## Math Manipulatives:

How do they aid student learning?

With insight from practicing teachers,

Dr. Julie Comay explores how math manipulatives — both physical and virtual — may help and hinder mathematical development .

Manipulatives — physical materials used to represent and solve problems — are ubiquitous in elementary mathematics classrooms. While their use is commonplace and their virtues sung by Canadian educators, both observation and research suggest a less straightforward picture. Studies show children regularly fail to draw critical connections between the materials and what they purport to represent. Their manipulations take on a life of their own, divorced from sense. As math education professor Jenni Back puts it: “The artefact becomes a prop to support the following of the procedure. It doesn’t have to be like that and indeed in other cultures it isn’t.”^{1}

A 2013 review of 55 studies (with more than 7,000 K-12 students) by Carbonneau et al. compared manipulative-supported learning with symbolic approaches. Though the results of individual studies were far from consistent with one another, the meta-analysis identified overall benefits to the use of manipulatives, with moderate effects on retention and smaller effects on problem-solving, transfer, and justification. However, these benefits were *least* apparent for the youngest age group (three to six year olds).^{2} The age caveat is surprising and interesting because manipulatives tend to be most prevalent in early years programs. It prompts rethinking of how we use manipulatives in kindergarten.

As a wide-eyed teacher candidate, I watch my host teacher lead her Grade 2 students through a multidigit addition problem. There’s something magical in her deft manipulations of popsicle sticks (single or bundled in groups of 10 and 100) in a series of bundlings and unbundlings that astonishingly replicate the conventional pen-and-paper algorithm for adding with regrouping. She reaches the end and asks the nearest child to count the sticks. When the result meshes with the written answer, it seems like pure sleight of hand – I am in awe, but then, I already know the answer. That it works is undeniable, but why it works perhaps less evident. Not having any idea what the answer should be, the children are rather less impressed, and as they dutifully arm themselves with popsicle sticks and elastics to try out a problem on their own, the magic dissipates in the fumbling with materials, shot elastics, misapplied procedures, and (disappointingly) incorrect answers. The teacher’s sure, fluid series of concept-laden actions have turned into another arbitrary, cumbersome school procedure to be replicated with little sense or meaning. It makes one think twice about what’s going on.

## What exactly are manipulatives?

As the etymology suggests, traditional manipulatives are intimately connected with the hands, manually manipulable tools used to represent mathematical abstractions such as number. They range from purpose-created materials designed to embody mathematical ideas and relationships (abaci, relational (Cuisenaire) rods, Base-10 blocks, Rekenreks) to materials such as interlocking cubes or pattern blocks used for a broader variety of purposes. Informal objects found in daily life (buttons, money, dice, paper clips) also play a large role in many classrooms as children count, compute, classify, and pattern with them. Regardless of the material, the rationale for its use, based on the developmental and constructivist theories of such thinkers as Jean Piaget, John Dewey, Jerome Bruner, Maria Montessori, and Friedrich Froebel, reflects the idea that children’s thinking only gradually moves from the concrete to the symbolic, and that abstract concepts originate in physical experience. “Kids need to have things in their hands to build number sense,” as one teacher said.

Developmentally and culturally, the first mathematical manipulatives may be – quite literally – the hands. Finger counting has a wide reach and a long history; its varied manifestations across geographic and ethnic

groups have interesting implications for understanding cultural differences in what has been called “embodied numerosity.” Researchers in this area claim that, “symbolic number magnitude processing is partially rooted in learned finger-counting habits, consistent with a motor simulation account of embodied numeracy… that argument is supported by both cross-cultural and within-culture differences in finger-counting habits”.^{3} For fascinating examples of cultural variations, see Bender and Baller.^{4} Exploring different ways in which students use their fingers to count to 10 may provide one fruitful starting place for a culturally responsive mathematics program.

The three-year-old who holds up three fingers to show her age is just beginning to use her fingers symbolically. Even after a child no longer overtly uses fingers to compute math problems, it’s common to see them attending to barely detectable finger twitches that may recall the sub-vocalizations of the early reader. Math educators Boaler and Chen cite evidence that training finger perception can improve math skill in this engaging 2016 Atlantic article: Why kids should use their fingers in math class. Included is a series of exercises designed to promote finger discrimination, including this PDF developed by YouCubed.

