What Math Knowledge Do Elementary Principals Need?

Elementary principals’ instructional decisions shape students’ experiences in mathematics. They influence what gets prioritized in classroom walkthroughs, what professional learning teachers receive, how intervention systems are organized, and what problems of practice become the focus of school improvement efforts.

Yet elementary principals are rarely expected to develop substantial mathematics knowledge for teaching as part of their leadership preparation. This raises an important question: how much mathematics do elementary principals actually need to know?

Not enough to become mathematics specialists. Not enough to teach every lesson in the building. But enough to recognize important patterns in student thinking, understand how mathematical ideas develop over time, and support teacher learning in meaningful ways.

What Principals Notice Matters

Imagine a principal visiting all of the third-grade classrooms. Students have been working on multiplication for almost 2 months. Students are solving 5 x 7 by drawing five circles and placing seven tally marks in each group. They count each tally mark one by one to determine the answer. At first glance, the classrooms appear successful. Students are engaged. They are using drawings and representations. Most students arrive at the correct answer. But underneath the surface is a more important question: Are the students making progress on developing a deeper understanding of multiplication?

Direct modeling by ones is an important starting point for learning multiplication. It helps students make sense of equal groups and the meaning of multiplication. But it is not where students should remain by the end of third grade. Students should begin developing more sophisticated reasoning strategies. A student might think, “Five groups of five is twenty-five, and five groups of two is ten, so five groups of seven is thirty-five.” Other students may simply know 5 x 7 = 35. Attending to the sophistication of the strategy use matters because future mathematics depends on it.

The Impact on Future Learning

In the following year, fourth grade, students will encounter multidigit multiplication problems such as 24 x 35. Students who still rely primarily on direct modeling by ones will struggle to make sense of the quantities in the larger problems. They need an understanding of groups, decomposition, place value, and the distributive property in order to reason efficiently about multiplication. If students leave third grade without these understandings, the effects compound for them over time. And if principals do not recognize the issue, it may never become visible as an instructional problem.

Instead, the problem might be constructed as needing to identify students for intervention, without first examining the learning opportunities they experienced in core instruction. The conversation shifts quickly to what students cannot do rather than examining the mathematics students were supported to understand and what opportunities in the classroom they have to understand them.

What Happens When Principals Do Notice?

When principals understand enough of the mathematical understandings we want for students in order to recognize these patterns, different kinds of questions emerge:

Do teachers understand the intent of the standard and the progression of multiplication understanding?
Do teachers know the mathematical goal but need support helping students move beyond direct modeling?
Do students have opportunities to discuss and compare strategies?
Do curriculum materials support this progression, or are there gaps in the materials? Or are there issues with how the materials are being enacted?

These are questions that instructional leaders ask. The principal does not need to enter the classroom as the expert with all the answers. But principals do need enough knowledge to participate meaningfully in conversations about teaching, learning, and student thinking.

Why This Knowledge Matters

What principals know shapes what they notice. What they notice shapes the conversations they lead. And those conversations influence both teacher learning and student learning.

When principals understand the mathematics underneath instruction and how children learn it, they are better positioned to support coherent systems of learning across classrooms and grade levels. They are more likely to identify instructional issues early, support teacher growth thoughtfully, and create conditions where students develop strong mathematical understanding over time.

Ultimately, mathematics knowledge matters for principals because instructional leadership matters for students. In fact, according to recent research (Grissom et al., 2021), principals are the second-most-important in-school factor affecting student learning, behind only classroom instruction — and their impact may have even been previously understated.

Looking Ahead

If mathematics knowledge for teaching matters for principals, the next question is: how does this knowledge develop?

In future posts, we will share what we have seen and learned in our work alongside principals as they develop knowledge of mathematics for teaching and learning. We will explore the kinds of experiences, structures, and design considerations that help principals build their capacity to notice student thinking, engage in instructional conversations, and support teacher learning in meaningful ways.

Coaching for Individuals or Systems: How You Set It Up Matters