Chapter 5: Developing Deep Mathematical Understanding

Key Recommendation:Middle schools should offer a common shared pathway grounded in the use of mathematical practices and processes to coherently develop deep mathematical understanding, ensuring the highest quality mathematics education for each and every student.

The mathematics and statistics explored in middle school includes some of the most important ideas individuals will use or encounter in their everyday lives. For too many students, these mathematical concepts are often not learned well or with enough depth. This chapter focuses on developing students’ deep understanding of important mathematical concepts in middle school in ways that are empowering and transformative.

The college and career readiness mathematics standards adopted and implemented during the past decade have led to important conversations regarding the mathematics that students learn in middle school. One result of these conversations was the positive shift in many teachers’ equitable instruction where students were being engaged with mathematics in more active ways. Even so, there still remains a considerable need for a more consistent, systematic, and widespread implementation of college and career readiness standards in the ways in which they were intended. At the middle school level, this implementation should not only prepare students for college and career but should also develop a deep mathematical understanding while cultivating a positive mathematical identity; help students use mathematics in order to understand and critique the world; and entice them to experience and celebrate the wonder, joy, and beauty of mathematics.

Students should have ample opportunities to engage in the learning of mathematical ideas that mirror how they will engage in mathematics as adults. Careful attention should be given to the potential relevance of any given context, as relevance is dependent on the prior experiences of an individual student or group of students (these types of considerations align to equitable instruction as described in chapter 4). Through the use of rich contextual and non-contextual tasks to explore mathematical ideas, educators engage students in ways that are transformative and empower them as confident thinkers and doers of mathematics. Such experiences foster mathematical literacy, statistical literacy, and, more broadly, STEM literacy, which equips students with the knowledge needed to make sound decisions and to solve problems arising in their professional and personal lives.

Each and every student should attain mathematical proficiency (NCTM 2014b), which is described in *Adding it Up: Helping Children Learn Mathematics* as consisting of the following intertwined strands: *conceptual understanding, procedural fluency*, *strategic competence*, *adaptive reasoning*, and *productive disposition* (NRC 2001). As teachers teach for mathematical proficiency, one emphasis they should pursue is that deep mathematical understanding is built upon and rooted in conceptual understanding, which leads to the development of true procedural fluency (as described in Smith, Steele, and Raith 2017). In the position statement “Procedural Fluency in Mathematics” (2014c), NCTM describes procedural fluency as more than fact acquisition through memorization or strictly following a procedure. Rather, procedural fluency is defined as the understanding that is built upon a foundation of conceptual understanding that comes both before and during instruction regarding procedures. Both strategic competence and adaptive reasoning describe the ways of mathematical thinking that need to be developed in students so they can be mathematical problem solvers (NCTM 2014b). Productive disposition is “the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics” (NRC 2001, p. 131). When teachers focus on developing their students’ mathematical proficiency as defined by these five interrelated strands, mathematics becomes more visible and accessible to each and every student. Furthermore, a high-quality mathematics curriculum that is relevant, exploratory, challenging, and integrative (Stevenson and Bishop 2012) should emphasize mathematical learning experiences that focus on middle school students’ development of mathematical concepts that began in elementary school in order to deepen, strengthen, and challenge their understanding of mathematical ideas as they progressively become more abstract and complex in middle school (McEwin and Dickinson 2012; NCTM 2000; NCTM 2006).

Solely focusing on content standards will not lead to transformative change in how mathematics is taught in middle school. Positive, transformative change in middle school mathematics teaching and learning will require embracing the mathematical proficiencies, practices, and processes as a means for building conceptual understanding and students who see themselves as doers of mathematics. The Strands of Mathematical Proficiency (NRC 2001), Standards for Mathematical Practice (NGA Center and CCSSO 2010), and Process Standards (NCTM 2000), figure 5.1, or other similar ideas adopted by a school, district, state, or province are instrumental to implementing mathematics instruction that is both accessible and equitable and should be represented in assessments at the classroom, school, district, and state or provincial levels. Additionally, the mathematical practices and processes should explicitly be considered through the lens of statistics (see *Statistical Education for Teachers* (SET; Franklin et al. 2015). The proficiencies describe aspects of what it means for a student to be mathematically proficient. The practices and processes embody how to do mathematics.

A mathematics teacher who plans at the outset with the proficiencies, practices, and processes in mind has the mindset to engage each and every student as active participants in their and their peers’ mathematics learning. The degree to which a mathematics program is accessible and equitable is directly related to how the proficiencies, practices, and processes are implemented and developed. Opportunities where students engage in the mathematical practices and processes while exploring relevant topics such as community issues (e.g., minimum wage versus living wage, transportation, shortage of natural resources) are instrumental in ensuring middle school students become mathematically literate and see the utility of mathematics in ways that empower students to understand and critique our world. Engaging students in the mathematics of relevant, often sensitive or controversial topics, requires careful attention and thoughtful implementation, but should and needs to be a part of students’ middle school mathematical learning experience.

The proficiencies, practices, and processes illuminate the vital role of planning, implementing, and assessing mathematics instruction. Implementing equitable instruction through the eight Mathematics Teaching Practices (NCTM 2014b) sparks movement toward the use of higher cognitive demand tasks, which emphasizes and values depth and quality of students’ mathematical learning experiences rather than quantity and speed. More specifically, when a teacher engages in practices that include strategies such as establishing norms for classroom participation, positioning students as capable, attending explicitly to race and culture, and pressing for academic success (for more, see Bartell et al. 2017), more students have access to the mathematics. Further, the eight Mathematics Teaching Practices, which represent what the *teacher* should be doing*,* and the student learning practices and processes, which represent what the *student* should be doing, go hand in hand in building a high-quality mathematics program, as the practices and processes represent the essence of the classroom culture and ultimately the depth of mathematics in which students are engaged.

Using mathematics when solving a problem that begins in the real world and ends with a solution in the real world is mathematical modeling (Niss, Blum, and Galbraith 2007; Pollak 2016). Students investigate a real-world problem and use mathematics as part of the process with a solution expressed in the real world. Both mathematical and statistical modeling are extremely valuable because they can provide a seamless link between making informed decisions in society and everyday life to developing a deeper and more coherent understanding of mathematics and statistics (Bargagliotti et al. 2020; Zbiek and Conner 2006). In addition, statistical models combine mathematical models with descriptions of variability.