## Manipulatives as symbols

Despite their tangible concreteness, manipulatives must also serve as *symbols* for mathematical ideas to carry mathematical meaning. Yet symbolic thinking isn’t a given for young children. This has been shown, for example, in their inability to transfer knowledge of a hidden object from a scale model to a full-size room until the representational connection between the two settings has been spelled out for them.^{5} To use manipulatives meaningfully, children must learn to view them in two ways, both as objects in themselves (popsicle sticks, wooden cubes) and as the ideas they represent (number, place value, etc). While this dual representation seems to happen naturally in pretend play contexts, where a block becomes a train in the blink of an eye, the connection is less obvious when objects represent immaterial ideas. Math educator Deborah Ball spells it out memorably: “Understanding does not travel through the fingertips and up the arm … Mathematical ideas really do not reside in cardboard and plastic materials”.^{6}

In their dual role, manipulatives make concepts visible and available for reflection.

“I can’t think of any area of math that doesn’t benefit from the use of manipulatives, certainly when introducing new concepts, but also regularly when children are challenged to work at and just beyond their zone of proximal development,” explains Kindergarten teacher Carol Stephenson. “They often successfully solve computations,

patterning challenges, algebraic problems, etc. with manipulatives beyond what they could accomplish without them. I imagine many adults are the same.”

I was reminded of Carol’s words the other day when a friend recalled her five-year-old daughter’s fascination with the small square tiles in their bathroom: “We were all puzzled and a bit annoyed when she started spending prolonged periods in the bathroom. Door closed, no running water… Finally we asked her what she was doing. ‘Come see’, she said, ‘it’s cool.’ It turned out she was using the tiles visually as a grid for figuring out times tables. She had never heard the word ‘multiplication’, but that’s what she was doing.”

Manipulatives are also used to illustrate or communicate thinking or, relatedly, prove a case, and it may be this communicative function that plays the strongest role in classrooms as children move up the grades. Even in kindergarten, “children can often show their understanding through the use of manipulatives before they can verbally articulate this understanding,” Carol adds.

Grade 3 teacher Michael Martins finds manipulatives “great to use for students to test their ideas or demonstrate/confirm their thinking about a concept or problem.”

## Concreteness and abstraction

While physical experience is likely instrumental in building basic number sense for young children, manipulatives may play a different role for older students engaged in complex reasoning. The recent inclusion of virtual manipulatives into the armory of available tools further highlights the question as to whether it’s primarily the physicality of manipulatives that supports mathematical thinking.

Sometimes drawing a simple picture is an easier way for a child to express and make sense of a problem.

Other times visualization, or written notation, may offer the most direct route to a solution. Julie Sarama and Doug Clements^{7} argue against oversimplifying a one-way trajectory from concrete to pictorial to abstract understanding, suggesting that concreteness is relative and learners continuously move among and connect those three levels of representation.

## Manipulatives in special education

Peltier and colleagues found a clear impact of manipulative use on attention, engagement, and math performance across all topics for students at-risk or identified with a learning disability^{8}. Unlike the 2013 Carbonneau analysis, this was true at all ages. Perhaps the focus on special education settings, in which teaching tends to be more direct and explicit, accounts in part for the increased early years effectiveness found in this recent meta-analysis. Indeed, its authors noted the almost universal use of explicit instruction when incorporating manipulatives across all 45 studies they examined.

Students with motor-coordination difficulties or challenges with spatial organization may initially need structured support around the physical use of manipulatives until they have learned protocols for their use. The time taken to build this familiarity is well spent because these are often the students for whom manipulatives can be most helpful. Once students gain facility with a material, it offers an external marker that can support working memory over a series of steps.

## 4 guidelines for using manipulatives in classrooms

Developed by Laski and colleagues, the following guidelines for manipulative use are based on cognitive science principles.^{9}

- Use each type of material consistently over an extended period of time. With repeated use, children deepen their understanding of how that material works in relation to the concepts explored. Their application becomes less rote, more flexible, and more concept-driven.
Mike Martins elaborates: “Familiarity with a material and how it relates to a concept helps to build confidence and frees students to make deeper meaning of concepts over repeated exposures. Also, having time to see peers interact with the material in ways that may be different from how a particular child engaged with it can help foster growth with concepts. However, offering the students different materials in order to illustrate a concept allows students to transfer the key learnings and make sense of the ideas with greater clarity. As they extrapolate the ideas and learnings which are similar between the materials, conceptual understanding can develop.”

- When introducing concepts, begin with materials that best (most transparently) depict the concept.
For example, a 10-frame filled with counters represents a unit of 10 more obviously than a dime. Over time, shifting to more arbitrary representations may support the move to conventional notation and abstract thinking.

