Why Problem-Solving Matters
A middle school class is tackling the problem, “A car travels 180 miles in 3 hours. At the same rate, how far will it travel in 5 hours?” Hands shoot up with different approaches. Some students divide 180 by 3 to get 60 miles per hour, then multiply by 5 to find 300. Others reason proportionally, “If 3 hours is 180, then 1 hour is 60, so 5 hours must be 300.” A few sketch a table or graph to show the relationship.
Every strategy works. And that’s the point. Problem-solving is where mathematics comes alive. It’s not about memorizing one “right way,” but about reasoning, exploring, and justifying. When students engage in problem-solving, they take ownership of their learning. They build perseverance, creativity, and confidence, qualities that connect mathematical thinking to real-life experiences well beyond the classroom.
What the Research Tells Us
Decades of research confirm that problem-solving isn’t just an outcome of learning math, it’s a driver of it (Kilpatrick, Swafford & Findell, 2001). Students who regularly tackle meaningful problems build deeper understanding, transfer skills to new situations, and begin to see themselves as capable mathematicians.
John Hattie’s Visible Learning research shows problem-solving approaches with effect sizes of 0.61–0.68, well above the average for accelerating achievement (Hattie, 2009; Hattie, 2023). He calls this the Goldilocks Principle: tasks that are “just right,” not too easy (or boring), not too hard, create the most considerable growth.
Professional bodies like the National Council of Teachers of Mathematics (NCTM) agree problem-solving should be both a process and a goal of instruction (NCTM, 2000; NCTM, 2014). In other words, it’s not the “extra challenge,” it’s the heartbeat of mathematics.
Frameworks like SOLO Taxonomy and Webb’s Depth of Knowledge show how problem-solving deepens learning. Students move from recalling facts (surface), to connecting and justifying ideas (relational), to applying knowledge flexibly in new contexts (extended) (Biggs & Collis, 1982; Webb, 1997; Marzano & Kendall, 2007).
Building a Problem-Solving Culture and Making Problem-Solving Central
If problem-solving is central, classrooms look different:
- Tasks are meaningful. Problems require reasoning, not just plugging into a formula.
- Multiple strategies are valued. Students compare approaches and debate efficiency.
- Struggle is normal. Mistakes are part of the learning process, not something to avoid (Bjork & Bjork, 2011; Kapur, 2016).
- Justification is expected. It’s not enough to answer, students explain why it works.
- Connections are made. Problems link math to real-world contexts students care about.
Example: Instead of assigning “Solve 2x + 3 = 11,” ask students to “Write three different equations that have a solution of x = 4. How are they similar? How are they different?” Suddenly, students aren’t just solving, they’re thinking like mathematicians. Discussing their reasoning with peers deepens understanding and supports transfer to new contexts
Practical Classroom Moves
1. Rich, Open-Ended Tasks
Instead of single-path exercises, use problems that allow for multiple strategies and solutions.
Example: “Design three rectangles with an area of 24 square units. What patterns do you notice?”
Some students list integer dimensions (3 × 8, 4 × 6, 2 × 12).
Others consider non-integer dimensions (1.5 × 16).
Discussion reveals the inverse relationship between length and width and lays the groundwork for area/perimeter reasoning.
2. Problem-Solving Routines
Use routines that structure thinking and invite all students into the problem.
Present a graph of a water tank filling over time. Before asking a formal question, prompt: “What do you notice? What do you wonder?” Students might respond, “It fills quickly at first,” or “I wonder how many gallons it holds.” Their observations and questions frame the lesson.
Three-Act Tasks (Dan Meyer style): Begin with a short, curious video or image (Act 1). Students brainstorm questions and estimate (Act 2). Finally, they calculate and justify solutions (Act 3). For example: show a video of popcorn spilling over a container and ask, “How many scoops will it take to fill the tub?”
3. Encourage Multiple Approaches
Highlight diverse strategies and compare their efficiency.
Example: Solve 36 ÷ 6.
- Student A: Uses repeated subtraction.
