For young children, developing number sense is critical, and manipulatives play an important part that process.
But are all manipulatives created equal? When we consider that true number sense involves understanding of quantity as it relates to place-value, then the answer may surprise you.
Look at the two different representations of the quantity eight:
Though it is important to give children experiences with a wide range of tools, which model would be a more useful tool for building understanding of number?
For many students, the first representation would be much too abstract. In order to verify the quantity, they would have to count each tile or cube one by one. However, when they see eight on a ten-frame, they are able to identify the quantity of eight as it relates to ten. This visual may be more helpful in developing and retaining the sense of ten.
Seeing quantities as instantly recognizable without having to recount from one is an important step in the stages of counting. Students develop cardinality when they can understand that the final number they counted represents the quantity of objects in a set. The ability to subitize comes later when students can instantly recognize a quantity of objects.
The use of a ten-frame helps children build mental images, develop the ability to subitize, and foster understanding of part/whole relationships, all important in strengthening a foundation for later work with place-value. The use of manipulatives such as cube-trains, though helpful in early states of counting, does not provide an instantly recognizable and distinct picture of the wholeness of ten.
The use of the ten-frame also aids in assessing a student’s ability to reason abstractly, construct viable arguments about mathematics, and look for and make use of structure – three of the Mathematical Practice Standards within the Common Core.
Consider these student’s responses to the following mathematical models of eight:
Ten-frames easily show the order and organization of our base-ten number system. They provide students with the flexibility of counting each dot individually, or seeing a quantity of dots as combinations of smaller groups of dots. Students can also see a particular quantity and it's relation to ten. More importantly, a student is able to develop multiple strategies for counting quantities and understand the connections between them.
If you have not made use of the ten-frame in your classroom, I encourage you to try it! You can download a free set here. Happy counting!!
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