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

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).

### Table 6.2. Connections to MET II (CBMS, 2012) Related to Ratios and Proportional Reasoning at the Middle Level

MET II describes the following essential ideas for ratios and proportional reasoning:

• “Reasoning about how quantities vary together in a proportional relationship, using tables, double number lines, and tape diagrams as supports.
• Distinguishing proportional relationships from other relationships, such as additive relationships and inversely proportional relationships.
• Using unit rates to solve problems and to formulate equations for proportional relationships.
• Recognizing that unit rates make connections with prior learning by connecting ratios to fractions.
• Viewing the concept of proportional relationship as an intellectual precursor and key example of a linear relationship.” (p. 41)

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).

### Table 6.3. Connections to MET II (CBMS, 2012) Related to the Number System at the Middle Level

MET II describes the following essential ideas for the number system:

• “Understanding and explaining methods of calculating products and quotients of fraction, by using area models, tape diagrams, and double number lines, and by reading relationships of quantities from equations.
• Using properties of operations (the CCSS term for the field axioms) to explain operations with rational numbers (including negative integers).
• Examining the concepts of greatest common factor and least common multiple.
• Using the standard U.S. division algorithm to explain why decimal expansions of fractions eventually repeat and showing how decimals that eventual repeat can be expressed as fractions.
• Explaining why irrational numbers are needed and how the number system expands from rational to real numbers.” (p. 41)

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.

### Table 6.4. Connections to MET II (CBMS, 2012) Related to Expressions, Equations, and Functions at the Middle Level

MET II describes the following essential ideas for expressions, equations, and functions:

• “Viewing numerical and algebraic expressions as “calculation recipes,” describing them in words, parsing them into their component parts, and interpreting the components in terms of a context.
• Examining lines of reasoning used to solve equations and systems of equations.
• Viewing proportional relationships and arithmetic sequences as special cases of linear relationships. Reasoning about similar triangles to develop the equation y = mx + b for (nonvertical) lines.” (p. 42)
• “Examining and reasoning about functional relationships represented using tables, graphs, equations, and descriptions of functions in words. In particular, examining how the way two quantities change together is reflected in a table, graph, and equation.
• Examining the patterns of change in proportional, linear, inversely proportional, quadratic, and exponential functions, and the types of real-world relationships these functions can model.” (p. 43)

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).

### Table 6.5. Connections to MET II (CBMS, 2012) Related to Geometry at the Middle Level

MET II describes the following essential ideas for geometry (including measurement):

• “Deriving area formulas such as the formulas for areas of triangles and parallelograms, considering the different height–base cases (including the ‘very oblique’ case where ‘the height is not directly over the base’).
• Explaining why the Pythagorean Theorem is valid in multiple ways. Applying the converse of the Pythagorean Theorem.
• Informally explaining and proving theorems about angles; solving problems about angle relationships.
• Examining dilations, translations, rotations, and reflections, and combinations of these.
• Understanding congruence in terms of translations, rotations, and reflections; and similarity in terms of translations, rotations, reflections, and dilations; solving problems involving congruence and similarity in multiple ways.” (p. 44)

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.

### Table 6.6. Connections to MET II (CBMS, 2012) and SET (Franklin et al., 2015) Related to Statistics and Probability at the Middle Level

MET II, SET, or both describe the following essential ideas for statistics and probability:

• Developing an understanding of the role of variability in statistical problem solving.
• Understanding ways to summarize, describe, and compare patterns in variability in univariate data, including frequencies, relative frequencies, and the mode (categorical data); measures of center and measures of variability (quantitative data); bar graphs (categorical data); dot plots, histograms, and box plots (quantitative data).
• Exploring patterns of association in bivariate data based on two-way tables (bivariate categorical data) and scatter plots (bivariate quantitative data).
• Understanding probability as a measure of the long-run relative frequency of an outcome, understanding basic rules of probability, and approximating probabilities through simulations.
• Understanding connections among probability, random sampling, and inference about a population.
• Comparing two data distributions and making informal inferences between two populations.

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).

### Vignette 6.1. A Sequence of Course Activities Focused on Proportional Reasoning and Student Thinking

The following sequence of activities (based on Steinthorsdottir, n.d.) was designed to develop students' deeper understandings of proportional reasoning, number choices, and their effects in missing-value proportion problems; ways in which middle level students think about missing-value proportion problems; and strategies for posing purposeful questions to support student learning and elicit student understanding. The sequence of activities described here was implemented as a subset of activities within a 2-week unit on proportional reasoning and posing purposeful questions.

Activity 1. Solve the Potion Problem. Teacher candidates solved the Potion Problem, a missing-value proportion problem, on their own, in as many ways as they could, trying to consider ways in which middle level students might approach the problem.

The Potion Problem. A potion calls for 4 drops of magical Bulbadox Juice for every two cat hairs. Neville squeezed the dropper too hard! If 44 drops of the magical Bulbadox Juice were in his pot, how many cat hairs should he use?

After working individually on ways to solve the problem, candidates shared strategies in small groups. Finally, the various strategies were shared with the whole class. For example, one strategy was written in words: There are half as many cat hairs as Bulbadox Juice, so Neville should use 22 cat hairs. Another strategy was presented numerically: 4n = 44; n = 11; 2(11) = 22. In a third explanation, candidates stated that 42 cat hairs, 2 fewer than the drops of juice, were needed. As a whole class, candidates discussed how the strategies compared, analyzing which were correct and which were alike conceptually. They came to an understanding of some generalized groupings of strategies for missing-value problems (multiplicative within measure space, multiplicative between measure space, build up (various versions), additive error, other errors). As a follow-up assignment, the candidates solved two new missing-value problems using three strategies: build up, within measure (scale factor), and between measure (invariance). Here are the two problems:

1. A group of aliens are planning a trip to Earth. For every 5 aliens they need seven Power Pouches to meet one day's food needs. If 40 aliens are traveling, how many Power Pouches will they need for each day of the trip?
2. Nina is making pillowcases to donate to a charity. She needs 18 yards of fabric for 16 pillowcases. A fabric store gave her their extra fabric. How many pillowcases can she make with 45 yards of fabric?

Activity 2. Compare Problem Numbers and Related Strategy Selection. During the next class, teacher candidates discussed The Potion Problem and the two new problems assigned with respect to numbers used in the problems and the effects of those values on potential middle level students’ strategies used to solve them (i.e., the given numbers influence which strategy is selected). In the first new problem, the within-measure scale factor is a whole number whereas the between-measures space ratio is a rational number, and in the second new problem, both the within-measure-space and between-measure-space comparisons yield rational numbers. Second, candidates analyzed middle level student work from the two new problems to see to what extent the numbers in the task influenced strategy selection. Third, the candidates viewed a video clip of a researcher interviewing a middle level student solving one of the problems they had just explored and discussed the student’s mathematical thinking as well as the interviewer’s posing of questions.

Activity 3. Interview a Middle Level Student. Each teacher candidate conducted an interview with a middle level student, posing a set of 15 missing-value problems (with varying number choices). To prepare, they solved the problems so that they were able to select the most appropriate problems to pose to their interviewees, on the basis of what they were observing. The interview was video-recorded, and afterward, candidates analyzed the video, focusing on the students’ mathematical thinking and their own posing of questions.

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.