Ch 9: Understanding Children's Thinking and Learning in Mathematics 🧠✨

 

Chapter 9: Understanding Children's Thinking and Learning in Mathematics 🧠✨

Welcome, PSTET Aspirants! 🌟

Teaching mathematics is not just about knowing the content—it's about understanding how children think, learn, and make sense of mathematical ideas. This chapter delves into the fascinating world of children's mathematical cognition. For PSTET (Paper 1), this pedagogical knowledge is just as important as mathematical content knowledge. Understanding how children learn helps you become not just a teacher, but an effective educator who can reach every child.

In this comprehensive chapter, we'll explore the constructivist approach to learning mathematics, analyze the reasoning patterns children use, and examine the crucial role of language in mathematical understanding. Let's step into the minds of young mathematicians! 🚀

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9.1 How Children Learn Mathematics: A Constructivist Perspective 🏗️

The way children learn mathematics has been profoundly influenced by constructivist theories of learning, particularly the work of Jean Piaget and Lev Vygotsky . Understanding these foundations is essential for PSTET.

🧒 Children as Active Learners and "Scientific Investigators"

Constructivism views children not as empty vessels waiting to be filled with knowledge, but as active constructors of their own understanding .

Traditional View	Constructivist View	Classroom Implication
Teacher transmits knowledge	Child constructs knowledge	Provide hands-on experiences, not just lectures
Children are passive recipients	Children are active investigators	Encourage exploration and discovery
Learning is memorizing facts	Learning is making meaning	Focus on understanding, not just correct answers
Errors are failures	Errors are learning opportunities	Analyze mistakes to understand thinking
One right way to solve	Multiple strategies valued	Celebrate different approaches

Children as Scientific Investigators: When children explore mathematical situations, they behave like little scientists :

• They observe patterns and relationships

• They form hypotheses about how things work

• They test their ideas through experimentation

• They revise their understanding based on results

• They generalize findings to new situations

Classroom Example: When introducing addition, instead of telling children "3 + 2 = 5," provide them with 3 blocks and 2 blocks and ask, "How many blocks in all?" Let them count, combine, and discover the relationship themselves. This is scientific investigation at the primary level! 🔬

🧱 Role of Prior Knowledge and Experience

Children do not come to mathematics as blank slates. They bring prior knowledge and everyday experiences that shape how they understand new mathematical concepts .

Source of Prior Knowledge	Example	Mathematical Connection
Home Experiences 🏠	Setting the table	One-to-one correspondence, counting
Shopping 🛒	Paying for items	Money concepts, addition, subtraction
Playing Games 🎲	Board games with dice	Counting, number recognition, chance
Sharing Snacks 🍪	Dividing cookies equally	Fair sharing, division concepts
Daily Routines ⏰	Bedtime, mealtime	Time concepts, sequencing

The Challenge of Prior Knowledge: Prior knowledge can both help and hinder learning .

Helpful Prior Knowledge	Problematic Prior Knowledge
Child knows counting from home games	Child thinks "bigger number always means bigger amount" (ignoring place value)
Child understands "more" from daily comparisons	Child thinks multiplication "makes things bigger" (fails with fractions)
Child can share equally with friends	Child thinks division always results in smaller numbers

Teacher's Role: Activate helpful prior knowledge and address misconceptions. Ask questions like, "What do you already know about this?" and "Have you seen something like this before?"

🔄 The Process of Assimilation and Accommodation

Piaget's theory describes how children's thinking develops through two complementary processes: assimilation and accommodation .

Process	Definition	Mathematics Example
Assimilation	Fitting new information into existing mental structures (schemas)	Child knows "adding makes bigger." When seeing 3 + 2 = 5, they assimilate this into their existing "adding makes bigger" schema.
Accommodation	Changing existing mental structures to fit new information	Child encounters 2 + 3 = 5 (same as 3 + 2). They must accommodate by creating a new understanding: order doesn't matter in addition.
Equilibrium	Balance between assimilation and accommodation	Child comfortably uses addition in various situations
Disequilibrium	Cognitive conflict that drives learning	Child discovers that 5 - 3 = 2 doesn't fit "subtraction makes smaller" (if starting number is smaller than subtracted number later)

The Learning Cycle in Mathematics:

text

    Existing Schema
          ↓
    New Experience
          ↓
Can I assimilate? → Yes → Schema confirmed
          ↓
         No
          ↓
    Disequilibrium (Cognitive Conflict!)
          ↓
    Accommodation (Change thinking)
          ↓
    New, More Sophisticated Schema
          ↓
    Equilibrium Restored

Classroom Example: Learning About Zero

Stage	Child's Thinking
Existing Schema	"Numbers are for counting things. 3 apples, 2 toys..."
New Experience	Teacher asks, "How many apples in an empty basket?"
Assimilation Attempt	Child tries to count... but there's nothing to count!
Disequilibrium	"How can there be a number for nothing?" 🤔
Accommodation	Child develops concept of zero as a number representing "none"
New Schema	"Zero is a number too. It means nothing."

