EARLY GRADES MATHEMATICS - MULUWA CHRISPIN

Early grades mathematics - by Muluwa Chrispin

Subtraction is a fundamental mathematical operation that children must understand deeply in order to develop strong number sense and problem-solving skills. It can be introduced in two main ways: as "take away" and as "difference or comparison." To help learners grasp these concepts effectively, teachers often use the Concrete-Pictorial-Abstract (CPA) approach, which moves from hands-on experiences with real objects to pictorial representations and finally to abstract symbols. This structured progression supports conceptual understanding and builds confidence in mathematical thinking. Understanding these strategies enables educators to cater to diverse learning styles and promote flexible thinking in the classroom.

To start with, subtraction can be understood in two main ways that includes; take away, this means removing a quantity from a whole. And difference or comparison, this is finding how much more or less one quantity is compared to another

To build deep understanding, the Concrete-Pictorial-Abstract (CPA) model is used. This model helps learners move from physical manipulation of objects (concrete), to visual representations (pictorial), and finally to solving using numbers and symbols (abstract). Each activity uses specific mathematical vocabulary that reinforces key subtraction concepts (Singapore Ministry of Education, 2001).

Subtraction as "Take Away"

At Concrete Level (Using Real Objects)

And activity can be carried out as follows; provide each learner with 10 small objects such as bottle tops, stones, or counters. Ask them to physically remove or "take away" a certain number (e.g., 4), and count how many remain.

For instance, "You have 10 bottle tops. If you take away 4, how many are left? Count the remaining bottle tops to find the answer."

This hands-on experience helps learners physically see and feel what happens when something is taken away.

Vocabulary to Use will be like; “take away”, “subtract”, “remove”, “left”, “remain”
These words help learners connect the physical act of removal to the concept of subtraction.

Pictorial Level (Using Drawings or Pictures)

An activity will be done as following; draw 8 apples on the board or worksheet. Ask learners to cross out 3 of the apples and count the number of apples that are not crossed out.

For Example; 🍎🍎🍎🍎🍎🍎🍎🍎 → Cross out 3 → 🍎🍎🍎🍎🍎
"Cross out 3 apples. How many apples are left that are not crossed out?"

This stage builds on the concrete experience and helps learners visualize subtraction without using real objects.

Vocabulary to use; “cross out”, “subtract”, “left”, “minus”, “remaining”

 

Abstract Level (Using Numbers and Symbols Only)

Present learners with subtraction equations such as 12 – 5 = ___ and ask them to solve mentally or by writing.

For Example; "12 minus 5 equals what?" → 12 – 5 = 7

This is the final stage where learners use only numbers and symbols to represent subtraction, based on their previous experiences.

Vocabulary to use; “subtract”, “minus”, “equals”, “difference”, “take away”

 

Subtraction as "Difference or Comparison"

Concrete Level (Comparing Two Sets of Objects)

Prepare two sets of pencils. Set A has 9 pencils, and Set B has 5. Place them side by side and ask learners, “How many more pencils does Set A have than Set B?”

For Example; "Set A has 9 pencils and Set B has 5. Count to find out how many more pencils are in Set A than Set B." Learners count up from 5 to 9 to find the difference.

This helps learners understand subtraction as the process of comparing two quantities.

Vocabulary to Use; “more than”, “fewer”, “compare”, “difference”, “how many more”

 

Pictorial Level (Comparing Drawing Heights or Lengths)

Draw two towers of blocks; one with 7 blocks and the other with 4 blocks. Ask learners, “Which tower is taller? How many more blocks does the taller tower have?”

For Example: Tower A: 🟫🟫🟫🟫🟫🟫🟫
                       Tower B: 🟫🟫🟫🟫
"What is the difference in height between the two towers?"

It encourages learners to make visual comparisons and determine the difference.

Vocabulary to use; “difference”, “taller”, “shorter”, “compare”, “more“, “less”

 

Abstract Level (Using Number Lines or Equations)

Use a number line to find the difference between two numbers. For example, place a marker at 9 and count up to 15. Ask: “What is the difference between 15 and 9?”

For Example: 15 – 9 = 6 Or use counting up: "Start at 9 and count up to 15: 10, 11, 12, 13, 14,

15    – that’s 6 jumps."

This concept, supports abstract thinking and builds fluency in recognizing subtraction as finding the distance or gap between numbers.

Vocabulary to use; difference, subtract, how many more, gap, between

 

Subtraction as "Difference or Comparison"

Concrete Level (Using Real Objects to Compare Two Sets)

Give learners two separate piles of real objects, such as pencils, bottle tops, or sticks. For example, Pile A has 9 pencils, and Pile B has 5 pencils. Ask the learner to compare the two piles by physically counting each one and identifying how many more objects are in one pile compared to the other (Charlesworth, and Lind, 2012).

