You've taught them the steps. They've memorized the keywords. They can circle the important numbers. But when faced with a math problem they haven't seen before, they freeze.
"I don't get it," they say. Or worse: "Just tell me what operation to use."
Here's the hard truth: keyword strategies and step-by-step procedures don't build problem solvers. They build students who can follow directions—but only when the problem looks familiar.
The DMT Framework offers a different path. Instead of teaching tricks, we teach students to think mathematically—to compose and decompose problems, to iterate through strategies, to partition complex situations into manageable pieces. This is how real mathematicians work.
In this post, you'll discover six research-backed problem solving strategies that transform how your elementary students approach math—starting Monday.
The Problem With Traditional Problem Solving Instruction
Research Finding: A landmark study by Carpenter et al. (2014) found that students taught with keyword-based problem solving strategies could solve familiar problems but failed when problems required genuine mathematical reasoning.
Source: Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2014). Children's Mathematics: Cognitively Guided Instruction (2nd ed.). Heinemann.
Traditional problem solving instruction follows a predictable pattern:
- Read the problem (often multiple times without understanding)
- Circle the numbers (without knowing why)
- Underline keywords ("total" means add, "left" means subtract)
- Choose an operation (based on the keyword, not the situation)
- Solve and check (if the answer seems reasonable)
This approach works—until it doesn't. When students encounter a problem without familiar keywords, or when multiple operations are needed, or when the problem requires conceptual understanding rather than procedural execution, they're lost.
The real issue: We're teaching students to perform problem solving, not to think mathematically.
The DMT Framework: Six Components for Mathematical Thinking
The DMT Framework transforms problem solving from a procedure into a thinking process. Each component builds students' capacity to approach unfamiliar problems with confidence:
1. Unit
Students learn to identify the complete unit in a problem—what's being counted, measured, or compared. This prevents the common error of operating on numbers without understanding what they represent.
2. Compose
Students build understanding by combining parts—seeing how smaller quantities create larger ones, how multiple steps connect, how different representations show the same relationship.
3. Decompose
Students break complex problems into manageable pieces—not by following steps, but by understanding the mathematical structure and choosing strategic entry points.
4. Iterate
Students develop repeated reasoning—trying strategies, checking results, adjusting approaches. This builds resilience and mathematical intuition.
5. Partition
Students learn to divide wholes into equal parts—essential for fractions, division, and proportional reasoning. They understand what "equal" really means.
6. Equal
Students develop relational thinking—understanding that the equal sign means "the same as," not "the answer comes next." This is foundational for algebraic thinking.
These aren't just vocabulary words. They're cognitive tools that students use to make sense of problems, choose strategies, and verify their thinking.
6 Problem Solving Strategies You Can Use Monday
Strategy 1: Three-Read Protocol
The Problem: Students rush to solve without understanding the situation.
The Strategy: Read the problem three times with different purposes:
- First read: What's the situation? (No numbers, no question)
- Second read: What are the quantities? (Identify numbers and what they measure)
- Third read: What's the question? (What are we trying to find?)
Example: "There were some cookies on a plate. Jake ate 3 cookies. Now there are 5 cookies left."
First read: Someone had cookies, ate some, and has some left.
Second read: 3 cookies (eaten), 5 cookies (remaining), unknown (starting amount)
Third read: How many cookies were there at the beginning?
Strategy 2: Visual Model First
The Problem: Students jump to equations without understanding relationships.
The Strategy: Require students to draw a visual model before writing any numbers or equations. Use tape diagrams, number bonds, or bar models to represent the situation.
DMT Connection: This builds Compose and Decompose thinking—students see how parts relate to wholes before manipulating symbols.
Strategy 3: Unit Language Routine
The Problem: Students operate on numbers without understanding what they represent.
The Strategy: Require students to state the unit for every number: "12 apples," not just "12." When solving, they track units: "12 apples minus 5 apples equals 7 apples."
DMT Connection: This builds Unit thinking—the foundation of all mathematical reasoning.
Strategy 4: Multiple Solution Pathways
The Problem: Students believe there's only one "right way" to solve a problem.
