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She attempts to steer the children toward the fraction that she wants them to make and reduce by rephrasing the question: “She spent eighteen of the twenty-four dollars. ” Unfortunately, now there is almost nothing left to solve. The context becomes irrelevant, and the children will sacriﬁce their own meaning making to accommodate what Susan wants. ” Annie, however, remains confused. Susan repeats the problem. “Think about it. Maria spends eighteen dollars out of the twenty-four she had. ” But Annie’s confusion is still apparent.
One fourth is more than one ﬁfth, and one eighth is more than one tenth—because you are dividing by more people. ” Michael, Ashleigh, and Gabrielle have made a huge jump in thinking as they have worked through this investigation. Starting with eliminating halves and comparing remainders as whole numbers (not in relation to the number of children), they progressed to unit fractions. As they attempted to represent their thinking, they combined the unit fractions into common fractions. They are now solid on understanding that the bigger the denominator, the smaller the fraction.
Teachers must have the horizons in mind when they plan activities, when they interact, question, and facilitate discussions. But horizons are not ﬁxed points in the landscape; they are constantly shifting. 3 depicts the landscape-of-learning framework. Thinking of teaching and learning as a landscape suggests a beautiful painting. But if learners can take so many paths and the horizons are constantly shifting, how do teachers ever manage? How do we help each child make the journey and still keep in mind the responsibility we have for the class as a whole?