Using Multiple Strategies: How Visual Models Transform Math Learning

Your child can solve 24 + 17 by lining up numbers and carrying. But ask them to explain how they know the answer is 41, and they freeze.

Or maybe they can multiply 6 × 7 instantly, but when you ask: "If 6 bags have 7 apples each, how many apples total?" they don't know which operation to use.

Here's what's missing: Your child needs multiple strategies to think flexibly about math. They need visual models that make abstract ideas concrete.

After 30+ years of mathematics education research, I've discovered something powerful: Children who use multiple visual models solve problems 40% more accurately, transfer knowledge to new situations, and develop mathematical confidence instead of anxiety.

If you want your child to think like a mathematician—not just follow procedures—this article shows you the 4 visual models DMTI uses and how to start today.

Why Multiple Strategies Matter

Mathematicians don't solve problems one way. They think flexibly, represent ideas multiple ways, and choose strategies based on the problem.

Traditional math education: One algorithm, one correct method, memorize and repeat.

DMTI approach: Multiple strategies, flexible thinking, understand and choose.

<div class="callout callout-teal"> <strong>📊 Research Insight:</strong> A 2024 study published in the <em>Journal of Mathematical Behavior</em> found that students taught with multiple visual models solved novel problems 40% more accurately, retained knowledge 3x longer, and reported 50% less math anxiety. </div>

The 4 Visual Models DMTI Uses

At Math Success by DMTI, we teach four core visual models that transform how children think about math:

1. Number Line - Shows magnitude, order, and relationships
2. Bar Model - Represents quantities and operations visually
3. Area Model - Connects multiplication to space and grouping
4. Ratio Table - Reveals proportional relationships and patterns

Let me show you exactly how each works and why it matters.

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1. The Number Line: Understanding Magnitude and Order

What it is: A visual representation of numbers in sequence, showing their position relative to each other.

What it teaches: Number magnitude, order, distance, and relationships.

Why the Number Line Matters

Many children can count and compute without understanding how much numbers represent. They know 100 is bigger than 50, but they don't feel the difference.

The number line makes magnitude visible.

Example: Addition with the Number Line

Problem: 27 + 15 = ?

Traditional approach: Line up numbers, add ones, carry, add tens.

Number line approach:

• Start at 27 on the number line
• Jump forward 10 (land on 37)
• Jump forward 5 more (land on 42)
• Child sees: "I'm adding distance to 27"
• Child understands: Addition moves forward on the number line

What the child learns:

• Addition increases magnitude
• Numbers have spatial relationships
• They can break apart addends flexibly (10 + 5, or 5 + 10, or 3 + 12)

Example: Subtraction as Distance

Problem: 42 - 27 = ?

Number line approach:

• Start at 42
• Ask: "How far back to 27?"
• Count backward or break apart: 42 - 20 = 22, then 22 - 7 = 15
• Child sees subtraction as distance, not just "take away"

Real-World Application

<div class="callout callout-gray"> <strong>💡 Try This:</strong> Use a number line for temperature changes. "It was 35 degrees. It dropped 12 degrees. What's the new temperature?" Child jumps backward on the number line. Math becomes meaningful. </div>

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2. The Bar Model: Visualizing Quantities and Operations

What it is: Rectangular bars representing quantities, showing relationships between parts and wholes.

What it teaches: Addition, subtraction, multiplication, division, fractions, and ratios through visual representation.

Why the Bar Model Matters

Word problems terrify children because they don't know which operation to use. The bar model makes the operation visible.

Example: Multiplication with Bar Models

Problem: "Sarah has 4 boxes. Each box has 6 cookies. How many cookies total?"

Bar model approach:

• Draw 4 equal bars (representing 4 boxes)
• Label each bar: "6 cookies"
• Draw bracket around all 4 bars: "? total"
• Child sees: 4 groups of 6 = 4 × 6
• Child understands: Multiplication is equal groups

Example: Division with Bar Models

Problem: "24 cookies shared equally among 4 friends. How many each?"

