*The high school mathematics curriculum needs to be reformed to encourage all students to understand the math that underlies the fabric of society.*

High school mathematics is not working for far too many students in the United States. Although the National Assessment of Educational Progress (NAEP) has shown significant and long-term positive trends in mathematics learning at the elementary and middle levels, high school NAEP scores have remained essentially flat for decades (National Center for Education Statistics, 2015, 2016). In addition, the Program for International Student Assessment shows that U.S. high school students trail their international peers in mathematical literacy, defined as the “capacity to formulate, employ, and interpret mathematics in a variety of contexts . . . to describe, explain, and predict phenomena” (Organization for Economic Cooperation and Development, 2016, p. 28).

Given these results, it is not surprising that young adults in the United States lack not only the quantitative and problem-solving skills necessary for success in the workplace and postsecondary education but also the numeracy and problem-solving skills necessary for “meaningful participation in our democratic institutions” (Goodman, Sands, & Coley, 2015, p. 5).

Urgent calls for reform in mathematics education date back at least four decades, including *An Agenda for Action* (National Council of Teachers of Mathematics [NCTM], 1980);* A Nation at Risk* (National Commission on Excellence in Education, 1983); *Curriculum and Evaluation Standards for School Mathematics* (NCTM, 1989); *Principles and Standards for School Mathematic*s (NCTM, 2000); the *Common Core State Standards for Mathematics* (National Governors Association [NGA] Center for Best Practice & Council of Chief State School Officers [CCSSO], 2010); and *Principles to Actions* (NCTM, 2014). These calls for change target various aspects of high school mathematics — from the structures of high school programs; to the knowledge, skills, and practices expected of students, to teaching and learning experiences to engage and support students. Yet an implementation gap exists between the recommendations for reform and the meaningful systemic change needed to ensure that mathematics works for each and every student.

The high school mathematics curriculum still contains a great deal of obsolete legacy content that needs to be eliminated to create time for more essential and relevant concepts.

The status quo in high school mathematics persists in part because the structure of high school mathematics remains nearly the same today as it was decades ago. Despite efforts in the 1990s to change the traditional sequential nature of high school mathematics through the development of five National Science Foundation–funded curricula (St. John et al., 2004), for the vast majority of students today the high school mathematics curriculum continues to begin with a year of algebra followed by a year of geometry and a second year of algebra. This sequence was first recommended by the Committee of Ten in 1892 and remains the course pathway at more than 90% of high schools in the United States (Dossey, McCrone, & Halvorsen, 2016).

The high school mathematics standards and curriculum also present challenges to teachers, many of whom find it difficult to teach at the desired level of rigor, given the sheer number of standards they are expected to teach. The high school mathematics curriculum still contains a great deal of obsolete legacy content — for example, traditional symbolic manipulation from Algebra 2 that is now more efficiently carried out with technology — that needs to be eliminated to create time for more essential and relevant concepts, such as data analysis and mathematical modeling (National Center on Education and the Economy [NCEE], 2013).

The answer to these challenges is not, as some would argue, simply to require less mathematics (Hacker, 2016); rather, it is necessary to identify, confront, and make long overdue changes to the structures, policies, instructional approaches, and focus and relevance of high school mathematics. To meaningfully address these issues, in April 2018, NCTM released *Catalyzing Change in High School Mathematics: Initiating Critical **Conversations, *meant to identify the challenges and engage stakeholders to begin the process of transforming high school mathematics education to ensure that every student has the mathematical experiences necessary for personal and professional success.

**Why learn mathematics? **

In the last few decades, policy makers and reformers have described the goal of boosting students’ college and career readiness (and, by extension, preparing those students to contribute to national defense and economic prosperity) as more or less the only purpose for learning high school mathematics, while other purposes, such as teaching students to think critically and participate actively in civic life, have been given short shrift (Tate, 2013). But in fact, mathematics can serve multiple purposes, and should be taught in ways that prepare students to “flourish as human beings” (Su, 2017, p. 483).

