The main mathematical categories in the relations and operations core were discussed in Chapter 2, and the steps through which our four age groups move were summarized in Box 5-2. These steps are elaborated in Box 5-10.

In the relations core, children learn to perceive, say, discuss, and create the relations more than, less than, and equal to on two sets. Initially they use general perceptual, or length, or density strategies to decide whether one set is more than, less than, or equal to another set. Gradually these are replaced by more accurate strategies: They match the entities in the sets to find out which has leftover entities, or they count both sets and use understandings of more than/less than order relations on numbers (see research reviewed in Clements and Sarama, 2007; Fuson, 1988). Eventually, in Grade 1, children begin to see the third set potentially present in relational situations, the difference between the smaller and the larger set (see research reviewed in Fuson, 1992a, 1992b). In this way, relational situations become the third kind of addition/subtraction situations: comparison situations.

In the operations core, children learn to see addition and subtraction situations in the real world by focusing on the mathematical aspects of those situations and making a model of the situation (called mathematizing these situations, as explained in Chapter 2). Initially such mathematizing may involve only focusing on the number of objects involved rather than on their color or their use (I see two red spoons and one blue spoon) and using those same objects to find the answer by refocusing on the total or counting it (I see three spoons in all). The three types of addition/subtraction situations that children must learn to solve were discussed in Chapter 2 and summarized in Box 2-4. These types are change plus/change minus, put together/take apart (sometimes called combine), and comparisons.

Addition and subtraction situations, and the word problems that describe such situations, provide many wonderful opportunities for learning language. Word problems are short and fairly predictable texts, so children can vary words in them while keeping much of the text. This enables them to say word problems in their own words and help everyone’s understanding. English language learners can repeat such texts and vary particular words as they wish, all with the support of visual objects or acted-out situations. Although children need to learn the special mathematics vocabulary involved in addition and subtraction, these problems also give them wonderful opportunities to integrate art (drawing pictures) and language practice and pretend play while also generalizing their growing mathematical knowledge.

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