- Avoid manipulatives that “resemble everyday objects or have distracting irrelevant features.” For example, if the purpose is counting, all objects counted should be uniform in size and shape.
That said, educators such as Marilyn Burns have claimed that objects with real-world correlates – such as bear counters – help children draw connections with their own experience

^{10}. There may be times where simple, uniform models of animals or people help to build a plausible narrative context and draw children into the story world of the problem. Whether there’s a difference between naming a wooden cube a person and using a humanoid object is an open question.

- Be explicit about the relation between the manipulative and the math concept – do not assume that it is self-evident.
When we already know how numbers work, it’s easy to see how an effective manipulative reflects that knowledge. But if, like children, we’re using manipulatives to

*discover how*numbers work, we will likely need some help to reflect on the meaning of our manipulations.

## Which kind of manipulative is best?

The 2013 Carbonneau et al. review showed a clear advantage to what they called “bland” as opposed to “perceptually rich” materials. Extraneous features and decorations added to make manipulatives more appealing to children tend to distract students from the relevant mathematical aspects and reduce the likelihood of conceptual connections.

Yet it may depend upon what you’re looking for. Studies on the use of money, for example, suggest that using real money (perceptually rich) as compared with less-detailed money-like (blander) materials may help with problem-solving but hinder accuracy. And ethnographic work showing the extraordinary prowess of street children engaged in buying

and selling is a reminder that decontextualized school knowledge and highly contextualized real-world understanding can be two separate universes, each with its own strengths and demands.

Clements and Sarama add that manipulatives without explicitly embedded math are generally more versatile and help to foster flexibility in thinking. For example, Cuisenaire rods without marked divisions or absolute values allow for a broader and more spatial grasp of number relationships. A single rod can represent infinitely many numbers – what matters is the relation of that rod to the other rods, as is beautifully demonstrated in this video.

There may be times when specialized manipulatives with limited applicability (such as Base-10 blocks) suit a lesson’s purposes. In general, though, it’s advised to stick to a small number of less explicit materials that embody a broad range of possibilities rather than try to introduce new purpose-built materials for each new concept.

“I generally prefer manipulatives that leave something to figure out, or multiple ways of considering things,” says

Carol Stephenson. “For example, I obviously use Cuisenaire rods quite a bit, but I like the ones that DO NOT have the notched lines. I think that steers children into using them more explicitly numerically, and may have them lean less into their spatial reasoning skills. However, some materials are perfectly suited to the task at hand. If you are working on measurement and weight, for example, precise weights and scales are immensely useful, and satisfying – even for five-year-olds!”

## What about play?

Many teachers, like Mike Martins in Grade 3, respect children’s drive to use manipulatives for their own purposes as well as to satisfy curricular demands. “I think students deserve to have time to play with the materials and find their utility for themselves. Once this time has been spent, outlining expectations about the materials so that the community can continue to learn from one another in a respectful way is essential.” After students have satisfied their curiosity about the affordances of a manipulative – creating symmetrical structures, building towers, designing beautiful patterns, incorporating them into storytelling – they will be much more ready and interested to work within the constraints of a math class.

Since the ultimate aim in using manipulatives is to move beyond the physicality of the object to the concept it represents – for the material to

become as transparent as possible – it makes sense that having sufficient opportunity to explore physical properties of the materials will free children up to discover their mathematical properties.

An initial period of play and exploration does more than forestall the kind of experimenting that distracts from the purposes of a lesson. If a teacher takes this play seriously in its own right, is attentive and interested and invites sharing of discoveries within the group, it can lead into wonderfully focused mathematical work and conversation. Though it has been argued that all undirected play with manipulatives compromises the mathematical purity of the materials, we would simply caution teachers not to rely on children’s play to engender mathematical understanding on its own.

## Which is better? Physical or virtual?

Doug Clements and Julie Sarama have long advocated on behalf of virtual manipulatives, claiming that their capacity to be tailored to represent a concept exactly (the ultimate bland manipulative, perhaps), allows for more precise and efficient work with mathematical concepts and relations^{11}. Working with virtual manipulatives may ease the challenges of dual representation for young children since the objects themselves are less physically salient. In addition, the neutrality of online feedback can be helpful for an anxious or self-conscious student. Secondary school teachers also suggest that working with online manipulatives may be a more socially acceptable way to support struggling math students as they revisit elusive concepts.

