collection of three things, real or imagined. Thus, the addition problem 3 + 2 = ? provides an abstract formulation for a vast number of actual situations in the world around one. The abstract nature of mathematics is part of its power: Because it is abstract, it can apply to a virtually limitless number of situations. But for children to use this mathematical power requires that they take situations and problems from the world around them and formulate them in mathematical terms. In other words, it requires children to mathematize situations.

Mathematizing happens when children can create a model of the situation by using mathematical objects (such as numbers or shapes), mathematical actions (such as counting or transforming shapes), and their structural relationships to solve problems about the situation. For example, children can use blocks to build a model of a castle tower, positioning the blocks to fit with a description of relationships among features of the tower, such as a front door on the first floor, a large room on the second floor, and a lookout tower on top of the roof. Mathematizing often involves representing relationships in a situation so that the relationships can be quantified.

For example, if there are three green toy dinosaurs in one box and five yellow toy dinosaurs in another box, children might pair up green and yellow dinosaurs and then determine that there are two more yellow dinosaurs than green ones because there are two yellow dinosaurs that do not have a green partner. With experience and guidance, children create increasingly abstract representations of the mathematical aspects of the situation. For example, drawing five circles instead of five yellow ducks or drawing a rectangle to represent the side of a box of tissues and, later, writing an equation to model a situation. Children become able to visualize these mathematical attributes mentally, which helps in various kinds of problem solving. Children also need eventually to learn to read and to write formal mathematical notation, such as numerals (1, 2, 6, 10) and other symbols (=, + , −) and to use these symbols in mathematizing situations. Thus, mathematizing involves reinventing, redescribing, reorganizing, quantifying, structuring, abstracting, and generalizing what is first understood on an intuitive and informal level in the context of everyday activity (Clements and Sarama, 2007).

Specific Mathematical Reasoning Processes

Mathematics learning in early childhood requires children to use several specific mathematical reasoning processes, also known as “big ideas,” across domains. These big ideas are overarching concepts that connect multiple concepts, procedures, or problems within or across domains or topics and are a particularly important aspect of the process of forming connections. Big ideas “invite students to look beyond surface features of



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