Motivating prospective elementary school teachers to learn mathematics by focusing upon children's mathematical thinking.

Elementary school children in the United States are not developing acceptable levels of mathematical proficiency (National Center for Education Statistics, 1999), and a major concern of teacher educators is that teachers lack the depth and flexibility of mathematical understanding and the corresponding beliefs they need to teach for proficiency (National Research Council [NRC], 2001). Although teachers' mathematical content knowledge plays a critical role in their instruction (Fennema & Franke, 1992; Hill, Sleep, Lewis, & Ball, 2007), teachers need more than content knowledge to be effective. Beliefs about mathematics, teaching, and learning affect not only the ways teachers teach mathematics (Philipp, 2007; Thompson, 1992) but also the ways prospective teachers learn mathematics. In California, the development of the mathematical content knowledge of prospective teachers takes place in undergraduate courses and is separated from their consideration of issues of teaching and learning, which often does not occur until students attend mathematics methodology courses as college graduates in a credential program. This article is based upon my assumption that, for prospective elementary school teachers, separating the learning of mathematics from the consideration of issues of mathematics teaching and learning is counterproductive to their development of mathematical content knowledge and to the development of their beliefs about mathematics teaching and learning. After presenting the theoretical underpinnings and summarizing the data in support of the claim that prospective elementary school teachers (PSTs) benefit by learning about children's mathematical thinking concurrently while learning mathematics, I describe four principles that serve as the focus of a mathematics laboratory developed and implemented for PSTs at San Diego State University and at local community colleges.

Why We integrate Children's Mathematical Thinking Into Mathematics Courses

Developing deep understanding of the mathematics of elementary school is far more difficult than was once thought (Ball, 1990; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998). Furthermore, even when PSTs attend a thoughtfully planned course designed to engage them in rich mathematical thinking, many PSTs react to the course in a perfunctory manner. Most PSTs do not know what mathematics they need to know to teach effectively, and many are not open to approaching the content anew in a deeper and more conceptual way than they experienced in elementary school because they hold a self-perpetuating belief that "if I, a college student, do not know something, then children would not be expected to know it, and if I do know something, I certainly don't need to learn it again." Furthermore, many PSTs believe that mathematics is a fixed set of rules and procedures, and when combined with their belief that children and adults learn mathematics by being shown how to solve problems in a prescribed, step-by-step fashion, these beliefs clash with the more conceptual, meaning-making goals many mathematics-course designers hold for PSTs (NRC, 2001). The approach my colleagues and I have taken is based upon our belief that by providing PSTs opportunities to develop more nuanced beliefs about mathematics, teaching, and learning early in their undergraduate experiences, we might launch them on a different growth trajectory that may orient them toward learning mathematics from a relational or meaning-making, rather than an instrumental, perspective (Skemp, 1978).

When my colleagues and I approached the issue of teaching mathematics to PSTs, we asked ourselves what it is that PSTs care about in relation to mathematics teaching and learning. We decided that fundamentally, PSTs entered teaching because they cared deeply about children, and rather than try to get PSTs to care about mathematics for mathematics sake, we took the approach that we wanted PSTs to care about mathematics for the sake of the children they would one day teach. Our Circles of Caring model (see Figure 1) highlights how their thinking about children may lead to PSTs' learning mathematics. The innermost circle, Children, reflects PSTs' initial concern for children, which is to protect children and keep them comfortable, safe, and happy. Many PSTs initially associate their caring for children with the belief that they should avoid challenging children. However, when PSTs are supported so that they engage children in mathematical problem solving or when they observe carefully selected video of children solving problems, many of the PSTs' circles of caring begin to expand to include children's mathematical thinking. Furthermore, when they learn about children's mathematical thinking, many PSTs begin to redefine caring as including challenging children so that they grapple with meaningful mathematics. Finally, when PSTs are supported so that they begin to engage with details of children's mathematical thinking, many realize that, to be prepared to support children's learning, they must themselves grapple with the mathematics, and their circles of caring extend to learning mathematics. In other words, by having PSTs look at mathematics through the lens of children's mathematical thinking, we help them come to care about mathematics, not as mathematicians, but as teachers. Our approach is based upon an old idea. John Dewey (1902/1990) noted, more than 100 years ago, that every subject might be thought of as having two aspects, "one for the scientist as a scientist; the other for the teacher as a teacher" (p. 351). He wrote, "[The teacher] is concerned, not with the subject-matter as such, but with the subject-matter as a related factor in a total and growing experience [of the child]. Thus to see it is to psychologize it" (p. 352). Note that we view our approach as a way, not the way, to support PSTs' learning of mathematics. Although we recognize that PSTs can become excited about learning mathematics and that a mathematical approach may work for some, we have chosen to take a different starting point in our work with PSTs.

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We tested our theory using a large-scale randomized experimental study, and the results showed that PSTs who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not (Philipp et al., 2007). I share two comments from students to highlight how learning about children's mathematical thinking supported PSTs in learning mathematics. The first is from Phil, a student who was relatively strong in the mathematics class but who came to recognize that ...