Hello, and welcome back to Math Universally Speaking! I’m your host, Ron Martiello, and today, we’re diving deeply into one of the most essential, yet misunderstood topics in math instruction: Numbers and Operations in Base Ten.

If you’ve listened to my other episodes or read Conquering Math Myths with Universal Design, you know we don’t promote tips and tricks just to get the answer. We believe in developing understanding through reasoning—not just calculating. Today, we’ll explore how to support students in building a strong foundation in Base Ten, giving them time to engage with the structure, and later, how to approach operations with understanding.

Understanding the Structure

Let’s begin with the structure of the Base Ten system. Each year, we increase the complexity of the numbers students work with, but the need for a strong Base Ten foundation starts early. This concept is part of the major work in grades K–2 (Student Achievement Partners, n.d.). In kindergarten and first grade, students begin counting by ones and tens. They use items like straws, sticks, or blocks to see how ones can be bundled into tens. Ten frames and double-frames also help anchor their understanding of ten.

By second grade, students are using key vocabulary—place value, digits, regrouping. Consistent use of this vocabulary helps reduce barriers. Phrases like “borrowing” or “knock next door and ask the neighbor for 10” may be well-intentioned, but they can cause confusion down the line. Instead, we want students to use this vocabulary as they attend to and make use of structure, as prescribed in Standard for Mathematical Practice #7.

Once students understand the Base Ten structure, they can manipulate numbers more flexibly. They use tools like base-ten blocks and number lines to explore concepts. This is where we can coach them on how to think flexibly.

For example, a second grader working with the number 365 might break it into 3 hundreds, 6 tens, and 5 ones—or regroup it into 2 hundreds, 16 tens, and 5 ones, or even 3 hundreds, 5 tens, and 15 ones. The ability to compose and decompose numbers multiple ways creates the groundwork for understanding how these numbers can be manipulated using operations.

From Understanding to Reasoning

Once students have a firm grasp of their understanding of numbers, they can begin to reason using the structure of Base Ten by rounding and comparing. They use the structure to justify why numbers are greater than or less than others. They begin to understand how rounding makes sense. In grades 3 and up students explore powers of ten, and how multiplying or dividing by ten can increase the value of certain digits.

These are rich opportunities for developing reasoning. We need to move beyond “answer-getting” and focus on why the math works. So please don’t rush these lessons. Spend time in conversation, giving students opportunities to reason through the Base Ten system.

Applying Understanding and Reasoning to Operations

Now, let’s talk operations. Once students grasp the Base Ten structure, we move to adding, subtracting, multiplying, and dividing. But there are many myths here—especially about rushing to the standard algorithm.

Let’s be clear: The standard algorithms belong in the curriculum, but we shouldn’t teach them prematurely. Over-proceduralizing the algorithm without connecting it to base ten creates silos between all four operations. Students view each of the algorithms as separate processes rather than a single structure that can manipulated four different ways. The empathetic intentions behind this decision aren’t necessarily bad, but rushing students through this process can do more harm than good.

For Example: Addition and Subtraction…The standard algorithm doesn’t need to be mastered until the end of 4th grade. That’s right. The END of 4th grade. Instruction in second and third grade builds toward that—through base-ten blocks, number lines, partial sums, and partial differences. These are all ways to scaffold students’ understanding of addition and subtraction. The traditional algorithm should come at the end of this journey—not the beginning. The goal here isn’t to simply “get the right answer,” but to understand how to add and subtract using Base Ten. Students may still need support to develop a deep understanding of place value and regrouping before they can successfully use the traditional algorithm for addition and subtraction.

The same goes for multiplication and division. The standard algorithm for multiplication is a 5th-grade standard. For division, it’s the end of 6th grade. Before that, students should use area models, partial products, partial quotients, and strategies that align with Base Ten reasoning.

Misconceptions about timing can lead to premature instruction and assessing students on standards that aren’t even in their grade level. We must align our expectations with grade-level standards to ensure students’ readiness. In fact, by teaching the right standards at the right time might actually save us time so we can teach concepts more deeply.

Opportunities for UDL

Student variability is real. Some students come in with a strong sense of these numbers. Others may compute quickly but lack structural understanding. And some are just beginning their journey.

No matter where students start, they are worthy of the climb. As educators, we can provide access, support the learning process, and honor agency even when it comes to large topics like Numbers and Operations in Base Ten.  The good news is that the standards leave room to be flexible during the learning process. If we think about Rigor as a balanced approach to math including conceptual understanding, procedural skills, and real world problem solving, it can help us design learning that is more responsive. 

For example, a student learning how to add and subtract using base ten in 2nd grade, may use base ten blocks. Some may prefer to draw the blocks on paper or whiteboards to represent the numbers they are working with. Some may think linearly and prefer to demonstrate their understanding an a number line. In the same way, 3rd and 4th graders begin to multiply and divide multi-digit numbers. Students can use base ten blocks or pencil and paper to create area models. There are also a number of digital tools online that students can utilize as well. I will add these to our website so you can explore some free options. 

As students develop their understanding, it is important to make connections to procedural skills related to their grade level. Teachers can make connections from conceptual understanding to procedural skills using strategies such as partial sums, partial differences, partial products and partial quotients. We can provide direct instruction when making these connections. However, we do not have to stop there. We can represent these connections through anchor charts, bulletin boards, and videos that students can reference as we support their journey from conceptual understanding toward procedural fluency. 

As students begin to apply and solve problems we can give them options for monitoring their progress. When it comes to Numbers and Operations in Base Ten, we want to make sure we are providing frequent and actionable feedback. We can correct conceptual misunderstandings for some while monitoring simple precision errors for others. Options may include setting up office hours during application activities so students can get the just in time feedback they need. Some students may want to work together with a trusted peer. These peers work together and discuss their progress and help each other check for  understanding and accuracy. Options for artificial intelligence are growing. In these cases, students are coached through application activities. Teachers can utilize settings so that the AI does not give answers but rather gives prompts to encourage the student’s own thinking to work through the process. 

In all of these cases, students can still achieve standards-based goals, while exercising their own agency to achieve them. 

A Little Help from My Friends

There’s only so much we can cover in a 10-minute podcast. Designing responsive instruction takes time and collaboration. But let me leave you with this:

Teaching Numbers and Operations in Base Ten is an investment in students’ futures. It takes patience. It takes trust—in your students and in the standards. And it takes partnership.

You’re not alone. Look to your colleagues above and below your grade level. Unpack the standards together. Build that collective efficacy.

Closing

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Together, we are reimagining math instruction and unlocking every student’s potential.

Until next time—take care!