Developing Numeracy: A Concrete Approach to Ones, Tens, and Hundreds

Imagine a child staring at the number “203” written on the board. When asked what number it is, they confidently answer “twenty-three.” This common error reveals something fascinating about how young children develop numeracy skills-and it reminds us why teaching place value requires much more than just showing children how to write numbers. Building a solid foundation in numeracy means helping children understand what numbers actually represent, not just memorize their symbols.

Why concrete materials matter in early numeracy 🔗

When we teach children about ones, tens, and hundreds, we’re asking them to make a significant conceptual leap. Young children naturally see numbers as collections of individual items-they understand “ten” as ten separate things. But place value asks them to think differently: to see ten individual items as one group of ten. This shift from counting individual objects to understanding grouped quantities is where concrete materials become essential.

Think of concrete materials like beads, straws, or matchsticks as bridges between a child’s everyday counting experiences and the more abstract concept of place value. When a child physically bundles ten straws together with a rubber band, something important happens. They’re not just following instructions-they’re experiencing the transformation of ten separate objects into one cohesive group. This hands-on process helps children understand that numbers can be organized in meaningful ways.

The power of grouping with everyday objects 🔗

Using materials children can manipulate offers distinct advantages. Straws or lollipop sticks are particularly effective because they’re groupable-children can physically bundle ten of them together, demonstrating the organizational structure of our base-ten system. This physical action of creating groups makes the abstract idea of “ten” concrete and memorable.

Consider starting with activities where children count out objects, then group them by tens. Place the individual items in one pile and the bundles of ten in another. Ask children to describe what they notice. Through this exploration, they begin to see patterns: ten ones equals one ten, twenty ones equals two tens, and so on. The beauty of this approach is that children discover these relationships themselves rather than simply being told about them.

Practice and repetition build lasting understanding 🔗

Learning place value isn’t a one-time lesson-it requires repeated exposure and practice in varied contexts. Games and activities that incorporate numeral formation help children internalize these concepts through enjoyable, purposeful repetition. When children play games involving place value, they’re not just practicing skills; they’re building mental models of how numbers work.

Making practice meaningful through games 🔗

Numeral games serve multiple purposes. They provide the repetition children need while maintaining engagement and motivation. Consider a simple dice game where children roll two dice-one for tens, one for ones-and build that number with concrete materials. Or create a trading game where ten individual counters can be exchanged for one ten-stick. These games embed practice within meaningful contexts, helping children see why understanding place value matters.

The key to effective practice is variety. Children benefit from encountering the same concept in different situations. One day they might bundle straws, another day they might use base-ten blocks, and later they might represent numbers with drawings. This varied practice helps children develop flexible thinking about numbers rather than rigid, procedural knowledge.

Common pitfalls in place value learning 🔗

Understanding common errors helps teachers address misconceptions before they become entrenched. When a child reads “203” as “twenty-three,” they’re revealing a fundamental misunderstanding about place value. They’re treating each digit as an independent number rather than understanding that the position of each digit determines its value. This type of error shows they haven’t yet grasped zero as a placeholder.

Why children reverse or misread numbers 🔗

Young children often write “14” when they mean “41” or read “307” as “thirty-seven.” These aren’t random mistakes-they follow predictable patterns that reveal how children are thinking. Often, children are listening to the sounds of number words and writing what they hear. “Sixteen” sounds like “six-teen,” so writing the six first seems logical to a young mind.

Similarly, when children see numbers like “203,” they may focus on the digits they recognize-the 2 and the 3-while glossing over the zero because it represents “nothing.” They haven’t yet understood that zero serves a crucial role as a placeholder, keeping other digits in their correct positions.

The challenge of abstract understanding 🔗

Another common struggle emerges when children treat place value as merely a label rather than understanding it as representing groups. A child might correctly identify that in the number 34, the 3 is in the “tens place,” but when asked what it means, they say “three” instead of “three tens” or “thirty.” This reveals they’ve memorized position names without grasping that position determines quantity.

