Beyond the math wars: Focus on teachers to improve instruction

To achieve in math, students need teachers equipped with the knowledge, skills, and mindsets to do and teach math well.

A tale of two classrooms: In one, students sit in neat, antiseptic rows, reciting math facts on demand and completing rote practice with no deep understanding. In the second, students buzz from task to task, exploring without guidance or instruction, chatting socially, and learning no math at all.

The entire field of math education agrees that neither of these extreme examples shows classrooms where students will learn math well. Despite that agreement, we continue to argue about whether math should be taught using systematic and explicit instruction or with more student-led constructivist tasks. This situation has led the field far from meaningful and desperately needed improvement.

This debate, which originated decades ago (see Schoenfeld, 2004), has reemerged with fervor. However, the explicit instruction versus constructivist thinking argument misses the point. Students need both kinds of instruction to help them make meaning in math, but they first need teachers with deep knowledge of the subject and how to teach it. They need teachers who can reason about standards, curriculum, and lesson goals.

What are the math wars, and why are they so divisive?

The U.S. became fertile ground for mathematics education reform following the release of A Nation at Risk (Gardner et al., 1983). The seminal report, which painted a grim picture of U.S. education, highlighted wide disparities in math performance between U.S. students and those in other countries. In response, the National Council of Teachers of Mathematics (NCTM) developed its Curriculum and Evaluation Standards (NCTM, 1989).

The standards prompted fierce backlash and a growing schism between those on opposite sides of the argument (Schoenfeld, 2004). On one side were those who agreed with the deeper conceptual mathematical understandings put forth by the standards and advocated a constructivist approach to teaching.

On the other side were those who called for more traditional curriculum, using systematic, explicit steps and repeated practice — also known as a transmission-oriented approach (Depaepe, Verschafel, & Star, 2020). Some of the fiercest opponents of the standards were pure and applied mathematics professors in California. With the support of the conservative contingent of the state board, they rewrote the standards to focus more on computation (Becker & Jacob, 2000).

Both sides sought to eliminate inadequate or ineffective math instruction. However, the battles became so divisive, in part, because of extremism on both sides. Finding compromise and common ground was deemed impossible (Mervis, 2006).

Decades later, the war continues with the California Mathematics Framework debate and the new Science of Math movement. Both sides firmly believe that the opposite paradigm is harmful. Constructivists note that explicit instruction tends to be used more frequently with minoritized students and students with special needs (Lambert, 2018; Naraian, 2019). They point out that marginalized learners are more likely to have teachers who frame mathematics as a fixed body of knowledge and to have little access to deep, creative, learning opportunities. They highlight how this teaching method unintentionally excludes some students from rich learning (Calabrese Barton & Tan, 2020; Louie et al., 2022).

When teachers don’t have deep, flexible, and fluent thinking, they can get locked into a single teaching philosophy and provide only superficial explanations of concepts.

Alternatively, advocates for explicit instruction note that passing students on to later grades without concrete, procedural knowledge poses an equity issue by leaving students underprepared for higher-order learning in the upper grades (Barshay, 2023). They suggest that constructivism reduces academic standards in favor of “watered-down curriculum and teaching approaches that … rely more on ideology than research” (Blume & Watanabe, 2023, p. 2).

The use of timed multiplication fact sheets is an example of the issues at work in the debate. Some research shows that these tests can produce math anxiety, which then leads to “low achievement, math avoidance, and negative experiences of math throughout life” (Boaler, 2014). Such tests, constructivists argue, send powerful messages to students that math is about speed and accuracy rather than complex thinking that requires time, trial, and error. Yet others argue that students who can “effortlessly retrieve declarative facts … will be much more likely to be successful on the target task” (Powell, Hughes, & Peltier, 2022, p. 8). Students who cannot complete simple calculations, they explain, will struggle with complex math problems because their working memory will be diverted to lower-level tasks. In such arguments, people take extreme positions, each side talks past each other, and there is little room left for rational conversation.

Students need both types of teaching

Despite the ongoing debate, evidence shows that elements traditionally associated with both approaches are important for student success (Knight, 2002). As people learn, they develop frameworks that help them make sense of the world (National Academies of Sciences, Engineering, and Medicine, 2018). They learn best when they use prior knowledge and incorporate new learnings (Sawyer, 2006).

Constructivist methods — such as posing complex problems, asking students to choose their own methods to solve them, and engaging in mathematical discourse — can enhance conceptual understanding and framework development. They also predict student achievement in mathematics (Banes et al., 2015; Blazar, 2015; O’Dwyer, Wang, & Shields, 2015).

At the same time, practicing repeatedly, associated with the explicit instruction side of the debate, helps to solidify long-term learning (Karpicke & Roediger III, 2008). A strong facility with numbers and the ability to manipulate them fluently similarly predicts high mathematics achievement (Jordan, Glutting, & Ramineni, 2010). Learners with only this procedural knowledge, however, struggle to apply what they know to novel situations (Sawyer, 2006).

For students to succeed in advanced math courses, they need both procedural knowledge and conceptual understanding — fluency and flexibility. Many researchers and practitioners understand that both are important to student learning (e.g., Horn & Garner, 2022; Schoenfeld, 2004), but that conversation often gets drowned out. These cyclical debates also miss the point that the pedagogical content knowledge and skill required to balance the two is complex. We currently lack enough teachers who can deliver such balanced instruction.

The importance of deep teacher content knowledge

Research shows that teacher knowledge plays a role in instructional quality and, thus, student learning (Bransford, Darling-Hammond, & LePage, 2005; Grossman & McDonald, 2008; Hill, Ball, & Shilling, 2008). Educators must have subject-matter knowledge (i.e., content); pedagogical knowledge (i.e., general teaching practices); and pedagogical content knowledge (Shulman, 1986, 1987).