On the other hand, a current review of 29 studies showed that spatial training that used concrete manipulatives was more effective than computerized training in building mathematical skill.^{12} In addition, some teachers and developmentalists express unease about prematurely detaching understanding from the hands and body. While Stephenson appreciates the online tools on a Smart Board that allow everyone to focus on the same thing, from the same perspective, “that was only after heaps and heaps of hand work… [overall] it just doesn’t make sense to me to replace experiences with three-dimensional objects with two-dimensional representations.”

And it doesn’t have to be one or the other. Jane Tom at Pegamigaabo School in North-west Ontario has developed a program that combines the benefits of each mode, using online representations of spatial problems on a Smart Board to stimulate hands-on work with physical materials.

Despite some differences, depending on the circumstances, empirical findings have generally shown the effectiveness of both modes. For better or worse, the past couple of years have provided us with a prolonged natural experiment into virtual learning, and it is likely that a great deal more evidence, both anecdotal and controlled, will be emerging in the near future.

Practically speaking, virtual manipulatives alleviate the challenges of managing large amounts of stuff in a crowded classroom, including issues of cost, accessibility, distribution, maintenance, and organization. Immature motor control matters less when pattern blocks, for example, snap into alignment and dice stay in their place instead of flying across the room. Noise is another reality of the material world, and a teacher opting for physical manipulatives will want to find ways to dampen the sounds of 35 students all rattling dice at the same time!

## Final thoughts

Though fine-grained contextual evidence is still emerging, there is general consensus among teachers and researchers that students at all levels can benefit from the spatial opportunities afforded by manipulatives for visualizing mathematical ideas and building the kinds of visual-spatial models that are widely used by practicing mathematicians. Used effectively, manipulatives create bridges to challenging concepts and stimulate students to draw connections and pose hypotheses. In their visibility, they externalize unarticulated lines of reasoning and generate reflection and discussion among students.

The aim for a teacher is not to teach by rote new procedures for

getting the right answer with manipulatives but to understand the potential of the materials to express mathematical principles and relationships. With this understanding, teachers are better able to make reasoned decisions about which manipulatives serve which purposes (and for whom), and how to best exploit them under their own particular set of circumstances.

Finally, and not least, working with manipulatives in a problem-centred environment can be a highly motivating experience for children, bringing in the kind of playful, aesthetic and creative dimensions that characterize much of the most interesting work in mathematics.

## References

1. Back, J. (2013). Manipulatives in the primary classroom. *Nrichmaths.org/10461.*

2. Carbonneau, K.J., Marley, S.C. and Selig., J.P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. *Journal of Educational Psychology, 105, *380-400.

3. Morrisey et al. (2016). Cross-cultural and intra-cultural differences in finger-counting habits and number magnitude processing: Embodied numerosity in Canadian and Chinese university students. *Journal of Numerical Cognition, 2(1), *1-19.

4. Bender, A. and Beller, S. (2012). Nature and culture of finger counting: diversity and representational effects of an embodied cognitive tool. *Cognition, 124(2), *156-82.

5. DeLoache, J.S., Uttal, D.H. and Pierroursakos, S.L. (1998). The development of early symbolization: Educational implications. *Learning and Instruction, 8(4), *325-339.

6. Ball, D. (__Summer, __1992). Magical hopes: Manipulatives and the reform of math education. *American Educator, 42.*

7. Sarama, J. and Clements, D.H. (2016). Physical and virtual manipulatives: What is ‘concrete’? In P.S. Moyer-Packenham (Ed.), *International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives, *Mathematics Education in the Digital Era 7, 71-93.

8. Peltier, C., Morin, K.L., Bouck, E.C., et al. (2020). A meta-analysis of single-case research using mathematics manipulatives with students at risk or identified with a disability. *Journal of Special Education, 54(1), *3-15.

9. Laski, E.V., Jamilah, R.J., Daoust, C. and Murray, A.K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. *Sage Open, *1-8.

10. Burns, M. (1996). How to make the most of math manipulatives. *Instructor*, 105, 45-51.

11. Clements, D.H. (1999). ‘Concrete’ manipulatives, concrete ideas. *Contemporary Issues in Early Childhood, 1(1), *45-60.

12. Hawes, Z.C.K., Gilligan-Lee, K.A. & Mix, K.S. (in press). Effects of spatial training on mathematics performance. *Developmental Psychology**.*

*13. *Clements, D.H. (1999). ‘Concrete’ manipulatives, concrete ideas. *Contemporary Issues in Early Childhood, 1(1), *45-60.

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