Content writer Siobhan Lawrence explains why visualising manipulatives is suitable for everybody.
Interlocking cubes, Cuisenaire rods, counters, dice, dominoes, geo-boards, to name but a few, are all examples of manipulatives and physical objects which you might associate with a primary classroom. Yet, in many secondary schools, these concrete ways of demonstrating a concept have often been left behind and are sometimes regarded by students as ‘babyish’. Indeed, these objects can sometimes be found buried in the deepest, darkest parts of the storage cupboard. Either that or teachers only find themselves encouraged to use manipulatives to aid their low attaining sets, whilst higher attainers are expected to understand the maths using a more abstract approach therefore harbouring a reputation that manipulatives don’t really belong in a secondary maths classroom. However, encouraging pupils to think about (and connect) a number of different representations of a concept is actually a really powerful learning tool, and one which should be encouraged more!
Manipulatives provide a depth of learning and mathematical understanding as they encourage ALL students to create multiple visual representations of the same concept. By representing these ideas in such a visual way allows for better consolidation as when there is less confusion, deeper understanding can develop and grow. Additionally, the use of manipulatives can begin to progress an ‘ownership’ of knowledge, which has other benefits too; if a student fully understands a concept, then a fear of the subject matter is reduced.
Finally, the use of manipulatives helps a working memory as students can keep track of what they are doing and display the process. This can be transferred in encouraging students to show their working out in exam questions. They also allow students to make use of reasoning skills and, once more, feeds into consolidation and depth of mathematical understanding; all of which are common and integral to a secondary classroom as well as a primary one.
There is no doubt that manipulatives enhance the abilities of students, at all levels, to reason and communicate mathematically. Working with manipulatives deepens understanding of concepts and relationships and makes learning meaningful: all key foci for the new mastery approach to the maths curriculum.
Top tips for Using Manipulatives in the Secondary Classroom
- Make sure the manipulatives are in the classroom (and not in storage!); they must be easily accessible.
- Incorporate them purposefully! Ensure they’re integral to the lesson to help remove the stigma that using manipulatives are for more low achievers and are, instead, fundamental in discovering patterns and making connections
- Challenge students to use the manipulatives, as an extension, to demonstrate the concepts covered in lessons.
- Use manipulatives often: as the use of them increases, students get used to the fact that the equipment is available. It will help reduce the stigma surrounding the objects and actually encourage them to explain their thinking, convince a friend or show their working out using them.
An Activity Idea Using a Manipulative
Build two consecutive triangular numbers from interlocking cubes – preferably in their ‘right angled’ form.
Now give the smaller one a turn so that it complements the next number to form a square.
This will allow students to begin exploring the relationship between triangular and square numbers: that when two consecutive triangular numbers are added together, it will make a square number. You could then extend your students further by discovering the formula for the nth triangular number.