Building Place Value Understanding with the Linear Abacus®: Beyond Counting by Ones

As parents and teachers, we've all been there: watching our child laboriously count each object one by one when faced with a simple addition problem. While counting is an essential first step in mathematical development, the transition to more efficient strategies is crucial for success in mathematics.

Place value—understanding that in our number system, the position of a digit determines its value—is the foundation of this transition. Many children struggle with place value concepts because traditional teaching approaches often emphasise procedural understanding ("carry the one") without building true conceptual understanding. The Linear Abacus® offers a unique approach through embodied learning, where children physically manipulate beads to discover how numbers work.

In the Stage 2 games "Add Up Beads" and "Subtract Down Beads," children move beyond basic counting to recognise number relationships, explore properties of addition, and use place value to subtract two-digit numbers. The Linear Abacus® makes these abstract concepts tangible by representing numbers as "names of a place in an order" that answers 'how many?'"—creating a foundation that will support your child's mathematical journey for years to come.

The Mathematics Behind the Games

Add Up Beads

Place value is arguably the most important concept in elementary mathematics. It's the understanding that in our base-10 system, each position represents a power of 10—ones, tens, hundreds, and so on. When children truly grasp place value, they can decompose numbers flexibly, making calculations much more efficient than counting by ones.

Research by mathematics education experts like Liping Ma shows that deep understanding of place value is a key differentiator between students who struggle with mathematics and those who excel. The Linear Abacus® approach aligns with research on embodied cognition, which demonstrates that physical manipulation of objects creates neural pathways that support abstract thinking.

Children often struggle with place value because traditional approaches rely heavily on symbolic representation before conceptual understanding is established. Common misconceptions include thinking of multi-digit numbers as individual digits rather than composed units (seeing 42 as "4" and "2" rather than 4 tens and 2 ones). The Linear Abacus® bridges this gap by providing a physical model where the arrangement of beads directly corresponds to place value structure, allowing children to literally see and touch the tens and ones.

The "Talk, Do, Write" Cycle in Action

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Talk: Mathematical Communication

When playing "Add Up Beads," effective questioning helps children articulate their thinking and discover mathematical properties. Try these probing questions:

• "How will you add these numbers together? Why did you choose that order?"

• "What combinations did you notice? Does the order matter?"

• "How can you use the colours on the beads to make number bonds to ten?"

• "Is there another way you could combine the three numbers?"

Student Reasoning Examples

Grade 3/4 students

Notice how creative these children are. The first students used the strategies known as doubles (4 + 4) and splitting numbers (5 = 2+ 3). The second students used the strategies known as place value and partitioning (17, 19 and 16b each have a 10), near doubles (7+6=13, and round and adjust (13+9 think 13+10 then -1).

A productive conversation might sound like:

Parent: "You rolled a 3, 5, and 7. How will you add these up?"

Child: "I'll do 5 plus 3 first, then add 7."

Parent: "Why did you decide to add 5 and 3 first?"

Child: "Because 5 and 3 make 8, and that's easier to remember."

Parent: "That's interesting! Could you add them in a different order? Would you get the same answer?"

This dialogue encourages the child to discover the commutative and associative properties of addition without explicitly naming them.

Do: Mathematical Gestures

The physical manipulation of beads on the Linear Abacus® is where much of the learning happens:

1. After rolling three numbers (e.g., 5, 3, and 7), the child slides beads to represent the first number (5 beads).

2. Next, they slide more beads to represent the second number (3 more beads), noticing they now have 8 beads (as they are 2 away from a full row of colour).

3. Finally, they add the third number (7 more beads which can be broken up into 2 and 5 using the colour pattern of the beads).

   
   

The colour coding of the Linear Abacus® (alternating every 10 beads) supports recognition of number bonds. For example, seeing that 2 more yellow beads are needed to complete a set of 10 helps children develop the crucial skill of "making ten" when adding.

If your child struggles with the physical actions, try guiding their hand initially or breaking down the movement into smaller steps, verbalising each step as you go.

Write: Mathematical Symbolism

Connecting physical actions to mathematical notation completes the learning cycle:

1. Start by asking your child to write the equation that represents their actions (5 + 3 + 7 = 15)

2. Then, help them use brackets to show the order of operations: (5 + 3) + 7 = 15

3. Compare this to other possible equations: 5 + (3 + 7) = 15

A supportive dialogue might include:

Parent: "Can you write down what you just did with the beads?"

Child: Writes "5 + 3 + 7 = 15"

Parent: "Great! Now, can you use brackets to show that you added 5 and 3 first?"

Child: Writes "(5 + 3) + 7 = 15"

Parent: "What if we added 3 and 7 first instead? How would we write that?"

Child: Writes "5 + (3 + 7) = 15"

Parent: "Do we get the same answer? What does that tell us about addition?"

This progression helps children connect the physical experience with mathematical notation, reinforcing the commutative and associative properties.

Subtract Down Beads

The "Talk, Do, Write" Cycle in Action

Talk: Mathematical Communication

When playing "Subtract Down Beads," effective questioning helps children articulate their place value understanding:

• "How did you slide the beads? Can you explain your actions?"

• "What strategy did you use to subtract? Can you write down these facts?"

• "How can you use place value to make this subtraction easier?"

• "How many tens and ones do you need to subtract?"

