Standard C.1. Mathematics Concepts, Practices, and Curriculum

Middle level teachers need strong conceptual understandings, procedural fluency, and factual knowledge of the mathematics studied in elementary, middle, and high schools and understandings of how to develop middle level content. Middle schools often offer a range of courses, including those that offer review of elementary content and those that offer high-school-course content (e.g., algebra and geometry). Additionally, middle school mathematics has a strong focus on fluency with rational numbers and algebraic thinking, which has traditionally been taught via a narrow school mathematics curriculum in which rote procedures, rather than mathematical practices and processes, were emphasized. The content and practices related to the well-prepared beginning teacher of mathematics at the middle level are described in elaborations in this section.

In the summaries below, we describe the significant concepts that a well-prepared beginning teacher of mathematics at the middle level must know to support middle level learners. These ideas connect to and reflect The Mathematical Education of Teachers II (MET II) (CBMS, 2012) and Statistical Education of Teachers (SET) (Franklin et al., 2015) content expectations, and, therefore, we include the related essential ideas from these reports in each section.

Ratios and Proportional Reasoning. Well-prepared beginning teachers of mathematics at the middle level have the following skills and dispositions (see also Table 6.2): (a) understand a ratio as a distinct entity representing a relationship different from the quantities it compares, (b) recognize the difference between proportional and nonproportional situations, (c) experience problem situations involving more than one variable with both direct and inverse variation, and (d) know and use a variety of strategies to solve problems involving ratios and proportions (CBMS, 2012; Lamon, 2012). Well-prepared beginners understand the content that precedes ratios and proportions (multiplicative comparisons and fractions) and content that follows ratios and proportions (linear functions). They are able to select tasks that lead to the use of multiple approaches, connect to relevant contexts (e.g., financial literacy), and connect to other content within mathematics (e.g., geometry and algebra). Well-prepared beginners therefore encourage students to employ a range of reasoning strategies, including rates and scaling, ratio tables, tape or strip diagrams, double number line diagrams, and equations (proportions) (Ercole, Frantz, & Ashline, 2011; Olson, Olson, & Slovin, 2015).

The Number System. Well-prepared beginning teachers of mathematics at the middle level have a unified understanding of the number system (an expected outcome for middle school students in the CCSS-M [NGO & CCSSO, 2010]). They understand number and the ordering of numbers in the system of rational numbers, recognize fractions, decimal fractions, and percentages as different representations of rational numbers. They understand algorithms, visual representations, and context-based problems associated with rational number operations. They are able to analyze students’ algorithms and representations to provide feedback that connects conceptual and procedural knowledge. Well-prepared beginning teachers of mathematics understand the properties of the operations and know that these properties provide access to efficient or novel solution strategies. MET II describes essential understandings for number systems (see Table 6.3).

Algebraic Thinking and Functions. Well-prepared beginning teachers of mathematics at the middle level have strong understandings of algebraic thinking, noticing the central role of generalization and the use of variables to represent numbers. MET II describes essential understandings related to expressions and equations, and for functions (see Table 6.4). In particular, these beginning teachers understand algebraic thinking as (a) the study of structures in the number system, including those arising in arithmetic; (b) the study of patterns, relations, and functions; and (c) the process of mathematical modeling (Kaput, 2008; Lloyd, Herbel-Eisenmann, & Star, 2011). Well-prepared beginners are aware that symbols such as equal signs, inequality symbols, and square-root symbols can be confusing for students and that connecting these symbols to their meanings is central to success in algebra. For example, they may ask such questions as “What answers make sense given the context of this problem?” or, “Does a solution set containing only the number 0 indicate having a solution?” They understand the critical importance of equivalence and approach the teaching of algebraic concepts by explicitly attending to equivalence, for example, asking students to justify methods for simplifying expressions by justifying that the new form is equivalent to the original.

