[Maths-Education] importance of physical materials for mathematics in elementary school

Mon Jun 25 11:01:00 BST 2007

Hi from stormy Nottingham,

I am reminded here of work I did with John Mason in the OU Course Developing Mathematical Thinking where the framework of "manipulate, get a sense of, articulate" was used to develop the notion that the learning or development should not be seen to lie in the artefact or in the manipulation but in the cognition and articulation arising form out of that - turning the activity and the artefact into something deeper. This is - as Basil Fawlty said stating "the bleeding obvious" but maybe the manipulative is often seen AS the embodiment rather than a mediation and transitional.

Peter 
*************************** 
Dr Peter Gates 
School of Education 
The University of Nottingham 
Nottingham NG8 1BB 
Tel: +44 0115 951 4432 (Direct line) 
Tel: +44 0773 0808 353 (Mobile) 
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-----Original Message-----
From: maths-education-bounces at lists.nottingham.ac.uk [mailto:maths-education-bounces at lists.nottingham.ac.uk] On Behalf Of Paul Ernest
Sent: Monday, June 25, 2007 10:53 AM
To: Mathematics Education discussion forum
Subject: RE: [Maths-Education] importance of physical materials for mathematics in elementary school

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Hi Keith (and everybody)

Your first point is an important one; any pedagogy or pedagogical tool can be performed ineffectively, without coherent aims and little gains

In the 80s we saw how 'investigatory maths' could be both great - in some cases - but also directionless, without conclusion or learning outcomes

Nevertheless, with regard to manipulatives: they can offer great potential benefits for a number of theoretical reasons

1. the enactive stage in Bruner's theory of curriculum progression.
Enactive experience can get deep into our embodied/body based learning and understanding. and provide a basis for Iconic, them Symbolic understanding

Eg, cutting out fractions, partitioning sweets; using pie/pizza diagrams for fractions; then symbolic fractions (various forms)

Bruner based his theory on a combination of Piaget and C S Peirce

Enactivism now has a big life of its own, using, eg, body based metaphors as a basis for understanding

2. SEN work emphasises learning styles - enactive/visual/auditory/written

enactive suits some learners very well (and this sortof ties in with Gardner's multiple intellgeive theory)

3 Structured concrete materials or concretely presented starting points can provide a shared, discussable, graspable (in both senses) basis for building mathematical activities (and concepts)

Admittedly careful attention needs to be directed at bridging such activities into more general understandings or the efforts can be wasted

4. Games often have a concrete element and these can be good learning media. Here's an old ref.

Ernest, P. ‘Games: A Rationale for their Use in the Teaching of Mathematics’, in P. Ernest, Ed. Mathematics Teaching: The State of the Art, London, Falmer Press, 1989

