There’s no magic in Maths MANIPULATIVES…

There’s no magic in that plastic! (Although, truth be told, we don’t do plastic around here.) 

As is common in mathematics education, the research suggests that manipulatives and representations are just TOOLS. It’s actually HOW they are used that is important. Maths manipulatives need to be used PURPOSEFULLY and APPROPRIATELY in order to have an impact.

It's only when learners use the manipulatives themselves to support their own SENSE MAKING that they will begin to see their power as TOOLS rather than just relying on them as crutches… I’ve written a blog on this already THINKING BOARDS - REPRESENTING MATHEMATICAL THINKING

For children to use concrete representations effectively without increased demands on their processing capacity, they must know the materials well enough to use them automatically (Boulton-Lewis, 1998) 

Moyer (2001) draws attention to the need for familiarity of the learner with the resource that is being used as a tool so as to reduce the cognitive demand of its use.

What to do? 

Let learners of ALL ages have a chance to play freely with a range of  MANIPLUATIVES for a while before setting a mathematical task (Black, 2019). Otherwise getting them to address the agenda (say, counting) can be at best inefficient, and at worst, a serious struggle! (Sarama and Clements, 2016). 

Although children can and do learn pre-mathematical foundations through their self-directed play, especially with structured manipulatives, such as pattern blocks or building blocks (Seo and Ginsburg 2004). However, these experiences are rarely mathematical without grown-up guidance (Sarama and Clements 2016).

Some research indicates the more manipulatives, the better (Moscardini, 2009; Carpenter 1999). However, there are opposing practices and evidence… 

Successful teachers in Japan tend to reuse the same manipulatives repeatedly (Uttal 1997). Research indicates that, indeed, deeper experience with one manipulative is more productive than equivalent experiences using various manipulatives (Hiebert and Wearne 1996).

The former is what successful Chinese teachers do (Ng and Rao 2010). A synthesis may be that multiple representations are useful (e.g., a physical manipulative and corresponding virtual manipulative, drawings, verbalizations, symbols), but many different manipulatives may be less useful.

These few manipulatives should be used for multiple tasks, so children do not view them as objects to play with but tools for thinking (Sowell 1989).

Do not neglect fingers as manipulatives, as they play a fundamental role (Crollen and Noël 2015).

For this we need to introduce RICH MATHEMATICAL TASKS which encourage children to use the manipulatives to investigate and understand the mathematical structures and processes and can support their mathematical truths and arguments. 

We want to discourage learners using them in a rote manner, without a sense of the ways in which the apparatus can reflect the mathematical structures. The learner needs opportunities to make sense of both the manipulative used and their relation to the mathematical idea which they are being used for. 

Our Number TOOLKITS contain RICH TASKS that encourage learners to use  TOOLS to demonstrate results and prove their mathematical truth in some sense (as this helps to develop mathematical thinking and support conceptual understanding).

You can download my favourite 3 TOOLS HERE for FREE... These few manipulatives can be used for multiple tasks I share on social media.

It should be noted that some teachers tend to use different manipulatives to increase “motivation” and “make math more fun” (Moyer 2000; Uttal 1997). By using manipulatives for ‘fun math,’ teachers artificially set up a classroom situation in which materials may not be used effectively. If “teachers act as if student interest will be generated only by diversions outside of mathematics” (Stigler and Hiebert, 1999).

FINAL WORDS...

Manipulatives are meaningful for learning only with respect to  learners activities and thinking:

To further inform your thinking  here are some useful links to literature that support the research and the pedagogy:

 Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education.