Mathematics Teaching in the 21st Century

“How should mathematics be taught in the 21st century?” This question affects every aspect of mathematics education discourse from conference topics, creation of STEM (science, technology, engineering, and mathematics) departments at universities, to the writing of the Common Core State Standards for Mathematics. To begin answering it, we need to examine “the grammar of mathematics education.”

 

David Tyack and Larry Cuban coined the phrase “the grammar of schooling” in their book Tinkering Toward Utopia, where they defined it as “the organizational forms that govern instruction.” It includes familiar schooling features such as age-grading of students and division of knowledge into separate subject areas. In essence, it delineates the acceptable rules and behaviors that a “real school” must follow. Tyack and Cuban argued that 20th-century educational reformers largely failed because they sought utopian change through large-scale systemic reform without regard for the grammar of schooling. Because those reforms did not work well in the classroom, assumed unrealistic resources, or increased teachers’ daily work routines without compensation, teachers modified the reformers’ original ideas. Hence, the history of educational reform is a story of “local, gradual, and piecemeal” change resulting from teachers acting as “tinkerers” who experimented with “practices that ripped through corners of the traditional pattern of schooling” implementing change that “preserves what is valuable and remedies what is not.”

 

What is the current grammar of mathematics education? The latest Trends in Mathematics and Science Study provides evidence that it is not different from that of the 19th century. Most mathematics classrooms in the U.S. still consist of students sitting in rows listening to a teacher explain, using rote procedures to solve specific problems while asking cognitively undemanding questions. If we want to answer the question “How should mathematics be taught in the 21st century?” we must change the grammar.

 

Two salient issues lie at the core of the current grammar. The first is K–5 mathematics. Once considered a place for “back to basics” teaching, research has shown that children are capable of more than arithmetic and that the foundation for advanced mathematics needs to be established here. The second is the question of what constitutes rigorous mathematical thinking and whether any one course, be it algebra or calculus, fulfills this need in the 21st century.

Lower school mathematics classes today should look and run differently than the ones we remember from our childhood. Just as health care facilities, government offices, and stock exchanges have evolved to meet the challenges of our society, so too has our understanding of the teaching and learning of mathematics. Preparing students to be confident participants in their communities and leaders in their fields requires mathematical literacy that involves more than getting correct test answers. Not only do all students need to grapple with the universal disciplines of the content of mathematics, but they and their teachers must also develop the skills and dispositions that will enable them to think flexibly, take risks, and work collaboratively in our modern global culture.

 

A recent piece on National Public Radio’s All Tech Consideredhighlighted five “movers and shakers” of the tech world. One of them, Babak Parviz, professor of electrical engineering at the University of Washington and project leader on Google’s Project Glass, pointed out at a recent TED talk, “I would hazard a guess that the era of the solo star scientist is probably over.” Reporter Steve Henn noted, “In fact, none of the men and women I just mentioned do much of anything alone. . . . Today’s big problems are so complex—so interdisciplinary—that all of these people make their marks working in teams.”

 

This echoes the work of Tony Wagner, the Harvard-based education expert. In his 2008 book The Global Achievement Gap,he explained that students need three basic skills if they want to thrive in a knowledge economy: critical thinking and problem solving, effective communication, and collaboration. In his 2012 book Creative Innovators: The Making of Young People Who Will Change The World, Wagner’s list grew into the “Seven Survival Skills.”

 

The teacher must, then, cultivate a classroom culture where students understand that autonomy and collaboration are equally important. If a teacher’s words and actions honor risk-taking, active investigation, and clear communication, students will sooner come to see themselves as competent mathematicians who thrive on cognitive challenges. However, if students are nurtured to believe that teachers are the keepers and distributers of mathematical knowledge, there is little evidence to suggest that students will rely on their own reasoning to solve future problems encountered inside and outside of the classroom.

 

Teachers are also working to promote effective mathematical discourse in the classroom, which requires students to organize their thoughts, formulate arguments, listen to and consider other students’ positions, and communicate their own positions. It is through discourse that the ideals of collaboration and autonomy intersect, are nurtured, and are celebrated. Today’s mathematics teachers must be willing to step out of the spotlight and think of themselves as “directors” rather than the “lead actors” in the classroom.

