Question: I’m previously Montessori trained and as a result believe that maths should remain as hands on, large -scale and tactile as long as possible through early and middle childhood – would you agree?

Alex Hodgkiss answered on 6 Mar 2018:

Great question! Concrete manipulatives and physical materials definitely have a place in maths education, but there seem to be some caveats. First, we know that young children’s thinking is not entirely ‘concrete’, as Piaget and some other theorists previously suggested, and it’s not the case that it’s best to choose concrete manipulatives for younger children, just because their thinking is more suited to these materials.

Also, it seems that in some cases, over reliance on concrete examples may lead to less transfer. E.g., Kaminski, Sloutsky, and Heckler (2009) found that 11 year olds and undergraduates exhibited less transfer (i.e. ability to use and apply knowledge to new contexts) when taught with concrete versus more abstract versions of concepts. Another related, interesting study by Novak et al (2014) compared the use of concrete objects and gestures (e.g. miming moving digits across the equals signs) to learning a maths concept, and found that gesture was better for transfer. Their argument here is that we want to push children along an action–> abstraction continuum, and that gestures are more abstract versions of action.

This all fits with Bruner’s idea that we first teach concepts with concrete manipulatives, before children form their own images and then symbolic representations (which interestingly seems to be part of the model for Shanghai mathematics method?). I agree with Rebecca on her point, here.

So in answer to your question, I think maths can and should be hands on, but, this needs to be paired with more abstract and symbolic representations, and the connections between these levels representations should be explicitly made by the teacher, i.e, children need specific guidance on the use of these manipulatives.