Various images of research-based mathematical modeling cycles and processes exist (see Cirillo et al. 2016 for overview). In one such example, mathematical modeling could be represented as a path through the following components with options to return to earlier components in order to revise and refine an emerging solution: formulating the problem, defining variables and stating assumptions, stating the problem in a mathematical way, solving the problem using a mathematical model, analyzing the mathematical model and analyzing and assessing the solution, refining the model and revisiting steps as needed, and reporting the results (see figure 5.2; NGA Center and CCSSO 2010; Consortium for Mathematics and Its Applications and the Society for Industrial and Applied Mathematics [COMAP and SIAM] 2019). Students should have ample opportunities to engage in mathematical modeling within and across mathematics content domains in middle school in a manner that has students investigating “… how different phenomena might perform or behave or different events might unfold given constraints or assumptions” (NCTM 2018, p. 41). This includes students learning about number, ratio and proportion, algebra and function, statistics and probability, and geometry and measurement. As students gain experience using mathematics in ways that empower them as informed decision makers, they see firsthand the utility of mathematics and how it relates to their ability to navigate situations in life. The open-ended possibilities for which mathematics are highlighted during modeling experiences makes implementation both exciting and challenging.

When students engage in mathematical and statistical modeling, it is important they investigate phenomenon that have relevance to their lives. Ideally, teachers will elicit scenarios or situations to investigate from their students. For example, consider the real-life scenario of charging a phone, which is likely to be relevant to many middle school students. Using a screenshot of a phone showing a very low battery can serve as a launch where students can identify the problem (phone about to run out of power) and pose questions they’d like to explore, such as “How long will it take to charge the battery?” or “How long until the phone battery dies?” (adapted from Fenton 2015).

To explore this scenario, students make assumptions about the phone and define the variables (in this case, the independent variable is time and the dependent variable is percent charged). Students can collect data on their own phone using different assumptions, such as not using the phone during charging or the type of charger used. By collecting data on charging their own phone such as the data set shown in figure 5.3, starting in this case with only a 5 percent charge, students can use mathematical models to make predictions, interpret and analyze the data, and form reasonable conclusions. This graph shows that the phone charge was linear for nearly 80 percent of the charging time but then became nonlinear during the approximately last 20 percent of the charging time. Students could then discuss why the model is not linear for the entire charging period, although it may intuitively seem to be in this context. Exploring nonlinear examples helps to develop students’ understanding of linear functions as a concept. Notably, this task provides a natural opportunity where a teacher and students can and should discuss together that not all functions that occur fit models nicely. This scenario also provides an opening to discuss that while the actual charging of a phone is continuous in nature, a graph that students might make could be a collection of discrete data points. Mathematics presents itself as messy and ill defined yet in interesting and curious ways. Middle school students need to experience mathematics in these ways. Furthermore, worthwhile extensions to this modeling example connecting to other STEM disciplines could include questions such as, “Why does a phone charge this way?” (the science behind this) and “In what other situations might a similar choice be made regarding how the battery charges?”

As described in *Developing Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Prekindergarten–Grade 8* (Lannin, Ellis, and Elliott 2011), students’ ability to reason mathematically becomes more complex in middle school. Mathematical reasoning includes the processes of conjecturing, generalizing, and justifying. During middle school, conjecturing becomes more general because students begin to understand that a generalization is not about one specific instance but applies to any case, which involves a high level of abstract thinking. For example, a student in middle school might be asked if a generalization they discover holds true for all whole numbers, integers, rational numbers, real numbers, or irrational numbers. Further, students’ justification in middle school becomes more sophisticated, using more formal symbolic notations (Lannin, Ellis, and Elliott 2011).

Students also move beyond explaining their answer and describing “what they did” to more formally justifying their reasoning using mathematical definitions and givens. The reasoning involved in conjecturing, generalizing, and justifying closely aligns with the mathematical proficiencies, practices, and processes highlighted earlier in the chapter. As teachers implement tasks that promote reasoning, teacher actions that support students’ mathematical reasoning include building on and extending students’ current understanding, using tasks that have multiple entry points, incorporating multiple representations, and using tools. Additionally, teachers should support, but not take over, student thinking; encourage a wide variety of mathematical paths, strategies, and approaches; and ensure tasks that promote reasoning are used on a consistent basis (NCTM 2014b). Mathematical reasoning in middle school focuses largely on informal reasoning and proving, setting the stage for proof in high school and beyond.

An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking (NCTM 2014b, p. 78).

Technology is a catalyst for change and innovation in our global society. Technology advancements have led to transformations in fields such as medicine, communication, and science, and allows for the exploration of large data sets in ways that seemed inconceivable even a decade ago. Such advances in technology must be reflected in middle school mathematics programs in ways that are thoughtful and keep the learning of mathematics at the forefront of students’ experiences. The NCTM Position Statement “Strategic Use of Technology in Teaching and Learning Mathematics” states that technology should be used strategically “… by students and teachers in thoughtfully designed ways and at carefully determined times so that the capabilities of the technology enhance how students and educators learn, experience, communicate, and do mathematics” (NCTM 2011, n.p.). According to Drijvers (2013), technology can serve three functions in the mathematics classroom: (1) developing concepts, (2) practicing skills, and (3) doing the mathematics. As such, teachers should provide their students ample opportunities to use mathematical technologies to investigate mathematics and to push forward their thinking across the content domains during middle school (Dick and Hollebrands 2011). For example, technology affords graphing applications across multiple representations that can be created efficiently and that are linked in ways that are dynamic, allowing for students to explore when changes in one representation are simultaneously shown in other representations. Similarly, technology can also be employed when teaching statistics by having students investigate, analyze, and interpret data using relevant graphical displays and numerical summaries that allow students to interpret a statistical analysis in context. It is crucial that teachers and students have access to current technologies with support in place for teachers providing guidance to implementing the use of technologies in ways that enhance student learning, are consistent with research, and make mathematical ideas more accessible. Productive beliefs about technology (adapted from NCTM 2014b, p. 82) include the following:

- Technology is an inescapable fact of life in the world in which we live, and it should be embraced as a powerful tool for doing mathematics. Using technology can assist students in visualizing and understanding important mathematical concepts and support students’ mathematical reasoning and problem solving.
- Technology not only changes how to teach but also affects what can be taught. Technology can assist students in investigating mathematical ideas and problems that might otherwise be too difficult or time-consuming to explore.
- All students should have access to technology that supports the teaching and learning of mathematics.
- Effective use of technology requires careful planning. Teachers need appropriate professional development to learn how to use technology effectively.