PSTET Tip: Questions about assimilation, accommodation, and Piaget's stages frequently appear in the Child Development section. Connect these concepts to mathematics learning!

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9.2 Children's Thinking and Reasoning Patterns 🧮

Children develop their own strategies for making sense of mathematics. Understanding these strategies helps teachers build on what children already know.

🖐️ Strategies Children Use to Make Meaning

Young children are remarkably creative in developing problem-solving strategies. Here are common approaches:

Strategy	Description	Example (8 + 5)	Developmental Level
Counting All 🖐️	Count both groups from beginning	Counts 8 objects, then 5 objects, then counts all 13	Early
Counting On ➡️	Starts with larger number and counts forward	"8... 9, 10, 11, 12, 13"	Developing
Counting Back ⬅️	For subtraction, count backward	For 13 - 5: "13... 12, 11, 10, 9, 8"	Developing
Using Fingers ✋	Physical representation of numbers	Holds up 8 fingers, then 5 more	Concrete
Drawing Pictures 🎨	Visual representation	Draws 8 circles and 5 circles, counts all	Pictorial
Making Tens 🔟	Rearrange to make a friendly number	Takes 2 from 5 to make 8 into 10, then 10 + 3 = 13	Advanced
Doubles/Near Doubles 👯	Uses known double facts	"8 + 8 = 16, so 8 + 5 is 3 less, so 13"	Advanced
Derived Facts 🧩	Uses known facts to figure out unknown	"I know 5 + 5 = 10, so 5 + 8 is 3 more = 13"	Advanced

The Progression of Counting Strategies:

text

Counting All → Counting On → Making Tens → Derived Facts
     ↓              ↓              ↓              ↓
   Concrete    Transitional    Strategic      Fluent/
                                      Flexible

Research Insight: Children who are allowed to develop and use their own strategies before being taught standard algorithms often demonstrate deeper conceptual understanding and greater flexibility in problem-solving .

🔍 Analyzing Common Thought Processes in Problem-Solving

Understanding how children think when solving problems helps teachers identify misconceptions and provide targeted support.

Common Reasoning Patterns:

Reasoning Pattern	Description	Example	Appropriate?
One-to-One Correspondence	Matching items between sets	Giving one cookie to each child	✓ Foundational
Conservation of Number	Understanding that number doesn't change when items are rearranged	8 blocks spread out still = 8 blocks	✓ Key milestone
Transitivity	If A > B and B > C, then A > C	Raj is taller than Riya, Riya is taller than Sam, so Raj is taller than Sam	✓ Logical
Reversibility	Understanding that operations can be undone	5 + 3 = 8, so 8 - 3 = 5	✓ Essential
Compensation	If one quantity increases, another must decrease to keep total same	4 + 4 = 8, so 5 + 3 = 8 (one up, one down)	✓ Advanced
Overgeneralization	Applying a rule where it doesn't belong	"Multiplication makes bigger" (fails with fractions)	⚠️ Misconception
Counting Errors	Skipping, double-counting, or mis-matching	Counts objects but points twice to one	⚠️ Needs practice

Common Misconceptions by Topic:

Topic	Misconception	Child's Thinking	Correct Understanding
Place Value	"14 and 41 are the same"	"Both have 1 and 4"	Position determines value
Addition	"Adding always makes bigger"	2 + 3 = 5 (bigger)	Adding zero: 5 + 0 = 5 (same)
Subtraction	"You can't subtract a bigger number from a smaller one"	Can't do 3 - 5	With negatives (later) or real-world: "You owe"
Multiplication	"Multiplication makes things bigger"	4 × 3 = 12 (bigger)	Multiplying by fraction: 4 × ½ = 2 (smaller)
Division	"Division always makes smaller"	12 ÷ 3 = 4 (smaller)	Dividing by fraction: 12 ÷ ¼ = 48 (bigger)
Fractions	"½ is smaller than ⅓"	"2 is bigger than 3, so ½ is bigger"	Need same whole, denominator size
Decimals	"0.5 is smaller than 0.25"	"5 is bigger than 25"	Place value, tenths vs hundredths

💭 The Importance of Asking "How Did You Get That Answer?"