These misconceptions often arise because children naturally try to fit new information into their existing understanding. When they first learned numbers, each digit represented its face value. The number 5 meant five things. Now we’re asking them to understand that 5 can mean fifty or five hundred depending on where it sits. This requires reconstructing their entire understanding of how numbers work.

Structured activities that address misconceptions 🔗

The good news is that targeted activities can help children overcome these challenges. The key is using activities that make place value concepts visible and explicit. Start by ensuring children have solid experience with bundling activities before moving to more abstract representations.

From concrete to pictorial to abstract 🔗

Effective teaching follows a progression from concrete hands-on materials, to pictorial representations, to abstract numerals. This Concrete-Pictorial-Abstract approach ensures children build understanding at each stage before moving forward.

Begin with children physically manipulating objects. Once comfortable, introduce drawings where they sketch bundles of ten and individual ones. For instance, to represent 25, children might draw two circles (each containing ten marks) and five individual marks. These drawings serve as transitional representations, bridging the gap between physical objects and abstract symbols.

Only after children are confident with both concrete materials and pictorial representations should we expect them to work comfortably with abstract numerals alone. Rushing this progression often leads to the misconceptions we discussed earlier.

Using place value mats and charts 🔗

Place value mats provide clear organizational structures that help children see how numbers are built. These mats typically have columns labeled “Hundreds,” “Tens,” and “Ones,” giving children designated spaces to place their materials. When working with the number 203, children place two bundles in the tens column, nothing in the ones column, and-crucially-two hundred-blocks in the hundreds column. The empty ones column visually demonstrates why we need the zero as a placeholder.

Encourage children to verbalize what they’re doing: “I’m putting two groups of ten here, which makes twenty. I have zero ones. And I have two hundreds.” This verbal rehearsal reinforces the connection between the physical materials, the positions on the mat, and the spoken number.

Supporting different learning paces 🔗

Not all children develop place value understanding at the same rate. Some grasp grouping concepts quickly, while others need extensive practice with concrete materials. The important thing is meeting children where they are and providing appropriate scaffolding.

For children who struggle, provide more time with hands-on materials. Don’t rush to pictorial or abstract representations. Consider using smaller number ranges initially-perhaps working with numbers only up to 30 or 50 before extending to 100. Some children also benefit from partner work, where they can explain their thinking to peers and hear different perspectives.

For children who quickly master basic concepts, offer extensions. Challenge them to represent three-digit numbers, explore patterns in the hundreds chart, or solve problems that require applying place value understanding in new contexts. The goal isn’t just speed but depth of understanding.

The long view: Building toward future mathematics 🔗

The time invested in developing strong place value understanding pays dividends throughout a child’s mathematical education. Place value isn’t just about reading and writing numbers-it’s the foundation for multi-digit addition and subtraction, multiplication, division, decimals, and even algebraic thinking.

When children truly understand that 34 represents three tens and four ones, they can more easily grasp why we sometimes need to regroup in addition, or why subtracting across zeros works the way it does. They develop number sense-an intuitive feeling for how numbers relate to each other and how they can be manipulated.

Perhaps most importantly, children who build genuine understanding through concrete experiences develop confidence in their mathematical abilities. They learn that mathematics makes sense, that they can figure things out, and that mistakes are opportunities for learning rather than signs of failure.

What do you think? How can we balance the need for hands-on exploration with the practical constraints of classroom time? What role should technology play in helping children visualize and manipulate numbers as they develop place value understanding?

References

1. https://thirdspacelearning.com/blog/common-errors-misconceptions-primary-maths-ks1-ks2/
2. https://mathsnoproblem.com/blog/teaching-maths-mastery/place-value-year-1
3. https://komodomath.com/blog/maths_mistakes_and_misconceptions
4. https://broadbentmaths.com/pages/place_value__representation_and_misconceptions_255363.cfm
5. https://files.eric.ed.gov/fulltext/ED620386.pdf