Why maths and more importantly why number and place value?
Over the years during our travels to developing countries, we have visited schools in local communities. We have noticed that all maths teaching takes place with the teacher at the front of the class teaching to the same concept to every child using a blackboard and chalk and with no manipulatives.
Place value is the most fundamental concept in maths. Children need to construct meaning for themselves and the most productive way to do this is by using manipulatives to represent concepts where discussions can occur and authentic co-operative activities can take place. Maths using the base 10 system is universal and manipulatives provide a common language to communicate. Ideally we want children to solve real world problems with hands on materials and learning aids. By volunteering our time in schools we hope that the teaching of mathematics by teachers will change.
What are manipulatives and what benefit do they have on learning?
Manipulatives are concrete objects that can be viewed and physically handled by children in order to demonstrate, illustrate, model or discover abstract mathematical concepts. Manipulatives increase the child’s motivation to learn by changing the child’s perspective of maths. Children need to use concrete materials first and eventually they will see how this connects with written procedures in mathematics.
Children do not have mental maturity to grasp abstract mathematical concepts in words or symbols alone and need many experiences with concrete materials for learning to occur. Children must understand what and why they are learning for it to be permanent.
Our manipulatives have both visual and tactile appeal and can be manipulated by learners through hands-on experiences and evidence suggests that kinesthetic experience can enhance perception and thinking in children.
The findings of much research has shown that students who use manipulatives during mathematics instruction outperform students who do not (Driscoll, 1983; Greabell, 1978; Raphael and Wahlstrom, 1989; Sowell, 1989; Suydam, 1986).
Luzic & O’Connell (2001) found that long-term use of manipulatives has a positive effect on student achievement by allowing students to use concrete objects to observe, model, and internalize abstract concepts.
The importance of place value, as explained my Ma and Pa Kettle.
The importance of retraining teachers
In developing countries many schools practice content based teaching – teaching according to the syllabus and delivering the curriculum in the easiest way they can. Many primary school teachers lack the training in the teaching of mathematics competencies to transform mathematical ideas into representations. Teachers need training in appropriate strategies for using manipulatives as their beliefs about how students learn mathematics may influence how and why they use manipulatives as they do. We need teachers to change their thinking and focus on teaching mathematical processes (what mathematicians do) rather than solely on mathematical content (what mathematicians know).
We know children can remember, for short periods of time, information taught through books and teacher directed sessions, deep understanding and the ability to apply learning to new situations requires conceptual understanding that is grounded in direct experience with concrete objects. Therefore the teacher has a critical role in assisting children to connect their manipulative experiences, through a variety of representations, to essential abstract mathematics. With well trained teachers and worthwhile experiences with hands-on learning, we can provide students with powerful learning in mathematics.
Piaget (1952) suggests that children begin to understand symbols and abstract concepts only after experiencing the ideas on a concrete level.
Dienes (1960) extended this to suggest that children whose mathematical learning is firmly grounded in manipulative experiences will be more likely to bridge the gap between the world in which they live and the abstract world of mathematics.