What Is a Math Problem in Education?
A math problem is more than a question with numbers. It’s a situation that requires students to make sense of relationships, choose a strategy, and explain their thinking.
In early math, students may see problems like 8 + 5 or 12 − 7. But the real work is not just calculating an answer. It’s understanding what the numbers represent and how they relate to each other.
As students progress, problems become more complex. They may involve multiple steps, real-world scenarios, or unfamiliar formats. At every level, though, the core idea stays the same. A math problem asks students to think, not just compute.
This is where many learners get stuck. If math is taught as a set of steps to memorize, students struggle when those steps no longer apply. Empower Math Coach Kristy Oliveira explains what happens when students rely too much on memorization. “Memorization can help in some areas, but when it becomes the main approach, students often struggle the moment a problem looks different. They may know a rule but not know when or why to use it.” Students will freeze when presented with word problems or unfamiliar questions.
- Over-reliance on memorization
Students may know facts but struggle to apply them in new situations. - Teaching only one method
When students are shown a single “correct” way, they may not develop flexible thinking. - Skipping visual and hands-on learning
Without concrete experiences, math becomes abstract too quickly. - Prioritizing speed over understanding
This can increase anxiety and discourage deeper thinking. - Avoiding struggle
When students are not given time to think through challenges, they miss opportunities to build resilience.
These patterns can lead to frustration and disengagement. Over time, students may begin to believe they are “not good at math,” when in reality they have not been given the right tools.
What's the biggest misconception teachers have about math problem solving?
One of the biggest misconceptions teachers have about math problem solving is that it’s about applying the correct procedure quickly and accurately. That sounds reasonable—but it misses how we actually learn math.
In reality, effective problem solving is much closer to sense-making than rule-following. When instruction overemphasizes procedures, a few things tend to happen:
- Students learn to look for clues that signal a formula, not to understand the situation.
- They believe every problem has a single “right method,” instead of multiple valid approaches.
- They struggle when a problem is even slightly unfamiliar because they have not built flexible thinking.
A more accurate view is that problem solving involves:
- Interpreting the situation (What’s really going on here?)
- Trying strategies, even imperfect ones
- Making mistakes and revising thinking
- Explaining reasoning, not just getting answers
Why Students Struggle With Math Problem Solving (and What Helps)
For years, math instruction has often emphasized speed. Timed tests, quick answers, and memorized facts have been treated as signs of success. The problem is that speed does not equal understanding.
Students who rely on memorization may perform well on familiar problems. But when they face something new, they often freeze. Without strategies, they don’t know how to begin.
Strong problem solvers approach math differently by:
- Looking for patterns and relationships
- Breaking problems into manageable parts
- Trying multiple approaches
- Checking whether their answer makes sense
- Learning from mistakes instead of avoiding them
This kind of thinking builds confidence. Students begin to trust their ability to figure things out, even when the problem is unfamiliar. Over time, this leads to real fluency. Not fast recall under pressure, but flexible, reliable understanding.
The Concrete-Representational-Abstract (CRA) Progression in Math
One of the most important elements of effective math instruction is the progression from concrete to abstract thinking.
Concrete → Representational → Abstract
At the concrete stage, students use physical objects. These might include counters, blocks, or everyday items. This stage helps them build meaning. Numbers are not just symbols. They represent real quantities. Laura Moore, a seasoned math coach, explains why starting with concrete is useful for students. “Concrete experiences create mental models they can rely on in new situations.” Without the concrete stage, students fall apart when things change.
Next comes the representational stage, where students move to drawings and visual models. They might use number lines, ten frames, arrays, or bar models. These tools help students see the math and make connections.
At the abstract stage, students work with numbers and symbols alone, without the support of models or objects. This is where traditional math instruction often begins, but it should not be the starting point. Without earlier experiences to build meaning, symbols can feel disconnected and confusing.
Coach Moore explains why it’s important not to skip the first two stages. “If you introduce symbols too early, they often learn to manipulate them without understanding what they represent. That’s when math starts feeling like memorizing mysterious rules.” They may be able to follow procedures, but they often lack a deep understanding of why those procedures work.
When students move through all three stages, they develop stronger number sense. They can move between models and symbols, which gives them more flexibility when solving problems.
Math Problem Solving Strategies: Quick Reference
| Strategy | What It Helps Students Do | When to Use It | Example |
| Count On | Start from a known number instead of counting everything | Early addition with small numbers | 5 + 3 → start at 5: 6, 7, 8 |
| Make Ten | Turn numbers into a friendly 10 to simplify thinking | When numbers are close to 10 | 8 + 5 → (8 + 2) + 3 = 13 |
| Doubles / Near Doubles | Use known doubles to solve similar problems | When numbers are the same or close | 6 + 7 → 6 + 6 + 1 = 13 |
| Decompose Numbers | Break numbers into parts to make them easier to work with | Larger numbers or multi-digit operations | 47 + 25 → (40 + 20) + (7 + 5) |
| Use Visual Models | Represent thinking with drawings or tools | When students need to “see” the math | Number lines, arrays, bar models |
| Explain Thinking | Clarify reasoning and learn from others | During discussion or reflection | Student explains steps aloud or in writing |
Give students multiple ways to approach problems so they can choose the most effective strategy based on the numbers.
