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Using Visual Models to Build Mathematical Ideas
Interacting with visual models is a key strategy for developing students' understanding and fluency with fractions. Moving to efficient and effective Fractional Strategies relies on understandings that are built through opportunities for students to construct their own models as well as interact with given visual models.
"Models are not the end, they are the means to the mathematics." (Petit, Laird, Marsden, & Ebby, 2016, p.6)
Initially, models are used to build an understanding of a concept (e.g., fractions are numbers). Through multiple opportunities to generate and interact with models coupled with teacher questioning, sharing solutions, and classroom discussions, students begin to transition away from reliance on visual models toward a deeper understanding of fraction concepts and fluency. Each time a new fraction concept is introduced the process begins anew.
The use of visual models (both teacher and student generated) should permeate instruction; not just be an incidental experience, but a way of thinking and learning for students.
- Students should have the opportunity to solve problems in which they interact with visual models
(e.g., shade 3/8 of an area model or find 3/8 on a number line);
- Students should have the opportunity to solve problems by generating their own visual models;
- Students should have the opportunity to use visual models to develop understanding of concepts (e.g., use a visual model to show that and are equivalent);
- Teachers should build instruction on student generated visual models to help students generalize mathematical ideas. Teachers should ask students to explain their visual models and respond to probing questions that capitalize on understandings in their visual models.
For an example of how one might use questioning to extend understanding from a visual model, see Kelyn's solution on page 5 of A Focus on Fractions: Bringing Research to the Classroom.
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Action Item
- Identify some OGAP questions that use visual models to help develop a concept (e.g., EO3, E26). Have students solve the problem(s). Use student solutions to further strengthen their understanding of the targeted concept.
- Identify some OGAP questions that have the potential to elicit student use of visual models to solve the problem (s) (e.g., CF06, 05, ASC03, ASC09). Analyze the evidence in their work and decide where on the OGAP Fraction Progression the solutions are and your next instructional step.
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