1.4 Mathematics Content Standards in the U.S.
The current standards for mathematics and assessment in the United States are derived from a variety of sources and past initiatives. David Klein -D. Klein (2003) gives a good summary of the major events and influential individuals during the 20th century. The current set of standards are strongly connected to a pair of reports from the 1980’s: one by NCTM -NCTM (1980), An Agenda for Action, and one by the U.S. National Commission on Excellence in Education -National Commission on Excellence in Education (1983), A Nation at Risk. Each provided a different view about what should occur to improve mathematics education in the United States. While these two documents were used as the basis for the ‘math wars’ of the late 20th century, the biggest difference between them is that the NCTM document focused primarily on standards for teaching methodologies, while A Nation at Risk focused on standards of content knowledge.
The NCTM -NCTM (1980) recommended “that
problem solving be the focus of school mathematics in the 1980s;
basic skills in mathematics be defined to encompass more than computational facility;
mathematics programs take full advantage of the power of calculators and computers at all grade levels;
stringent standards of both effectiveness and efficiency be applied to the teaching of mathematics;
the success of mathematics programs and student learning be evaluated by a wider range of measures than conventional testing;
more mathematics study be required for all students and a flexible curriculum with a greater range of options be designed to accommodate the diverse needs of the student population;
mathematics teachers demand of themselves and their colleagues a high level of professionalism;
public support for mathematics instruction be raised to a level commensurate with the importance of mathematical understanding to individuals and society” (p. 1).
The National Commission on Excellence in Education -National Commission on Excellence in Education (1983) recommended that
“The teaching of mathematics in high school should equip graduates to: (a) understand geometric and algebraic concepts; (b) understand elementary probability and statistics; (c) apply mathematics in everyday situations; and (d) estimate, approximate, measure, and test the accuracy of their calculations. In addition to the traditional sequence of studies available for college-bound students, new, equally demanding mathematics curricula need to be developed for those who do not plan to continue their formal education immediately” (p. 25).
“Standardized tests of achievement (not to be confused with aptitude tests) should be administered at major transition points from one level of schooling to another and particularly from high school to college or work. The purposes of these tests would be to: (a) certify the student’s credentials; (b) identify the need for remedial intervention; and (c) identify the opportunity for advanced or accelerated work. The tests should be administered as part of a nationwide (but not Federal) system of State and local standardized tests. This system should include other diagnostic procedures that assist teachers and students to evaluate student progress” (p. 28).
“Persons preparing to teach should be required to meet high educational standards, to demonstrate an aptitude for teaching, and to demonstrate competence in an academic discipline. Colleges and universities offering teacher preparation programs should be judged by how well their graduates meet these criteria” (p. 30).
“Substantial nonschool personnel resources should be employed to help solve the immediate problem of the shortage of mathematics and science teachers. Qualified individuals including recent graduates with mathematics and science degrees, graduate students, and industrial and retired scientists could, with appropriate preparation, immediately begin teaching in these fields. A number of our leading science centers have the capacity to begin educating and retraining teachers immediately. Other areas of critical teacher need, such as English, must also be/addressed” (p. 31).
NCTM followed it’s 1980 report with the publication of the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). This document focused on the standards for teaching methodologies and mathematical practices. Content standards were sketched out, with overviews of the recommended content knowledge for students in 4-year grade bands. NCTM expanded this work with additional texts focusing on teaching standards and assessment: Professional Teaching Standards (1991) and Assessment Standards (1995). In 2000, NCTM released an updated version the 1989 standards with the Principles and Standards for School Mathematics (NCTM, 2000). During this same time, many states developed more specific content standards reflecting the guidelines recommended by the 1989 document (Raimi & Braden, 1998). Although the NCTM document did not make content recomendations by specific grade levels, many of the state standards still resided at the grade-band level (Reys & Lappan, 2007, p. 677).
1.4.1 Common Core State Standards-Math
In 2002, Congress passed the No Child Left Behind Legislation. This bill required that states determine measurable content standards for each grade level and develop assessments based on these standards to be given to all students at specific grade levels (generally fourth, eighth, and twelfth grade) in order to receive federal school funding. Since each state developed their own standards, the level and specificity of these standards varied greatly between States (Reys & Lappan, 2007).
It was in this environment that the National Governors Association Center for Best Practices and the Council of Chief State School Officers launched an effort in 2009 to develop “a common core of internationally benchmarked standards in math and language arts for grades K-12” (NGA-CCSSO, 2010). These standards emphasize content knowledge, while also encouraging some of the teaching methodologies proposed by the NCTM documents. However, these standards to do instruct teachers how to teach or what curriculum to use. They instead focus on what students should know and be able to do at each grade level.
In order to help teachers better understand how the topics in this book relate to the actual teaching and learning of their students, we include related content standards from the Common Core State Standards for Mathematics throughout the text. While not all states have adopted the Common Core State Standards for their school systems, these standards are a good representation of what is expected from students at different stages in their mathematical education.
Related Content Standards
- (8.F.1) Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
- (HSF.IF.7) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Standards that align with the content of each section are provided in blocks with the title “Related Content Standards.” One such box is given above as an example. The sequence of numbers and letters proceeding each standard is its standardized abbreviation. The first part of the abbreviation identifies the grade band. Since the Common Core State Standards for high school do not have a recommended grade level for each of the standards, they are denoted by “HS” and the general area of mathematics. The second part of the abbreviation refers to the general domain of mathematics, and the third part refers to the location of the standard in that grade band and domain.
