Problem-Solving with Heuristics Part 2

Jessica Kaminski
May 16
4 min read

As students move into upper elementary and middle school, the problems they encounter become more complex and layered. The strategies required to solve them aren’t just about choosing the right operation. They often require students to reason through multiple steps, organize information, and make sense of unknown. At this stage, students will need more than computation skills. They will need an organized plan of attack that is efficient and appropriate for the specific type of problem.

In Part 1 of this series, we introduced eight foundational heuristics typically taught in Grades K–2, such as Act It Out, Draw a Picture, and Look for Patterns. 🧰 These early strategies lay the groundwork for building strong habits of reasoning and representation. If your students are new to heuristics, or need a refresher, it’s worth revisiting those tools before diving in. Be sure to check out this blog post to watch video samples and learn some teaching tips to help implement them.

In this post, we’ll explore the heuristics introduced in Grades 3–8, where students are expected to apply prior knowledge, select strategies purposefully, and solve increasingly sophisticated word problems. Explicitly teaching heuristics help students analyze structure, plan efficiently, and think flexibly when a straightforward path isn’t available.

🧠 What Are Heuristics?

Heuristics are problem-solving strategies that help students make sense of unfamiliar or multi-step word problems. Instead of relying on a memorized formula or guessing what to do, students learn to analyze the structure of a problem and choose a thoughtful approach. These strategies are especially helpful when there isn’t an obvious path to take to solve the problem. By teaching heuristics, we are adequately preparing students to use logic to solve problems in the real world.

Using Singapore's approach to math, heuristics are grouped into four main categories that guide how students think and respond:

🎨 Give a Representation – Students draw a model, diagram, or visual to organize information and show relationships between quantities.
🔎 Make a Calculated Guess – Strategies like guess and check, trial and error, or working backward promote logical reasoning and estimation based on the conditions of the problem.
📋 Go Through a Process – By looking for patterns, making a systematic list, or using the before-and-after concept, students follow a clear path toward solving.
🧩 Change the Problem – Students simplify, restate, or solve a similar problem to make it more manageable and reveal underlying structure.

Some problems may be solved using just one heuristic, while others might require a combination of strategies. The goal is not just to solve the problem, but to solve it efficiently, logically, and with a clear understanding of why the strategy works. That’s the heart of mathematical thinking.

🔍 Polya’s 4-Step Process

One of the most effective frameworks we can teach students to help them reason through challenging problems is George Polya’s 4-Step Problem-Solving Process. This approach gives students a structure they can rely on, especially when they feel unsure of how to begin.

Here’s a quick reminder of Polya’s four steps (but you can find out more at this blog post):

🔍 Understand the Problem – What do I know? What am I being asked?
🧠 Devise a Plan – What strategy or heuristic could help me solve it?
✏️ Carry Out the Plan – Apply the strategy and work through the steps.
🔁 Look Back and Check – Does my answer make sense? Could I solve it a different way?

This process does more than guide students through solving a problem. It helps them evaluate which heuristic to use and why. As problems become more abstract or multi-step, students who are familiar with Polya’s method are better equipped to pause, reason, and respond thoughtfully rather than rushing to an answer.

Now, let’s dive into the strategies introduced in Grades 3–8 that will continue to build students’ confidence and competence in mathematical problem solving.

🧰 Heuristics to Add to Your Toolbox for Grades 3-8

9 Solve Part of the Problem

This strategy helps students break down complex problems by solving one piece at a time, using each part to move closer to the final answer.

Try This: Invite students to make a list of the steps needed to solve the problem. Encourage them to check off each step as they go to stay organized and build confidence.

10 Use Before–After Concept

This helps students track what changes from one state to another in multi-step or change-related problems. Consider this example from Grade 4 from Primary Mathematics 2022.

11 Make a Supposition

Students assume a value in the problem and test whether it meets the requirements of the problem, adjusting as needed. Check out this Grade 3 example from Primary Mathematics.

12 Tabulating

This strategy helps students organize information in a table, especially when two or more things are changing at the same time. It’s great for keeping track of patterns or combinations clearly and accurately. Try out this problem from Primary Mathematics 2022 Grade 4.

13 Restate the Problem

Putting the problem into their own words helps students clarify what’s being asked and focus on key information. This strategy also encourages them to reframe the question to solve for a different part of the problem first in order to find the unknown value. Consider this 5th grade example from Primary Mathematics 2022.

What's Next?

In this post, we explored the heuristics introduced in Grades 3–8, including strategies like solving part of the problem, using the before–after concept, and making suppositions. These tools build upon the foundation laid in earlier grades and help students approach more complex, non-routine problems with confidence and clarity.

Teaching these heuristics explicitly, through modeling, guided practice, and real-world examples is key to building independent problem solvers who can think flexibly and justify their reasoning. When students are equipped with both Polya’s 4-Step Problem-Solving Method and a toolbox of strategies, they’re prepared to tackle any problem that comes their way.

🎥 Enjoyed the video examples? The Math with Purpose Video Library is your next step! With over 800 step-by-step videos aligned to Singapore-based textbooks, this resource supports teachers, parents, and students alike making it easier to teach complex problem-solving strategies with clarity.