Mathematics underlies much of the fabric of society, from polling and data mining in politics, to algorithms used in targeting advertisements, to complex mathematical models of financial instruments and policies that affect the lives of millions of people. Students should leave high school with the quantitative literacy and critical-thinking processes necessary to determine the validity of claims made in scientific, economic, social, and political arenas (Brelias, 2015). Students should have an appreciation for the beauty and usefulness of mathematics and statistics. And students should see themselves as capable lifelong learners and confident doers of mathematics and statistics.

To support these broadened purposes for learning mathematics, *Catalyzing Change *offers 41 Essential Concepts that every student should learn before graduating from high school. Culled from the domains of number, algebra and functions, statistics and probability, and geometry and measurement, these concepts are built on a careful review of recommendations from several organizations and major reports, including the *Common Core State Standards for School Mathematics* (NGA Center & CCSSO, 2010); the *Guidelines for Assessment and Instruction in Statistics Education* (GAISE) and the *Statistical Education of Teachers* reports (Franklin et al., 2007; Franklin et al., 2015); *Principles and Standards for School Mathematics* (NCTM, 2000); the *Guidelines for Assessment and Instruction in Mathematics Modeling Education* (Garfunkel & Montgomery, 2016); *What Does It Really Mean to Be College and Work Ready?* (NCEE, 2013); and *A Common Vision for Undergraduate Mathematical Sciences Programs in 2025* (Saxe & Braddy, 2015).

The Essential Concepts, however, do not represent yet another set of standards to adhere to or a list of disconnected topics to be covered. They represent a distillation of the critical concepts and skills that students should develop, retain, and be able to use after high school, regardless of a state’s, province’s, or district’s standards. These understandings are important for students to remember long after they have forgotten how to carry out specific techniques or apply particular formulas.

For example, the Essential Concepts in Statistical Inference include the need for students to understand the roles of randomization in statistical studies, bias and sample size in statistical studies, and statistical inference. When people misunderstand these concepts, they can easily be persuaded to adopt all sorts of mistaken ideas, such as the false but widespread belief that childhood vaccines are associated with autism, a belief that has resulted in fewer parents vaccinating their children and therefore directly led to an increase in the number of cases of measles in the United States during the last decade (Deer, 2011; Olive et al., 2018; Taylor, Swerdfeger, & Eslick, 2014). It is not necessary for all students to know how to use the algorithms associated with statistical inference; more critical is that every student leaves high school with an understanding of these concepts. As members of society, they will be presented with data-based claims daily, and they need to be capable of reasoning critically and questioning the statistical investigation process behind the claims they encounter.

When taught effectively, with an emphasis on critical thinking and mathematical reasoning, high school mathematics has the potential to help combat the increasing problem of truth decay in American society (Kavanagh & Rich, 2018). And when mathematics assignments and classroom activities address meaningful topics — such as the analysis of public health problems, income inequality, and environmental sustainability — students gain “access to rich, rigorous mathematics that offers opportunities and self-empowerment for them to understand and use mathematics in their world” (Wager & Stinson, 2012, p. 10). To broaden the purposes of learning mathematics and to make the curriculum more focused and relevant, *Catalyzing Change* recommends that “Each and every student should learn the Essential Concepts in order to expand professional opportunities, understand and critique the world, and experience the joy, wonder, and beauty of mathematics” (NCTM, 2018, p. 9).

To ensure that the push for a richer and more relevant curriculum does not lead simply to adding more content, *Catalyzing Change* includes guidelines to help determine which mathematics courses and content should count toward high school graduation. In short, such courses must:

- require clarity and precision in mathematical reasoning;
- have focused and significant mathematical learning standards;
- maintain the integrity of the mathematical standards;
- be a nonterminal part of a coherent mathematical learning progression; and
- approach mathematics in an instructionally balanced way that includes attention to conceptual understanding, procedural fluency, and problem solving as well as mathematical reasoning and critical-thinking practices.

Under these guidelines, for example, it would make little sense to count a computer science or engineering course for mathematics credit (as is sometimes proposed) if that course does not build the mathematical foundation necessary for students to further their study of STEM fields.

