles they embody (Uttal et al, 2009). Virtual manipulatives may be helpful in this respect, as they seem to reduce the demands of this dual representation.

Sharing the cognitive load reduces the strain on attention and memory (Mix, 2009). Virtual manipulatives appear to have the advantage over their physical counterparts by constraining the user to focus on particular actions and limiting possible responses (Moyer-Packenham & Westenkow, 2013). This reduces perceptual distraction, scaffolds thinking and can also draw students attention to mathematical features of which they were previously unaware. Applets also generally provide both visual and verbal information simultaneously which appears to improve recall (Moyer-Packenham et al, 2008).

**HINT: **Where can I find this interactive? *Part-whole rods 1 – Mathletics Year 3 Course*

Because students can literally do more with virtual manipulatives they seem to elicit a more creative variety of responses (Samara & Clements, 1999). Students can:

- test variability, leading to generalisations
- experiment, promoting conjecture and problem-solving
- make surprise discoveries and connection

Virtual manipulatives can also construct phenomena that can not be easily modelled physically, such as graphs (Petit, 203). This makes them a powerful teaching tool, particularly for older learners who perceive virtual manipulatives to be more sophisticated than the physical versions (Moyer et al, 2002). Applets also appeal to tinkerers, students who learn by exploration and experimentation (Resnick & Rosenbaum (2013).

*Limitations*

However, it is worth reiterating that unlimited freedom is generally not productive, with most students requiring a level of teacher guidance. It is a tricky juggling act for teachers and one that requires them to closely consider both the learning goals of the activity and the capabilities of their students. Boedemer et al also note that learners may have to process large amounts of information on a screen at one time, adding to the cognitive load (2004).

Applets are now emerging that respond to student responses and adjust the scaffolding and representations according, gradually withdrawing help. Research suggests students using this model outperform other students significantly (Sedig 2009, in Samara & Clements, 2009).

### 4. Promotes mathematical precision and dialogue

As well as demonstrating conceptual understanding, many curriculums now require students of all ages to develop fluency, ‘carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily’ and to reason, ‘analysing, proving, evaluating, explaining, inferring, justifying and generalising’ (ACARA 2013).

*Strengths*

Many studies that found students working on computers tend to work with more precision, efficiency and exactness than those working with pencil and paper (Samara & Clements, 2009). Compared to physical manipulatives, virtual manipulatives provide:

- An infinite supply of precise and stable mathematical models that can be quickly replicated, altered and deleted
- Ease of manipulation, organisation, storage and retrieval

These functions appear to offer these advantages:

- Students can produce more solutions more quickly
- More methodical and purposeful action on the part of the student, leading to more mathematically accurate examples
- The creation of mathematical generalisations
- Motivation for students to engage with, and persist at, mathematical tasks

(Moyer- Packenham & Westenkow, 2013)

This effect on student motivation is worth highlighting. Multiple studies show that students perceive virtual manipulatives as more enjoyable and game-like and that they are more likely to persist with tasks while using them (Hunt et al, 2011). Increased engagement seems to lead to a greater sense of self-efficacy, resulting in greater learning outcomes (Moyer- Packenham & Westenkow, 2013).

Samara & Clements also found that compared to physical tools, virtual manipulatives bring maths ideas to consciousness more easily with students being able to explain both their thoughts and the actions with greater precision (2009).

*Limitations*

While virtual manipulatives have the potential to enable greater mathematical precision and dialogue, it cannot be assumed that this actually happens. Teachers play an important role by:

- Facilitating the mathematical discussions that contribute to the development of mathematical understanding, providing students with the language they need at the point of need
- Modelling actions and behaviours
- Explicitly explaining what models mean and how they related to symbols and other representations, particularly to students who appear to be disengaged or non-goal oriented in their use of the manipulative

Group exploration using interactive white boards or other shared learning technologies is important, though teachers should ensure students are in control of the process and not watching passively. Virtual manipulatives also vary in how easy they are to use. Ones that are hard to navigate or manipulate may lead to frustration and thwart the learning process.