Previously on ALTA Blog:
Last time out we explored both the perilous nature of engaging with the research into the use of concrete resources and established our first guiding principle: The more salient a resource, the less likely transfer becomes. In this piece we continue our journey through the available research and establish our second principle…
When exploring the use of manipulatives with other teachers, I like to set the following problem and ask them to make a quick drawing of how they might represent the underlying structure of the problem.
Task One: If you have time, why not try this now. Represent the underlying structure of this problem in a drawing.
When I pose this problem to teachers, the representations produced will vary (as is fully expected) and it acts as a great point from which to approach our second evidence-informed principle and somewhat of a bridge between principles one and two.
Built into the context of the problem is the structure of comparison. For me, the most effective way to model this structure is to use the cubes to create two horizontal bars through which to compare the value of each group. As the numbers involved become larger, a bar model is probably necessary, as I think there might be a point of diminishing returns when the number of cubes needed rises. Though where this point actually rests is up for debate.
In this example, the representation chosen was determined by the structure of the problem and used to model that structure to pupils. This is, however, just one way in which we might opt to use manipulatives. Martin and Schwartz (2005) outline four possible effects we might see (offloading, repurposing, induction and physically distributed learning) through the use of manipulatives which are, in turn, brought about through varying degrees of stability in the environment and ideas explored. I often say that this is the study I’m least sure I understand and that uncertainty grows with each passing day but it acts, if nothing else, as a demonstration of the fact that there are multiple reasons and potential outcomes from the use of manipulatives.
At the heart of every teacher’s decision to use manipulatives is a desire for pupils to make meaning through the manipulation of objects (designed to support pupils in developing their understanding of complex and often abstract ideas). But where does this meaning come from?
In the example above, the meaning is borne not from the cubes themselves but from their arrangement in a form which allows for direct comparison. In the case of the additive structure aggregation (the combination of two distinct groups) we might also opt to use cubes when composing our representation but it is the movement which distinguishes this structure from others (such as augmentation). The meaning, therefore, is constructed with, not by the manipulatives.
| Principle | Guidance |
| Meaning is constructed with, not by the manipulatives. | Use fewer, carefully chosen manipulatives. |
Task Two (If you’re playing along at home): What mathematical idea/concept could this image of a five-frame with two yellow and one red counter represent? How many different ideas can you collect?
So what does this mean for us in the classroom? When I ask teachers, from primary and secondary, what it might be possible to represent using the image of the five-frame above, we generate a multitude of possibilities. This image can effectively represent a wide array of concepts that span the entirety of mathematics. Therefore, it is crucial to approach its utilisation with great care. While we may not dictate the precise meaning that students derive from it, we do have the capacity to explicitly guide their attention and focus to the aspects of the representation that are most relevant and important for the concept at hand. This is where our second principle comes in: meaning is constructed with, not by the manipulatives.
This principle reminds us that manipulatives are not magic. They do not automatically convey meaning or understanding to pupils. They are tools that can be used in different ways for different purposes, depending on the mathematical idea we want to explore and the level of abstraction we want to achieve. Therefore, we need to be careful and deliberate in our choice and use of manipulatives, and avoid the temptation to use them indiscriminately or superficially.
We also need to be aware of the potential pitfalls and limitations of manipulatives, such as over-reliance, distraction, confusion, or misinterpretation. We need to scaffold pupils’ engagement with manipulatives, provide clear instructions and expectations, model and explain how to use them effectively, monitor and assess pupils’ understanding and reasoning, and gradually withdraw or replace them with more abstract representations when appropriate.
By doing so, we can help pupils construct meaning with manipulatives, rather than expecting them to do so by themselves. We can also help them make connections between different representations, and ultimately develop a deeper and more flexible understanding of mathematics.
To sum up, in this post we have established our second evidence-informed principle for using manipulatives in mathematics teaching: meaning is constructed with, not by the manipulatives. This principle highlights the importance of being intentional and purposeful in our use of manipulatives, and of supporting pupils’ transition from concrete to abstract thinking. In the next blog post, we will explore our third and final principle, which will address the question of environmental factors at which we are at the behest.
Until next time, thanks for reading.
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