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

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

Having strong subject-matter knowledge is critical for all well-prepared beginning teachers of mathematics, especially those at the high school level [C.1.1]. Intensive subject-matter preparation is necessary, given the level of the material; a more detailed outline of the essential mathematical understandings well-prepared beginning teachers of high school mathematics need is given below.

Many high school mathematics teacher candidates will have experienced success with a narrow school mathematics curriculum that did not promote conceptual knowledge or emphasize mathematical practices and processes. Thus, they must gain personal experiences with those practices and the ways they can support deeper knowledge of important mathematical concepts so that they can fully support their students’ mathematical development [C.1.2]. For example, Vignette 7.1 demonstrates the close connection between the choice of a mathematical task and the teacher’s role in facilitating discourse that is centered on mathematical ideas. This vignette is intended to illustrate how beginning teachers of mathematics at the high school level have deep knowledge of mathematics and skill in mathematical communication, reasoning, and sense making and move instruction beyond a focus on writing algebraic equations to making connections to other mathematical representations.

### Vignette 7.1. Meaningful Algebraic Expressions

The instructor of a methods class gave her students the following problem.

The height (in feet) of a ball t seconds after being thrown into the air is given by the following equation: What is the maximum height the ball will reach? (Adapted from NCTM, 2009, pp. 32–33).

The instructor first asked the students to connect the function to models of projectile motion from physics, and the students discussed the meanings of the variables and coefficients in the equation. They also discussed how a function like this might be derived from data.

To answer the question, many students thought of taking the first derivative and solving for 0, knowing that the maximum height will occur at a critical point. While acknowledging that this approach is correct, the instructor urged her students to consider other ways that they could find the maximum height. While the students worked in small groups to explore the problem further, they devised a wide range of strategies including the following:

Group 1 drew a graph of the function using a graphing utility and used it to estimate the maximum.

Group 2 made a table of values to estimate the maximum. However, they further noticed that the function has zeroes at t = and t = . They reasoned that, because a parabola is symmetric, the maximum value will occur at the midpoint between those 2 zero values, t = . So the maximum is .

Group 3 students remembered that the method of completing the square could be used to find the vertex of a parabola, which they felt should be the maximum value. But they admitted that their understanding of completing the square was limited to having memorized a procedure, and they had forgotten the procedure.

In a class discussion, the instructor asked each group to present their solution to the class, carefully explaining their reasoning; the other students were encouraged to analyze the approach presented. After Group 1 presented its solution, another student asked, “Just because it looks like the maximum on the graph, are you sure that is the exact maximum height?” The Group 1 members admitted that they were approximating, so the value might not be exact. This exchange was an opportunity for the instructor to guide the discussion toward foundational aspects of rational and irrational numbers and precision, with the goal that the students articulate that the maximum might be irrational or it might be rational with a periodic decimal expansion that exceeds the numerical precision of graphing utility.

Group 2’s solution was met with some amazement by the other students, who had not thought of taking such a simple approach. The instructor led a brief discussion concerning why the graph is symmetric.

Group 3 showed their progress but admitted that they were not entirely sure how to complete the square. The instructor allowed the other groups to discuss briefly among themselves, and the class reached agreement on the algebra . The instructor encouraged the students to discuss why this computation would reveal the maximum; that is, -16 times a square must be 0 or negative, so h(t) must be less than or equal to . She also challenged the students to represent completing the square geometrically: “After all, doesn’t it sound geometric?” She provided algebra tiles and asked the students to explore several simpler problems to better understand the method; she then asked them to use Desmos to represent this problem. She led a discussion of the underlying structure of the method of completing the square (cf. CBMS, 2012, p. 55). Finally, she explained that when they take their abstract algebra course the following semester, the students will explore connections between the complex numbers and the set of all polynomials in a variable x with real coefficients (cf. CBMS, 2012, p 59).

The instructor concluded the lesson by assigning the students to write a reflection on mathematics content and mathematical practices that they had investigated in the lesson.

Finally, high school mathematics teacher candidates need to gain productive dispositions toward engaging in mathematics, recognizing that even in the absence of a known solution method, they have resources that can help them make progress toward a solution. Despite challenges they may face in their advanced mathematics classes, they still see mathematics as an exciting and interesting endeavor [C.1.3]; without such dispositions, they will have little success in convincing their students of the value and importance of mathematics.