Technology provides opportunities to investigate mathematical relationships, make and check conjectures, develop conceptual understanding, test models of mathematical relationships, simulate situations to explore chance, and do the routine work in many problems, allowing students to focus on mathematical interpretations and reasoning. Furthermore, when considering the role of technology more broadly, the mathematics education community should take a leadership role in making computational thinking an integral part of middle school students’ educational experience. Computational thinking means evaluating information to solve complex problems using data and logic and is “… increasingly seen as a set of broadly valuable thinking skills that helps people solve problems, design systems, and understand human behavior …” (National Science and Technology Council [NSTC] 2018, pp. 23–24).

The ways in which students experience learning mathematics content in middle school is critical and is inextricably linked to developing students’ mathematical identities. It is the responsibility of middle school educators to ensure that mathematics content is taught through engaging students using the mathematical practices and processes in ways that focus on depth and coherence, where mathematics is not overproceduralized. Such meaningful mathematical learning experiences equip and empower students with the awareness and ability to use mathematics as a way to make sense of the world and make informed decisions as members of a democratic society. School mathematics should be a gateway to students’ success, not a gatekeeper.

During the past three decades, the PK–12 mathematics standards movement has been underway and ever evolving. By and large, existing standards build upon earlier standards initiatives (e.g., *Curriculum and Evaluation Standards for School Mathematics* [NCTM 1989], *Principles and Standards for School Mathematics* [NCTM 2000], *Curriculum Focal Points* [NCTM 2006], and *Common Core State Standards for Mathematics* [NGA Center and CCSSO 2010]) and bring more focus to middle school mathematics curriculum. This focus provides a foundation for students that is grounded in understanding and sense making and should not be skipped or diminished to jump ahead to an algorithmic approach. One important consideration is how the standards, regardless of which set are adopted, are implemented. Taking current standards and overproceduralizing them loses the intent of college and career readiness standards that promote a balance of conceptual understanding and procedural fluency along with an emphasis on coherence. Documents such as, but not limited to, the various standards documents provide guidance to middle school stakeholders regarding the mathematics content teachers should know so they can provide students with a high-quality mathematics program.

Learning progressions, which often provide a foundation for standards used in schools, districts, states, or provinces, are important because they support a coherent development of mathematics within and across grade levels. Other documents such as the recent paper from OECD “A Synthesis of Research on Learning Trajectories/Progressions in Mathematics” (Confrey et al. 2019) provides needed guidance for how the mathematics content should progress across grade levels (e.g., vertical articulation). Notably, learning progression documents highlight the need for middle school mathematics teachers and stakeholders to develop a coherent understanding of the mathematics taught across grade levels. Usiskin (2014) argues that from grades 7–10, at least seven important mathematical transitions take place for students. In the earlier grades of middle school, examples of those include the transition from whole number to real number, from number to variable, and from properties of individual figures to general properties of classes of figures. Knowing how mathematics content (and learning) progresses over time involves acquiring an understanding of the mathematics content and its logical interconnectivity across grade levels as well as how students develop an understanding of the mathematics. In short, it is not enough for a teacher to only know the standards taught at their grade; they must possess horizon knowledge of how mathematical ideas evolve over a span of time (Ball, Thames, and Phelps 2008). Coherence happens when concepts are connected across grades and students see how the mathematics they are learning builds on their previous mathematical knowledge and prepares them for future mathematics learning.

The goal of *Catalyzing Change in Middle School Mathematics* is not to provide a comprehensive list of essential concepts for middle school; rather it is to spark conversations around mathematics content guided by existing documents. To organize this conversation, an examination of NCTM’s *Curriculum Focal Points* (2006) (which is NCTM’s most recent document providing direction for PK–8 mathematics content coherence) and current college and career readiness standards used by states and provinces, including the Common Core State Standards for School Mathematics (NGA Center and CCSSO 2010), identified five content domains:

- Number
- Ratio and Proportion
- Algebra and Function
- Statistics and Probability
- Geometry and Measurement

A discussion of these five domains serves to illustrate how deep mathematical understanding might be developed in middle school students. Within each of the five domains, the following topics are discussed: (1) essential content in middle school and how students should experience the content focused on deep mathematical understanding; (2) an example showcasing deep understanding; and (3) how the content in this domain connects to what students learn in elementary and high school. The intent of the examples is to provide an idea of how depth and coherence can be accomplished within each domain, but the examples do not in any way address all of the content of the domain, and they are not intended to be prescriptive but instead to initiate collegial and critical conversations.

Students in middle school are in the process of developing an expanded and connected understanding of number, operations, and associated contexts. Previously explored number systems are broadened to include integers and irrational numbers while emphasizing properties including commutative, associative, and distributive properties. Highlighting and exploring the connections among the structures, properties, relationships, operations, and representations of number systems being explored is critical. As described by the National Research Council (2001 p. 72), number and operations are “abstractions that apply to a broad range of real and imagined situations.” Representations of number and operations whether physical, pictorial, in words, or written symbolically can assist students in communicating about these situations. A strong understanding of the content of the Number domain is needed for work in other mathematics content domains.