**The need for equal access **

Besides ensuring that mathematics curricula include the right content, we must also address and dismantle the structures that stand as barriers to positive mathematical experiences for students. Two widespread structural barriers and unjust practices identified in *Catalyzing Change *include tracking students into course pathways that do not prepare them for the continued study of mathematics and tracking teachers in ways that deny students with the highest need access to teachers with significant experience. Reform efforts that have focused largely on standards have been unable to address and alleviate these concerns.

Besides ensuring that mathematics curricula include the right content, we must also address and dismantle the structures that stand as barriers to positive mathematical experiences for students.

Student tracking is insidious because it places some students into qualitatively different or lower levels of a mathematics course. In some cases, it puts them into pathways that are not mathematically meaningful and do not prepare them for any continued study of mathematics or effective participation in a democratic society.

For example, the practice of placing students in a two-year version of an Algebra 1 course, as opposed to a double-period version of algebra, effectively ensures they will not progress rapidly enough in the curriculum to remain on the path to college and career readiness. However,* Catalyzing Change* draws a distinction between tracking and acceleration, arguing that acceleration may be appropriate for students who have demonstrated deep understanding of grade-level or course-based mathematics standards beyond their current level.

Like mathematics students, mathematics teachers themselves are often tracked, with the most experienced or effective teachers assigned to upper-level mathematics courses and the least experienced assigned to entry-level courses (Strutchens, Quander, & Gutiérrez, 2011). *Catalyzing Change* recommends that, whenever possible, teachers in the same department have balanced teaching assignments, including both upper-level and entry-level courses. This deepens teachers’ knowledge of the overall curriculum, reduces burnout among new teachers, and makes it easier to establish collaborative teams of teachers with different levels of experience and expertise (Gutiérrez, 2002; Strutchens et al., 2011).

**Improving instruction **

Mathematics teaching should not only engage and support students as they learn concepts and develop skills and understanding but also empower them to see themselves as capable of participating in and being *doers of *mathematics. For example, consider one well-established, research-informed instructional strategy: “Effective teaching of mathematics uses purposeful questions to assess and advance student reasoning and sense making about important mathematical ideas and relationships” (NCTM, 2014, p. 10). As important as this is for teachers to know, they must recognize that simply asking students questions during instruction is not enough. *How *teacher questioning unfolds in the classroom can have a significant influence on how students see themselves as members of the mathematics learning community (Boston et al., 2017). For example, when questioning students, teachers should also ask themselves these questions:

- Am I making sure all student ideas and questions are heard, valued, and pursued?
- Who am I calling on to answer questions? Whose ideas do I select for further inquiry, and whose thinking do I tend to disregard? (Boston et al., 2017).

Teachers ultimately shape the classroom environment, in part through their questioning techniques, which in turn shape how students see themselves and their competence (Goffney, 2018). Students who feel a strong mathematical identity and sense of agency are more likely to pursue the study of mathematics and use mathematics effectively in their personal lives. To improve the mathematical experience of students, *Catalyzing Change *recommends that “classroom instruction should be consistent with research-informed and equitable teaching practices” (NCTM, 2018, p. 25).

**A common learning experience **

To maximize students’ opportunities after high school and prepare them to actively engage in democratic society, *Catalyzing Change* recommends that high schools require students to enroll in mathematically demanding (non-terminal) mathematics courses every year in high school.

In addition, the expectation in* Catalyzing Change *is that a single curricular model be used to deliver the common pathway to all students in a school, ensuring that every student has access to a high-quality mathematics education, while avoiding the creation of separate and unequal tracks. Specifically, it recommends that:

High schools should offer continuous four-year mathematics pathways with all students studying mathematics each year, including two to three years of mathematics in a common shared pathway focusing on the Essential Concepts, to ensure the highest-quality mathematics education for all students. (NCTM, 2018, p. 83)

*Catalyzing Change* envisions that the Essential Concepts will be part of a common shared pathway that will provide the equitable educational experience that every student deserves. For example, the common pathway could be organized to begin with the Essential Concepts of geometry and measurement in grade 9, followed by the Essential Concepts of statistics and probability. Then in grade 10 and the first half of grade 11, students would continue the study of algebra that they began in middle school, moving from their previous work with coordinate geometry and linearity to quadratics and exponential equations. This example pathway would move through the first 2.5 years of high school. Students’ mathematics coursework beyond the Essential Concepts will be determined by each student’s own needs, goals, interests, and aspirations, rather than by any difference in mathematical ability perceived by anyone else. Coursework could include a traditional pathway directed toward calculus, or a different pathway focused on statistics or mathematical modeling.

