Problem-Solving with Heuristics Part 1

When faced with a complex word problem, many students aren’t sure where to begin. It’s not that they can’t do the math; it’s that they haven’t yet developed a strategy for how to approach the problem. That’s where heuristics come in.

In Singapore's approach to math, heuristics are specific strategies students use to make sense of and solve challenging word problems. They’re not one-size-fits-all tricks, but tools students can select based on the structure of the problem. The goal is for students to evaluate a situation, choose a helpful strategy, and apply it with confidence. This partners well with the mathematical practices in today's standards as students have to determine how to solve a problem.

It's vital that we intentionally teach and model these heuristics so that we’re giving students a problem-solving toolbox they can carry with them into any math situation. While we try to provide them with common examples to solve, we can't possibly expose students to every single problem they will face. It's important that students see non-routine problems and consider which strategies can help them make sense of the problem and solve it. Combined with Polya’s 4-Step Problem-Solving approach, these strategies help students tackle non-routine word problems with clarity and purpose.

🧠 What Are Heuristics?

Heuristics are strategies that guide students through the problem-solving process, especially when the solution isn’t immediately obvious. Rather than relying on a set formula, students learn to think critically and choose an approach based on the structure of the problem. Heuristics help students solve novel or non-routine problems that prepare them for using mathematics in the real world.

Heuristics are typically grouped into four main categories:

1. Give a Representation – Drawing a bar model, diagram, or other visual helps students represent the relationships in a problem and make sense of what’s happening.

2. Make a Calculated Guess – Using strategies like guess and check, trial and error, or working backward encourages logical reasoning based on the information provided.

3. Go Through a Process – Students apply patterns, make systematic lists or tables, or use before-and-after concepts to follow a step-by-step path toward a solution.

4. Change the Problem – By simplifying, restating, or solving a similar problem, students reduce complexity and find a clearer path forward.

Some complex problems may be solved using more than one heuristic, and often the goal is to determine the most efficient strategy. In some cases, a single problem may require multiple heuristics working together; just like real-world problem solving.

Heuristics introduced in Grades K-2

These strategies aren’t introduced all at once. Instead, students begin developing their heuristic toolbox as early as Kindergarten, starting with simple representations and acting out problems. As they move through the grades, new heuristics are added and earlier ones are refined. By Grade 6, students have been exposed to the full range of heuristics and are equipped to select, combine, and apply them with intention.

🔍 Polya’s 4-Step Process

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Before we dive into the heuristics, let’s recall George Polya’s 4-Step Problem-Solving Process and how it's used to help students determine which heuristics to choose. This process encourages students to comprehend the problem and use logic to decide how to proceed. You can find out more information by checking out this blog post here.

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Look back and check

Polya’s framework provides the steps to help students comprehend a problem, while the heuristics provide the strategies students can use within that structure. Let's take a moment to study the heuristics students would learn in Grades K-2 to develop a strong foundation. Note that the remaining heuristics will be introduced in Part 2 for Grades 3-6.

🧰 Heuristics to Add to Your Toolbox for Grades K-2

1. Act It Out

Especially helpful in lower grades, acting out a problem with objects or movement helps kinesthetic learners grasp what’s happening before they represent it on paper. This is the foundation of the concrete-pictorial-abstract approach as students physically represent a problem using math manipulatives or by telling a story.

Try This: Use counters, blocks, or students themselves to act out a story problem.

2. Draw a Picture

A simple drawing or sketch helps students visualize the scenario and begin identifying relationships between quantities. This can be a literal picture of puppies or cats that might be represented in the problem or an array that models the problem. This heuristic introduced in Kindergarten will set the stage for students drawing more complex bar models beginning in Grade 2.

Try This: Ask students to draw what’s happening in the story before choosing an operation.

3. Look for Patterns

Spotting repeated reasoning or numerical patterns helps students predict and generalize in problems involving sequences or relationships. This encourages students to look at the problem as a whole and explore similarities and differences. Check out this example from Grade 1 Primary Mathematics 2022 where I model solving a problem using the strategy of looking for a pattern.

4. Make a Systematic List

Creating an organized list or chart ensures students consider all possibilities without repetition. It helps students to organize information and describe the options easily. Check out this example from Primary Mathematics 2022 Grade 2 where I model using Polya's 4-Step Problem-Solving method to make a list to solve the problem.

5. Guess and Check

While this may not seem like an effective strategy, there are times when the best option is to make a reasonable guess and refine it based on feedback from the problem. This is a great strategy to help students develop logic and consider what next steps to take. Consider this problem from Grade 1 Primary Mathematics 2022. Why do we have to use guessing and checking to solve the problem?

6. Draw a Model/Bar Diagram

Singapore is best known for their bar models that help students organize known and unknown values visually, especially for part-whole or comparison problems. Many imitators try to break this down into a procedure, but truly understanding the heuristic shows that students should be reasoning and considering what should be drawn. Take a look at this example of a Grade 2 problem from Primary Mathematics 2022 where students would just be learning about bar models for this first time.

7. Work Backwards

When the final result is given, students can reverse the steps to figure out what came before using inverse operations. This requires students to change their thinking from "What's the final answer?" to "How did we get to the final answer?" Consider this Grade 2 example from Primary Mathematics 2022 where students use their understanding of operations to solve.

8. Simplify the Problem

Students reduce a complex problem to a simpler version to better understand its structure before scaling it up. This often proves to be a more challenging skill, because it requires students to wade through the information to find the components that would help to solve the problem.

Try This: Ask, “What if we used smaller numbers or fewer people?” "What if we used a different number?"

🔁 Putting It All Together

When we combine Singapore’s heuristics with Polya’s 4-Step Problem-Solving process, we’re giving students two things: structure and strategy. They learn how to approach a problem, and what tools to use along the way. That’s what real problem-solving is about: thinking flexibly, applying understanding, and justifying their process. These 7 heuristics that are introduced in Grades K-2 set the foundation for more complicated problems in the upper grades. Consider how you might challenge students to add tools to their toolbox by solving the problems in the videos above.

✏️ Quick Classroom Tip:

Anchor your problem-solving time with a routine. Post Polya’s 4-Step Problem-Solving process in your room, and develop an anchor chart of the heuristics learned in the classroom. As you model problems, name the heuristic you’re using and tie it to the step in Polya’s process. Over time, your students will begin to do the same.

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What’s Next?

In this post, we explored the foundational heuristics that students begin building in the early grades; strategies like acting it out, drawing pictures, and looking for patterns. These tools lay the groundwork for deeper problem-solving as students progress through elementary school.

In Part 2, we’ll take a closer look at the heuristics introduced in Grades 3–6, including strategies like drawing bar models, making suppositions, and using the before–after concept. You’ll see how these tools build upon earlier skills and support students as they tackle more complex, multi-step problems.

🎥 Loved the video examples? You will love the Math with Purpose Video Library—a growing collection of over 600 videos aligned to Singapore-based textbooks, created by a certified teacher and textbook author (that’s me!). Whether you're a teacher, parent, or school leader, these videos offer clear, step-by-step support for helping students master core concepts and problem-solving strategies.

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