According to The Mathematical Education of Teachers II (MET II) (CBMS, 2012), well-prepared beginning teachers of mathematics at the high school level need to learn mathematical content that is “tailored to the work of teaching, examining connections between middle grades and high school mathematics as well as those between high school and college” (p. 54). In addition, they need to build strength with mathematical processes and practices; again according to MET II, they need a “full range of mathematical experience themselves: struggling with hard problems, discovering their own solutions, reasoning mathematically, modeling with mathematics, and developing mathematical habits of mind” (CBMS, 2012, p. 54) [see C.1.2]. As the Mathematical Association of America advocated for all who pursue mathematics majors, they must develop “effective thinking and communication skills” (Tucker, Burroughs, & Hodge, 2015, p. 1).  Finally, they need productive dispositions toward mathematics, recognizing that mathematics is inherently a human activity [see C.1.3]. Many aspects of mathematics are rooted in the development of useful tools to address problems in the real world, and recognizing the impulse to find practical uses in mathematics provides both meaning and motivation to the study of mathematics. Acknowledging that mathematics has been created by a wide range of civilizations across history reinforces that it does not belong to a particular group.

The elaboration of mathematical content standards across this document relies on the expertise and thorough treatment of the topics as provided in the MET II report (CBMS, 2012). The structure of the MET II chapter related to high school mathematics differed from that of the other grade-band chapters in that it was organized around courses in a beginning teacher’s undergraduate mathematics major that leads to teaching licensure, whereas the elementary and middle grades chapters were related to the content standards of the Common Core State Standards – Mathematics (CCSS-M [NGA & CCSSO, 2010]) as a framework. Thus, the structure of this chapter, relying on MET II, also differs from the structures of the early childhood, upper elementary, and middle level grade-band chapters in this volume. In the following, we provide four essential ideas about high school mathematics content for beginning teachers, including understanding algebra as generalized arithmetic, the role of functions in mathematics, the role of diagrams and definitions in geometry, and statistical models and statistical inference as well as the connections among these mathematical areas. While the high school mathematics curriculum continues to evolve, additional experiences in mathematical modeling and computer programming or coding may also be needed. Each of these areas is discussed below; the NCTM CAEP Mathematics Content for Secondary Addendum to the NCTM CAEP Standards 2012 (NCTM & CAEP, 2012b) has provided more detailed descriptions of the mathematical knowledge that beginning mathematics teachers should have and also reiterated the importance of mathematical processes and practices.

Reasoning in algebra. Well-prepared beginning teachers of mathematics at the high school level understand that students in the elementary grades focus on the meanings of numbers and operations on them as foundations for developing tools and techniques in arithmetic and that in the middle school grades, students move from this focus on arithmetic to a focus on algebra as generalized arithmetic (Usiskin, 2004). These teachers have strong understandings of the foundational concepts in Pre-K–8 number and operations and how they lead into algebraic thinking. Without a purposeful focus on this perspective, teachers can begin their careers relying on naive views of high school algebra as symbol pushing—rules performed on symbols without applying underlying reasoning. The CCSS-M mathematical practice “look for and express regularity in repeated reasoning” (NGA & CCSSO, 2010, p. 8) captures a broader purpose of high school algebra: to enable students to build expressions, functions, and equations to model situations. Embedded in an understanding of algebra is an understanding of the roles and nature of variables as a part of the language of mathematics. Students begin to develop this perspective at the middle level, and it is addressed in courses designed specifically for high school mathematics teacher candidates. This perspective is also appropriate in mathematics courses with broad audiences, from calculus, to introduction to proofs, to linear or abstract algebra.

Functions in mathematics. Well-prepared beginning teachers of mathematics at the high school level understand big ideas about functions (Lloyd & Beckmann-Kazez, 2010):

The concept of function is intentionally broad and flexible…. Functions provide a means to describe how related quantities vary together…. Functions can be classified into different families, each with its own unique characteristics…. Functions can be combined by adding, subtracting, multiplying, dividing, and composing them… Functions can be represented in multiple ways, including algebraic, graphical, verbal, and tabular representations. (pp. 7–8)

These ideas have their foundations in middle-level mathematics, and well-prepared beginning teachers of high school mathematics understand the connections across grades. In addition to understanding the nature of functions, well-prepared beginning teachers of high school mathematics must have facility with the foundational functions that are studied in high school mathematics: linear, quadratic, and other polynomials; rational functions; exponential, logarithmic, and trigonometric functions; and functions that are recursively defined. This understanding of functions permeates the calculus sequence; although the tools of calculus are required of beginning teachers, it is not to be assumed that a preservice teacher who has completed the calculus sequence has investigated functions adequately to build an understanding of functions to teach them well. The functions of precalculus-level mathematics are then revisited in courses for middle and high school mathematics from an advanced perspective and are studied from a teaching perspective in methods courses.