For Example; Count Pile A: 1, 2, 3, 4, 5, 6, 7, 8, 9
                       Count Pile B: 1, 2, 3, 4, 5
Ask: “Which pile has more?” Then, “How many more does Pile A have than Pile B?”
To find the answer, take away 5 pencils from the 9 in Pile A:
9 – 5 = 4
So, Pile A has 4 more pencils than Pile B.

Vocabulary to use and emphasize; “More than” – used to describe the larger quantity. “Fewer than” – used for the smaller quantity. “Compare” – to look at how amounts are different. “Difference” – the result of subtracting one amount from another.

 

Pictorial Level (Using Drawings or Visuals to Compare)

Draw two vertical towers (bars or columns) using blocks or squares on the board or paper. One tower has 7 blocks, the other has 4 blocks. Ask learners to visually compare the height of the towers and identify how many blocks taller one is than the other.

For example: draw; Tower A: 🟥🟥🟥🟥🟥🟥🟥 (7 blocks).       Tower B: 🟦🟦🟦🟦 (4 blocks)
Ask: “Which tower is taller?” (Tower A). Then ask: “How many more blocks does Tower A have?” Count the extra blocks above Tower B → 3 So, the difference in height is 3 blocks.

Vocabulary to use and emphasize; “Taller” – the tower with more blocks. “Shorter” – the one with fewer blocks. “Difference” – the amount by which one is more or less than the other. “More or Less” – general comparison words

 

Abstract Level (Using Numbers and Symbols Only)

Use a number line or write equations to help learners calculate the difference between two numbers. For example, ask, “What is the difference between 15 and 9?” Encourage learners to solve it by subtracting.

For example; on a number line, start at 9 and count up to 15:

·         From 9 to 10 = 1

·         From 10 to 15 = 5
Total jumps = 1 + 5 = 6
So, the difference is 6             Or solve directly using an equation: 15 – 9 = 6 This shows that the number 15 is 6 more than 9.

Vocabulary to use and emphasize; “Difference” – the result of subtraction. “Subtract” – the operation used. “How many more” – used to ask about the comparison between two numbers. “Equals” – shows the result.

Finally, the three Learners who worked out 67 - 23 the procedures can be explained as follows;

Learner A: Compensation Strategy

This learner used a method called the compensation strategy, where numbers are adjusted to make the subtraction easier to perform mentally. The idea is to round the second number to a "friendly" number (usually a multiple of 10) and compensate by adjusting the first number equally. This keeps the subtraction balanced and still gives the correct result.

The learner noticed that 23 is not easy to subtract mentally because it is not a round number. To make it easier, they added 7 to both 67 and 23, turning the problem into 74 – 30. Since both numbers were increased by the same amount, the final answer remained the same:
(67 + 7) – (23 + 7) = 74 – 30 = 44

This method relies on the principle of equality: adding the same number to both the minuend (the number being subtracted from) and the subtrahend (the number being subtracted) keeps the difference unchanged.

For Example; ask learners to solve 52 – 19 using compensation. Guide them to think: “19 is close to 20. If I add 1 to 19 to make 20, I must also add 1 to 52 to keep the problem fair.” So, the problem becomes: (52 + 1) – (19 + 1) = 53 – 20 = 33. By turning the subtraction into round numbers, learners can subtract faster and with less mental effort.

Key Vocabulary includes; “Compensation” - compensation means adjusting numbers either up or down to make mental subtraction easier. For example, adding 7 to both numbers in 67 – 23 changes the problem to 74 – 30, which is easier to calculate mentally. This technique helps learners work flexibly with numbers and reduces calculation errors.

“Balance” - balance refers to keeping the subtraction fair by applying the same change to both the minuend and subtrahend. It ensures that the difference between the two numbers remains the same, preserving the answer. Understanding balance is important to avoid mistakes during mental adjustments.

“Minuend” - the minuend is the number you subtract from in a subtraction problem. In 67 – 23, 67 is the minuend because it is the starting quantity before taking away. Recognizing the minuend helps learners understand the direction of subtraction.

“Subtrahend” - the subtrahend is the number being subtracted from the minuend. In 67 – 23, 23 is the subtrahend because it represents the amount taken away. Knowing this term helps learners identify the parts of a subtraction problem clearly.

“Friendly numbers” - friendly numbers are easy-to-work-with numbers, often multiples of 10 like 10, 20, or 30. They simplify mental calculations because they reduce complexity in subtraction or addition. Adjusting numbers to friendly numbers is a common strategy in mental math.