In middle school, students’ understanding should build on number and equivalence concepts from elementary school and be based on generalizations carefully tied to the properties of whole numbers. Students extend their understanding as they continue to work with rational numbers by including integers and further cultivate their understanding of equivalence by working with numerical and algebraic expressions and equations. Students’ sense making as they move between number and algebra runs counter to the sometimes well-practiced habits of memorization without understanding. Students’ deep understanding includes having a variety of strategies and approaches from which to choose from and use when working within number systems and knowing why the procedures they use work. Without having these conceptual underpinnings, students may rely solely on memorized steps without meaning. A strong conceptual understanding of number systems enables students to acquire both comfort and flexibility when working within and between systems with the benefit of procedural fluency.

Exploring and understanding how mathematics content connects vertically across grade levels is essential. For example, consider students’ use of visual representations, such as area models, to work with multiplication and to develop an understanding of the distributive property. In the upper elementary grades, students should learn to multiply multidigit whole numbers using an area model to represent partial products (see figure 5.4). Later in elementary and in middle school, when students are multiplying fractions written as mixed numbers, they again may use an area model to represent partial products and further advance their understanding of the distributive property (see figure 5.5).

The symbolic and visual representations of the distributive property as shown in figures 5.4 and 5.5 highlight a coherent connection between the use of an area model with whole number multiplication and with the multiplication of mixed numbers. Understanding this explicit connection is critical, yet can sometimes be overlooked as students transition from numerical to algebraic expressions during middle school. Middle school teachers can leverage their students’ numerical understanding of the distributive property and connect it to maintaining equivalence when simplifying algebraic expressions, such as shown in figure 5.6.

This visual depiction allows students to see the multiplication of two binomial expressions as an application of the distributive property, using what they know to understand what they do not yet know. Carefully linking the use of the distributive property with both whole numbers and rational numbers to its new algebraic application allows students to develop an understanding of the distributive property as a beautiful mathematical structure that holds true from whole numbers to mixed numbers to algebraic expressions. With this deep understanding of the distributive property, students do not need to rely on rote steps or trivial memorizing devices such as the mnemonic F.O.I.L., which is sometimes used to keep track of the terms being multiplied but falls apart when students encounter a trinomial. This example showcases how the distributive property connected to an area model first introduced during elementary school is an essential foundation for work in middle school mathematics and beyond.

The number domain is essential across all grade levels. In elementary school, students develop a foundation of number with a focus on whole numbers, fractions, and decimals. In middle school, students continue to advance their understanding of number through work with rational numbers, including integers, as well as how numerical ideas apply algebraically. In high school, students further expand their knowledge of the number domain with irrational and rational numbers as well as, more broadly, their ability to reason quantitatively and abstractly.

In middle school, ratios, proportions, and proportional relationships comprise a major area of school mathematics that is crucial for students to learn and that is sometimes challenging for teachers to teach. Students in middle school need to understand proportionality conceptually to ensure their future mathematical success. One possible starting point is to maintain equivalence when increasing or decreasing the size of a recipe that pairs quantities that form equivalent ratios, such as 2 cups of sugar for every 3 cups of water (e.g., 4 cups of sugar for every 6 cups of water, 6 cups of sugar for every 9 cups of water). Students can work with the distinct quantities using equivalent ratios in tables, often referred to as ratio tables, graph pairs of quantities as ordered pairs, and begin to establish the foundation for proportional reasoning. A focus should be placed on equivalent ratios being multiples of each other and understanding why such comparisons are not mathematically additive. As students become more comfortable reasoning proportionally, they can build comprehension of finding a unit rate, connect that to the proportional relationship, and understand the idea of constant of proportionality. Students should have mathematical learning experiences where they examine how proportional relationships are connected to and different from linear relationships. Students use their knowledge of ratios and foundational understanding of proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent of increase or decrease. Furthermore, students can also extend this understanding to scale drawings by relating the ratio of corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.

In middle school, students’ understanding of ratios and proportional relationships has strong connections to the other content domains, including algebra and function and statistics and probability. For example, conceptual understanding of ratio and proportion includes the understanding that a ratio is a relationship between quantities that multiplicatively co-vary together, recognizing that equivalent ratios can be generated by multiplying each of the quantities by a common factor. Students should be fluent in using multiple representations (e.g., ratio tables, double number lines) and in solving problems involving ratios. Further, students develop conceptual understanding of a rate as a set of infinitely many equivalent ratios and that a unit rate is a rate in which the comparison is to one unit. Teaching ratios and proportional relationships deeply, coherently, and with understanding runs counter to having students work primarily on an algorithmic solution (e.g., cross multiplication) for problems in which they are searching to find the missing value without any understanding of the meaning of a ratio.

Consider the following example task (adapted from Fey et al. 2018) where a survey of middle school students and their lunch choices (healthy pizza or healthy chicken bites) was conducted. Here are four statements that people in the community made about the two types of lunches served in the middle school cafeteria:

- Students who preferred pizza outnumbered those who preferred chicken bites by a ratio of 15,411 to 5,137.
- There were 10,274 more students who preferred pizza.
- Seventy-five percent of the students preferred pizza.
- Students who preferred pizza outnumbered those who preferred chicken bites by a ratio of 3 to 1.

The language of ratio is often introduced in middle school as a way of making comparisons. Such claims are typical of what is seen in advertisements, and students are inundated with comparisons from the internet and other media. In the example statements above, students are exposed to different forms of comparing quantities, including ratios and percentages in the four different comparisons provided to students. Teachers should challenge students to make sense of the situations and discuss questions such as what they notice from each form of comparison given, what information is missing from each comparison, how accurate and useful each comparison is, and whether all the comparisons could have come from the same survey. A powerful aspect of this example task is to emphasize to students the rewriting of one survey statement into another comparison form.

Proportional reasoning is linked conceptually to multiple topics in elementary, middle, and high school mathematics (Lobato and Ellis 2010). As a prerequisite to proportional reasoning, students need to develop deep and flexible meaning of numbers and operations, especially for fractions and multiplication, and move from additive to multiplicative thinking in grades 3–5. In middle school, students work extensively with proportional relationships, often through experiences connected to measurement and algebra, such as examining proportionality in their work with slope, examining constant of proportionality in linear functions, and algebraic equations (Lobato and Ellis 2010). In high school and beyond, students continue to hone their ability to reason proportionally in more complex and formal ways.