- Student B: Thinks multiplicatively (6 × ? = 36).
- Student C: Draws groups of 6.
Discussing these approaches highlights efficiency, connections, and flexibility—core aspects of problem-solving.
4. Scaffold Perseverance
Break big problems into manageable phases to motivate persistence.
Example: For a ratio problem:
- Step 1: Identify what is known (e.g., 2 cups of flour for 3 cups of sugar).
- Step 2: Brainstorm possible strategies (tables, scaling, equations).
- Step 3: Attempt a solution.
- Step 4: Compare and reflect.
5. Reflection and Metacognition
Close tasks by asking students to reflect on their strategies.
Prompts include:
- “Which strategy made the most sense to you and why?”
- “What mistake actually helped you understand more clearly?”
6. Use Non-Routine Problems
Offer problems where the pathway is not immediately obvious or incorporate AI to generate alternative approaches for comparison.
Example: “A farmer has 24 fence posts and wants to build a rectangular pen. What shapes could they make? Which has the largest area?”
Teacher and Student Voices
“At first, my students resisted open-ended tasks, they just wanted me to hand them the formula. But as we continued, I saw them begin to take risks, debate strategies, and even devise methods I hadn’t taught. That’s when I knew math had become real for them.”
– Year 8 Teacher
Students notice the difference too: “When we solve problems, I feel like I’m actually doing math, not just memorizing. Even if I get stuck, I know I can try another way.”
Why This Works
Cognitive science shows that struggle, when supported, strengthens memory and understanding (Bjork & Bjork, 2011; Kapur, 2016). Problem-solving engages both concepts and procedures, weaving them into flexible knowledge that transfers.
It also builds what researchers call a productive disposition—the belief that mathematics is sensible, useful, and doable (Kilpatrick, Swafford & Findell, 2001). Students who see themselves as problem-solvers are more likely to persist, take risks, and apply their learning in new contexts.
Example: A student who once relied only on the formula for percent now reasons about discounts, tips, and ratios in everyday life because they understand the relationships—not just the steps.
Reflection for Teachers
Think back to your last math lesson. Were students applying a set of steps you provided, or were they reasoning, making choices, and justifying their work?
What’s one way you could reframe an exercise into a problem-solving opportunity this week? Could you turn a routine computation into an open-ended task? Could you add a justify your reasoning prompt?
Join the Conversation
This blog is part of an ongoing conversation among educators. I’d love to hear from you:
- Share your reflections and stories using #MathSuccessJourney.
- Tag @CengageSchool, @BigIdeasLearning, and @SophieSpecjal
- Join the mid-series webinar for interactive Q&A and teacher spotlights.
- Contribute classroom examples for future spotlights by contacting me at [email protected]
Your experiences bring these strategies to life. Together, we can build a community where students develop fluency that goes beyond memorization, giving them the tools to reason and apply knowledge with confidence.
Looking Ahead
Mathematical modeling transforms math from abstract exercises into a tool for understanding and improving the world. When students connect mathematics to real-life problems, they discover its power to create change and develop the confidence to think critically and creatively.
Next blog: Strategy #6: Mathematical Modeling.
References
Biggs, J. & Collis, K. (1982). Evaluating the Quality of Learning: The SOLO Taxonomy. New York: Academic Press.
Bjork, R. A. & Bjork, E. L. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning. Psychology and the Real World, 2, pp. 56-64.
Hattie, J. (2009). Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement. London: Routledge.
- Hattie, J. (2023). Visible Learning: The Sequel. London: Routledge.
Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), pp. 289-299.
- Kilpatrick, J., Swafford, J. & Findell, B. (eds.) (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press.
- Marzano, R. J. & Kendall, J. S. (2007). The New Taxonomy of Educational Objectives. 2nd ed. Thousand Oaks, CA: Corwin Press.
- National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
- National Council of Teachers of Mathematics (NCTM). (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.
- Webb, N. L. (1997). Criteria for Alignment of Expectations and Assessments in Mathematics and Science Education. Washington, DC: Council of Chief State School Officers.