**Next steps **

Mathematics education at the high school level is part of a complex system of policies, traditions, and societal expectations. This system and its structures (school district policies, practices, and conditions) must be critically examined, changed, and improved. All stakeholders — school, district, and state administrators; instructional leaders and coaches; classroom teachers; counselors; curriculum and assessment developers; higher education administration and faculty, and policy makers at all levels — will need to be part of the process of reexamining long-standing beliefs, practices, and policies. This work is critical for all of us to undertake. It is also long overdue.

Francis Su, past president of the Mathematical Association of America, has argued that answering the “why we teach mathematics” question is critical because the answer will have a strong influence on who we think should learn mathematics and how we think mathematics should be taught (Su, 2017). *Catalyzing Change* answers the *why* question in ways that support every student’s learning of essential mathematics concepts in equitable environments that build each student’s mathematical identity and agency. The work of making this happen will not be easy because the challenges are real and long-standing. But we owe this effort not only to our students but also to ourselves as we work together to create and nurture the society we wish to inhabit.

**References**

Boston, M., Dillon, F., Smith, M.S., & Miller, S. (2017). *Taking action: Implementing effective mathematics teaching practices grades 9–12. *Reston, VA: National Council of Teachers of Mathematics.

Brelias, A. (2015). Mathematics for what? High school students reflect on mathematics as a tool for social inquiry.* Democracy & Education, 2*3 (1), 1–11.

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Dossey, J.A., McCrone, S.S., & Halvorsen, K T. (2016). *Mathematics education in the United States, 2016: A capsule summary fact book. *Reston, VA: National Council of Teachers of Mathematics.

Franklin, C., Bargagliotti, A., Case, C., Kader, G., Schaeffer, R., & Spangler, D. (2015). *The statistical education of teachers.* Arlington, VA: American Statistical Association.

Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheaffer, R. (2007). *Guidelines for assessment and instruction in statistics education (GAISE Report): A pre-K–12 curriculum framework.* Alexandria, VA: American Statistical Association.

Garfunkel, S. & Montgomery, M. (Eds.). (2016). *Guidelines for assessment and instruction in mathematical modeling education (GAIMME). *Philadelphia, PA: Consortium for Mathematics and Its Applications and Society for Industrial and Applied Mathematics.

Goffney, I. (2018). Where do we go from here? Next steps in rehumanizing mathematics for Black, Indigenous, and Latinx students. In R. Gutiérrez, I. Goffney, & M. Boston (Eds.), *Annual perspectives in mathematics. education 2018: Rehumanizing mathematics for Black, Indigenous, and Latinx students* (pp. 159–170). Reston, VA: National Council of Teachers of Mathematics.

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Strutchens, M.E., Quander, J.R., & Gutiérrez, R. (2011). Mathematical learning communities that foster reasoning and sense making for all high school students. In M.E. Strutchens & J.R. Quander (Eds.),* Focus in high school mathematics: Fostering reasoning and sense making* (pp. 101–113). Reston, VA: National Council of Teachers of Mathematics.

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Tate, W.F. (2013). Race, retrenchment, and the reform of school mathematics. In E. Gutstein & B. Peterson (Eds.),* Rethinking mathematics: Teaching social justice by the numbers* (2nd ed., pp. 42-51). Milwaukee, WI: Rethinking Schools.

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Wager, A.A. & Stinson, D.W. (2012). A sojourn into the empowering uncertainties of teaching and learning mathematics for social change. In A. Wager & D. Stinson (Eds.), *Teaching mathematics for social justice: Conversations with educators* (pp. 3–18). Reston, VA: National Council of Teachers of Mathematics.

**Citation:** Berry III, R.Q. and Larson, M.R. (2019). The need to catalyze change in high school mathematics. *Phi Delta Kappan, 100 *(6), 39-44.

## One Comment

All of these ideas may be well and good, but why does the NCTM charge $40 for the book that details and expands these ideas? Should this material be made available for free??