Jones D.K. wrote:
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> Arthur Baroody once wrote that "manipulatives don't come with 
> guarantees" and more recent work continues that theme (see references 
> below); for example Patricia Moyer concludes her study of 10 
> middle-grade teachers (in the US) by saying that the teachers' uses of 
> manipulatives "was little more than a diversion in classrooms where 
> teachers were not able to represent mathematics concepts themselves".
>
> There's also research in other fields (such as science education) 
> where physical activities are being compared with computer-based 
> activities (see references below). For example, Klahr and colleagues 
> recently concluded from their study that "children were able to learn 
> as well with virtual as with physical materials, the inherent 
> pragmatic advantages of virtual materials in science may make them the 
> preferred instructional medium in many hands-on contexts".
>
> So maybe the question is about how, and in what contexts, experience 
> with physical materials can be suitably combined with computer-based 
> activity...
>
> And, in terms of work in design technology (and maybe in 
> mathematics?), there are going to be interesting research questions 
> when (or if?) school students get to use the upcoming range of 3D 
> printers - see, for
> example:
> http://en.wikipedia.org/wiki/3D_printing
>
> Keith
> ---------
> Keith Jones
> University of Southampton
> School of Education
> UK
>
> UoS Mathematics Education Research Group http://www.crme.soton.ac.uk
>
>
> --------------
> Klahr, D., Triona, L.M. and Williams, C. (2007) Hands on what? The
> relative effectiveness of physical vs. virtual materials in an
> engineering design project by middle school children. Journal of
> Research in Science Teaching, 44(1), 183-203.
>
> Triona, L. M., and Klahr, D. (2003). Point and click or grab and heft:
> Comparing the influence of physical and virtual instructional materials
> on elementary school students' ability to design experiments. Cognition
> and Instruction 21: 149-173.
>
> ------------
> Baroody, A. J. (1989). Manipulatives don't come with guarantees.
> Arithmetic Teacher, 37(2), 4-5.
>
> Clements, D. H. & McMillen, S. (1996). Rethinking "Concrete"
> Manipulatives. Teaching Children Mathematics, 2(5), 270-279
> Online at:
> http://investigations.terc.edu/relevant/RethinkingConcrete.html
>
> Gilbert, R. K., & Bush, W. S. (1988). Familiarity, availability, and use
> of manipulative devices in mathematics at the primary level. School
> Science and Mathematics, 88, 459-469.
>
> Merseth, K. K. (1978). Using materials and activities in teaching
> addition and subtraction algorithms. In M. N. Suydam & R. E. Reys
> (Eds.), Developing Computational Skills (pp. 61-77). Reston, VA: NCTM.
>
> Moyer, P. S. (2001). Are we having fun yet? How teachers use
> manipulatives to teach mathematics. Educational Studies in Mathematics,
> 47, 175-197.
>
> Prigge, G. R. (1978), The differential effects of the use of
> manipulative aids on the learning of geometric concepts by elementary
> school children, Journal for Research in Mathematics Education, 9(5),
> 361-367.
>
> Raphael, D. and Wahlstrom, M. (1989), The influence of instructional
> aids on mathematics achievement, Journal for Research in Mathematics
> Education, 20(2), 173-190.
>
> Ross, R. & Kurtz, R. (1993). Making manipulatives work: A strategy for
> success. Arithmetic Teacher, 40, 254-257.
>
> Sowell, E. J. (1989). Effects of manipulative materials in mathematics
> instruction. Journal for Research in Mathematics Education, 20, 498-505.
>
> Thompson, P. W. (1992), Notations, conventions and constraints:
> contributions to effective uses of concrete materials in elementary
> mathematics, Journal for Research in Mathematics Education, 23(2),
> 123-147.
>
>> -----Original Message-----
>> From: maths-education-bounces at lists.nottingham.ac.uk
>> [mailto:maths-education-bounces at lists.nottingham.ac.uk] On
>> Behalf Of Katherine Mackrell
>> Sent: 25 June 2007 04:10
>> To: maths-education at lists.nottingham.ac.uk
>> Subject: [Maths-Education] importance of physical materials
>> for mathematics in elementary school
>>
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>> This message has been generated through the Mathematics
>> Education email discussion list.
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>> I have just met a Serbian teacher who informs me that there
>> is pressure to
>> remove the use of physical materials in design technology in
>> elementary
>> schools and to replace these with computer software.
>>
>> Much as I love 3D computer software, I share his concern with
>> regard to
>> students' basic understanding of 3D space and the impact that
>> this change
>> might make on students' mathematics.  Could people please
>> send me arguments/
>> references, etc that would summarise the importance of using physical
>> materials for developing mathematical understanding at the
>> elementary level?
>>   It's so obviously important to me that I'm not quite sure
>> where to start!
>>
>> Thanks
>>
>> Kate
>>
>>
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>>
>
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Best wishes

Paul
__________________________________________________
Paul Ernest
Emeritus Professor, Philosophy of Maths Education
Leader, Doctoral Programme in Maths Education
University of Exeter
School of Education and Lifelong Learning
St Lukes, Heavitree Road
Exeter  EX1 2LU, UK

Office:   Holnicote 107
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Check out Philosophy of Mathematics Education Journal, #20 (Special Issue
on Social Justice), at this web address

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