In middle school, students begin to engage more formally with algebraic concepts, building upon the foundation of algebraic thinking of the elementary grades where they explored patterns, began to make generalizations, explored properties of numbers, and continued to develop an understanding of equivalence. In this domain, the focus in middle school is highly algebraic, with students beginning to explore functions later in their middle school experience. Students should engage in key mathematical ideas, including writing, interpreting, using, and evaluating algebraic expressions and equations; developing an understanding of linear equations that includes systems of equations and work with the relationships in bivariate data; and understanding the concept of a function that includes the ability to identify those that are linear and those that are nonlinear. Students should have ample opportunities to use technology to investigate concepts such as a variable, equivalent expressions, a solution to an equation, and multiple representations (e.g., graphs, tables, equations, or verbal) of linear relationships. Students should explain and justify why two expressions are equivalent, what it means to say a solution to an equation satisfies the equation, what makes a relationship linear, and interpret the slope of a linear relationship in a given context. “Beyond specific techniques, algebra should be seen as a collection of unifying concepts that enable one to solve problems flexibly” (NCTM 2018, p. 45). Students should not walk away from middle school thinking of algebra as a set of steps to follow to get an “answer.” Instead, students should be engaged in the learning in a manner that affords them opportunities to engage in problems that occur in everyday life.

In middle school, students’ understanding of algebra and function benefits from a strong conceptual understanding of the properties of operations and equivalence with respect to expressions, equations, and inequalities. Students with this conceptual understanding can describe relationships algebraically and use models to make predictions. Students also feel comfortable moving between different representations (e.g., symbolic, table, graph, verbal, and pictorial) as they use, interpret, and communicate their thinking. Students understand that a single line may be represented with multiple equations in different forms and that different forms convey different information about the line. Students also develop an understanding of linearity and are comfortable determining whether relationships are linear or nonlinear. Building middle school students’ deep understanding in the algebra and function domain runs counter to students rotely memorizing steps to solve multistep problems without attending to conceptual understanding, sense making, and exploring relationships and patterns.

Consider the example task, likely most appropriate later in a students’ middle school experience, representing an algebraic model from data. In this example, students are told that the highway speed limit is 65 miles per hour (mph) and are given a table (see figure 5.7) of information. The table displays the speed in miles per hour that the vehicle was traveling and the corresponding fine for each of the five speeds.

Students are asked to find a model that could be used to determine the total fine of any speeding ticket. They must define each variable and then use their algebraic model to predict the fine for a person driving 20 mph over the speed limit. Students can be challenged to use their model to predict the speed that would result in a $200 fine and to state the relationship between the domain and range.

Writing an equation provides a means for expressing relationships among variables, such as speed and ticket fine. Near the end of middle school, students use algebraic concepts learned to explore functional relationships between quantities. Functions can be represented in multiple ways—in algebraic symbols, graphs, tables, and words—and classified into different families (e.g., linear and nonlinear or in other forms) by examining their patterns of change. In the speeding ticket example, the data are provided in a table and students could produce a graph or represent the situation using an algebraic equation where a Fine = $85 + $5 (speed − 65). It is important for students to know this function is based on an assumption of a linear relationship; in this case, for every one additional mile per hour over the speed limit, the fine increases by $5.

The fines for speeding in another state are listed in figure 5.8. By graphing the data (see figure 5.9), students can see that the data does not result in a graph of a linear function. The context provides an opportunity for students to be introduced to step functions, which are studied in more depth in high school.

The same ticket and fine can be merely an annoyance for some individuals and a serious financial burden for others. Moreover, states could levy additional fines for people unable to pay. States could also offer an option for drivers to pay additional fees to lessen the ticket. When students engage in and respond to tasks such as the speeding ticket example, they are using mathematics to better understand and examine inequities in policies. When engaged in the Speeding Ticket task, students have the opportunity to carefully define the meaning of variables, model a relationship between two quantities, and connect multiple representations.

In upper elementary school, students begin to use variables to describe patterns, representing them in symbols as well as with words, graphs, and tables (providing experiences using multiple representations), and this algebraic work expands throughout middle school. In middle school, students build on their understanding of addition-subtraction and multiplication-division as inverse relationships as well as their knowledge of the meaning of equality as preserving equivalence on both sides of the equals sign, and extend these ideas to algebraic settings. When students transition to high school, they explore a wider variety of function families (e.g., quadratics, exponential, logarithm, trigonometric, etc.) while continuing to use multiple representations.

With many fields increasing attention on large sets of data (“big data”) and basing decisions on inferences derived from data analysis (Bargagliotti et al. 2020), data literacy and statistical thinking are becoming ever more critical skills that students must have to understand their world and to expand their opportunities. Students need to be able to critically assess statistical and databased information they encounter in the media and other sources. “Who sponsored the study?” “What questions were investigated and how was the data collected?” “How many participants were in the study?” “What was the data analyzed?” “Were the conclusions appropriate for the study design?” are some typical questions students should raise when confronted with such information.

The study and application of statistics in middle school and PK–12 education has undergone a radical transformation. This transformation has moved from a focus on calculation of statistical and numerical summaries such as means, medians, and range to students developing their ability to use statistical thinking to understand and describe variability within data and to interpret statistical summary measures and graphical representations within the context in which they are working. Two important distinctions between statistical and mathematical thinking are that “(1) statistical thinking focuses on engaging in a process that is centered on understanding and describing variability in data; and (2) context plays a critical role in the practice of statistics. Creating a graph or computing a mean without context is not statistics. In the practice of statistics, the context of the problem under study drives the method of data collection, the analysis of the data, and the interpretation of results” (Kader and Jacobbe 2013, p. 7).