The Strategy: Present a problem and ask: "How many different ways can we solve this?" Have students share multiple approaches. Validate all mathematically sound strategies.
DMT Connection: This builds Iterate thinking—students learn to try, check, and adjust their approaches.
Strategy 5: Problem String Sequences
The Problem: Students treat each problem as isolated, missing patterns and relationships.
The Strategy: Present a sequence of related problems that build on each other:
Problem 1: 5 + 3 = ?
Problem 2: 15 + 3 = ?
Problem 3: 15 + 13 = ?
Problem 4: 115 + 13 = ?
Ask: "How did Problem 1 help you solve Problem 4?"
DMT Connection: This builds Iterate and Compose thinking—students see how mathematical ideas connect and build.
Strategy 6: Error Analysis Discussions
The Problem: Students see mistakes as failures rather than learning opportunities.
The Strategy: Present a solved problem with a common error. Ask: "What mistake was made? Why might someone make this error? How do we fix it?"
DMT Connection: This builds Equal thinking—students learn to verify that both sides of an equation represent the same quantity.
Teacher Transformation: From "Just Tell Me" to "Let Me Think"
"I used to spend my entire math block walking around, telling students which operation to use. They'd finish five problems, but ask me the same question on problem six. I was exhausted, and they weren't learning."
Sarah Martinez, a 4th-grade teacher in Boise, Idaho, struggled with the same problem many teachers face: her students could follow procedures but couldn't think independently.
"When I started using the DMT Framework, everything changed. I stopped giving them the steps and started asking them to show me their thinking. The first week was chaotic—they hated it. But by week three, something shifted."
"One student, Marcus, had always said 'I'm bad at math.' After two months of DMT problem solving strategies, he raised his hand during a difficult multi-step problem and said, 'I think we need to decompose this first, then we can compose the parts.' I nearly cried."
Sarah's students showed 47% growth on problem-solving assessments compared to the previous year. But more importantly, they stopped asking "What operation?" and started saying "Let me think about this."
"They're not just solving problems anymore. They're thinking like mathematicians."
What the Research Says
Boaler (2016) found that students taught with conceptual, thinking-based approaches outperformed procedurally-taught students on both conceptual and procedural assessments.
Source: Boaler, J. (2016). Mathematical Mindsets: 10 Strategies to Ignite Student Learning. Jossey-Bass.
Hattie (2009) meta-analysis identified problem-solving teaching strategies as having an effect size of 0.61—well above the average educational intervention (0.40).
Source: Hattie, J. (2009). Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement. Routledge.
NCTM (2014) emphasizes that problem solving should be the vehicle through which all mathematics is taught, not a separate topic to be covered occasionally.
Source: National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. NCTM.
Getting Started: Your First Week
Don't try to implement all six strategies at once. Start small, build momentum:
Week 1 Plan:
-
M
Monday:
Introduce Three-Read Protocol with one problem. Model it explicitly.
-
T
Tuesday:
Students practice Three-Read in pairs. Add Unit Language requirement.
-
W
Wednesday:
Require visual models before any computation.
-
Th
Thursday:
Present one problem with multiple solution pathways. Celebrate different approaches.
-
F
Friday:
Error analysis discussion. Normalize mistakes as learning opportunities.
Key principle: Consistency beats intensity. Five minutes of thinking-based problem solving daily is better than one 45-minute session weekly.
Ready to Transform Your Math Problem Solving Instruction?
The strategies in this post are just the beginning. The Free Foundations Course walks you through all six DMT Framework components with video lessons, classroom examples, and ready-to-use resources.
Join thousands of teachers who've transformed their math instruction. No credit card required.
The Bottom Line
Math problem solving isn't about teaching students to follow steps. It's about building mathematical thinkers who can approach unfamiliar problems with confidence, flexibility, and reasoning.
The DMT Framework gives you the language, the strategies, and the research-backed approach to make this transformation in your classroom. Your students don't need more tricks. They need to think like mathematicians.
Start Monday. Pick one strategy. Watch your students shift from "Just tell me what to do" to "Let me think about this."
That's the moment everything changes.