CBI in Mathematics

Introduction

Mathematics might seem an unlikely candidate for Concept-Based Inquiry. Many view math as a collection of procedures to master—algorithms, formulas, and techniques that require practice more than inquiry. Yet mathematics is fundamentally conceptual. Every procedure embodies mathematical ideas; every formula expresses relationships. When students understand the concepts underlying procedures, they gain power that no amount of rote practice can provide.

This chapter reimagines mathematics instruction through a CBI lens. We'll explore how mathematical concepts create coherence across topics, how inquiry reveals mathematical understanding, and how conceptual focus transforms students from procedure-followers to mathematical thinkers.

12.1 Mathematics as Conceptual Discipline

The Conceptual Nature of Mathematics

Mathematics is built on concepts:

Mathematical Objects as Concepts The entities mathematicians work with are conceptual:

• Numbers (and their many forms)
• Operations (and their properties)
• Relationships (and their representations)
• Shapes (and their properties)
• Patterns (and their structures)

Procedures as Concept Applications Every algorithm embodies conceptual understanding:

• Long division expresses the concept of partitioning
• The quadratic formula applies the concept of completing the square
• The Pythagorean theorem expresses a relationship between sides of right triangles

The Danger of Procedure Without Concept When math instruction focuses solely on procedures:

• Students can perform without understanding
• Minor variations confuse learners
• Transfer to new situations fails
• Mathematics feels arbitrary and meaningless

Core Mathematical Concepts

Mathematics education involves several types of concepts:

Content Concepts: Mathematical objects and relationships

• Number, operation, equation, function, variable
• Ratio, proportion, rate, percentage
• Shape, angle, area, volume, dimension
• Pattern, sequence, series, recursion

Process Concepts: Mathematical practices

• Representation, modeling, proof, conjecture
• Estimation, precision, approximation
• Generalization, abstraction, specialization
• Algorithm, efficiency, optimization

Structural Concepts: How mathematics is organized

• Equivalence, relationship, structure, transformation
• Properties (commutative, associative, distributive)
• Inverse, identity, composition
• Infinity, continuity, discreteness

Mathematical Generalizations

Strong mathematical generalizations express enduring relationships:

About Numbers and Operations:

• "Operations and their inverses undo each other, returning to the starting point."
• "Any number can be decomposed and recomposed in multiple ways, enabling flexible computation."
• "The base-ten system uses place value to represent quantities of any size with only ten symbols."

About Relationships:

• "Proportional relationships maintain equivalent ratios across all values."
• "Functions express consistent relationships between inputs and outputs."
• "Linear relationships have constant rates of change."

About Mathematical Practice:

• "Multiple representations of the same mathematical idea reveal different aspects of its structure."
• "Mathematical proofs establish certainty through logical chains of reasoning."
• "Patterns that hold for specific cases may suggest generalizations that hold universally."

12.2 Conceptual Understanding vs. Procedural Fluency

The False Dichotomy

Much debate in mathematics education pits conceptual understanding against procedural fluency. CBI rejects this false choice:

Both Are Essential

• Procedures without concepts are fragile and limited
• Concepts without procedures lack power and efficiency
• Mathematical proficiency requires both, connected

Sequence Matters Research suggests: concept → procedure → fluency

When students understand WHY a procedure works:

• They can reconstruct it if forgotten
• They recognize when it applies
• They can modify it for new situations
• They develop fluency more efficiently

Concepts First, Procedures After

Example: Multi-Digit Multiplication

Traditional Approach: Teach the standard algorithm, practice until fluent Limitation: Students can multiply but don't understand why it works or when to use it

CBI Approach:

Concept: Multiplication represents area or repeated groups; multi-digit multiplication uses place value and the distributive property

Investigation: Students explore multiple strategies:

• Area models (visual representation)
• Partial products (explicit place value)
• Standard algorithm (compressed efficiency)

Generalization: "Multi-digit multiplication decomposes factors using place value, multiplies parts, and combines results."

