2   Introduction DOUBLE TAP TO ZOOM WITH PHONE OR TABLET be seen as environments, both physical and intellectual, that are created to bring mathematics alive for children. Teachers who establish micro-​math cultures in their classrooms mathematize daily routines, activities, play, ex- plorations, and investigations that children encounter through relationships with their friends, families, and environments. Within any chosen task or activity, mathematizing teachers can help chil- dren make sense of the math concepts they are learning. Take for example infant and toddler children who love to fill and dump water into cups, bowls, and other containers at the water table. A teacher with a mathematizing eye observes and identifies the mathematics that the children are investigating, in this instance the concept of conservation and its accompanying variables (volume, capacity, adding, subtracting). Adults can promote critical thinking by having children explore water with cups, sponges, and many other tools and materials. Teachers can also support children’s linguistic development by incorporating language-​modeling techniques such as labeling, expand- ing/extending language, and parallel and spiral talking (see chapter 4) to help children develop language and conversation skills. A teacher’s math language at the water table could include statements such as these: • “You are pouring the water into the cup!” • “You are squeezing the sponge and filling the cup!” • “The cup is full!” • “You are dumping the water from your cup!” • “Your cup is empty now. The water is all gone!” At the preschool level, the same mathematical conversations could con- tinue, with the opportunity of including more advanced math vocabulary and inquiry to the exchanges. This will help children develop and make meaning of the math concepts they are investigating. Preschool teachers can use language such as the following: • “Your cup is half full!” • “The water is reaching the top of your cup!” • “I wonder how many cups of water it will take to fill the tall container?” • “Why do you think it took the same number of cups to fill the tall con- tainer and the short, wide container?” The Mathematizing for Learning Process (MLP) approach (see chapter 1) is an effective framework that can support teachers in their interpretation, COPYRIGHTED MATERIAL