Strengths of this strategy includes; encourages flexible thinking about numbers,
This strategy pushes learners to think beyond fixed procedures by adjusting numbers creatively. It develops number sense and helps learners see subtraction as a flexible process, not just a rule to follow. Flexible thinking supports problem-solving in different contexts.

Makes use of round numbers which are easier to subtract mentally, Using round or friendly numbers simplifies calculations by removing complicated digits. This approach reduces cognitive load and speeds up mental math, making subtraction more accessible, especially for larger numbers.

Helps learners who are strong in mental math and number relationships, learners with good mental calculation skills can benefit greatly from this strategy, as it builds on their understanding of how numbers relate to each other. It encourages them to use these relationships to simplify problems quickly (National Council of Teachers of Mathematics, 2000).

Supports estimation and quicker calculations; the compensation method helps learners estimate answers faster and perform calculations more efficiently. By rounding numbers to the nearest ten or friendly number, learners develop skills useful for real-life situations that require quick approximations.

On the other hand, the weaknesses of the strategy; it requires a clear understanding of number relationships and equality, for compensation to work, learners must understand that adding or subtracting the same amount to both numbers keeps the difference unchanged. Without this knowledge, learners may apply the method incorrectly, leading to wrong answers.

Learners may forget to compensate both numbers, which leads to incorrect answers, a common mistake is adjusting only one number in the subtraction, which changes the actual difference. This oversight shows the importance of teaching balance and double-checking steps during mental calculations.

May confuse younger learners who are still mastering basic subtraction steps, for beginners, the idea of changing numbers before subtracting can be confusing. They often need concrete experiences with simple subtraction before moving to compensation, as premature introduction might hinder understanding.

Less effective if the adjustment is a large number, which can complicate the thinking, if learners add or subtract a large number to compensate, the mental load increases and can cause confusion. Large adjustments reduce the simplicity of the strategy and may slow down problem-solving instead of helping.

 

Learner B: Decomposition / Place Value Strategy

This strategy involves decomposing or partitioning both numbers into tens and ones to make subtraction easier. The learner first separates the numbers into their place value components, then subtracts the tens from tens and the ones from ones individually. After performing both subtractions, they combine the results to find the final answer (Bruner, 1966). This method is especially useful for learners who are still developing their mental math skills and need a step-by-step process.

Activity; Give learners base-ten blocks or place value charts. Represent 67 as 6 tens and 7 ones, and 23 as 2 tens and 3 ones.
Ask them to:  Remove 2 tens from 6 tens → 4 tens

Remove 3 ones from 7 ones → 4 ones
Then combine: 4 tens + 4 ones = 44

 

For example;

67 – 23
Break each number into its place value parts:
67 = 60 + 7 and 23 = 20 + 3
Now subtract each part separately:
(60 – 20) + (7 – 3) = 40 + 4 = 44

This hands-on activity helps reinforce the link between number structure and subtraction.

Vocabulary; “decompose” - to decompose a number means to break it into smaller parts, such as splitting 67 into 60 and 7. This helps learners focus on manageable pieces of a problem, making calculations easier and more understandable.

“Place Value” - place value refers to the worth of a digit depending on its position within a number. For example, in 67, the 6 represents 60 because it is in the tens place, while the 7 represents 7 ones.

“Tens and Ones” - two-digit numbers consist of two main parts: tens and ones. Understanding these components helps learners to accurately read, write, and manipulate numbers during operations like subtraction.

“Subtract” - to subtract means to take away one number from another. This operation finds the difference or remainder after removing a quantity from a total.

“Combine” - combining means putting parts back together after breaking them down. For example, after subtracting tens and ones separately, learners combine the results to get the final answer.

Therefore, strengths of this strategy include; encourages a deep understanding of place value - using decomposition in subtraction helps learners grasp the importance of each digit’s position. This foundational skill supports all future math learning by building a strong number sense.

Makes the subtraction process more visual and organized, breaking numbers into parts and subtracting each part separately allows learners to see how numbers are structured. This organized approach reduces confusion and helps clarify each step.

Supports mental calculation by training learners to think in smaller parts, decomposing numbers into tens and ones simplifies the subtraction process mentally. It encourages learners to solve problems piece by piece instead of dealing with large numbers all at once.

Helps build confidence and flexibility in working with numbers, mastering this method boosts learners’ confidence, as they gain flexible strategies to approach subtraction. This flexibility makes it easier to tackle a variety of problems with ease.

Weaknesses include; it can be time-consuming with larger or more complex numbers, for big numbers, breaking them down and subtracting parts individually can take more time than using other faster strategies, which might slow down problem-solving.