6 Math Problem-Solving Strategies Students Can Use
Effective problem solving is built on a set of flexible strategies. These are not tricks or shortcuts. They are ways of thinking about numbers that make math more manageable and meaningful.
1. Count On Instead of Count All
Early learners often count every object from the beginning. For example, to solve 5 + 3, they might count: 1, 2, 3, 4, 5, 6, 7, 8.
Counting on is more efficient. The student starts with the larger number and counts up: 6, 7, 8.
This shift may seem small, but it represents a big step in understanding. Students begin to see numbers as starting points, not just totals. Watch this YouTube video to see the strategy in action.
2. Make Ten
Making ten is one of the most powerful early math strategies. Students use combinations that add to ten to simplify problems.
For example:
8 + 5 becomes (8 + 2) + 3 = 10 + 3 = 13
This strategy works because ten is a benchmark number in our base-ten system. It reduces the mental effort required to solve problems.
Students who are comfortable with combinations of ten often develop stronger mental math skills overall. This video provides a good visual of the strategy.
3. Use Doubles and Near Doubles
Doubles facts, like 6 + 6 or 7 + 7, are often easier for students to remember. Near doubles build on this knowledge.
For example:
6 + 7 can be thought of as 6 + 6 + 1 = 13
This strategy helps students use what they already know. Instead of memorizing every possible fact, they rely on relationships between numbers.
4. Decompose Numbers
Decomposing, or breaking numbers apart, allows students to work with smaller, more manageable pieces.
For example:
47 + 25 can be broken into (40 + 20) + (7 + 5)
This reinforces place value understanding. Students see that numbers are made up of tens and ones, not just single units.
It also supports mental math. Large problems become easier when they are broken into parts.
5. Use Visual Models
Visual models help students represent their thinking. Common models include number lines, arrays, area models, and bar diagrams.
These tools make abstract ideas more concrete. They allow students to track their thinking and check their work.
For example, a number line can show how addition involves movement forward, while subtraction involves movement backward.
Visuals are especially helpful for students who struggle with traditional methods. They provide another way to access the math.
6. Explain Thinking Out Loud
One of the most overlooked strategies is also one of the most effective. Students should be encouraged to explain how they solved a problem.
This can happen through discussion, partner work, or written explanations.
When students explain their thinking, they:
- Clarify their understanding
- Identify gaps or errors
- Learn from others’ strategies
- Build confidence in their reasoning
Classrooms that prioritize math talk tend to produce stronger problem solvers.
- Focus on Reasoning: Effective problem solving requires sense-making and flexible thinking, not just quick, procedural computation.1
- Prioritize Understanding: True math fluency is built on comprehension and strategy choice, not on speed or rote memorization.1
- Use the CRA Progression: Anchor meaning by moving students through Concrete, Representational, and Abstract stages to prevent fragile knowledge.1
- Teach Multiple Strategies: Encourage flexibility by modeling approaches like Make Ten, Decomposing Numbers, and using visual models.1
- Encourage Explanation: Students clarify their understanding, identify errors, and build confidence by explaining their thinking aloud or in writing.
Frequently Asked Questions
Yes, problem solving can significantly improve math confidence. When students learn strategies, they feel more capable of figuring things out on their own. Success becomes tied to understanding rather than speed. This shift reduces anxiety and increases engagement over time.
Visual models help students represent abstract ideas in a more concrete way. They make it easier to see relationships between numbers and track problem-solving steps. Tools like number lines and arrays support both accuracy and understanding. Over time, students rely less on visuals as their mental math skills improve.
The most effective approach is to teach multiple strategies and give students opportunities to explore them. Teachers should use visual models, hands-on learning, and discussion to support understanding. Moving from concrete to abstract helps students build a strong foundation. Consistent practice with reasoning is more valuable than repetition alone.
Many students struggle because they are taught to focus on speed and memorization instead of understanding. When they encounter a new type of problem, they may not know how to adapt. Without a toolkit of strategies, math can feel rigid and confusing. Building number sense helps students approach problems more effectively.
Math problem solving strategies are methods students use to understand and solve math situations. These include approaches like making ten, using doubles, drawing models, or breaking numbers apart. The goal is to help students think flexibly rather than rely on memorized steps. Strong strategies allow students to approach unfamiliar problems with confidence.