In the January of 2021, Christopher Such and I were, as I’m sure everyone does from time to time, casually discussing the validity of claims made in a paper about the use of digital manipulatives. And while I cannot remember whether we reached agreement on the matter at the time, the conversation ended with a resolution to conduct a thorough analysis of the available evidence by following the sources in the paper. Over the course of the next six months, I painstakingly explored upwards of 70 papers as the task at hand shifted from mere claim-testing towards the altogether more useful task of developing a set of guiding principles for the utilisation of concrete resources in the mathematics classroom. What follows over the next three posts is a summary of that analysis, which I hope all teachers of mathematics will find useful in some small way.

Education research is an imperfect science. A point to which anyone involved in the field will attest. Yet the philosophical lenses applied to much of the research into the use of manipulatives make it especially difficult for teachers to navigate. Full disclosure, on the great philosophical continuum from the objective reality of Auguste Comte’s positivism to the highly subjective realm of William James’ pragmatism, I probably fall somewhere close to subscribing to a post-positivist viewpoint in as much as I believe there is an objective reality that we should try to measure but I’m not convinced we will ever truly be able to do so. 

This is important because it outlines the lens through which my analysis of the available research papers was conducted. In my ideal scenario the scientific method is central to education research design and that’s what I’m hoping to see when I probe the claims made in the papers I read. Hands up, I know this isn’t the prevailing view of many education researchers and I’m aware that many do not agree with me but this was the ‘jumping off point’  for my analysis nonetheless.

I noticed very quickly that three key themes crop up time and again in studies designed to test the efficacy of manipulatives in the mathematics classroom: small sample sizes, short time-frames, recycled references. Or as I like to call them, The Unholy Trifecta of Education Research. 

Principle	Guidance
Small sample sizes	Inflated effect sizes
Short time-frame	Performance happens in the moment, Learning happens over time
Recycled references	Potentially hazardous for teacher workload if left unchecked – see Dale’s Cone of Experience

Given that these features were commonplace in the papers analysed, the claims made herein as a result of their analysis must be taken with a pinch of salt. There exists, to my knowledge, no robust body of research conducted utilising the scientific-method and there appears little appetite to rectify this at present. 

If, like me, this wasn’t what you hoped to find, I’m sorry to say that things get worse before they get better.

There is a general sense amongst many teachers, particularly those teaching in the primary phase, that the use of manipulatives is an effective way to support pupils in developing their understanding of mathematical concepts. Yet, as early as 1960s (and perhaps even earlier) studies comparing the efficacy of concrete and symbolic modes of instruction have found that the utilisation of manipulatives can be considerably less effective than symbolic alternatives. Examples span the last seven decades but the most frequently cited and, in my opinion, well-designed of these is a 1972 study by Elizabeth Hammer Fennema in which the use of concrete resources to teach repeated addition was compared to teaching the same concept with just symbolic notation. The study found that “...children who used the symbolic model performed at a higher level than those who used the Concrete model…” and it is just one of many where similar findings were reported.

This might, on the surface, appear quite bleak but we should not fret for too long because we know that engagement with education research consists of more than just taking note of the findings presented in a collection of papers. To truly engage we must embark on a cycle of reading, thinking and trialling so that our classroom practice might be informed and refined by what we read and think, with particular consideration given to working out how applicable any presented findings are to our own context.  

If you’ve spent time teaching in Key Stage 1 (particularly Year 2/Primary 1/First Grade), or supported colleagues there as maths lead, there’s a high likelihood you’ll have pondered the following conundrum at some point: The pupils understand when using manipulatives but they just can’t do it in their books. It’s a common theme, a situation that occurs on a daily basis in classrooms around the world, and one so prevalent that studies have been conducted in an attempt to find out the possible reasons for its manifestation. 

The hands-down greatest of these is a study conducted in 2009 by David Uttal and colleagues at Northwestern University in the United States that I like to call “The Snoopy Investigation”. In this study they took a toy Snoopy and put it in a room in a doll’s house. They showed it to some children before hiding a teddy Snoopy in the equivalent room in a real house. Even though the children were, essentially, told where Snoopy was, they struggled to find the teddy in the real house. 

This reminds me of ‘the surgeon and the general conundrum’ in Dan Willingham’s, “Why don’t students like school?”. In that study, only 30% of participants could solve the second problem when told the solution to the first, despite them sharing an identical deep structure, and it seems like something similar is going on here with ‘The Snoopy Investigation’. The conclusion drawn by Uttal (and reflected in other studies) is that the salience of the representation blocked transfer – or certainly made it more difficult to achieve. Thus, when we consider our choice of manipulative/representation and when to use them, we must always have in mind whether our decision is likely to benefit transfer or whether transfer, at that moment in time, is worth considering at all. 

If transfer is of concern, then we can be aided in our efforts by giving consideration of the term ‘concrete’. Often it is taken to mean a physical, tactile object that can be manipulated but the original etymology of concrete – a sense of growing together – is altogether much more helpful and alludes to an interpretation that might be of greater benefit in the classroom. Returning to “Why don’t students like school?” – Willingham describes ‘concrete’ as the knowledge or understanding a pupil currently has. Their understanding is real to them, they can navigate it in their minds, for all intents and purposes it might as well be made of cement. In the research into the use of manipulatives this is referred to as integrated concrete knowledge and it moves us beyond a linear, two-dimensional interpretation of ‘concrete’ towards something altogether more robust. Our understanding of ‘concreteness’ grows, and gains strength, as we combine the sensory experience with the integration of knowledge to create a rounded, deep and useful interpretation of one of the most powerful ideas in education. 

Once we make the leap and broadened our interpretation of ‘concreteness’ we allow ourselves the opportunity to utilise one potential solution to the issue of transfer. Virtual manipulatives (such as those expertly crafted and freely available on www.mathsbot.com) are less salient than their physical counterparts, are often more mathematically accurate representations of certain concepts – something to which we will turn our attention in subsequent blogs – and can act as a stepping stone between resources whose salience could prevent transfer towards agile and proficient manipulation of fundamental mathematical ideas.

Principle	Guidance
The more salient a resource, the less likely transfer becomes.	Make generalisation a priority in task design and utilise virtual manipulatives.

The question then becomes when do we move from using actual cupcakes to cubes than represent cupcakes and beyond? 

Sadly, this is not a question which has a singular answer. As I wrote in “Thinking Deeply about Primary Mathematics” we must be critical consumers of the advice we receive about the use of manipulatives and the first step is thinking about why we make the decisions we make and what our intentions are. Don’t just use them for the sake of it. Be deliberate, be considerate, allow yourself the chance to be more effective.