If learners misplace parts, it may lead to incorrect answers, mistakes can happen if learners subtract tens from ones or vice versa. This shows the need for careful attention to place value to avoid errors (McClure, 2003).

Requires good attention to detail and a solid grasp of place value, without understanding place value deeply, learners may get confused during decomposition, which affects accuracy. This method demands focus and clear understanding.

May not be efficient for quick calculations in higher-level math, as learners progress, they need faster methods. Unless this strategy is well internalized, it might be too slow for advanced math tasks requiring quick mental subtraction.

 

Learner C: Number Line/Counting On Strategy

Instead of subtracting 23 from 67 directly, the learner asks, “How many more do I need to add to 23 to reach 67?” This strategy focuses on adding up from 23 to 67 rather than directly subtracting, which is particularly useful when learners find subtraction less intuitive. Subtraction as Addition: The process is viewed as finding the difference by adding rather than subtracting. This approach helps the learner conceptualize subtraction as a distance between numbers on a number line.

For example; The learner is given the subtraction problem 67 – 23 and chooses to solve it using the number line strategy (or counting on).

On a number line, they start at 23 and count up (or "jump") until they reach 67. The number line could be represented as:    23 → 24 → 25 → 26 → ... → 67

The learner makes 44 jumps forward to get from 23 to 67. Answer: 67 – 23 = 44 (the number of jumps taken from 23 to reach 67).

Vocabulary include; “Counting on” - the process of starting from a smaller number and counting forward until you reach the larger number. “Number line” - a visual representation of numbers in order, which helps in understanding their relative positions and distances. “Jumps” - the individual steps or increments counted on the number line, used to represent the addition process in the counting on method. “Difference” - the result of subtraction; it represents the gap or space between two numbers. “Addition of the difference” - in this strategy, the learner adds the difference (i.e., the gap between numbers) to the smaller number to find the larger one. “Distance” - the concept of how far apart two numbers are, which in this case is expressed by the number of jumps from 23 to 67.

Strengths of this strategy include; visualizing subtraction as a difference, this method helps learners see subtraction not just as “taking away” but as a measure of how much more one number is compared to another. It visualizes subtraction as the distance between numbers on a number line (Haylock, and Cockburn, 2017).

Real-life contexts, the strategy is helpful in everyday situations such as time and money. For example, if a learner needs to figure out how much time has passed from 3:00 PM to 5:00 PM, they can "count on" from 3 to 5 (2 hours), or if they are adding money (e.g., counting up from $23 to $67).

Building number sense, Counting on is a natural strategy that builds understanding of number relationships and helps students get comfortable with larger numbers.

On the other hand weaknesses; time-consuming with large gaps, when the numbers are large (e.g., from 23 to 67), this method can be slow because the learner has to count each individual jump or step, which may not be efficient for larger numbers (Anghileri, 2006).  For example, counting from 23 to 100 would require many jumps, making it cumbersome.

Less efficient for advanced learners, for learners who are already comfortable with subtraction, counting on might seem inefficient because they would typically use other strategies, like decomposition or compensation, to perform calculations faster.

Error-prone in complex problems, in problems where the difference is large or involves multiple steps, the learner might lose track of their place or make mistakes when counting up. This can become problematic when working with large numbers or more complex mathematical problems (Van de and et al, 2019).

In conclusion, teaching subtraction through the Concrete-Pictorial-Abstract (CPA) approach and exposing learners to multiple mental strategies—such as compensation, decomposition, and counting on provides a solid foundation for mathematical understanding. These methods help learners move beyond rote procedures to develop meaningful concepts of subtraction as both "take away" and "difference or comparison." By using appropriate vocabulary and allowing learners to engage with numbers in flexible and visual ways, teachers can support diverse learning needs and promote confidence, accuracy, and deeper reasoning in mathematics.

 

REFERENCES

Anghileri, J. (2006). Teaching number sense. Bloomsbury Publishing.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Charlesworth, R., & Lind, K. K. (2012). Math and science for young children (7th ed.). Cengage

Learning.

Haylock, D., & Cockburn, A. D. (2017). Understanding mathematics for young children: A guide

for teachers of children aged 3–7 (5th ed.). SAGE Publications.

McClure, L. (2003). Mental calculation strategies. NRICH Project, University of Cambridge.

https://nrich.maths.org

NCTM (National Council of Teachers of Mathematics). (2000). Principles and standards for

school mathematics. Reston, VA: NCTM.

Singapore Ministry of Education. (2001). Primary Mathematics Syllabus. Curriculum Planning &

Development Division.

Van de Walle, J. A., et al (2019). Elementary and middle school mathematics: Teaching

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