Then: Practice develops fluency, with conceptual understanding supporting retention

Productive Struggle

CBI embraces productive struggle—the intellectual work of making sense of mathematics:

What Productive Struggle Looks Like:

• Students wrestle with problems before receiving methods
• Multiple approaches are explored and compared
• Errors are analyzed for conceptual insight
• Understanding builds through effort

What It Doesn't Look Like:

• Frustration without support
• Floundering without purpose
• Time-wasting for its own sake

Teacher Role:

• Provide problems worth struggling with
• Offer strategic support (questions, not answers)
• Facilitate discussion of approaches
• Help students consolidate insights

12.3 Problem-Based Mathematical Inquiry

Problems as Provocations

In CBI mathematics, problems serve as provocations for conceptual exploration:

Good Mathematical Problems:

• Are accessible but challenging
• Allow multiple solution paths
• Reveal mathematical structure
• Connect to conceptual targets
• Create need for new understanding

Example: "How many different rectangles can you make with a perimeter of 24 units? What do you notice about their areas?"

This problem:

• Is accessible (students can start immediately)
• Allows exploration (many rectangles to find)
• Reveals structure (relationship between dimensions and area)
• Develops concepts (perimeter, area, dimension, optimization)
• Creates need (for systematic approach)

The Problem-Based Lesson Structure

Launch (5-10 minutes)

• Present the problem
• Ensure understanding of the task
• Activate prior knowledge
• Resist the urge to show methods

Explore (15-25 minutes)

• Students work individually or in pairs
• Teacher circulates, observes, questions
• Note different approaches for discussion
• Provide strategic support as needed

Discuss (15-20 minutes)

• Share and compare approaches
• Sequence strategically (less to more sophisticated)
• Draw out mathematical ideas
• Build toward generalizations

Connect (5-10 minutes)

• Name the concepts and strategies
• Make connections explicit
• Preview how this connects to future work
• Assign practice that reinforces understanding

Three-Act Math Tasks

Dan Meyer's Three-Act structure creates engaging mathematical inquiry:

Act 1: The Hook Present an intriguing image, video, or scenario. Students notice and wonder.

Act 2: The Information Students identify what information they need to solve the problem. Provide data as requested.

Act 3: The Reveal Share the actual answer. Compare with predictions. Discuss approaches.

Sequel: Extend the problem to deepen or transfer understanding.

12.4 Multiple Representations

Why Multiple Representations Matter

Mathematical concepts exist across representations:

• Concrete: Physical objects and manipulatives
• Visual: Diagrams, graphs, pictures
• Symbolic: Numbers, variables, equations
• Verbal: Words and explanations
• Contextual: Real-world applications

Conceptual Understanding = Flexible Movement Between Representations

Students who truly understand can:

• Translate between representations
• Choose appropriate representations for different purposes
• See how representations connect
• Recognize the same idea in different forms

Teaching Through Multiple Representations

Proportional Relationships Example:

Inquiry Approach:

• Present the context
• Students create representations
• Compare and connect representations
• Identify what each representation reveals
• Develop generalizations about proportional relationships

Representation Translation Tasks

Regularly ask students to translate between representations:

"Here's a graph. Write an equation that matches." "Here's an equation. Describe a real-world situation it could model." "Here's a table. Sketch a graph that shows the same relationship."

These tasks reveal and develop conceptual understanding.

12.5 Mathematical Discussion and Discourse

Talking About Mathematics

Mathematical discussion develops conceptual understanding:

Why Discussion Matters:

• Articulating thinking clarifies understanding
• Hearing others' approaches reveals mathematical structure
• Defending reasoning develops precision
• Questions deepen thinking

Discourse Moves:

Revoicing: "So you're saying that..." Pressing for reasoning: "Why does that work?" Connecting: "How does this relate to what Maria said?" Challenging: "Does this always work? What if...?" Generalizing: "What can we say that's always true?"

Number Talks

Brief, regular discussions about computation develop conceptual understanding:

Structure (10-15 minutes):

1. Present a computation (no pencils)
2. Students solve mentally
3. Share strategies
4. Record and compare approaches
5. Discuss efficiency and connections

Example: 99 × 6

Student strategies might include:

• 100 × 6 - 1 × 6 = 600 - 6 = 594 (compensation)
• 99 × 3 × 2 = 297 × 2 = 594 (factoring)
• 100 × 6 - 6 = 594 (adjusting)

Discussion Focus:

• What strategies are similar?
• What properties make these work?
• Which is most efficient for this problem?
• When might different strategies be better?

Generalization: "Numbers can be decomposed and recomposed flexibly, using properties to simplify computation."

Math Talks and Reasoning

Beyond number talks, facilitate reasoning about mathematical ideas:

Would You Rather? "Would you rather have 25% of $80 or 80% of $25? Why?"

Students reason, debate, and discover mathematical relationships.