Well-prepared beginners have strong conceptual and procedural understandings of linear equations, systems of linear equations, linear functions, and slope of a line. They understand that functions describe a relationship or situation in which one quantity determines another and are able to help learners understand the meaning of function and use appropriate notations and representations. Well-prepared beginners have strong understandings of how mathematical representations such as graphs, tables, and equations support and influence algebraic and functional thinking. They have strategies for integrating representations into instruction in ways that help students move flexibly among representations and connect each representation to the given context/situation.

Geometry and Measurement. Well-prepared beginning teachers of mathematics at the middle level see the value of geometry for middle level learners; they are able to connect geometry to measurement and algebra to showcase the importance of geometry across the content while understanding that geometry itself is relevant and important for middle level learners (Sinclair, Pimm, & Skelin, 2012a). They have strategies for connecting geometry to ratios, proportions, and algebraic thinking when they explore scale drawings and transformations. Another critical connection between algebra and geometry important to middle grades mathematics is the Pythagorean Theorem; well-prepared beginners can explain why the Pythagorean Theorem is true (e.g., by decomposing a square in two ways), apply the Pythagorean Theorem, and select high-quality tasks and facilitate lessons in which their students explain and use the Pythagorean Theorem.

Well-prepared beginners understand that measurement is an important and useful mathematics content strand and therefore should be applied to authentic, culturally relevant contexts. Additionally, measurements are interrelated and must be understood and taught in ways that showcase the connections, such as connections within area formulas and connections among length, area, and volume concepts and formulas. Table 6.5 lists the essential understandings provided in MET II for Geometry (in which they include concepts related to measurement).

Statistics and Probability. Well-prepared beginning teachers of mathematics at the middle level know the content described in Statistical Education of Teachers (SET) (Franklin et al., 2015) and MET II (see Table 6.6). They have experience with the important process of doing statistics, as explained in the American Statistical Association's (ASA's) Guidelines for Assessment and Instruction in Statistics Education (GAISE) (2007), which includes formulating questions, collecting and analyzing data, and interpreting results. Well-prepared beginners are committed and able to engage students in statistical thinking and data-analysis processes beyond computation of values, calculation of statistical measures, and rote creation of data displays that draws focus away from in-depth analysis. They have solid understandings of variability, and are able to describe the center and spread of data, understand which measures might be used in which situations, and engage students in the selection of measures to describe data. Well-prepared beginners understand, and help students understand, that different types of graphs and other data representations provide different information about the data, and, therefore, the choice of graphical representation can affect how well the data are understood. Well-prepared beginners have strong understandings of bivariate data, how to represent them, and how to help students see their connections to proportional and algebraic reasoning. Additionally, they are able to compare two data sets, noticing differences and making inferences about the two populations.

Well-prepared beginners have deep understandings of chance and the misconceptions that people have about chance (e.g., that the chance occurrence of five heads on a coin toss has no effect on whether another head will occur on the next coin toss). They appreciate that experiments, including simulations, can help students understand probability concepts as well as understand how probability is used in real-life contexts beyond games. Additionally, well-prepared beginners understand and help their students understand connections among probability, random sampling, and inference about a population.

Mathematical Process and Practices. Well-prepared beginning teachers of mathematics at the middle level must be able to demonstrate mathematical practices and processes while they are learning (or relearning) the mathematics and statistics content. In these demonstrations they are able to model such things as considering various options for solving a problem; selecting an efficient strategy given the numbers or variables in the problem; using appropriate representations, models, and tools; and noticing underlying structures or patterns that can provide students insights into solving problems. Additionally, well-prepared beginners know what mathematical practices or processes are described in their state’s middle level mathematics standards (e.g., the Mathematical Practices in the CCSS-M [NGA & CCSSO, 2010]) and know that these standards are reflected in the middle level content and are not additional, discrete topics.

