Linear Abacus
The Games Book Series · Stage 1 · Number Sense
30 Beads Down & 30 Beads Up
Blog 2 of 12
What happens when a child stops counting every bead — and starts reasoning instead.
You've sat down with your child. There's a dice on the table, beads on the string. They roll a five and start counting: one, two, three, four, five — moving each bead across one at a time. It works. But you wonder: is there more to this than counting?
Do you ever watch your child struggle with basic addition and subtraction? Perhaps they rely heavily on counting on their fingers, or they haven't yet developed efficient mental strategies for working with numbers. You're not alone! Many children find it challenging to move beyond counting by ones to more sophisticated strategies like making tens, using doubles, or working with place value.
The "30 Beads Down" and "30 Beads Up" games from the Linear Abacus® Games Book directly address these challenges through hands-on, embodied learning. The purpose of our book is to help children build their skills in reasoning, critical thinking, and calculating, to support their journey towards solving word problems. Unlike traditional counters, the Linear Abacus® uniquely positions numbers as "names of places in an order," allowing children to physically experience how numbers relate to each other. Through these complementary games, children develop a robust understanding of addition and subtraction as related operations, building mental strategies that will serve them throughout their mathematical journey.
Addition and subtraction are fundamental operations that form the foundation for all future mathematical learning. While they may seem straightforward to adults, these concepts are complex for young learners who are still developing number sense.
Research consistently shows that children who develop multiple mental strategies for working with numbers have stronger mathematical outcomes than those who rely solely on counting by ones. The "30 Beads" games specifically support the development of strategies such as:
Many children struggle with these concepts because traditional teaching approaches often emphasise procedural understanding without building conceptual foundations. The Linear Abacus® addresses this gap by making abstract number relationships physically tangible and visible through coloured beads and meaningful actions.
The Framework
TALK DO WRITE is the framework behind every Linear Abacus® game. It is not three steps that happen in sequence. It is three things happening at once.
When your child says where they will land before moving the beads, talks through their reasoning as they slide them, and you write the arithmetic together afterwards — the language, the gesture, and the notation are all expressing the same mathematical structure. They are not separate activities. They are the same act, in three registers.
The game is the occasion. The conversation is where the mathematics lives.
We have separated Talk, Do, and Write below to help you ask the right questions, match them to gestures, and write what you see and do together — but in the game itself, all three happen at once.
When playing these games, asking the right questions can dramatically enhance mathematical thinking. Here are key questions to use during both games:
A productive dialogue might sound like:
Parent
So, you're on the number 8 and you rolled a 7. How will you add that to your beads?
Child
I'll just count: 1, 2, 3, 4, 5, 6, 7.
Parent
That works! I wonder if there's another way that might be faster? What if you think about the colours on the abacus?
Child
Well, I could use the 2 of the leftover blue beads and half a row of yellow which is 5.
Parent
Great strategy! That's using partitioning, which means you're learning to split numbers to help you add efficiently. Notice that the colour pattern on the beads encourages children to see 7 as 2 and 5.
Child
Yep, I split 7 into 2 and 5 because it's easier to talk about where I land. 8 + 2 is 10, 10 + 5 is 15. I know 8 + 7 is 15. I can see that I land on 15 because the colours on the abacus string help me.
Parent
Well done, we can even fold the beads into groups of tens and ones to see 15. Let's try it.
The physical actions with the Linear Abacus® are crucial for developing number sense:
When subtracting, children should slide beads from left to right, creating a visible gap between what remains and what has been subtracted
Encourage children to move beads in meaningful groups rather than one at a time
The colour pattern of the beads carries mathematical meaning that children begin to read intuitively through the games
For example, if they are sitting on 25 and need to subtract 8, they might move 5 yellow beads first, then 3 blue beads
When adding, children slide beads from right to left
The colour coding helps children use benchmarks of 5 and 10
For example, when adding 7, we saw how they moved 2 blue beads first then 5 more yellow beads
Place pegs or markers at key positions to help track where they landed. This will help them transfer their thinking onto number lines
Connect the physical actions to mathematical symbols by:
Writing number sentences that match the bead movements (e.g., 8 + 7 = ?)
Drawing the bead configurations with annotations showing the actions
Using arrow notation to show movement on the beads
An example — after adding 7 to 8 you might annotate:
Encourage children to explain their annotations: "I drew this arrow on top of the beads to show how I moved from 10 to 15 by adding 5 more beads. 5 is an action — counting on."
Take these concepts into everyday life:
Other Linear Abacus® games that develop related concepts include:
When your child consistently explains their strategies without needing to count by ones, they're ready for more challenging activities like "Make it Balance" or "Reach 100 Beads."
To assess your child's understanding, consider:
Can they use strategies beyond counting by ones?
Do they recognise and use the colour patterns on the Linear Abacus®?
Can they explain their thinking process clearly?
Are they becoming more efficient with their calculations?
Watch for these common misconceptions:
The Linear Abacus® approach is firmly grounded in research on embodied cognition, which shows that physical experiences are fundamental to how we develop mathematical understanding. By physically manipulating beads, children create neural connections that link concrete actions to abstract number concepts. Making connections between facts, models and procedures has many advantages over remembering isolated bits of information.
Studies by researchers like Jo Boaler and James Hiebert demonstrate that children who develop multiple mental strategies and can explain their mathematical thinking outperform those who learn procedures without understanding. The "Talk, Do, Write" cycle of the Linear Abacus® activities directly supports this development of mathematical reasoning and communication skills.
Beyond mathematics, these games also develop executive function, turn-taking, communication skills, and logical reasoning.
Boaler, J. (2015). Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching. Jossey-Bass.
Boaler, J., & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adults' lives. Journal for Research in Mathematics Education, 48(1), 78–105.
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age.
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251–283.
30 Beads Down and 30 Beads Up are complementary games that build addition and subtraction strategies through manipulating beads on the Linear Abacus®. Use questions like "How did you know that?" and "Is there another way you could do that?" to promote strategic thinking.
Key Mathematical LanguageUse this blog alongside the Games Book for the full set of probing questions, illustrated student reasoning, and diagrams that bring each game to life.
The Games Book contains thirty games, complete instructions, probing questions, student reasoning examples, and diagrams — everything you need to sit down and play with confidence.