Always, Sometimes, Never "x² is greater than x. Always, sometimes, or never?"

Students investigate, find counterexamples, and develop nuanced understanding.

Which One Doesn't Belong? Present four mathematical objects. Students argue why each could be the one that doesn't belong.

All students can participate; multiple valid answers exist; reasoning is central.

12.6 Assessment in Mathematics

Assessing Conceptual Understanding

Move beyond procedure-checking to assess conceptual understanding:

Explanation Tasks "Explain why division by a fraction is the same as multiplication by its reciprocal."

This requires conceptual understanding, not just procedure.

Comparison Tasks "Leila says 3/4 > 5/8 because 3/4 is 3 parts out of 4, which is more than half. Marcus says 3/4 > 5/8 because 3/4 = 6/8 and 6/8 > 5/8. Both are correct. What concept does each student's explanation use?"

Students analyze different conceptual approaches.

Error Analysis "A student solved 3.4 × 10 = 3.40. What misconception does this reveal? How would you help this student?"

Understanding others' errors requires deep conceptual understanding.

Representation Tasks "Represent 3/4 ÷ 1/2 in at least two different ways. Explain how your representations show the quotient."

Multiple representations reveal conceptual understanding.

Formative Assessment in Math

Observation During Work Time Circulate and listen for conceptual understanding:

• What strategies are students using?
• What connections are they making (or missing)?
• What misconceptions appear?

Exit Tickets with Explanation Don't just ask for answers; ask for reasoning:

• "Solve and explain your strategy."
• "Which method would you use and why?"
• "What does this tell you about [concept]?"

Student Self-Assessment Students evaluate their own understanding:

• "I can do this procedure but don't understand why it works."
• "I understand the concept and can apply it flexibly."
• "I can explain this to someone else."

Classroom Snapshot: 6th Grade Mathematics

Unit: Ratios and Proportional Relationships Duration: 4 weeks Concepts: Ratio, Rate, Proportion, Equivalence, Scaling Generalization: "Proportional relationships maintain equivalent ratios, enabling prediction and comparison across quantities."

Week 1: Building Ratio Understanding

Day 1: Provocation

Present a paint-mixing scenario with colored water:

• 2 cups blue + 1 cup yellow makes one shade of green
• 4 cups blue + 2 cups yellow makes another batch

Question: "Will these be the same shade of green? How do you know?"

Students predict, then watch demonstration.

Days 2-3: Exploring Equivalent Ratios

Hands-on investigation: Students mix actual solutions (colored water) to discover when shades match.

Key Questions:

• What makes two mixtures the same shade?
• What different combinations would make this shade?
• How can you predict if a mixture will match?

Emerging Concept: Equivalent ratios represent the same relationship.

Days 4-5: Multiple Representations of Ratios

Students represent ratios in multiple ways:

• Concrete (physical objects)
• Pictorial (diagrams)
• Tables
• Written descriptions
• Symbolic (a:b, a/b, "a to b")

Discussion: What does each representation help us see?

Week 2: Rates and Unit Rates

Day 1: Rate Provocation

Present two job scenarios:

• Job A: $45 for 3 hours of work
• Job B: $72 for 6 hours of work

Question: "Which job pays better?"

Students struggle with comparison, creating need for common basis.

Days 2-3: Unit Rate Investigation

Students discover that finding "per 1" enables comparison.

Activities:

• Compare unit prices for different product sizes
• Compare speeds given distance and time
• Compare rates in various contexts

Generalization Building: Unit rates express "how much per one" and enable comparison.

Days 4-5: Unit Rate Applications

Real-world applications:

• Best buy problems (unit price)
• Speed and distance
• Recipe scaling

Connecting Representations:

• Unit rates on graphs (what does the slope represent?)
• Unit rates in tables (constant change per row)
• Unit rates in equations (coefficient)

Week 3: Proportional Reasoning

Day 1: Proportion Provocation

Problem: "A photograph is 4 inches wide and 6 inches tall. You want to enlarge it to be 10 inches wide. How tall should it be?"

Students explore multiple strategies:

• Scale factor approach
• Equivalent ratio approach
• Unit rate approach

Days 2-3: Recognizing Proportional Relationships

Investigation: Which relationships are proportional?