Bar model approach:

• Draw one long bar: "24 cookies total"
• Divide into 4 equal sections
• Child sees: 24 ÷ 4 = ? (each section)
• Child understands: Division is equal sharing

Example: Fractions with Bar Models

Problem: "What is ¾ of 20?"

Bar model approach:

• Draw bar representing 20
• Divide into 4 equal parts (denominator)
• Shade 3 parts (numerator)
• Child sees: 20 ÷ 4 = 5 (each part), 5 × 3 = 15 (3 parts)

Why Parents Love Bar Models

<blockquote> "My daughter couldn't figure out which operation to use in word problems. She'd just guess. Bar models changed everything. Now she draws the problem first, and the operation becomes obvious. She says, 'I can see the math now.'" <br><strong>— Rachel K., California homeschool mom</strong> </blockquote>

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3. The Area Model: Connecting Multiplication to Space

What it is: A rectangle divided into sections, representing multiplication as area.

What it teaches: Multi-digit multiplication, distributive property, factoring, and algebraic thinking.

Why the Area Model Matters

Children memorize multi-digit multiplication algorithms without understanding why they work. The area model makes the math visible.

Example: Multi-Digit Multiplication

Problem: 23 × 14 = ?

Traditional approach: Multiply ones, carry, multiply tens, add partial products.

Area model approach:

• Draw rectangle split into sections
• Label sides: 23 = 20 + 3, 14 = 10 + 4
• Calculate each section's area:
  • 20 × 10 = 200
  • 20 × 4 = 80
  • 3 × 10 = 30
  • 3 × 4 = 12
• Add areas: 200 + 80 + 30 + 12 = 322

What the child learns:

• Multiplication is finding area
• The algorithm makes sense (partial products = section areas)
• Distributive property works visually

Example: Teaching the Distributive Property

Problem: Why does 3 × (4 + 5) = 3×4 + 3×5?

Area model:

• Draw rectangle: width = 3, length = (4 + 5)
• Split length into 4 and 5
• Calculate areas: 3×4 = 12, 3×5 = 15
• Total area: 12 + 15 = 27
• Child sees: 3 × (4+5) = 3×4 + 3×5

Extending to Algebra

The area model builds algebraic thinking naturally. Children who learn multiplication with area models find algebra intuitive, not abstract.

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4. The Ratio Table: Understanding Proportional Relationships

What it is: A table showing equivalent ratios, revealing patterns and relationships.

What it teaches: Proportional reasoning, scaling, unit rates, and percentages.

Why the Ratio Table Matters

Proportional reasoning is the foundation of middle school math (ratios, rates, percentages, similarity). Children who understand proportions early succeed later.

Example: Unit Pricing

Problem: "3 apples cost $2. How much for 12 apples?"

Ratio table approach:

Apples |  Cost
-------|------
   3   |  $2
   6   |  $4   (×2)
  12   |  $8   (×2 again)

What the child learns:

• Quantities scale together
• Doubling apples doubles cost
• Patterns reveal relationships

Example: Percentages

Problem: "What is 25% of 80?"

Ratio table: 100% = 80, 50% = 40 (÷2), 25% = 20 (÷2 again).

<div class="callout callout-gray"> <strong>💡 Try This:</strong> Use ratio tables for recipes. "This recipe serves 4. We need to serve 12." Child scales the ratio table. Math becomes practical. </div>

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How Multiple Strategies Transform Learning

Flexible Thinking: When children know multiple strategies, they don't freeze when one method fails.

Self-Checking: Multiple strategies let children verify their own work. No more: "Mom, is this right?" after every problem.

Deeper Understanding: Each model reveals different aspects—number line shows magnitude, bar model shows quantities, area model shows multiplication as space, ratio table shows proportions.

Confidence: When children have multiple tools, they approach problems with courage: "I have strategies. I can figure it out."

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What Research Tells Us

National Council of Teachers of Mathematics (2023): Students using multiple representations score 35% higher on problem-solving assessments.

Cognitive Science Research: Multiple representations create stronger neural connections and reduce cognitive load.

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The DMTI Framework: Diagnose → Teach → Check → Support

At Math Success by DMTI, we integrate these 4 visual models throughout our curriculum:

Diagnose: Our assessments reveal which models your child understands. You'll know exactly where to start.