Secondary math teachers obviously must have a deep understanding of math content. However, math conceptual knowledge starts in the early grades. Even for 1st-grade teachers, subject-specific content knowledge is important (Copur-Gencturk & Tolar, 2022; Hill, Rowan, & Ball, 2005) because it enables teachers to understand students’ mathematical thinking (Copur-Gencturk & Li, 2023). Early-grade teachers must possess sufficient math knowledge to understand students’ thinking so they can support their growth as mathematicians. Once they solidify their foundational content knowledge, teachers need time to practice using a variety of pedagogical tools that engage students in different settings and learning objectives.

Rather than fanning the flames of decades-old wars by taking an either/or approach to math instruction, we should recognize that students need math teachers who can determine when a given approach is appropriate.

Only with mastery of content knowledge and an expanded pedagogical toolset can teachers develop math-specific pedagogical content knowledge — often called mathematical knowledge for teaching (MKT; Ball, Thames, & Phelps, 2008). This knowledge allows them to design learning experiences that meet different mathematical goals and objectives. MKT plays an important role in teacher expertise in the math classroom (Copur-Gencturk & Tolar, 2022; Phelps, Howell, & Liu, 2020; Schoenfeld, 2020). It is a prerequisite for teacher effectiveness (Depaepe, Verschafel, & Star, 2020) and a significant teacher-level predictor of student outcomes (Ball, Lubienski, & Mewborn, 2001; Brewer & Goldhaber, 2000; Hill, Rowan, & Ball, 2005). Mathematical knowledge for teachers could break the cycle of student math struggle. Developing this skill set, however, requires intensive investment and time. The either/or debate over fluency versus flexibility does not help.

If we know what’s important, why don’t we do it?

Too many teachers in math classrooms don’t have the pedagogical content knowledge for this kind of deep flexible thinking. One reason is the widespread shortage of math teachers (Sutcher, Darling-Hammond, & Carver-Thomas, 2019). Even when math positions are filled, however, teachers often do not have the knowledge or confidence to teach it well, particularly in early grades.

Our societal attitudes toward math contribute to the problem. Teachers, like other adults, are subject to the historical view that being good at math requires an innate talent (Li & Schoenfeld, 2019). Math anxiety continues to be prevalent (Luttenberger, Wimmer, & Paechter, 2018), including among teachers entering the profession (Schwartz, 2023). This impacts educators’ expectations for their own success in teaching the subject (Bursal & Paznokas, 2006). Teachers of lower elementary grades (versus upper elementary) and those who believe in fixed instruction and the transmission of facts hold higher levels of anxiety than those without such views (Ganley et al., 2019). They tend to lack the expertise required to teach key mathematical strands such as algebraic thinking (Pincheira & Alsina, 2021).

Teachers’ attitudes about and abilities in math influence students’ learning in the earliest grades, particularly for those already marginalized in math spaces, such as girls (Schaeffer et al., 2021). Furthermore, this lack of mathematical foundation among early elementary educators yields teachers who are unable to explain deeper concepts. Instead, they tend to rely on memorized rules without coherence and understanding.

When teachers don’t have deep, flexible, and fluent thinking, they can get locked into a single teaching philosophy and provide only superficial explanations of concepts. Their students will find it hard to gain a deeper understanding of, for example, fraction division, even if they can follow a procedure to solve such problems. Students need teachers to take a standard, unit, or lesson and ask themselves questions like, “What is this requiring of students? Where do students need to be fluent and where do they need to be flexible in their thinking?”

Researchers have shown that teacher expertise in the mathematics classroom plays a role in improving instruction (Copur-Gencturk & Li, 2023; Copur-Gencturk & Tolar, 2022; Schoenfeld, 2016, 2020; Yığ, 2022). But the flexibility versus fluency debate is drowning that out in our public discourse and stymieing real transformation in the field.

We are not seeking to demonize or blame teachers. This is a structural and societal problem, not an individual one. We need to recognize that the persistent either/or argument about how students should learn math is moot without a teaching force equipped with the knowledge, skills, and mindsets to do and teach math well.

How do we create deep teacher pedagogical content knowledge?

Rather than fanning the flames of decades-old wars by taking an either/or approach to math instruction, we should recognize that students need math teachers who can determine when a given approach is appropriate. Getting there will require a systemic overhaul, considering the pervasiveness of math anxiety, beliefs about innate math ability, and beliefs about math as linear and fact-based (as opposed to inquiry-based). We continue to describe math as a subject to be feared, rather than enjoyed, and we view people who are “good at math” as gifted or more intelligent. Society’s discursive view of math must change.

Here are some ideas to get us started:

• Increase the number of math content courses required for teacher candidates, including for teachers of our youngest students.
• Invest in teacher induction programs that center on math pedagogical content knowledge development in the context of the math curriculum teachers will be using.
• Create ongoing math learning communities for teachers, where they have time to examine student work, reflect on practice, and plan together for upcoming lessons.
• Engage building-level leaders in developing their own mathematical knowledge for teaching alongside teachers.
• Include in all teacher learning spaces mindset work about the nature of math and who can be good at it.

Schools, districts, and teacher preparation programs must pour into all educators the ability to be fluent and flexible mathematicians. Students do not need an argument about fluency versus flexibility. Instead, they need teachers who understand how both operate to support students in becoming confident and knowledgeable mathematicians. They need a national conversation about how we help teachers (and administrators) see themselves as mathematicians.

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This article appears in the December 2024 issue of Kappan, Vol. 106, No. 4, p. 8-13.