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Young children aged 3 to 7 can learn a great deal about numbers. In a home or daycare environment, this learning can occur as children experience daily routines. Young children will learn to count, match, see, and compare numbers if caregivers or older children count, show objects, and point out small numbers of things. Such informal teaching can be done while children play, eat, get dressed, go up and down stairs, jump, and otherwise move through the day. These activities are engaging and fun but need to be encouraged and modeled by adults or more advanced children in the group. In larger day-care or school settings, numerical understanding results from similar informal learning opportunities combined with more structured experiences that enable all children to engage in supported learning activities with adult and peer modeling and help.

In this article, Laura Grandau traces a self-study research project focused on teaching algebra to fourth-grade students. Facing a new curriculum and a new grade level, Grandau considers what good instruction and "good habits of practice" may be. Through journaling, videos, observations, analysis of students' verbal and written responses, and consultations with her "critical friend" Sean (also a teacher), Grandau develops a "critical distance" from her teaching. While doing this reflective work, she negotiates the ambiguities of teaching and learning. Although Grandau's work is of interest to teachers of mathematics, those in critical friend or peer mentoring relationships, and teacher researchers, it also serves as a model of how inquiry teaches, fostering growth in teachers, students, and communities of practice.

Research on the learning and teaching of algebra has recently been identified as a priority by members of the mathematics education research community (e.g., Ball 2003; Carpenter and Levi 2000; Kaput 1998; Olive, Izsak, and Blanton 2002). Rather than view algebra as an isolated course of study to be completed in the eighth or ninth grade, these researchers advocate the reconceptualization of algebra as a strand that weaves throughout other areas of mathematics in the K–12 curriculum.

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