This simple question is one of the most powerful tools in a mathematics teacher's repertoire .

Purpose	What It Reveals	Teacher Response
Understand Thinking	The child's strategy, whether correct or incorrect	"I see you counted on from the bigger number. That's efficient!"
Identify Misconceptions	Where thinking went wrong	"Ah, you thought 14 and 41 were the same because they have the same digits. Let's look at place value."
Validate Multiple Strategies	Different approaches to same problem	"Rohan used counting on, but Priya made a ten. Both work!"
Encourage Metacognition	Children think about their own thinking	"Why did you choose that strategy?"
Build Mathematical Communication	Explaining reasoning to others	"Can you explain your method to the class?"

Sample Dialogue:

Teacher	Student	What Teacher Learns
"What is 8 + 6?"	"14."	Correct answer
"How did you get that?"	"I counted on my fingers. I said 8, then 9, 10, 11, 12, 13, 14."	Using counting on strategy; still relies on fingers
"Is there another way?"	"I could do 8 + 2 = 10, then 10 + 4 = 14."	Can use making ten strategy—more advanced!
"Which way is faster?"	"Making ten, I think."	Developing efficiency awareness

Classroom Routine: Make "How did you solve it?" a regular part of mathematics discussions. Create a "Strategy Share" time where students present different approaches to the same problem .

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9.3 The Language of Mathematics 🗣️🔢

Mathematics has its own specialized vocabulary and ways of expressing ideas. For many children, the language of mathematics can be a significant barrier to learning .

📖 Understanding Mathematical Vocabulary

Mathematical words often have specific meanings that differ from everyday usage.

Word	Everyday Meaning	Mathematical Meaning	Potential Confusion
Sum	Total amount	Result of addition	"What's the sum?" might be unclear
Difference	How things are not alike	Result of subtraction	Child might describe how things are different
Product	Something made or produced	Result of multiplication	"What's the product?"—confusing without context
Table	Furniture	Arrangement of data or multiplication facts	Multiple meanings confuse
Volume	Loudness	Amount of space	Science vs. math confusion
Mean	Unkind	Average	Emotional vs. mathematical
Set	Collection	Well-defined collection of objects	Seems simple but technical
Face	Part of head	Flat surface of 3D shape	Body vs. geometry
Net	Mesh for catching fish	2D pattern that folds into 3D shape	Highly specialized
Prime	Main, best	Number with exactly two factors	Value judgment vs. mathematical property

Teaching Strategy: Word Banks

Create classroom mathematics word banks with:

• The word

• A student-friendly definition

• A picture or example

• The word used in a sentence

Word	Definition	Picture/Example	Sentence
Sum	The answer when you add	5 + 3 = 8	"The sum of 5 and 3 is 8."
Difference	The answer when you subtract	8 - 3 = 5	"The difference between 8 and 3 is 5."
Product	The answer when you multiply	4 × 3 = 12	"The product of 4 and 3 is 12."

🤔 Challenges Posed by the Language of Word Problems

Word problems combine mathematical understanding with reading comprehension—a double challenge for many students .

Language Challenges in Word Problems:

Challenge Type	Example	Why It's Difficult
Unfamiliar Vocabulary	"Rohan purchased 3 dozen erasers."	Child may not know "purchased" or "dozen"
Complex Sentence Structure	"If Riya had 15 marbles and gave 7 to her friend, how many more does she need to have 20?"	Multiple steps, complex relationships
Implied Information	"A bus can carry 40 children. How many buses for 125 children?"	Need to interpret "carry" and understand division with remainder
Irrelevant Information	"There are 25 boys and 18 girls. Each child has 2 pencils. How many children total?"	Extra information (2 pencils) distracts
Keywords Can Mislead	"Rohan had 15 stamps. He gave away 7. How many in all?"	"Gave away" = subtract, but "in all" misleads toward addition
Cultural References	"If a dozen gulab jamuns cost ₹60..."	Child may not know the sweet
Multiple Interpretations	"How many more?"	Could mean difference or additional amount needed

The Keyword Strategy: Helpful but Limited

Teaching keywords can be a useful starting point, but over-reliance causes problems .

Operation	Keywords	Problem
Addition ➕	in all, total, altogether, sum, more	"More" can also appear in comparison subtraction
Subtraction ➖	left, remain, difference, fewer, less	"Difference" is mathematical vocabulary
Multiplication ✖️	each, every, times, product	"Times" can be confusing in everyday language
Division ➗	share, each, per, divided by	"Each" appears in multiplication too

Example of Keyword Misleading:

• Problem: "Riya has 5 more stamps than Raj. Raj has 8 stamps. How many does Riya have?"