Diagrams and definitions in geometry. Well-prepared beginning teachers of mathematics at the high school level understand that the role of geometry in high school differs from its role in middle school and that the objects of study in geometry in high school are “general properties of classes of figures” rather than “properties of individual figures” (Usiskin, 2004). They understand geometry from the perspective of transformations. Sinclair, Pimm, and Skelin (2012b) stated that “working with diagrams is central to geometric thinking” (p. 9). They further noted, “Geometry is about working with variance and invariance, despite appearing to be about theorems” (p. 22). “Working with and on definitions is central to geometry” (p. 36), and “a written proof is the endpoint of the process of proving” (p. 48). The ability to use dynamic geometry software to investigate and understand variance and invariance of geometric objects is essential.

Statistical models and statistical inference. Statistics holds a unique place in the high school mathematics curriculum. Statistics and mathematics as disciplines share a common foundation, but they are distinct areas, each with its own methods and traditions. The most notable distinction is that statistics is built on inference from inductive processes, whereas mathematics is decidedly deductive. However, statistics is generally included as part of the mathematics curriculum. Beginning teachers might easily, mistakenly believe that statistics is just another course in mathematics (like algebra, or geometry, or calculus) and undervalue the important distinctions between the two subjects.

The Statistical Education of Teachers (SET) (Franklin et al., 2015) described the problem-solving experiences and habits of mind that statisticians have identified as important to their discipline: A “modern data-analytic approach to statistical thinking, a simulation-based introduction to inference using appropriate technologies, and an introduction to formal inference” (p. 8) are the appropriate introductions to statistics for high school teacher preparation. The authors recommended attention to both randomization and classical procedures for comparing two parameters based on both independent and dependent samples (small and large), the basic principles of the design and analysis of sample surveys and experiments, inference in the simple linear regression model, and tests of independence/homogeneity for categorical data. (p. 8)

They further recommended a focus on statistical modeling “based on multiple regression techniques, including both categorical and numerical explanatory variables, exponential and power models (through data transformations), models for analyzing designed experiments, and logistic regression models” (p. 8). High school mathematics teachers hold the bulk of responsibility for ensuring the integrity of the statistics taught and learned in schools. Statistics, data analysis, and modeling offer students unique opportunities to address real-world problems that affect them in their communities.

Peck, Gould, and Miller (2013) identified five big ideas that are central to teaching high school statistics:

Data consist of structure and variability…. Distributions describe variability…. Hypothesis tests answer the question ‘do I think this could have happened by chance?’… The way in which data are collected matters…. Evaluating an estimator involves considering bias, precision, and the sampling method. (pp. 10–11)

This perspective on statistics and statistical reasoning is present in the statistics coursework of effective programs; a modeling course can address the distinctions between mathematical models and statistical models and offers an appropriate forum for discussing the distinction between mathematics and statistics to prepare teachers to address it in their high school classrooms.

New emphases. New emphases in the high school curriculum are part of the fluid nature of curriculum and its responses to the needs of society. NCTM (2016b) is embarking on a project to define with clarity and specificity pathways for high school mathematics. Well-prepared beginning teachers of mathematics at the high school level must be aware of changes to high school mathematics that are likely in the coming years.

One new emphasis is on mathematical modeling. Authors of Guidelines for Assessment and Instruction in Mathematical Modeling Education (Consortium for Mathematics and Its Applications [COMAP] & Society for Industrial and Applied Mathematics [SIAM], 2016) advocate the importance of mathematical modeling, “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena” (p. 8). Mathematical modeling is different from using manipulatives to “model mathematics” and from “direct modeling,” both of which pertain to how mathematics is represented (Hirsch & Roth McDuffie, 2016, p. ix–x). Investigating mathematical concepts through engineering design or a basic modeling cycle provides not only opportunities to integrate other disciplines but also a means for students to improve their mathematical understandings (Usiskin, 2015). Well-prepared teachers have substantive experiences engaging in mathematical modeling so that they can understand mathematical modeling and its potential place in the curriculum.

