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American students are struggling with math, but what’s really to blame? Some blame the pandemic. Others point to overreliance on technology or a broader cultural attitude that treats math anxiety as acceptable.
But new research I led found that difficulties with advanced topics often stem from earlier gaps in understanding. Because mathematics is cumulative, students who struggle with algebra, for example, may be facing unresolved challenges with fractions, number sense or other skills typically developed in earlier grades. When these deficiencies go unaddressed, they persist and create bigger problems down the road.
These deficiencies are shaped by instructional choices made in classrooms every day. Chief among them is the ongoing debate over whether students are being equipped with a genuine understanding or merely trained to follow steps.
In reality, effective math learning requires both. Students must know how to carry out procedures, but also need to understand why they apply to specific problems. Like a chef, mastering math is not just about following a recipe or executing techniques correctly; it is about understanding how elements work together so that, when faced with something new, students know how to reason through the problem and build on previous knowledge.
And this imbalance often begins before middle school. For example, in early elementary grades, the pressure to focus on rote memorization of addition facts and subtraction “tricks” can occur at the expense of number sense. When memorization is prioritized over understanding the quantities involved, we set the stage for the conceptual disconnect that becomes a crisis in later grades.
In many traditional math curricula, procedural knowledge dominates. Students memorize steps and by middle school, that can become their entire conception of math. When students understand the steps through conceptual knowledge, they can explain and justify their work. In a classroom, this may look like a student understanding that the area of a triangle is half that of a rectangle because they can visually decompose the shapes, rather than simply reciting the formula A=12bh. This can help them make connections and understand the “why” behind the process. Research even points to conceptual understanding improving procedural knowledge more than vice versa.
The focus on procedural knowledge could be driven in part by the need for schools to meet goals for standardized test scores. Standardized testing rewards correct answers more than understanding, which may reinforce the imbalance of conceptual and procedural learning. Many states and districts reduce teacher performance to student test scores, despite these scores failing to capture a complete picture of student learning. Under this pressure, many teachers may feel compelled to “teach to the test,” prioritizing procedural accuracy to ensure their students can answer the basic multiple-choice questions that dominate these exams.
Let’s Rethink How We Teach Early Math – Starting with Teacher Prep
Declines in NAEP scores may intensify the urgency, fueling a climate where short-term gains matter more than long-term mathematical understanding. In a standardized testing-focused environment, conceptual knowledge can feel like a risk.
Addressing this imbalance does not require eliminating standardized tests, nor does it demand that every lesson become an exhaustive explanation. Instead, it requires an intentional approach to integrating conceptual knowledge into math instruction. Procedural knowledge remains essential, but it should be grounded in meaning and understanding, not memorization alone.
For this to happen, educators must be supported in teaching conceptually. Professional development that emphasizes conceptual explanations, student reasoning, and common misconceptions can bridge this gap.
Teachers also need tools that make conceptual knowledge manageable within real classroom constraints. Diagnostic assessments, formative checks and student work analysis can reveal where understanding breaks down, allowing teachers to target specific concepts not well understood. When instruction focuses on the ideas students struggle with most, conceptual knowledge can become feasible.
Tools that use diagnostic questions to identify where students’ understanding of math concepts falls short – what researchers call “misconceptions” or “instructionally relevant errors” – can help educators gain insight into how students think about and approach math problems. Rather than spending valuable instructional time trying to infer misunderstandings, educators can focus on addressing them.
A randomized controlled trial across 20 schools and 3,000 students found that students who used one such tool achieved two to four months of additional learning gains in a single school year. The tool was developed with support from the Learning Engineering Virtual Institute, which assists learning engineers in the development of AI-based tools that will significantly improve middle school math learning.
Math is hard — but perhaps that is because it is often taught without meaning. Many students learn the steps to solving a problem without ever understanding why it works. Procedures alone are not enough. Memorization can only take students so far; true learning happens when knowledge can be applied. If we want students to reason, problem-solve and build their math knowledge, conceptual knowledge cannot be optional.
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