• Keyword strategy: "more" → add → 5 + 8 = 13 ✅ (works here)

• Problem: "Riya has 5 stamps. Raj has 8 more stamps than Riya. How many does Raj have?"

• Keyword strategy: "more" → add → 5 + 8 = 13 ✅ (still works)

• Problem: "Riya has 13 stamps. She has 5 more than Raj. How many does Raj have?"

• Keyword strategy: "more" → add → 13 + 5 = 18 ❌ (should be subtract: 13 - 5 = 8)

The problem: "More" can signal both addition (when comparing) and subtraction (when finding the smaller quantity in a comparison).

🎯 Strategies to Bridge Everyday Language and Mathematical Language

Effective teachers actively help students navigate between everyday language and mathematical language .

Strategy 1: Contextualize Problems

Abstract Problem	Contextualized Version
15 - 7 = ?	"Simran had 15 stickers. She gave 7 to her friend. How many stickers does Simran have left?"
24 ÷ 3 = ?	"24 children are divided into 3 equal teams. How many children in each team?"

Strategy 2: Use Multiple Representations

Representation	Example
Words	"Three plus two equals five."
Symbols	3 + 2 = 5
Pictures	[Drawing of 3 apples + 2 apples = 5 apples]
Manipulatives	3 blocks + 2 blocks = 5 blocks
Real Situations	"You have 3 cookies and I give you 2 more."

Strategy 3: Teach Mathematical Vocabulary Explicitly

Everyday Language	Mathematical Language	Bridge
"Put together"	Add	"When we put together numbers, we add them."
"Take away"	Subtract	"Taking away is called subtraction."
"Answer"	Sum, difference, product, quotient	"The answer in addition is the sum."
"How many left?"	Remainder	"What's left over is the remainder."

Strategy 4: Deconstruct Word Problems Together

The R.U.D.E. Method for Language:

Step	Action	Language Focus
R - Read	Read the entire problem	Identify unfamiliar words
U - Unpack	Rewrite in own words	Simplify complex sentences
D - Decide	Identify knowns and unknown	What do we need to find?
E - Execute	Solve and check	Does answer make sense with the story?

Example Deconstruction:

"A shopkeeper had 85 apples. He sold 37 apples in the morning and 28 apples in the evening. How many apples are left?"

1. Read: Understand it's about apples, selling, and what's left.

2. Unpack: "Start with 85. Sell some in morning. Sell more in evening. Find what's left."

3. Decide: Need to subtract morning sales (37), then subtract evening sales (28). Or add both sales first, then subtract.

4. Execute: 85 - 37 = 48; 48 - 28 = 20 apples left.

Strategy 5: Create Language-Rich Mathematics Classrooms

Strategy	Description	Example
Think-Pair-Share 💭	Think individually, discuss with partner, share with class	"Tell your partner how you solved the problem."
Math Talk 🗣️	Structured discussions about mathematical thinking	"I agree with Priya because..."
Sentence Starters 🏁	Provide frames for mathematical communication	"My strategy was to..." "I know my answer is correct because..."
Math Journals 📓	Write about mathematical thinking	"Today I learned that..." "A problem I solved was..."
Vocabulary Games 🎮	Fun practice with mathematical terms	Bingo, matching games, charades with math words

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Chapter 9 Summary: Quick Revision Notes 📝

Topic	Key Points
Constructivist Approach	Children actively construct knowledge; they are scientific investigators of their world
Prior Knowledge	Children bring experiences that shape learning; can help or hinder
Assimilation	Fitting new info into existing schemas
Accommodation	Changing schemas to fit new info
Children's Strategies	Counting all, counting on, making tens, derived facts, using fingers, drawing
Common Misconceptions	Overgeneralizations like "multiplication makes bigger"
Ask "How Did You Get That?"	Reveals thinking, identifies misconceptions, validates multiple strategies
Mathematical Language	Specialized vocabulary with meanings different from everyday usage
Word Problem Challenges	Unfamiliar vocabulary, complex sentences, misleading keywords
Bridging Strategies	Contextualize, multiple representations, explicit vocabulary, deconstruction, language-rich classrooms

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Chapter 9 Exercises: Test Your Understanding 🧪📝

A. Concept Check (Fill in the Blanks) ✍️

1. According to Piaget, when children fit new information into existing mental structures, they are using ________.

2. When children must change their thinking to accommodate new information, they experience ________, which drives learning.