The statistical problem-solving process described in the *American Statistical Association’s Guidelines for Assessment and Instruction in Statistics Education* (Bargagliotti et al. 2020) involves four interrelated components:

**Formulating a statistical question**—an investigative question that can be addressed with data**Collecting and considering data**—designing a plan for collecting or considering appropriate data, implementing the plan, and collecting data**Analyzing the data**—creating and exploring various representations of the distribution to identify and describe patterns in the variability in the data and summarize various features of the distribution**Interpreting the results**—providing a statistical answer to the statistical question posed and investigated that takes the variability in the data into account

Within the statistical problem-solving process, probability offers the mathematical tools to create models for predicting long-run behavior in random situations. Probability provides both a key tool in statistical thinking for describing outcome patterns that emerge in the long run, and it is important for development and refinement during the statistical problem-solving process (Bargagliotti et al. 2020). Specifically, a probability is a measure of the chance that something will happen on the basis of the relative frequency of an occurrence when some random process is carried out a large number of times. It is a measure of certainty or of uncertainty that describes the variability of the context (Bargagliotti et al. 2020). This use of probability is different from determining the theoretical probability. In middle school, experimental probability allows students to perform engaging experiments and make their own conjectures about the probable outcome of events, perhaps of interest to them, and assign probabilities on the basis of their experimentation. The more experiments they perform, the closer the experimental probability becomes to a theoretical probability.

In statistics in middle school (and PK–12), the statistical problem-solving process should be at the forefront of students’ statistics education and experience. Within that frame, students should develop an understanding of statistical variability, an ability to summarize and describe distributions for both categorical and quantitative variables, the skill to compare two or more groups with respect to the distribution for a categorical variable or for a quantitative variable, and the capability to investigate patterns of association in bivariate categorical or bivariate quantitative data. Students with conceptual understanding recognize the need to describe distributions of quantitative data using shape, center, and measures of variability, and they can articulate how these are related for a given distribution. Students also recognize that measures of center and measures of variability for a quantitative numerical data set (both univariate and bivariate data sets) are both necessary to understand the story in a set of data, and they can interpret these measures in a context and can use them to compare distributions of different data sets. In probability, students use simulation to draw informal comparative inferences about two populations, use random sampling to draw inferences about a population, use technology to investigate chance processes, and develop, use, and evaluate probability models that help make interpretations and draw conclusions. This deep understanding of all aspects of statistics and probability run counter to working with scenarios that have predetermined results rather than variability, focus primarily on computing measures of center, and concentrate on computing probabilities of events without reasoning about the context.

Now consider two middle school content examples: one on statistics, adapted from Kader and Mamer (2008), and one on probability. The first example focuses on modeling the statistical problem-solving process. There is a local recreational soccer league that needs to select one team to represent the league at a regional tournament. The league decides to base the decision on games played by each team over the course of a season. The league has narrowed down the options to two teams, Team A and Team B, with both of these teams scoring a mean average of 6 goals during the last 9 games of the season. The league believes the safer choice is to select the team that has the least variability in goals scored—an indication of the team being more consistent from game to game. To try to figure this out, one league official wants the group to investigate the following question: “Is the distribution of goals scored less variable for Team A than for Team B?” This question represents the first component of the statistical problem-solving process; that is, a statistical question that can be investigated using data has been formulated.

Next, students engage in the second component of the statistical process, which includes designing a plan for collecting and considering useful data, implementing the plan, and collecting data. During this phase, students consider what data would be most useful in making a determination regarding the variability in goals scored. They consider whether they should use all the games played in the season rather than just the last 9 games that league officials used to narrow down the choices to Team A and Team B. After much discussion, students decide that considering only the last 9 games is best because it would better reflect each team’s most current level of performance at games. The students decide they will ask for the raw data from the league officials. After students have collected the data for both teams, they are ready to analyze the data.

Figure 5.10 represents the scoring for the two soccer teams for the last nine games of the season. Students decided to visually display each team’s scores on a dot plot. Both teams scored a mean of six goals during the nine games. In the two dot plots, the number within each circle represents the distance each score is from the mean score of 6.

Students should notice that the distances for the values below the mean are balanced by the distances for the values above the mean; that is, the mean value for a collection of numeric data indicates the balance point of the data distribution as well as the numerical average. Thus, one way to judge the variability in a distribution of data is to measure the distance of the observations from the mean. Considering that Teams A and B both have the same mean of 6, students could be asked whether they think that—

- both teams are equally likely to score more than 9 goals or less than 3 goals; and
- how the teams are alike and how they are different with respect to their scoring.

To answer these questions, students could consider the total distance from the mean for both teams. For Team A, the total distance, or SAD—Sum of the Absolute values of Deviations—is the sum of the distances between the mean of 6 and all nine values. The SAD for Team A is 16. For Team B, the SAD is 18. Revisit the question being investigated: “Is the distribution of goals scored less variable for Team A than for Team B?” Students can utilize their measures of variability from the mean as a way to answer this statistical question. At 18, Team B has a greater absolute sum of the distances of each observation from the mean compared to Team A with a SAD of 16. Thus, using the SAD as our measure of variability, Team A has less variability in the distribution of goals than Team B. In this example, the number of game goals is the same for each team, so comparing the total of the distances is acceptable. However, if the number of game goals is different for each team, then comparing total distance is not appropriate. If this were the case, the distances could be compared by dividing the SAD by the number of observations in the distribution (which adjusts for the difference in group sizes) giving the Mean of the Absolute Deviations (MAD):

$\begin{array}{l}\text{\hspace{1em}}\text{\hspace{1em}}\text{MAD}={\displaystyle \frac{\text{SAD}}{\text{numberofvalues}}}\\ \text{TheMADforTeamAis}\text{\hspace{0.17em}}{\displaystyle \frac{16}{9}}=1.\overline{77}.\\ \text{TheMADforTeamBis}\text{\hspace{0.17em}}{\displaystyle \frac{18}{9}}=2.\end{array}$

The MAD tells us that on average, Team A’s game goals vary from its mean of 6 goals by 1.7̄7̄ goals while on average Team B’s game goals vary from its mean by 2 goals. Thus, according to the MAD, Team A has less variability and more consistency in scoring goals.