A productive conversation might sound like:

Parent: "You rolled a 5 and then a 2. What number does that make?" (*Note that the first roll represents the ones and the second roll represents the tens).

Child: "That's 25."

Parent: "How will you subtract 25 from 100 using the Linear Abacus®?"

Child: "I'll move 2 tens first, that's 20. Then I'll move 5 ones."

Parent: "Great thinking! How many beads will be left after you subtract?"

This dialogue reinforces the concept that two-digit numbers can be broken down into tens and ones, making subtraction more manageable.

Do: Mathematical Gestures

The physical manipulation in "Subtract Down Beads" highlights place value understanding:

1. After rolling to create a two-digit number (e.g., 25), the child first identifies the tens digit (2) and the ones digit (5).

2. Starting with 100 beads, they slide 2 full rows of 10 beads (20 beads total) to the right.

3. Then they slide 5 individual beads to complete the subtraction (they may see this straight away as half of a row of colour which is 10).

The colour patterns on the Linear Abacus® help children visualise groups of ten, reinforcing the decimal structure of our number system. Children can see that a two-digit number consists of some number of complete tens plus some number of ones.

If your child struggles with this concept, try using language first to describe what they see to to reinforce the connection between the physical model and the symbols used in the place value system.

Write: Mathematical Symbolism

Documenting the subtraction process helps solidify place value understanding:

1. Encourage your child to write the full number sentence: 100 - 25 = 75

2. Then, help them break down the subtraction using place value: 100 - 20 - 5 = 75

A supportive dialogue might include:

Parent: "Let's write down what you just did with the beads."

Child: Writes "100 - 25 = 75"

Parent: "Can you show how you broke down 25 into tens and ones when you subtracted?"

Child: Writes "100 - 20 - 5 = 75"

Parent: "Excellent! You used place value to make the subtraction easier."

This progression helps children connect the physical experience with mathematical notation, reinforcing place value concepts.

Extending Learning Beyond the Game

The concepts explored in these games can be reinforced through everyday activities:

• Grocery Shopping: Ask your child to estimate the total cost of items by rounding to the nearest ten and adding. "This costs $24 and this costs $17. About how much altogether?"

• Cooking: When doubling or halving recipes, discuss how the quantities change. "If we need 3 cups of flour for one batch, how many do we need for two batches?"

• Card Games: Play "Make Ten" with a deck of cards, where players try to find pairs that sum to 10.

Related Linear Abacus® games that develop these concepts include "Make It Balance" and "Reach 100 Beads." As your child becomes more confident with place value concepts, look for opportunities to extend to three-digit numbers in everyday situations.

Parent Reflection Guide

To assess your child's understanding of place value through these games, look for:

• Grouping by Tens: Does your child automatically group beads by tens rather than counting each bead individually?

• Flexible Number Breaking: Can they break numbers apart in different ways (e.g., seeing 25 as 20 + 5 or as 10 + 15)?

• Reasoning About Operations: Do they explain their thinking using place value language ("I added 2 tens and 5 ones")?

• Efficiency: Are they moving beyond counting by ones to more efficient strategies?

Common misconceptions to watch for include:

• Treating multi-digit numbers as separate digits rather than as tens and ones

• Always counting from the beginning rather than using the structure of the number system

• Difficulty understanding that the position of a digit determines its value

Your child is ready for more challenging activities when they consistently use place value to solve problems and can flexibly decompose numbers in multiple ways.

Research Connection

The approach used in the Linear Abacus® aligns with research on embodied cognition, which suggests that mathematical understanding is grounded in physical experience. Studies by researchers like Martha Alibali and Susan Goldin-Meadow have shown that children's gestures during mathematical problem-solving often reveal understanding that precedes their ability to verbalise concepts.

The "Talk, Do, Write" cycle implements recommendations from Jo Boaler's research on mathematical mindsets, which emphasises the importance of multiple representations and discussion in developing conceptual understanding. Additionally, the focus on exploring mathematical properties through guided discovery aligns with findings from cognitive science about how children construct mathematical knowledge.

Beyond place value understanding, these games develop executive function skills such as cognitive flexibility (thinking about numbers in different ways) and working memory (keeping track of parts of numbers during calculation). Research by Clancy Blair and others has shown that these executive function skills are strong predictors of long-term mathematical achievement.

Quick Reference Box

Add Up Beads & Subtract Down Beads: Key Concepts

Add three numbers using various strategies and explore the properties of addition. Subtract two-digit numbers using place value understanding. Key questions: "How can you break up the numbers to make them easier to work with?" "Can you explain how you're using tens and ones?"

Mathematical Language:

• Place value: The value of a digit based on its position in a number

• Commutative property: The order of addition doesn't change the sum (a + b = b + a)

• Associative property: How numbers are grouped doesn't change the sum ((a + b) + c = a + (b + c))

• Number bonds: Pairs of numbers that add up to a given number (e.g., 3 and 7 are a number bond of 10)

References

Alibali, M. W., & Nathan, M. J. (2012). Embodiment in mathematics teaching and learning: Evidence from learners' and teachers' gestures. Journal of the Learning Sciences, 21(2), 247-286.

Blair, C., & Razza, R. P. (2007). Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Child Development, 78(2), 647-663.

Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.

Goldin-Meadow, S., Cook, S. W., & Mitchell, Z. A. (2009). Gesturing gives children new ideas about math. Psychological Science, 20(3), 267-272.

Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Routledge.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.