Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Whole-Number Concepts and Operations

IntroductionConsider a prospective elementary teacher (PT] solving 527 - 135, using the standard algorithm and explaining regrouping as follows:You put a 1 over next to the number and that gives you 10____I don't get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from? This PT clearly followed the correct procedure and arrived at the correct answer, but she was not able to provide an explanation for why this solution method results in a correct answer. Figure 1 shows her written work.Now consider another PT's reflection describing her inability to explain regrouping:I learned [at the beginning of my elementary mathematics methods class] that there was a lot more to the concept [of number and place value] than I was aware of. I am able to use math effectively in my everyday life, such as balancing my checkbook, but when I was presented with questions as to why I carry out such procedures as carrying1 and borrowing in addition and subtraction, I was stuck. I could not explain why I followed any of these procedures or rules. I just knew how to do them. This came as a huge shock to me considering I did well in most of my math classes. I felt terrible that I could not explain simple addition and subtraction.Both of these PTs have determined that they want to teach children, yet at this point neither of them would be able to conceptually help an elementary-aged child make sense of why regrouping works when using the standard algorithms taught in the United States. Moreover, solving a problem using the algorithms is not sufficient knowledge for teaching mathematics to children. In the United States, the National Council of Teachers of Mathematics (NCTM, 2000a] and the Common Core State Standards for Mathematics (CCSSM] (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010] call for children to develop a conceptual understanding (Hiebert & Lefevre, 1986] of the mathematics they encounter. Procedural fluency is one of several aspects of being mathematically proficient (National Research Council, 2001]; the other four aspects are conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. In order to be equipped to support students' development of mathematical proficiency, inservice teachers and PTs also need such an understanding of mathematics. Researchers have highlighted the need for teachers to have a deep and multifaceted understanding of the mathematics they teach (Hill, Ball, & Schilling, 2008; Ma, 1999]. Less clear, however, is how improvement in teachers' knowledge can be accomplished.At the core of elementary school mathematics is the teaching of number concepts and operations. NCTM (2000a] stressed that all pre K-12 students should "[a] understand numbers, ways of representing numbers, relationships among numbers, and number systems; [b] understand the meanings of operations and how they relate to one another; [and c] compute fluently and make reasonable estimates" (p. 32]. A conceptual understanding of number and operations underlies learning of all future mathematics and other STEM subjects. "Number pervades all areas of mathematics. The other four Content Standards [other than Number and Operations] as well as all five Process Standards are grounded in number" (NCTM, 2000b, HI]. In the CCSSM, "Number and Operation in Base Ten" is one of the focal domains in each grade from K through 5, followed by "The Number System" in Grades 6-8 and "Number and Quantity" in high school.Even with this strong focus on number throughout the K-12 curriculum, children in the United States and other countries "experience considerable difficulty constructing appropriate number concepts of multidigit numeration and appropriate procedures for multidigit arithmetic" (Verschaffel, Greer, & De Corte, 2007, p. …