• Students analyze tables, graphs, equations, contexts
• Identify characteristics of proportional relationships
• Distinguish proportional from non-proportional

Characteristics Discovered:

• Constant ratio between quantities
• Graph is a straight line through the origin
• Equation has form y = kx
• "Times as many" relationships

Days 4-5: Solving Proportion Problems

Multiple strategies:

• Equivalent ratios
• Scale factor
• Unit rate
• Cross-multiplication (connected to WHY it works)

Emphasis: Choose efficient strategies; understand why they work.

Week 4: Synthesis and Transfer

Day 1-2: Connecting Representations

Major task: Given a proportional relationship in one representation, create all other representations.

Example: Start with "A car uses 3 gallons of gas to travel 90 miles."

• Create a table
• Create a graph
• Write an equation
• Find the unit rate
• Predict for other values

Day 3: Generalization Development

Class discussion building toward target:

Questions:

• What's always true about proportional relationships?
• How can you recognize one?
• How do proportions help us solve problems?

Class Generalization: "Proportional relationships maintain equivalent ratios, enabling prediction and comparison across quantities."

Day 4: Transfer Assessment

New context: Students apply proportional reasoning to scenarios they haven't seen:

• Scale drawings
• Similar figures
• Percentage problems
• Sampling and prediction

Day 5: Reflection and Connection

Students complete reflections:

• How has my understanding of ratios changed?
• What strategies do I now have for ratio problems?
• Where do I see proportions in my life?
• What questions do I still have?

Preview: How this connects to upcoming work on percentages and similarity.

Templates

Template 12.1: Problem-Based Mathematics Lesson Plan

Lesson Topic: _________________ Grade Level: _____ Duration: _________

CONCEPTUAL FOCUS

Target Concept(s): ____________________________________________ Generalization: _______________________________________________ Connection to Prior Learning: ___________________________________ Connection to Future Learning: __________________________________

THE PROBLEM

Problem Statement:

Why This Problem?

• Multiple solution paths: ________________________________________
• Reveals mathematical structure: __________________________________
• Connects to concept: __________________________________________
• Accessible yet challenging: _____________________________________

LESSON STRUCTURE

LAUNCH (_____ minutes)

• How will I present the problem? __________________________________
• What prior knowledge will I activate? ______________________________
• How will I ensure understanding without giving away methods? __________

EXPLORE (_____ minutes)

• Individual or partner work? _____________________________________
• What will I observe for? _______________________________________
• What questions will I ask without giving answers? ____________________
• What approaches do I anticipate? ________________________________

DISCUSS (_____ minutes)

• What sequence will I use to share strategies? _______________________
• What connections will I highlight? ________________________________
• What questions will I ask? _____________________________________
• How will I build toward the generalization? _________________________

CONNECT (_____ minutes)

• What vocabulary/notation will I introduce? _________________________
• How does this connect to prior and future learning? __________________
• What practice will students do? _________________________________

ANTICIPATED STRATEGIES

ASSESSMENT

How will I know students understand the concept, not just the procedure?

Template 12.2: Multiple Representations Task Designer

Concept: ___________________ Grade Level: _____

TARGET GENERALIZATION:

THE MATHEMATICAL SITUATION

Contextual (real-world scenario):

REPRESENTATION DEVELOPMENT

TRANSLATION TASKS

"Given the table, create a graph that shows the same relationship." "Given the equation, describe a real-world situation it could represent." "Given the graph, write an equation."

DISCUSSION QUESTIONS

"What does each representation help us see?"

"How do you know these representations show the same relationship?"

"Which representation would be most helpful for [specific purpose]? Why?"

GENERALIZATION PROMPT

"What's always true about [concept] based on what we've seen in these representations?"

Template 12.3: Mathematical Discussion Planning Guide

Problem/Topic: _______________________________________________ Concept Focus: ______________________________________________ Generalization Goal: __________________________________________

ANTICIPATED STUDENT STRATEGIES

DISCUSSION SEQUENCE

Order for Sharing (typically less to more sophisticated):

1. _________________________ (why share first?)
2. _________________________ (what does this add?)
3. _________________________ (how does this build?)
4. _________________________ (most sophisticated; why save for last?)

CONNECTING QUESTIONS

"How is [Strategy A] similar to [Strategy B]?"

"What mathematical idea makes both of these work?"

"Which strategy would be most efficient for [different problem]? Why?"

GENERALIZING QUESTIONS

"What's always true about [concept] based on these strategies?"

"Would this work for any numbers? How do we know?"