To conclude the investigation, students interpret the results of their analysis. Having determined the variability of the scores using the SAD and MAD for both Team A and Team B, students conclude that Team A’s distribution has a lower MAD, meaning there is less variability in the distribution of scores for Team A than for Team B. Students can then prepare a report providing a statistical answer to the original question (i.e., Is the distribution of goals scored less variable for Team A than for Team B?) that takes the variability in the data into account to share with the league official. Before providing the report, students can also be asked whether they believe only using a measure of variability, such as the MAD, is all the league officials should consider. The league officials made a decision to use one criterion for selecting the team. The students had the benefit of examining the distributions of game goals for the two teams. Do they have new statistical questions they believe should be investigated about the teams before making a final decision? Questioning throughout the statistical problem-solving process is at the core of statistical reasoning.

The MAD provides a measure of average variation in the data compared with the mean, and it is a precursor for the standard deviation, the more commonly used measure for the variation in data from the mean that is studied in high school and beyond. By understanding how to interpret the MAD, when students are first introduced to standard deviation and its messy formula, they already know how to interpret the standard deviation that essentially has the same interpretation as the MAD. The mean alone is an incomplete measure; the mean and the MAD give some sense of the data distribution. For example, suppose the mean temperatures for San Francisco and St. Louis are both 87. Do you think their MADs are the same? The MAD provides important information about the amount of variability within a distribution of quantitative data. Follow-up questions for students could be: “Is it possible to create a soccer team with the mean score of 6 for 9 games with a smaller value for the MAD?” “A greater value for the MAD?”

Now, consider a probability-focused example. In middle school, students investigate the meaning of randomness and use simulation to investigate informal inference. For example, suppose Dorian, a player on the school’s basketball team, typically makes about 60 percent of her attempted baskets. So far this year, however, she has only made 9 baskets in 20 attempts. Is she really in a slump? Outcomes vary by chance as well as for other reasons. If Dorian’s percentage of baskets made is actually 60 percent, is making only 9 baskets in 20 attempts surprising—or is it just variability due to chance? Given 20 attempts with a 60 percent probability of making a basket, the expected number of baskets is 12. On average, we expect Dorian to make 12 baskets out of 20 attempted baskets. However, in real life the actual number of baskets made can vary. Is 9 a plausible number of made baskets due to chance variation if 12 baskets are expected? Figure 5.11 shows a simulation of the distribution of 1,000 counts for made baskets in 20 attempts with a probability of 0.6 of making a basket. In the simulation, Dorian would make 9 or less baskets in 136 of 1,000 replications of attempting 20 baskets, giving a probability of 0.136. Is this surprising or just due to chance? The answer often depends on the context, but typically having something occur 13.6 percent of the time by chance would not be considered surprising.

This example illustrates how probability is used to make a decision on the likelihood of an event occurring by chance as well as how simulation can be used to investigate the variability around a situation in which the expected outcome would be 60 percent of 20, or 12. Chance variability suggests that making anywhere from 5 to 18 baskets might happen but that 9 to 15 baskets happening by chance is fairly common. When each student simulates the same situation, they will get about the same results, establishing a pattern in the random behavior of chance outcomes in this context. Such examples lay the foundation for more formal inference in high school.

Statistics and probability are linked conceptually to a range of elementary, middle, and high school topics. In the elementary grades, students organize and analyze data collected to address a statistical question by classifying and categorizing objects, representing the distribution of data in tables, and depicting the distribution of data with various graphical displays, including picture graphs, bar graphs, and dot plots. When students progress to middle school, they continue their study of statistics and probability by interpreting linear models and distinguishing between correlation and causation; using simulation to create sampling distributions to move from descriptive statistics to making inferences and justify conclusions from sample surveys, experiments, and observational studies; and reasoning with categorical data through the use of simulation to make inferences. In high school, students work with contingency tables and estimate conditional probabilities and consider relative risks, which are important for making informed decisions in everyday life.

In middle school, students engage in geometry and measurement in ways that are highly interconnected as well as closely related to the number, ratio and proportion, and algebra and function domains. Students continue to develop an understanding of area, surface area, and volume of two- and three-dimensional shapes (e.g., circles, cones, cylinders, and spheres) as well as reasoning about the relationships among geometric shapes through actions such as rearranging, decomposing, composing, transforming, and examining cross sections. When connecting geometry to proportional relationships, students work with scale drawings of two-dimensional real-world and mathematical problems to analyze figures and situations as well as to investigate the relationships between similar figures. Later in middle school, students work with geometric transformations and develop ideas about congruence and similarity of two-dimensional figures. Students develop an understanding of the Pythagorean theorem and its converse, and then use it to work with distances on the coordinate plane and to analyze polygons and polyhedra. Students also begin to work with angles in a more formal way, such as exploring a transversal intersecting parallel lines. All of this work has students deeply engaged in length, area, and volume measurements in ways that substantially extend, rather than repeat, their prior experiences working with shapes, formulas, and measurable attributes in the elementary grades.

Students in middle school should experience geometry and measurement in a manner that is integrated and active, and capitalizes on the wonder, joy, and beauty of examining the world in which they live. For example, students should work with physical geometric models so that they can manipulate their orientation and explore their attributes to fully “see” the figure. Students should also engage in this domain in ways that are dynamic by using technology applications that allow shapes and figures to be constructed precisely and manipulated efficiently. Students should also consider the relationship between those manipulations and the resulting impact on lengths, angles, areas, surface areas, and volumes. Students should be able to make conjectures and form generalizations, preparing them for proof in high school and beyond. It is also important for middle school students to have ample opportunities to use tools to explore this domain, including drawings, grid and dot paper, informal constructions through paper folding and cutting, building and constructing, and technology. Such experiences could investigate building and design; art and aesthetics; visualization; distance and travel; and everyday applications of distance, angles, area, surface area, volume, and transformations that can easily be connected to careers such as architecture, medicine, engineering, construction, art, and much more.

In middle school, students’ understanding of geometry and measurement allows them to develop formulas for dimensions such as area, surface area, and volume of shapes as well as cultivate conceptual understanding of why the formulas make sense and why a change in one measurement impacts the outcome. Students who have conceptual understanding can identify patterns in geometric measurement formulas, leading to conjectures and generalizations. A deep knowledge of the geometry and measurement domain includes seeing connections to other domains, such as exploring scale factors with approaches that develop proportional reasoning by analyzing figures and situations and that connect to the concept of similarity. Angles, shapes, and distances should provide opportunities to understand and apply similarity and congruence. A deep understanding of geometry and measurement based on reasoning about relationships and context is contrary to student learning focused on memorization and rote application.

