Building Number Sense with the Linear Abacus®: 30 Beads Games

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.

The Mathematics Behind the Games

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:

- Making tens (recognising that 7+3=10)

- Using place value (seeing 25 as 2 tens and 5 ones)

- Partitioning numbers (breaking 8 into 5+3 to make calculation easier)

- Using doubles (knowing that 6+6=12)

- Working with complements (understanding that if 6+4=10, then 10-4=6)

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 "Talk, Do, Write" Cycle in Action

Talk: Mathematical Communication

When playing these games, asking the right questions can dramatically enhance mathematical thinking. Here are key questions to use during both games:

For 30 Beads Down:

- "How have you moved the beads? Did you move them one at a time or in groups?"

- "Why did you move the beads that way?"

- "How will you work out how many beads remain without counting each one?"

- "Will you use place value to help you know where you landed?"

For 30 Beads Up:

- "How many beads do you have altogether? How do you know?"

- "What strategy did you use to find the total?"

- "Could you think of another way to find the total?"

- "How are you splitting numbers to make calculations easier?"

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."

Do: Mathematical Gestures

The physical actions with the Linear Abacus® are crucial for developing number sense:

In 30 Beads Down:

1. When subtracting, children should slide beads from left to right, creating a visible gap between what remains and what has been subtracted

2. Encourage children to move beads in meaningful groups rather than one at a time

3. The colour patterns (groups of 10) help children see place value relationships

4. For example, if they are sitting on 25 and need to subtract 8, they might move 5 yellow beads first, then 3 blue beads

In 30 Beads Up:

1. When adding, children slide beads from right to left

2. The colour coding helps children use benchmarks of 5 and 10

3. For example, when adding 7, we saw how they moved 2 blue beads first then 5 more yellow beads

4. Place pegs or markers at key positions to help track where they landed. This will help them transfer their thinking onto number lines

If children struggle with physical manipulation, start with smaller numbers and emphasise the connection between the movement and the mathematics.

Write: Mathematical Symbolism

Connect the physical actions to mathematical symbols by:

1. Writing number sentences that match the bead movements (e.g., 8 + 7 = ?)

2. Drawing the bead configurations with annotations showing the actions

3. Using arrow notation to show movement on the beads

An example, after adding 7 to 8 you might annotate:

8 + 2 + 5

= 10 + 5

= 15

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"

Extending Learning Beyond the Game

Take these concepts into everyday life:

- When shopping, ask: "If we need 10 apples and have 7, how many more do we need?"

- During cleanup: "We have 18 blocks to put away. If we put away 9, how many will be left?"

- During cooking: "If the recipe needs 24 chocolate chips and we've added 15, how many more should we add?"

Other Linear Abacus® games that develop related concepts include:

- "Add Up Beads" for exploring the commutative and associative properties

- "Subtract Down Beads" for subtraction with larger numbers

- "What's the Difference?" for comparing numbers

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."

Parent and Teacher Reflection Guide

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:

- Always counting from the beginning rather than recognising groups

- Not making connections between addition and subtraction

- Struggling to decompose numbers strategically

If your child consistently uses only counting by ones, try focusing on the 10 colour patterns on the Linear Abacus® and explicitly model using them to make calculations easier.

Research Connection

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.

Quick Reference

Games at a Glance: 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 Language:

- Place value: Understanding that in our number system, the position of a digit determines its value

- Partitioning: Breaking numbers into parts to make calculations easier

- Mental strategies: Approaches to calculation that go beyond counting by ones

- Number bonds: Pairs of numbers that add up to a given number (e.g., 6+4=10)

References

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.