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  1. Models • "thinker-toys" • help children explore ideas and make sense of them. • manipulatives, calculators and computers should be readily available for student use as a regular part of your classroom environment Jamar Pickreign, Ph.D. 2005

  2. Models • many types of models can be used to represent mathematical concepts. • concrete models, or manipulatives, are frequently used to initiate understanding in mathematics. • remember that the math is in the relationships that the learner constructs through use of representational models • learners must actively construct these relationships Jamar Pickreign, Ph.D. 2005

  3. "Commercial" Manipulatives • numerous commercially manufactured manipulatives available • examples: base blocks, connecting blocks, centimeter rods, various counters (colored tiles, two-sided counters, etc.), attribute blocks, pattern blocks, fraction sets, geoboards, rulers, scales, thermometers, measuring cups, play money, tangrams, pentominos, geoblocks, dice, spinners, dominos, and calculators. Jamar Pickreign, Ph.D. 2005

  4. "Handcrafted" Manipulatives • many manipulative models can be made by teachers and learners using commonly available material • having learners create their own manipulatives may provide a sense of ownership in the model and opportunity to gain conceptual insight Jamar Pickreign, Ph.D. 2005

  5. "Virtual" Manipulatives • there also exist “virtual” models or “on-line thinker tools.” • many make use of current technologies allowing users to manipulate computer generated graphics similarly to traditional models. • there are a number of these models available on the internet ( http://illuminations.nctm.org/,http://standards.nctm.org/document/eexamples/index.htm,http://matti.usu.edu/nlvm/nav/index.html) Jamar Pickreign, Ph.D. 2005

  6. "Virtual" Manipulatives • As with any model, care must be exercised to select those that will facilitate thinking and learning. • A good rule of thumb is to decide whether a model allows the user to think, or if it does the thinking for her or him. Jamar Pickreign, Ph.D. 2005

  7. Using Concrete Manipulatives-Some Guidelines • Successful use of manipulatives can be positively affected by the way they are introduced and incorporated into the classroom. • Joyner (1990) offers some simple "rules." As you plan lessons that include the use of manipulatives, think about the following: Jamar Pickreign, Ph.D. 2005

  8. Packaging the manipulative. • Re-sealable plastic bags, or boxes can be used to store enough of the model for an individual learner to use • ease of distribution at the start and collection when finished. • encourages learners to be responsible for their tools. Jamar Pickreign, Ph.D. 2005

  9. Free exploration or play • allow the learners some time to explore and familiarize themselves with the material. • they gain comfort with the material, • diminishes the novelty of the material, • can reduce the "distractibility" when using them to model mathematics concepts. Jamar Pickreign, Ph.D. 2005

  10. Demonstrate • Show the learners how the material can be used to model mathematics concepts. • Remember that the mathematics is not in the material; a bean is a bean, but a bean can be a model of the number one. Jamar Pickreign, Ph.D. 2005