3. The constructivist approach views children as ________ ________ of their own knowledge .

4. Asking "________ ________ ________ ________ ________?" is a powerful tool for understanding children's mathematical thinking .

5. The keyword "more" can sometimes indicate ________ and sometimes indicate ________, causing confusion .

B. Match the Following (Strategies and Descriptions) 🔗

Column A (Strategy)	Column B (Description)
1. Counting All	A. Starts with larger number and counts forward
2. Counting On	B. Uses known facts to figure out unknown
3. Making Tens	C. Counts both groups from beginning
4. Derived Facts	D. Rearranges numbers to make a friendly number

C. True or False? ✅❌

1. According to constructivism, children are passive recipients of mathematical knowledge.

2. Prior knowledge always helps children learn new mathematical concepts.

3. The process of accommodation involves changing existing mental structures.

4. Using fingers to count is a sign of mathematical weakness and should be discouraged.

5. Word problems are challenging partly because of their complex language .

D. Analyze the Student's Thinking 🧐

For each student response, identify:

• The strategy the student is using

• Whether the strategy is appropriate

• What question you would ask next

Scenario 1:
Teacher: "What is 9 + 6?"
Student: "15. I said 9... 10, 11, 12, 13, 14, 15."

Scenario 2:
Teacher: "What is 15 - 7?"
Student: "I don't know. You can't do it because 7 is bigger than 5."

Scenario 3:
Teacher: "What is 8 + 7?"
Student: "15. I know 8 + 8 = 16, so 8 + 7 is one less."

E. Word Problem Language Analysis 📖

Identify the potential language challenges in this word problem:

"Rohan purchased 3 dozen bananas. Each banana costs ₹2. He gave the shopkeeper a ₹100 note. What amount will he get back?"

1. Unfamiliar vocabulary words: ________

2. Mathematical concepts involved: ________

3. Potential confusion points: ________

F. Reflective Questions 🤔

1. Why is it important for teachers to understand the difference between assimilation and accommodation? How would this knowledge influence your teaching?

2. A student consistently adds 12 + 8 by counting on fingers. Another student uses making ten (10 + 10). How would you support both learners?

3. Design a brief classroom activity that would help students bridge everyday language and mathematical language for the terms "sum," "difference," and "product."

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Answer Key 🔑

A. Concept Check

1. Assimilation

2. Disequilibrium

3. Active constructors

4. How did you get that answer

5. Addition, subtraction (in comparison problems)

B. Match the Following

1-C, 2-A, 3-D, 4-B

C. True or False

1. ❌ False (Constructivism views children as active constructors)

2. ❌ False (Prior knowledge can help or hinder)

3. ✅ True

4. ❌ False (Fingers are developmentally appropriate and show mathematical thinking)

5. ✅ True

D. Analyze the Student's Thinking

Scenario 1:

• Strategy: Counting on from the larger number

• Appropriate? Yes—efficient strategy for this level

• Next question: "Can you think of another way to solve it?"

Scenario 2:

• Strategy: Attempting to apply whole number subtraction rules

• Appropriate? Shows a misconception—doesn't understand borrowing

• Next question: "Let's use blocks. If you have 15 blocks and take away 7, how many are left?"

Scenario 3:

• Strategy: Using near doubles (derived facts)

• Appropriate? Yes—advanced, efficient strategy

• Next question: "Great strategy! What other doubles facts help you?"

E. Word Problem Language Analysis

1. Unfamiliar vocabulary: purchased, dozen, amount

2. Mathematical concepts: Multiplication (3 × 12 × 2), subtraction (100 - total)

3. Potential confusion points: "Dozen" = 12; multiple steps; "amount" = change

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PSTET Success Tips 🌟

1. Connect Theory to Practice: When answering questions about Piaget or constructivism, always provide classroom examples to show application.

2. Know the Stages: Piaget's stages (sensorimotor, pre-operational, concrete operational, formal operational) are essential—especially concrete operational for primary grades .

3. Emphasize Questioning: The importance of asking "How did you get that answer?" is a recurring theme—remember it! 

4. Language Matters: Be prepared to discuss the challenges of mathematical language and strategies to address them .

5. Misconceptions Are Learning Opportunities: Frame errors as windows into children's thinking, not failures.

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Remember: Understanding how children think is the foundation of effective teaching. Every wrong answer tells a story about a child's thinking. Every strategy, whether efficient or not, reveals a child's attempt to make sense of mathematics. Your job as a teacher is not just to correct, but to understand, guide, and celebrate the journey of mathematical discovery. 🌟

Happy Studying, Future Teachers! 📚🍎