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The manipulative experience is an important part of how children learn math. I would actually call it interacting with blocks rather than playing, but regardless of what you call it, it is a necessary first step for learning. There are stages of learning through which children pass. These stages start with children physically interacting with objects, growing into representing the objects with pictures or diagrams, and finally maturing into abstract thinkers who can communicate their thinking with math notation. Teachers and parents need to be careful when using manipulatives, however. In the United States, the manipulatives most commonly used with young children are single objects that can be counted, Unifix cubes, bottle caps, chips, or buttons. While these manipulatives have great benefits in the very early stages of counting and modeling problems, they do little to support the important strategies needed for automaticity. In fact, at a certain point they begin to reinforce low-level counting strategies. For example, to solve 6 + 7 with Unifix cubes, children need to count out six, then seven, and then either “count on” as they combine or (as is most common) count three times – first the two sets, then the total. Because the materials have no built-in structure, they offer little support for the development of alternative strategies. Building structure into manipulatives is not always beneficial by itself, however. For example, base ten blocks have the base ten structure built in, the problem is that while the structures may be apparent to adults they are not always apparent to children” (Fosnot and Dolk, pp. 104-106. Young Mathematicians at Work Constructing Number Sense, Addition, and Subtraction, 2001). The assumption is that if children just use the materials enough, they will “take in,” or “come to see,” the arithmetical structure. This doesn’t always happen without guidance in “seeing” the structures. That is why talking about how they got the answer is so important. These conversations are when adults have the opportunity to point out and name the structures the children are describing. Adults working with children need to listen closely as they work with the tools to gain insights into the child’s thinking. That allows you to know where the child is and you can build on from there. Manipulatives are a great tool for showing what children are thinking; however, be careful not to assume that the child sees the math structures in the same way that you see them as an adult. For instance, you could be working on the place value idea of tens and ones and decide to use dimes and pennies as the manipulative. On the surface this seems very appropriate, but any Kindergarten or First Grade teacher will tell you that not all students at these ages can use coins in the way you intend. If the student has not developed the ability to “unitize” (see the one dime as having a value of ten cents), they are not seeing the coins the same way you are as an adult. If you put out two dimes and four pennies and ask the child how many, you are just as likely to get the answer of 6 (2 silver and 4 brown or 6 coins) as you are to get the intended answer of 24 cents. The materials cannot transmit knowledge; the learner must construct the relationships and not all students will construct those relationships, at the exact same time or in the exact same way. The most widespread misuse of manipulative materials occurs when the teacher tells students, “Do as I do.” There is a natural temptation to get out the materials and show children exactly how to use them. Children will blindly follow the teacher’s directions, and it may even look as if they understand. It is just as possible to get students to move blocks around mindlessly as it is to teach them to “invert and multiply” mindlessly. Neither promotes thinking or aids in the development of concepts (Ball, 1992). Physical models are important instructional tools but they are not substitutes for lessons that promote reflective thinking. (John Van de Walle, page 33 Elementary and Middle School Mathematics Teaching Developmentally, 2007) Children usually develop concepts in a fairly predictable manner, but the time it takes each individual can vary greatly. The path most students progress through can be summarized as: Concrete -> Representative-> Math Notation When students are learning about computation and our base-ten number system Carpenter, Fennema et al have discovered that this development is greatly enhanced when the numbers are presented in a problem so that the numbers have contextual meaning. For instance, in the addition sentence: 4 + 7 = __, the numbers would be housed in a simple story to help build context. If you had 4 cookies and I gave you 7 more, how many cookies would you now have? In this way, the child has a mental image or can create a pictorial one to represent the action, and the numbers and symbols make more sense because they are in a context to which children can relate. This theory is known as Constructivism and is based on the work of Jean Piaget and Jerome Bruner. For more about Constructivism and the implications of Bruner’s theories on the classroom: Click Here
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