 

Some of the behaviors and metacognitive disciplines that teachers in the Lower School work to nurture are listed below. You might recognize some of the examples from students’ work, or witness them in action when visiting the classroom.

 

 

The National Math Advisory Panel’s report Foundations for Successtargeted algebra as the most critical mathematics topic and renewed the question, “Should all 8th graders take algebra?” The question originated in the 1980s, when policymakers and educators concluded that algebra was a gatekeeper to coursework needed for a middle-class income and was mathematical training all students needed. However, because of the current narrow definition of algebra as symbolic manipulation, the question is inadequate.

 

As experienced mathematics educators, we know that “algebraic thinking” (see Driscoll, 1999) involves acquiring the “habits of mind” of “doing-undoing, building rules to represent functions, and abstracting from computation.” Mathematician Lynn Steen recognizes algebra as the language of the information age not because of its symbolic rigor but because “it is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.” Research shows that preparation for algebra requires developing algebraic habits of mind and strong proportional reasoning skills (see Harel & Confrey, 1994; Lamon, 2007). Therefore, the question should be: “How do we develop algebraic thinking throughout K–12 education, how do we know when students are cognitively ready for algebra, and how will algebra courses develop students’ flexibility in mathematical thinking?”

 

In short, we need to move beyond the notion that students need to pass an antiquated version of 20th-century algebra and toward the mathematical sciences. In a talk at the Research in Undergraduate Mathematics Education Conference, Confrey defined the mathematical sciences as “An umbrella term embracing the techniques of mathematics, numeric analysis, visualizations, and statistics cast in an appropriate formalism. It recognizes the importance of mathematics and statistics in modeling and analyzing phenomena.” Students need these skills to be successful 21st-century citizens.

 

As for the question of “rigorous mathematics,” that debate has shifted from algebra to calculus. However, as Steen (2007) argues, calculus is not the only type of rigorous mathematics: “Aiming school mathematics for calculus is not an effective strategy to achieve the goal of improving all students’ mathematical competence. Good alternatives exist. They can be found by looking carefully at all ways in which mathematics appears in postsecondary contexts. Notwithstanding other purposes and pressures, secondary education does not respond to the demands of higher education. If colleges say that calculus is what everyone needs, or that good students are those who can quickly manipulate algebraically intricate expressions, then parents will demand, and schools will focus on, this type of mathematics. But programs with these mathematical requirements represent only the one-third of postsecondary education encompassed by STEM disciplines. Moreover, these kinds of courses, which rely on very specific skills, have the effect of filtering out many otherwise interested and able students.” Indeed, probability and statistics is more relevant in the current job market, where nearly every field uses data-driven decision-making.

Developing 21st-century mathematics skills requires changing the extant grammar. Beyond fluency in symbolic manipulation, students must learn to think flexibly, take risks, develop algebraic habits of mind, engage in mathematical discourse, and connect various disciplines together to solve complex problems. At Catlin Gabel, we constantly “tinker” to achieve these goals. In the Lower School, teachers work on implementing best practices by studying current research, discussing, and planning in grade level teams on a weekly basis. They constantly weave innovative research more deeply into the study and discourse of their classrooms; this year, for example, the focus is on measurement. In the Middle School, a wide selection of mathematics courses prepare students for deep algebraic thinking based on their cognitive development level. And in the Upper School, problem-based courses develop students’ discourse abilities, authentic problems are embedded in the curriculum, and two statistics courses are offered as an alternative or in addition to calculus.

 

We are in a unique position at Catlin Gabel because, as a progressive school, we are privileged to define our own grammar of schooling. Working together as pioneering tinkerers, not naive agents who throw new pedagogy against the wall to see what sticks, let’s bring our knowledge and experiences to seek unconventional solutions to unique problems. We hope this edition of the Callerignites discussion in the community, and we look forward to jointly defining a progressive Catlin Gabel grammar of schooling.

 

Courtney Nelson has been the Lower School math specialist since 2011. She holds a BS in landscape architecture from the University of Massachusetts–Amherst and an MA in elementary education from Lewis & Clark College. Kenny Nguyen has been an Upper School math teacher since 2012. He holds a BA in mathematics from the University of Chicago, an MA in learning technologies from the University of Michigan, and a PhD in mathematics education from North Carolina State University.