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Many students just seem to look for some numbers and do something with them, hoping they solve the problem. Alexander had recently learned about using modeling for word problems in a workshop he had attended. Alexander presented the following problem: Lily and her brother, Scotty, were collecting cans for the recycling drive. Alexander went over the problem and drew a rectangular bar divided into two parts on the board, explaining that each part of the rectangle was for the cans collected on one of the weekends and the bracket indicated how many cans were collected in all. Alexander asked students what was not known, and where the given numbers would go and why.He began to share the model diagrams with his teammates and they were excited to see how students might respond to this approach. One weekend they collected 59 cans and the next weekend they collected 85 cans. This resulted in the following bar model: The class then discussed what equations made sense given the relationship of the numbers in the bar model. Alexander's teammates mentioned that they noticed a much higher degree of interest and confidence in problem solving when Mr. Everyone noticed that many more students were successful in solving problems once modeling was introduced and encouraged.While students need to experience many real-life situations they will get bogged down with the "noise" of the problem such as names, locations, kinds of objects, and other details.
Interestingly, it also informed the development of curriculum in Singapore, as they developed the "Thinking Schools, Thinking Nation" era of reforming their educational model and instructional strategies (Singapore Ministry of Education, 1997).
The bar model is a critical part of "Singapore Math." It is used and extended across multiple grades to capture the relationships within mathematical problems.
Forsten, 2010, p.1Students often have regarded each word problem as a new experience, and fail to connect a given problem to past problems of a similar type.
Students need to sort out the important information in a word problem, and identify the relationships among the numbers involved in the situation.
Research has also validated that students need to see an idea in multiple representations in order to identify and represent the common core (Dienes, undated).
For word problems involving the operation of addition, students need to experience several types of problems to generalize that when two parts are joined they result in a total or a quantity that represents the whole.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.Are the parts similar in size, or is one larger than the other?Once students are comfortable with one kind of diagram, they can think about how to relate it to a new situation.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.They even practiced several model diagrams among themselves as no one had ever learned to use models with word problems. This time many students wrote the equation, 59 85 = ? As the class continued to do more word problems, the diagrams appeared to be a helpful step in scaffolding success with word problems.Since part of their PLC work freed them up to observe lessons in each others' rooms, they decided they would watch Mr. Word problems require that students have the skills to read, understand, strategize, compute, and check their work. Following a consistent step-by-step approach-and providing explicit, guided instruction in the beginning - can help our students organize their thoughts and make the problem-solving task manageable.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.