Using tools such as connectable blocks to teach maths concepts to primary-age children has been commonplace in classrooms for more than a century. But for the past 40 years or so, the digital revolution has meant that physical manipulatives like these have, in some cases, given way to virtual, on-screen equivalents.

But is there any evidence that these “e-manipulatives” are any better or worse than their tangible equivalents?

E-manipulatives are usually referred to as digital or virtual manipulatives - although the letter can also refer to physical objects that have been augmented using digital technology. So, instead of snapping tangible blocks together to learn how a number can be formed, pupils manipulate blocks on screen - dragging and dropping them virtually.

For some schools, the use of these next-generation manipulatives is proving helpful, particularly when tailoring maths activities to different pupils.

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Graham Macaulay, director of innovation and growth at the LEO Academy Trust - which comprises seven primary schools in Sutton, south London - says teachers in his school have had success using virtual tools to teach maths.

“If you are using physical manipulatives in the classroom, then you are probably limited by the number you’ve got, and if every child wanted to go and use the same physical manipulative for a lesson, then that may provide resourcing challenges,” he says.

At LEO, all pupils have their own device, which they can take home with them - something which allows the child to access resources that are tailored to their individual needs.

“One thing that I’m really passionate about is letting children select the right resource that supports them with their learning,” says Macaulay.

“There are so many manipulatives out there - whether they be physical or whether they be digital - and, actually, I think different children really respond well to different ones, and that’s why, in our teaching, we will use a combination of the two.”

Using digital manipulatives also allows LEO to offer a library of resources to pupils, rather than relying on the ones that are physically available in the school, he adds.

“If you’ve got a lower-attaining child who relies on that manipulative and requires the visual association of a number and the physical representation, they’ve got a bank of them - they can just pick the ones that are most appropriate to them. But I think with physical manipulatives, you can’t prepare for every eventuality, so you might end up using the same one for every child.”

In some schools, then, virtual manipulatives have already taken off. But what does the research say?

E-manipulatives: the advantages

Andrew Manches, director of the Children and Technology Group at the University of Manchester, explains that research on virtual manipulatives began in the 1980s, and that many of the advantages identified back then remain pertinent.

“One is the fact that you can record what you’re doing, so that’s really good from a teacher’s point of view regarding assessment, but also a good way for children to look at what they’ve done,” explains Manches, who is also co-director of the Digital Education Research Centre at Manchester.

“With digital manipulatives, you can often go back a stage and see what you did previously, which you can’t do in the physical world, so you’ve got this ability to keep a record. However, when we talk about tangible, physical blocks, for example, we sometimes say they ‘live in the eternal present’, meaning you don’t know what they looked like a minute ago.”

Other advantages include the ease of sharing digital manipulatives, Manches says - provided everyone has access to a screen. Sharing a finite number of physical objects between a whole class can be considerably more problematic.

Jeff Bush, a research fellow in the Institute of Cognitive Science at the University of Colorado Boulder, is a former maths teacher who has written extensively on the use of virtual manipulatives to teach fractions. He also worked at Woot Math, a company that produces such resources.

“Digital manipulatives are one of many great tools for teachers to use but, like most pedagogies, they aren’t a panacea,” he says. 

“[They allow] some types of interactions that would be extremely challenging to do in a physical environment. A simple example is that of a function machine. Students put something in and the machine does a unique operation, like rotating it or cutting it in half.”

While these types of displays can be effective using physical aids, digital function machines “can take this further, showing how images can be turned to black and white or mirrored”, Bush says. “Then, the idea of a function can be slowly transitioned into more quantitative representations.”

What are the disadvantages?

There are, however, some pitfalls to avoid when using virtual manipulatives, says Bush.

“Students spend lots of time in front of a screen - arguably, too much time,” he points out. “If maths time becomes synonymous with screen time, then students miss out on valuable tactile experiences, especially when those lead to productive mathematical interactions with their teacher and peers.”

Another issue Bush flags is that not all digital manipulatives are made the same or are accessed in the same way - and while those that utilise a touch screen tend to be more dynamic, those that rely too much on a keyboard and mouse can be less effective at giving students a dynamic, tactile experience.

“Also, tools often omit features like snapping objects together,” he adds. “When I worked at Woot Math, a developer there told me a story about how the CEO valued the snapping feeling and click so much that she allocated over a month of engineering time to make it work.

“While this was, no doubt, expensive, I think it was well worth the effort since their digital manipulatives definitely have a tactile feel, especially when used with a tablet.”

However, when developers skip steps like this, the tools are “less engaging” and can leave students - especially those who have had less success at maths in the past - behind. “This is the opposite of what we want, since digital manipulatives, when designed and implemented right, can be a great equaliser in maths classrooms,” Bush says

Physical manipulatives

One advantage that tangible manipulatives have over their virtual equivalents is the increased potential to offer students the chance to learn by making mistakes, says Manches. While digital interfaces allow pupils to undo their errors, there is pedagogical value in those mistakes - value that might be missed when only operating virtually.

“There’s a lot of research showing that the messiness in learning - when things don’t happen exactly as planned - is sometimes of real benefit to children,” he says.

One example might be presenting blocks to a class and asking pupils to show the different ways to break up the number 10, and separate them into two different groups.

“If a child puts them into three different groups, then the programme might view that as wrong, even if the answers are correct and, actually, that’s a new learning opportunity,” Manches says. “So, there’s a lot of research about teaching moments and opportunities, and how materials like this create those moments.

“Sometimes, if you digitalise everything, you get rid of that interaction. It can become a bit too automated and a bit too much about practising certain things rather than getting to a deeper, conceptual understanding.”

Having said that, Manches adds that if there is “one teacher and 30 kids, it can be asking a lot for a teacher to be there all the time”, and virtual manipulatives offer the opportunity for some respite.

But there is also a risk that some teachers might not be on top of what is required to utilise digital manipulatives. “As you would imagine, manipulatives are only as good as the person who is presenting them,” Manches says. In addition, there is a risk that, in trying to create digital manipulatives that pupils can use independently, the nuances of teaching might be lost.

“There’s an extent to which you can get a computer to actually be a teacher, and that’s been a tension in using virtual manipulatives - the extent to which you can programme a teacher. As good as a computer can be, it is never going to be as good as a good teacher because what a good teacher can do is watch what the child is doing, they can ask them questions, and pick up on what they misunderstood and explain it.”

It is clear that digital manipulatives have lots of potential if applied the right way. However, says Bush, this should never be in a way that completely replaces the “still valuable” physical manipulatives.

Macaulay agrees. Ultimately, he believes that this is not a choice between the tangible and the virtual - and that schools can continue to have the best of both worlds. 

“The most important thing is that, from my experience, I don’t think it is a question of choosing either digital manipulatives or physical manipulatives,” he says.

“Actually, they both have value in the classroom in different ways and have different uses that they bring to the table.”