Now consider an example related to key ideas in geometry and measurement in middle school. In this example, students in a class are working to scale up and proportionally enlarge a piece of artwork (for the full example of this implementation, see Bush and colleagues 2013). The teacher has printed a scaled down 8 × 10-inch photograph of a compelling piece of artwork and cuts the 8 × 10-inch photograph into twenty 2 × 2-inch squares. Each student gets one 2 × 2-inch square (numbered on the back) and a blank 4 × 4-inch piece of paper (students are not shown the whole piece of artwork), and are asked to recreate what is on their 2 × 2-inch square onto the 4 × 4-inch blank square by making a proportional outline and using the same colors to color in the sections. After all the students complete their scaled-up drawing on their 4 × 4-inch square, the class recreates the artwork by piecing together the students’ twenty 4 × 4-inch squares on numbered chart paper (using the numbers the teacher had written on the back for placement). See figure 5.12 for an example.

As students engage in this mathematics task connected to art, the teacher can maximize the opportunity to involve students in the following key mathematical learning related to geometry and measurement, notably attending to new ideas rather than what is already familiar to students:

- As students draw on their 4 × 4-inch square, they use visualization, measurement estimation, and measurement tools to best scale up their intricate 2 × 2-inch piece. The importance of precision is illuminated because if the image is not scaled up precisely, it will have a negative effect on the overall recreation of the artwork.
- Before the students’ 4 × 4-inch squares are assembled to recreate the artwork, students explore what the size of the photograph of the 2 × 2-inch squares would be (8 × 10 inches) and conjecture how much greater their scaled-up re-creation will be. A common incorrect response is that the final version would be double the size of the original because 4 inches is double 2 inches; however, the discovery and later generalization is made that when the length and width are both doubled, the area is actually quadrupled.
- The length and area generalization leads to further discussion regarding what might happen if three-dimensional figures were investigated. This can result in students making conjectures and, ultimately, arriving at the generalization that if the length, width, and height of a cube are each doubled, the volume of the new cube is eight times the volume of the original.
- These generalizations can lead to a discussion related to similarity and transformations, specifically,
*dilations*. For example, now that students know that the area of their re-created artwork is four times the area of the original photograph, they can make a conjecture about the scale factor of the dilation. Once they determine the dilation is a scale factor of only two (8 × 10-inch photograph to 16 × 20-inch re-creation), students can reason that the dilation is the number of times the length and width dimensions increased (rather than the number of times the area increased).

Students’ middle school geometry experiences are built from the foundational knowledge from the elementary grades that includes an understanding of properties of different classes of shapes (e.g., triangles, quadrilaterals) and identifying and naming a range of two-dimensional and three-dimensional shapes (Sinclair, Pimm, and Skelin 2012). Students learned about different measurement attributes and how to use measuring tools effectively and how to precisely engage in measuring angles and lengths, including those of simple two-dimensional and three-dimensional figures, capacities, temperatures, time, and so on. They developed meaning of area, surface area, and volume. In middle school, students expand upon these ideas to explore additional two- and three-dimensional shapes and work with geometry and measurement in more complex ways, including transformations, examination of cross sections, and applications that integrate with algebraic thinking and proportional reasoning. After middle school, students move to more formalized applications of geometry and measurement, including proof and trigonometry.

The availability of resources for teaching mathematics has never been so easily and freely accessible. Online resources, social media, and print resources provide teachers with the opportunity to not solely rely on district-selected textbooks, as was often the case in the past. However, teachers should not be expected, nor do they have adequate time, to pull resources at random or from what turns up in a search engine. The result may be inconsistent with school- or district-wide structures and would likely lead to instruction that is not deep, coherent, or aligned with a carefully crafted learning trajectory. *Catalyzing Change in Middle School Mathematics* does not prescribe what curriculum should be used but rather to advocate that teachers be supported in collectively planning, implementing, and reflecting on the use of instructional materials. Such instructional materials should be coherent, systematic, vetted, and high quality, and embody the ideas in chapters 4 and 5.

Shulman (1986) proposed that the content knowledge needed for teaching was unique, describing it as pedagogical content knowledge. Decades later, Ball, Thames, and Phelps (2008) extended this work and further described domains of pedagogical content knowledge needed for teaching that include knowledge of content and student, content and teaching, and content and curriculum. Supporting teachers’ ongoing acquisition of content knowledge and pedagogical content knowledge is essential to the successful implementation of the middle school mathematical practices, processes, and content as described in this chapter. The content knowledge of middle school mathematics teachers is a shared responsibility of both their teacher preparation programs and the continued professional learning support they should receive from their schools and districts as well as their continuing higher education.

The Association of Mathematics Teacher Educators (AMTE 2017) provides standards to guide the content knowledge middle school mathematics teachers need during their preparation programs that are aligned with *The Mathematical Education of Teachers II* (MET II; CBMS 2012) and the *Statistical Education for Teachers* (SET; Franklin et al. 2015). The AMTE standards recommend “at least 15 semester hours (or equivalent) of mathematics and statistics courses designed specifically for future middle level teachers, including courses that engage middle level candidates in opportunities to demonstrate the mathematical practices” as well as “at least 9 semester hours (or equivalent) of mathematics and statistics courses beyond the precalculus level, including at least one statistics course” (p. 111).

Schools and districts can further support teachers’ content development and specialized content knowledge for teaching through professional learning opportunities focused on deepening teacher content knowledge in ways that help teachers to see connections across the mathematical ideas they teach as well as how the mathematics of middle school connects across disciplines and to the real world. Building teacher comfort and confidence in their mathematical content knowledge can help teachers make and support better choices regarding curriculum and resources, and engage in the teaching of mathematics that embody the ideas in chapters 4 and 5. Resources such as NCTM’s *Essential Understanding Series* (2010–2013), which includes four books focused on grades 6–8 as well as a K–8 book on mathematical reasoning, are specifically designed to build teacher content knowledge.