Name It and Connect It

Part 1 of this series was all about feeling the structure of mathematics. We looked at what fluency actually is — and what it compresses.

In this post, I’ll be discussing the three big structures behind all of the strategies for K-5, and how they connect to fluency and numeracy.

Across all of K–5 mathematics, three big structures do most of the heavy lifting. They don’t belong to any single grade level or content area. They run through multiplication, fractions, place value, and early algebraic thinking like a spine. Once you see them, you can’t unsee them.

Place Value 

Our number system is built on a simple but powerful idea: position determines value, and each position is ten times the one to its right. That structure is what makes 27 × 4 manageable — we can split 27 into 20 and 7 precisely because of how our number system is organized. It’s also what makes the multi-digit addition and subtraction with whole numbers and decimals work. An algorithm isn’t the structure. It is the structure, compressed.

Properties of Operations

This is the one students use constantly without knowing it has a name. When a student solves 27 × 4 by doing 25 × 4 + 2 × 4, they’re using the distributive property. When they flip (3 × 8) x 5 to 3 x (8 x 5) because it’s easier to think about, they’re using associativity. These properties aren’t rules to memorize — they’re descriptions of how numbers actually behave. And they work the same way whether the numbers are whole numbers, fractions, or decimals.

Equivalence 

This one tends to get underestimated. Equivalence isn’t just “same value, different form.” It’s the engine behind some of the most important mathematical moves students make. Finding a common denominator before adding fractions? That’s equivalence. Rewriting 28 as 20 + 8 before multiplying? That’s equivalence. Recognizing that

describe the same relationship? Also equivalence. Students who own this idea deeply have a tool that works across every content area they’ll encounter.

Here’s the thing about these three structures: they don’t stay in their lanes. Place value only does its full work because of equivalence. The distributive property only works with fractions because equivalence lets us rewrite them. Early algebraic thinking is what happens when students start to see all three as a unified system of structures rather than separate topics.

That’s not a coincidence. It’s the structure.

Fluency compresses the structures — place value, properties of operations, equivalence. Those structures are the shared foundation of both numeracy and computational thinking.

In my next post, I’ll connect these three structures to computational thinking — and why the connection has been there all along.