Manipulatives are objects that can be handled and moved and are used to develop learners’ understanding of a mathematical situation (Griffiths et al., 2017). The popular concrete-pictorial-abstract (CPA) heuristic advocates starting with more practical, hands-on modes of learning before progressing to the use of images then finally formal mathematical symbols and equations. Many teachers can talk confidently about how manipulatives fit into this process but fewer can justify why they select one manipulative over another. Indeed, a pedagogical misunderstanding in this respect can lead to ambiguities such as using pictorial representations of manipulatives in text books as a replacement for the physical resource itself (Merttens, 2012). An abundance of literature spanning the best part of a century provides six areas for consideration that teachers using manipulatives may find helpful:

Have a clear rationale

Teachers need to ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept (EEF, 2020). For example, would using inter-linking blocks or moving counters between two circles be more effective when calculating within 10? It is recommended that, after the focused learning objective is clear, the teacher then chooses or devises a manipulative-rich activity that he or she concludes will best help students attain the desired objective.

Provide the appropriate level of guidance

Conflicting recommendations have been provided to teachers concerning the level of instructional guidance a teacher gives (Carbonneau et al., 2013). Supporters of high levels of instructional guidance argue that without teacher guidance, students left to their own devices will not learn concepts or, worse, learn the wrong concepts (Horan & Carr, 2018). Those in favour of low instructional guidance suggest that control of decisions relating to mathematical tools should not be claimed solely as the teacher’s domain (Moyer & Jones, 2004). Ideas of ‘transitioning’ guidance advocate moving to lower guidance as students gain skill and fluency (Horan & Carr, 2018), such as removing concrete counting resources as students demonstrate increased fluency.

Allow sufficient time

Within his vision of mastery learning, Benjamin Bloom identified the time allowed for learning as a key strategy (Bloom, 1968). The consensus among researchers seems to be that it is most effective to use a specific manipulative consistently over a long period of time (Laski et al., 2015). Allowing sufficient time is crucial to promote deeper understanding and reasoning. In lessons which support the deepest levels of thinking and reasoning, students are given plenty of time to work with the manipulatives (Stein & Bovalino, 2001), often utilising a whole lesson to guide students in the operational use of a certain manipulative.

Bland or rich?

The perceptual richness of a manipulative has an impact on its efficacy depending on the content being delivered and the conceptual understanding being promoted. Results tend to be in favour of learning with the use of perceptually bland manipulatives which typically have a simplistic form (Carbonneau et al., 2020) if the manipulative is to embody the idea of dual representation; bland manipulatives allow children to focus more on the mathematical concept. However, the notion of a simplistic ‘one size fits all’ approach should not be applied when weighing the merits of using a rich manipulative or bland one.

‘The notion of a simplistic “one size fits all” approach should not be applied when weighing the merits of using a rich manipulative or bland one.’

Link to the abstract ideas

Linking manipulatives to abstract symbols and ideas is a key pedagogic principle for their effective use (Griffiths et al., 2017), and students must understand the links between the manipulatives and the mathematical ideas they represent (EEF, 2017). Assuming students will interpret an abstract resource in the same way as their teacher without explicit exemplification is a dangerous game; some students might not understand that a place value counter with ‘10’ on it represents 10 ones. To use a symbolic object effectively, teachers must focus more on what the symbol is intended to represent and less on its physical properties (Uttal et al., 2009).

Organisation is key

Good lessons using manipulatives do not just happen; they are the product of much advance thought and preparation (Stein & Bovalino, 2001). Teachers who are more successful with manipulatives both design their own lessons and prepare the classroom and the manipulatives for the activity. Routines such as grouping students, bagging their manipulatives and having them ready to go on their tables may sound obvious to some, but negating these simple steps can lead to a detrimental effect on both behaviour and engagement. This preparation included rehearsing with the manipulatives to preempt any misconceptions by anticipating problems the students might encounter.

How do these practical and pedagogical factors combine to form your preference when it comes to choosing a manipulative for your lesson? Manipulatives in themselves are not a panacea (Chinn, 2004) so only by considering this question carefully will you ensure that they can support your teaching and learning in an effective way.

References

Bloom, B. S. (1968). Learning for mastery. Instruction and curriculum. Regional education laboratory for the Carolinas and Virginia, topical papers and reprints, number 1. Evaluation Comment, 1(2). https://files.eric.ed.gov/fulltext/ED053419.pdf

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. https://doi.org/10.1037/a0031084

Carbonneau, K. J., Min Wong, R., & Borysenko, N. (2020). The influence of perceptually rich manipulatives and collaboration on mathematic problem-solving and perseverance. Contemporary Educational Psychology, 61, 101846. https://doi.org/10.1016/j.cedpsych.2020.101846

Chinn, S. (2004). The trouble with maths: A practical guide to helping learners with numeracy difficulties (1st ed.). Routledge.

Education Endowment Foundation (EEF]. (2017). Improving mathematics in key stages 2 and 3. https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3//

Education Endowment Foundation (EEF]. (2020). Improving mathematics in the early years and key stage 1. https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/early-maths/

Griffiths, R., Back, J., & Gifford, S. (2017). Using manipulatives in the foundations of arithmetic. Nuffield Foundation. https://www.nuffieldfoundation.org/sites/default/files/files/Nuffield%20Main%20Report%20Mar%202017web(1).pdf

Horan, E. M., & Carr, M. M. (2018). How much guidance do students need? An intervention study on kindergarten mathematics with manipulatives. International Journal of Educational Psychology, 7(3), 286. http://dx.doi.org/10.17583/ijep.2018.3672

Laski, E. V., Jor’dan, J. R., Daoust, C., & Murray, A. K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. SAGE Open, 5(2). https://doi.org/10.1177/2158244015589588

Merttens, R. (2012). The ‘Concrete-Pictorial-Abstract’ heuristic. Mathematics Teaching, 228, 33–38. https://atm.org.uk/journal/archive/mt228files/atm-mt228-33-38.pdf

Moyer, P. S., & Jones, M. G. (2010). Controlling choice: Teachers, students, and manipulatives in mathematics classrooms. School Science and Mathematics, 104(1), 16–31. https://doi.org/10.1111/j.1949-8594.2004.tb17978.x

Stein, M. K., & Bovalino, J. W. (2001). Reflections on practice: Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6(6), 356–359. https://doi.org/10.5951/MTMS.6.6.0356

Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual representation and the linking of concrete and symbolic representations. Child Development Perspectives, 3(3), 156–159. https://doi.org/10.1111/j.1750-8606.2009.00097.x