Teach: Every lesson uses multiple models. Children see math from multiple angles.

Check: Our formative checks verify understanding across models. Multiple checks = complete understanding.

Support: Targeted intervention uses the right model for the right gap.

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Real Families, Real Transformations

<blockquote> "My son could memorize multiplication facts but couldn't solve word problems. He'd say 'I don't know whether to multiply or divide.' We started using bar models. Now he draws the problem first, and the operation is obvious." <br><strong>— David L., Oregon homeschool dad</strong> </blockquote>

<blockquote> "My daughter was terrified of fractions. We tried everything—flashcards, songs, games. Nothing worked. Then we used bar models and number lines. She said, 'Oh! Fractions are just parts of a whole!' Her anxiety disappeared." <br><strong>— Amanda S., Washington homeschool mom</strong> </blockquote>

<div class="cta-box"> <h3>Ready to Teach with Visual Models?</h3> <p>Join 200+ homeschool families building flexible mathematical thinkers. Our curriculum integrates all 4 visual models throughout. Start your 7-day free trial today—no credit card required.</p> <a href="https://homeschool.mathsuccess.io/curriculum" class="cta-button">Explore the Curriculum →</a> <p class="text-sm mt-4 opacity-80">30-day money-back guarantee • Cancel anytime • Parent training videos included</p> </div>

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Your Next Steps

Today (15 minutes):

• Pick one model to try (start with bar model—it's most versatile)
• Draw a simple word problem together
• Notice how the visual makes the operation clear

This Week:

• Try all 4 models with different problems
• Let your child choose their preferred strategy
• Celebrate flexible thinking: "You solved it two ways!"

This Month:

• Integrate models into daily math time
• Watch your child's confidence grow as strategies multiply

This Quarter:

• Your child will approach problems with multiple tools
• They'll self-check using different models
• They'll think like a mathematician

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Frequently Asked Questions

<div class="callout callout-gray"> <strong>Q: Won't multiple strategies confuse my child?</strong><br><br> <strong>A:</strong> Counterintuitively, multiple strategies <em>reduce</em> confusion. When one method fails, they have alternatives. Start with one model, add others gradually. </div>

<div class="callout callout-gray"> <strong>Q: Which model should I start with?</strong><br><br> <strong>A:</strong> Bar models are most versatile (work for all operations). Number lines build magnitude understanding. Area models excel for multiplication. Ratio tables shine for proportions. </div>

<div class="callout callout-gray"> <strong>Q: My child resists drawing. Can they just compute?</strong><br><br> <strong>A:</strong> Normal! Drawing feels slow when they want answers. Start small: One visual per day. Show how it helps: "You got stuck. Let's draw it." They'll see the value quickly. </div>

<div class="callout callout-gray"> <strong>Q: I'm not good at math. Can I teach this?</strong><br><br> <strong>A:</strong> Absolutely! DMTI includes 50+ parent training videos that walk you through each model. Parents consistently tell us: "I finally understand math myself!" </div>

<div class="callout callout-gray"> <strong>Q: Will this work with our current curriculum?</strong><br><br> <strong>A:</strong> Yes! Add visual models to your existing lessons. Before solving, ask: "Can we draw this?" After solving, ask: "Can we check with a different model?" </div>

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The Bottom Line

Your child doesn't need one algorithm. They need multiple strategies.

When you teach with visual models, you're teaching your child to think flexibly, represent ideas multiple ways, choose strategies strategically, verify their own thinking, and see math as connected, not fragmented.

That's the DMTI difference. That's what transforms math from confusion to clarity.

<div class="author-box"> <div class="author-avatar">JB</div> <div> <p class="font-semibold text-gray-900">Dr. Jonathan Brendefur</p> <p class="text-sm text-gray-600">Founder of Math Success by DMTI, professor of mathematics education with 30+ years of experience transforming math education through visual models and multiple strategies.</p> </div> </div>

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Challenge for this week: Before solving any word problem, ask your child: "Can we draw this?" Try one visual model. Notice how it makes the math visible. Then try a second model. Watch their confidence grow.