*Young students often solve beginning word problems, acting them out, and modeling them with the real objects of the problem situation, e.g. Over time they expand to using representational drawings, initially drawing pictures that realistically portray the items in a problem, and progressing to multi-purpose representations such as circles or tally marks.*After many concrete experiences with real-life word problems involving joining and separating, or multiplying and dividing objects, teachers can transition students to inverted-V and bar model drawings which are multi-purpose graphic organizers tied to particular types of word problems.An increase in nonlinguistic representations allows students to better recall knowledge and has a strong impact on student achievement (Marzano, et. In classic education research, Bruner (1961) identified three modes of learning: enactive (manipulating concrete objects), iconic (pictures or diagrams), and symbolic (formal equation).

*Young students often solve beginning word problems, acting them out, and modeling them with the real objects of the problem situation, e.g. Over time they expand to using representational drawings, initially drawing pictures that realistically portray the items in a problem, and progressing to multi-purpose representations such as circles or tally marks.After many concrete experiences with real-life word problems involving joining and separating, or multiplying and dividing objects, teachers can transition students to inverted-V and bar model drawings which are multi-purpose graphic organizers tied to particular types of word problems.An increase in nonlinguistic representations allows students to better recall knowledge and has a strong impact on student achievement (Marzano, et. In classic education research, Bruner (1961) identified three modes of learning: enactive (manipulating concrete objects), iconic (pictures or diagrams), and symbolic (formal equation).*

Whether the items are bears, balloons, or cookies no longer matters as the students see the core idea of two subsets becoming one set.

Dienes discovered that this abstraction is only an idea; therefore it is hard to represent.

Diagrams can capture the similarity students notice in addition/joining problems where both addends are known and the total or whole is the unknown.

Diagrams will also be useful for missing addend situations.

A model can help students organize their thinking about a given problem, and identify an equation that would be helpful in solving the problem.

Models are a kind of graphic organizer for the numbers in a word problem, and may connect to students' work with graphic organizers in other subjects.

Using models is a critical step in helping students transition from concrete manipulative work with word problems to the abstract step of generating an equation to solve contextual problems. Alexander and teachers from his grade level team were talking during their Professional Learning Community (PLC) meeting about how students struggle with word problems.

By learning to use simple models to represent key mathematical relationships in a word problem, students can more easily make sense of word problems, recognize both the number relationships in a given problem and connections among types of problems, and successfully solve problems with the assurance that their solutions are reasonable. Everyone felt only a few of their students seem to be able to quickly generate the correct equation to solve the problem.

A student who has become proficient with using a part-part-whole bar model diagram when the total or whole is unknown, (as in the collecting cans problem in Mr.

Alexander's class), cannot only use the model in other part-part-whole situations, but can use it in new situations, for example, a missing addend situation.

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