1.4.2 NCTM-CAEP Standards
Another requirement of the No Child Left Behind legislation was that states had to create standards for teachers to receive certification. These standards for certification needed to be measured with a blend of standardized assessments, portfolios, and coursework. As a result, certification in secondary mathematics in most states requires a major in mathematics (or its equivalent), the teaching of and about standards during certification coursework, and achievement of certain scores on a content specific standardized test (such as the PRAXIS). In addition, the NCTM partnered with the Council for the Accreditation of Educator Preparation (CAEP) to develop standards for what beginning secondary mathematics teachers should know and be able to do in the teaching of mathematics (NCTM, 2020).
The NCTM-CAEP content standards are built upon the standards developed by the Conference Board of Mathematical Sciences in -Conference Board of the Mathematical Sciences (2001) and -Conference Board of the Mathematical Sciences (2012). The content of this book builds upon the contents of the first standard of “Knowing and Understanding Mathematics”. In particular:
Candidates demonstrate and apply, with the incorporation of mathematics technology, conceptual understanding, procedural fluency, and factual knowledge of major mathematical domains: Number; Algebra and Functions; Statistics and Probability; Geometry, Trigonometry, and Measurement; Calculus; and Discrete Mathematics.
We list the details of the NCTM-CAEP content standards below and discuss how this book connects to each of these standards.
Essential Concepts in Number. Candidates demonstrate and apply conceptual understanding, procedural fluency, and factual knowledge of number including flexibly applying procedures, using real and rational numbers in contexts, developing solution strategies, and evaluating the correctness of conclusions. Major mathematical concepts in Number include number theory; ratio, rate, and proportion; and structure, relationships, operations, and representations.
Number systems are primarily covered in Chapters 4, 7, and 8. We introduce initial concepts of the number systems in Chapter 4, focusing on operations and the basic properties of common number systems, including whole numbers, integers, rational, real, and complex numbers. Chapters 7 and 8 expand these ideas through the added structure of rings and fields.
Essential Concepts in Algebra and Functions. Candidates demonstrate and apply understandings of major mathematics concepts, procedures, knowledge, and applications of algebra and functions including how mathematics can be used systematically to represent patterns and relationships including proportional reasoning, to analyze change, and to model everyday events and problems of life and society. Essential Concepts in Algebra and Functions include algebra that connects mathematical structure to symbolic, graphical, and tabular descriptions; connecting algebra to functions; and developing families of functions as a fundamental concept of mathematics. Additional Concepts should include algebra from a more theoretical approach including relationship between structures (e.g., groups, rings, and fields) as well as formal structures for number systems and numerical and symbolic calculations.
Many of these concepts are spread throughout the text, as algebra and functions are fundamental to the secondary curriculum. That said, Chapters 5, 7, 8, and 9 specifically focus on topics of particular importance in understanding algebra and functions. In particular Chapter 9 combines the themes of the earlier chapters together using real-valued functions.
Essential Concepts in Calculus. Candidates demonstrate and apply understandings of major mathematics concepts, procedures, knowledge, and applications of calculus including the mathematical study of the calculation of instantaneous rates of change and the summation of infinitely many small factors to determine some whole. Essential Concepts in Calculus include limits; continuity; the Fundamental Theorem of Calculus; and the meaning and techniques of differentiation and integration.
The essential concepts in calculus are currently excluded from this text with the understanding that most pre-service teachers are already required to take at least two semesters worth of calculus courses that focus on this content standard.
Essential Concepts in Statistics and Probability. Candidates demonstrate and apply understandings of statistical thinking and the major concepts, procedures, knowledge, and applications of statistics and probability including how statistical problem solving and decision making depend on understanding, explaining, and quantifying the variability in a set of data to make decisions. They understand the role of randomization and chance in determining the probability of events. Essential Concepts in Statistics and Probability include quantitative literacy; visualizing and summarizing data; statistical inference; probability; and applied problems.
The essential concepts of statistics and probability are the focus of Part IV of the text, Data Analysis. In that section we focus on statistics and a study of variability, looking at different ways to measure, communicate, and understand variability in the context of statistical problems. We also discuss general principles of data analysis studies including data collection, hypothesis testing, and methods of reporting results.
Essential Concepts in Geometry, Trigonometry, and Measurement. Candidates demonstrate and apply understandings of major mathematics concepts, procedures, knowledge, and applications of geometry including using visual representations for numerical functions and relations, data and statistics, and networks, to provide a lens for solving problems in the physical world. Essential Concepts in Geometry, Trigonometry, and Measurement include transformations; geometric arguments; reasoning and proof; applied problems; and non-Euclidean geometries.
Part III on Geometry looks at the subject of Geometry from the perspectives of constructional, transformational, analytic, and algebraic. Each perspective helps us to better understand the essential concepts in geometry, trigonometry, and measurement, along with the interactions between geometry, algebra, functions, and number systems.
1.4.3 Exercises
Reflect on your K-12 mathematics education. Based on the brief history described, what documents were being used to guide your curriculum?
Those who study mathematics curriculum describe the changing focus of what is written in school standards for mathematics as a pendulum. Over time, the pendulum swings between a focus on procedural fluency such as A Nation at Risk -National Commission on Excellence in Education (1983) describes and the more process oriented standards as outlined by NCTM (1989).
- How does the Common Core State Standards attempt to unify the two extremes of the pendulum swing?
- How does treating the process standards as separate from the content standards provide an opportunity to continue to let the pendulum swing?
- Regardless of which documents were currently in vogue while you were in school, do you think your mathematics education was more procedurally focused or more process standard focused?
- Is how you were taught what you hope your own teaching will be like? Why or why not?
- K-12 mathematics students often think that mathematics is a set of unrelated procedures that they have to memorize how to do. In contrast, people who do mathematics for a living think mathematics is a conceptual and logical system where everything fits together. As you currently understand them, do you think the Common Core State Standards could support students in moving from thinking about mathematics as a set of unrelated procedures to a more conceptual system? Why or why not?