Dynamic Representations

A dynamic representation is a visual representation of a mathematical object that changes in time. 

Why use Base-10 (Dienes) blocks — and why explore other bases?

Base-10 blocks give students a geometric model of place value, making the multiplicative structure of our number system visible. The proportional relationships between “ones”, “tens”, and “hundreds” are not just stated—they are seen and handled. This supports conceptual place-value understanding, especially for learners who rely on concrete and visual representations…

When we allow students to explore base 2–10 using consistent block structures, we go further: we make the general structure of positional number systems explicit. Students begin to notice that

• each place is worth “one group of the base”,
• the laws of arithmetic (e.g., distributive and associative laws) behave consistently across any base.
• carrying and borrowing is simply exchanging units for a larger or smaller block.

This helps reduce misconceptions that arise when students think place value is arbitrary or unique to base 10.

Building conceptual understanding and reducing cognitive load

Using the same virtual manipulative across different bases provides the benefit of variation without overload: only the base changes, while the visual grammar stays consistent. This aligns with cognitive-load principles: …

• Intrinsic load (the complexity of the mathematical idea) is managed because the representation remains familiar.
• Extraneous load is reduced because students don’t need to learn a new model for each base.
• Germane load is increased as students focus on noticing structure, patterns, and generalisations.

This allows teachers to direct students’ attention to the key mathematical idea: place value is a multiplicative structure based on the size of the base, not on the digits themselves.

Connecting bases to algebra tiles and the laws of arithmetic

When students explore bases conceptually, they are primed to understand base 𝑥—which is the foundation of algebra tiles…

• A “unit” becomes 1
• A “long” becomes 𝑥
• A “flat” becomes 𝑥²

Because the structure mirrors Dienes blocks, students can more readily transfer their understanding. They see that the operations they perform in base 10 (exchanging, grouping, multiplying blocks) align with the laws of arithmetic that govern algebraic manipulation.

This makes algebra feel like a generalisation of something they already understand, rather than a new symbolic game.

The role of virtual manipulatives

Digital blocks bring powerful advantages:

• They remove practical barriers (limited resources, uneven sets, physical tidying).
• They allow rapid switching between bases without introducing new physical shapes.
• They support iterative exploration, letting students test hypotheses and see immediate visual feedback.
• They fit naturally into whole-class teaching on the IWB as well as individual exploration.

Summary

Using Dienes blocks across multiple bases—and extending the idea to base 𝑥—helps students:

• build deep conceptual understanding of place value,
• recognise the general structure underpinning arithmetic,
• experience lower cognitive load through consistent representations,
• make smooth transitions into algebra through familiar geometric models,
• and benefit from the flexibility, clarity, and interactivity of virtual manipulatives.