Well-prepared beginners are aware of physical and technological tools that support learning of middle level mathematics. Selecting appropriate tools is a critical mathematical practice for middle level learners while they transition from more concrete content to more abstract mathematical representations. Technology tools enable teachers and students to connect differing representations of mathematical concepts and build learner knowledge within and among the representations (NCTM, 2014a). Algebraic thinking, for example, includes significant use of equations, tables, and graphs. Well-prepared beginners use virtual tools such as graphing utilities (e.g., data-graphing tools, dynamic geometry software, and electronic spreadsheets) as well as physical models and tools in various ways (e.g., for enrichment, whole-class instruction, review, new instruction) to develop students’ knowledge and understandings of mathematics.

Although beginning teachers of mathematics will not see all connections between mathematical practices and content, well-prepared beginners are able to articulate strong connections between practices and content for at least one major middle school concept as well as describe some connections across the middle school topics described in Elaboration ML.1. For example, they understand the importance of contextualizing and decontextualizing with regard to mathematical problems presented through authentic situations. They apply that understanding to content instruction (e.g., of ratios) through using contextualized settings in which they have students explore the ways in which solution strategies or procedures can be derived. Vignette 6.1 provides a series of course activities as examples of support for candidates' developing deeper understanding of proportional reasoning while modeling the mathematical practices (and considering teaching practices).

Several aspects of this vignette are important to note. First, this carefully sequenced series of activities, along with carefully selected tasks, provides candidates opportunities to simultaneously develop and deepen content knowledge, mathematical practices, and teaching practices. Because candidates are just beginning to anticipate student thinking, they benefit from the scaffolding of activities as in the series described here. Second, at the center of these activities are actual middle level learners. Candidates considered how middle level learners might solve problems, analyzed how they actually solved the problems, and experienced first-hand how middle level learners think about these tasks. Third, candidates enter their fieldwork prepared and with a focus on student learning. This experience is quite different from sending students to field placements in which the amount they learn and what they learn about is dependent on their assigned classroom. Because each candidate interviewed a student using the same collection of tasks, the class could collectively discuss and learn about the content, mathematical practices, and teaching practices.

Understanding how content builds on other content is a critical component of a middle level teachers’ content knowledge. As discussed in Chapter 2, available content progressions provide considerably more detail on how content develops in sophistication over time than is apparent in most standards documents. Well-prepared beginners recognize that content progressions and learning trajectories are important resources, while also recognizing that each learner is unique, with different prior knowledge and different ways he or she might approach solving problems and engaging in mathematics. For example, the progression of proportional reasoning is critical knowledge for middle level teachers. Ratios are grounded in multiplicative comparisons learned in elementary school. In middle school, students explore unitizing and rates. Students may reason about ratios and rates in many ways, from informal strategies to the use of diagrams, ratio tables, graphs, and equations. Students’ ways of reasoning may make sense to them personally or reflect the way they learned about multiplicative comparisons or ratios in school previously or at home. Well-prepared beginners are able to determine and support the individual ways students reason about ratios and rates, while also helping each student deepen his or her knowledge by understanding other ways of solving ratio and proportion problems. Vignette 6.1 (above) provides an example of how course experiences can be designed to help teacher candidates deepen their understandings of student thinking related to proportional reasoning and how that reasoning grows in sophistication over time and through carefully sequenced experiences.

A solid foundation in proportional reasoning leads to developing the concept of slope as based on ratio and proportional reasoning with respect to linearity, which similarly can be conceptualized in a variety of ways. Valuing different ways of thinking about ratios, rates, proportions, slope, and so on is important in developing each student’s mathematical identity.

When well-prepared middle level candidates seek to solicit students' thinking to determine where students might be on a learning trajectory, they recognize that middle level learners may feel awkward in sharing their unique thinking. Although having a collection of strategies for assessing each learner’s unique understandings may be beyond the beginner's scope, well-prepared beginners understand the importance of looking for students’ unique mathematical reasoning and have strategies for soliciting, understanding, and respecting the mathematical representations and explanations of their students. For example, a teacher may notice a novel strategy of a student and invite that student to project their solution, inviting the class to see how that strategy is like and different from their own and eventually asking students when they might use one strategy over another.