A recent publication that considered the future of mathematics, The Mathematical Sciences in 2025 (National Academies Press, 2013), states that “over the years there have been important shifts in the level of activity in certain subjects—for example, the growing significance of probabilistic methods, the rise of discrete mathematics, and the growing use of Bayesian statistics” (p. 72). The book identifies two new drivers of mathematics–computation and big data, and for both of these drivers it describes how discrete mathematics plays an important role. Well-prepared beginners should be aware of major concepts in discrete mathematics and recognize its future importance.

Finally, calls for increasing emphasis on computer science and coding are often delegated to mathematics teachers, given “similarities, connections, and intersections between the fields of computer science and mathematics” (NCTM, 2016a, p. 1). Indeed, writing computer code can be a powerful tool for solving mathematics problems, and computer coding can include interesting mathematics related to the design and analysis of algorithms—for example, understanding why the time required for a program designed to solve a problem can grow exponentially as the problem size increases, indicating that it may not be a fruitful approach. However, as acknowledged in NCTM’s (2016a) position statement on Computer Science and Mathematics Education, well-prepared mathematics teachers must understand that computer science is not merely a subfield of mathematics but is a field in its own right that requires specialized knowledge to teach.

Use of technology is an expected part of society, and ability to use technology is expected in the workforce. Technology is also an important tool for doing mathematics, and well-prepared beginning teachers are proficient in its use. Technology is more than a computational aid.  Well-prepared beginners are able to guide students in exploring how technology can be used to explore patterns, shape, transformations, and sequences. Technology can assist one in making connections between multiple representations, and it can help students communicate their mathematical ideas to their classmates. Well-prepared beginning teachers are particularly prepared to use “mathematical action technologies” (cf. NCTM, 2014, p. xi) useful for high school, including spreadsheets, dynamic geometry software, function-graphing utilities and graphing calculators; computer algebra systems; statistics simulation software; and other applets and technological tools that can enhance students’ conceptual understanding of mathematical concepts. These are powerful tools for doing mathematics that will be a part of the lives of the students they teach.

In effective programs, use of these tools is embedded throughout candidates’ preparation – in their content preparation, methods courses, and clinical experiences, including in courses not specifically designed for teachers, so that they can use them in meaningful ways. As stated in the MET II (CBMS, 2012),

Teachers should become familiar with various software programs and technology platforms, learning how to use them to analyze data, to reduce computational overhead, to build computational models of mathematical objects, and to perform mathematical experiments. The experiences should include dynamic geometry environments, computer algebra systems, and statistical software, used both to apply what students know and as tools to help them understand new mathematical ideas—in college, and in high school. Not only can the proper use of technology make complex ideas tractable, it can also help one understand subtle mathematical concepts. At the same time, technology used in a superficial way, without connection to mathematical reasoning, can take up precious course time without advancing learning. (p. 57)

Well-prepared beginning high school mathematics teachers are comfortable using technology to engage in mathematics and to effectively support meaningful mathematics learning. They develop dispositions toward critically evaluating the appropriate use of emerging technologies and are prepared to respond to new technological tools when they become available.

On the one hand, although generic conveyance software such as PowerPoint and interactive whiteboards can provide valuable support for classroom instruction, such software is not inherently mathematical and does not provide the experiences essential for learning high school mathematics with technology. On the other hand, technology can provide powerful tools to enhance communication about mathematics within the classroom, including blogs and interactive platforms such as teacher.desmos.com. Effective beginning teachers view technology as an expected and vital part of the classroom, not as a replacement for effective teaching but as an embedded tool that students use to explore mathematics.

Additionally, well-prepared beginning high school teachers view online communications, such as the MathTwitterBlogosphere (#MTBoS), as an essential professional resource to increase their understanding of mathematics teaching, to share their triumphs and seek insights into their conundrums, and to build professional relationships with other mathematics teachers across the nation. At the same time, they maintain a critical ability to ascertain the value of content available on the Web and in social media and are prepared to respond to misinformation found there.

Effective beginning teachers are also prepared to use other nonelectronic tools such as manipulatives in their classrooms. Contrary to common beliefs, high school students benefit from using algebra tiles, 3D models, and other physical manipulatives (cf. NCTM, 2014). Well-prepared beginners have the knowledge needed to “make sound decisions about when such tools enhance teaching and learning, recognizing both the insights to be gained and possible limitations of such tools” (NCTM, 2012, p. 3).