DISCOURSE MOVES TO USE

• Revoicing: "So you're saying..."
• Pressing for reasoning: "Why does that work?"
• Connecting: "How does this relate to what [student] said?"
• Challenging: "Would this always work?"
• Generalizing: "What can we say is always true?"

EXIT TICKET

AI Prompts for Mathematics CBI

Prompt 12.1: Problem Design for Conceptual Understanding

I'm teaching [mathematical topic] to [grade level] students. The concept I want to develop is [concept], leading to this generalization: [generalization].

Design a rich mathematical problem that:
1. Is accessible to students at this level (has an entry point)
2. Allows multiple solution strategies
3. Reveals the underlying mathematical structure
4. Creates productive struggle
5. Leads naturally to the target generalization

Include:
- The problem statement
- Anticipated student strategies (at least 3)
- Discussion questions to connect strategies to concepts
- A suggested sequence for sharing strategies
- Extension questions for students ready for more

Prompt 12.2: Multiple Representations Development

I want to develop [grade level] students' understanding of [concept/topic] through multiple representations.

Design a lesson sequence that:
1. Begins with a contextual situation students can engage with
2. Guides students to create each representation (concrete, visual, table, graph, symbolic)
3. Includes tasks where students translate between representations
4. Makes connections between representations explicit
5. Builds toward this generalization: [generalization]

Include discussion questions that help students see how the same mathematical idea appears across representations.

Prompt 12.3: Number Talk Design

I want to design number talks that develop [grade level] students' understanding of [concept—e.g., place value, properties of operations, fraction relationships].

Create a series of 5 number talks that:
1. Build on each other across the week
2. Encourage mental math strategies
3. Reveal mathematical structure
4. Lead to generalizations about how numbers work
5. Are accessible yet promote higher-level strategies

For each number talk:
- The computation presented
- Anticipated student strategies
- Questions to ask during discussion
- The mathematical concept being developed

Prompt 12.4: Assessing Conceptual Understanding

I'm concluding a unit on [topic] with [grade level] students. The target concepts are [concepts] and the generalization is [generalization].

Design assessments that measure conceptual understanding, not just procedural skill:
1. Explanation tasks (students explain why procedures work)
2. Representation tasks (students translate between representations)
3. Error analysis tasks (students diagnose and explain misconceptions)
4. Novel application tasks (students apply understanding to new contexts)
5. Comparison tasks (students analyze different approaches)

Include rubrics or criteria that distinguish conceptual understanding from procedural knowledge.

Prompt 12.5: Connecting Procedures to Concepts

I need to teach [specific procedure—e.g., long division, solving equations, finding area] to [grade level] students.

Help me design instruction that:
1. Builds conceptual understanding BEFORE introducing the standard procedure
2. Uses multiple representations to develop understanding
3. Shows WHY the procedure works, not just HOW to do it
4. Connects the procedure to underlying mathematical concepts
5. Develops the target generalization: [generalization]

Include:
- Prerequisite concepts students need
- Concrete/visual representations to use
- Questions that reveal mathematical structure
- How to introduce the standard procedure after concepts are established

Key Takeaways

1. Mathematics is conceptual: Every procedure embodies concepts; CBI makes these concepts explicit and transferable

2. Concepts first, then procedures: Understanding why enables efficient learning of how; concepts provide the foundation for fluency

3. Problems drive inquiry: Rich problems that allow multiple approaches reveal mathematical structure and develop understanding

4. Multiple representations matter: Conceptual understanding means flexible movement between representations of the same idea

5. Discussion develops thinking: Mathematical discourse—sharing strategies, defending reasoning, questioning—builds conceptual understanding

6. Assess understanding, not just performance: Look for conceptual understanding through explanation, representation, and transfer tasks

7. Productive struggle is essential: Students need to wrestle with mathematical ideas; this struggle develops understanding

Reflection Questions

1. Think about a procedure you teach. What concepts underlie it? How could you make those concepts explicit?

2. Examine a lesson you've taught. Where did students engage with concepts? Where was focus purely procedural? How might you adjust?

3. How do you currently use mathematical discussion? What discourse moves could you incorporate?

4. How often do students work with multiple representations of the same mathematical idea? What opportunities could you create?

5. Look at a recent assessment. What did it measure—procedural knowledge or conceptual understanding? How could you add conceptual assessment?

In the next chapter, we explore CBI in Science, where inquiry has long been central and concepts like energy, systems, and change create coherence across scientific disciplines.