Physical Models of SubtractionCarrying on from out last week's 'AHA' moment, where we realized we needed to support our students understanding of the concept of subtraction, even if they could already use the algorithm (how we were taught to subtract - or the stacking method as the students call it). From the book, Solving for Why: Understanding, Assessing, and Teaching Students Who Struggle With Math, by John Tapper, we know that we want our students to develop a broad conceptual understanding of mathematics as well as a procedural understanding. "Mathematical models are mental constructions that we all use to understand complex ideas (Schoenfeld 1994), Without models on which to ground our mathematical thinking, we may only attain the kind of knowledge that comes from practicing every kind of mathematical procedure we ever need. This understanding is not flexible; rather, it is bound to particular contexts. We can see this with our students when they are only able to solve problems that are matches for the instruction they've just received. They can, for example, solve problems with multi-digit multiplication, but only after they've just practiced it. Without a model to ground mathematical understanding, students are unlikely to generalize what they know (Ryan and Williams, 2007)." from Tapper, J. 2012. Solving For Why. Math Solutions; Sausalito, CA. As we explored physical models to represent the subtraction situation, 82 - 57, we realized that some models were just re-writing the numbers using manipulatives, for example playing cards. So, we worked on ways to represent the actual number of objects. We then realized it was important to organize our objects so that people could easily see the numbers we were subtracting. Various strategies were developed including; Grouping by 5s or 10s, colour-coding, using other manipulatives as dividers. Our next challenge was for students to use the manipulatives to show the regrouping required to solve this problem. A couple of groups demonstrated their thinking by using colour-coding and a series of steps. They were able to verbally explain what was happening in each step of the subtraction process. This group explained their process on video. Our next step was to have students demonstrate their thinking in a different context; by using representational or drawn models of the subtraction process. Several models showed the students clearly understood and could visually represent the process of subtracting 26 from 53.
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AuthorsMrs. Montanaro, Mr. Messer and Mrs. Austman teach grades 3/4 at Elbow Park School in Calgary AB. Archives
April 2017
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