Why Do Children Solve Problems Before Learning the Formula?

Understanding problem-based mathematics teaching

“If the teacher knows the method, why not just show it first?”

That is a fair question, especially when a child seems unsure how to start. Traditional lessons often begin with the teacher demonstrating a procedure, followed by practice. A problem-based lesson may begin with a situation that children can make sense of using ideas they already know.

The goal is not to make learning unnecessarily difficult. It is to help children understand why a method works, when it is useful and how it connects to other ideas.

AN EXAMPLE: SHARING 18 COOKIES

Suppose a teacher asks: “We have 18 cookies to share equally among 3 children. How many does each child receive?”

Before teaching the formal division symbol, the class might draw three plates and distribute cookies one at a time. Another student may reason that 3 groups of 6 make 18. Another might start with 18 and subtract 3 repeatedly until none remain.

Each approach reveals a relationship among equal sharing, repeated groups, multiplication and division.

When the teacher introduces 18 ÷ 3 = 6, the equation has meaning: it describes a problem students have already represented.

Original Math School diagram illustrating the example above.

PROBLEM-BASED DOES NOT MEAN 'NO INSTRUCTION'

A good lesson has a clear mathematical destination. Teachers select tasks that students can enter, even if they cannot immediately see the whole solution.

They may offer drawings, manipulatives, a simpler related example or a guiding question. They observe which strategies arise, invite students to compare them, and then explicitly consolidate the mathematical idea.

Illustrative Mathematics describes a structure of warm-up, instructional activities, lesson synthesis and cool-down. Within activities, students experience a launch, work time and synthesis. The synthesis is especially important: it is where the teacher helps students connect their discoveries with precise mathematical language and efficient methods.

WHAT DOES 'PRODUCTIVE STRUGGLE' REALLY MEAN?

Some thinking time is useful. A child who tests an idea, revises a diagram or discusses an unexpected result can gain a deeper understanding of the problem.

But struggle is not valuable simply because it is uncomfortable. If a student cannot get started, the teacher should reduce an unnecessary barrier. Support can include clarifying a context, making quantities visible or using a simpler number.

The aim is a challenge that invites reasoning—not prolonged confusion or embarrassment.

WHY TALKING ABOUT MATH MATTERS

Imagine two children solve 18 ÷ 3 and get 6. One draws three groups. The other reasons from 3 × 6 = 18.

When they explain their methods to each other, they begin to see that division can mean sharing and that multiplication and division are related. They can also discover when one strategy is more efficient.

Mathematical talk is not a distraction from learning mathematics. It gives teachers a window into student understanding and gives children practice making their thinking precise.

DO CHILDREN STILL NEED PRACTICE AND FLUENCY?

Absolutely. Understanding and fluency are partners, not rivals.

Children benefit from recalling number facts, using written methods accurately and practicing previously learned ideas. Once a strategy has been understood and made explicit, repeated application helps it become more reliable.

A sensible course blends exploration, direct clarification, guided practice, independent work and cumulative review. The balance changes with the child, the topic and their prior knowledge.

WHAT SHOULD PARENTS LOOK FOR IN A LESSON?

Listen to what your child can explain after class, not only how many exercises they completed. Can they draw a model? Can they say why they chose an operation? Can they check a solution against the story?

A helpful question at home is: “How did you think about that?” Another is: “Could there be a different method?”

If a child struggles, ask them to describe what the numbers represent before rushing to demonstrate a shortcut.

IS THIS APPROACH BETTER FOR EVERY STUDENT IN EVERY SITUATION?

No single teaching method is best for every learner and task. Young children and students with gaps may need additional modeling, explicit teaching and carefully chosen practice. More fluent learners may be ready for increasingly open-ended investigations.

The strength of a problem-based approach lies in deliberate design: the task, support, discussion and synthesis must work together. Illustrative Mathematics grounds its approach in research on learning through meaningful problem solving, discourse, explicit synthesis and coherent progressions; this is not a claim that problem-based lessons automatically outperform every alternative in every setting.

THE TAKEAWAY

When children encounter a meaningful problem before the final formula, they have a chance to build the idea the formula expresses. A knowledgeable teacher then helps transform those early approaches into clear, accurate and efficient mathematics.

At Math School, our goal is to develop students who can calculate, reason and communicate—not merely remember what to write on the next line.

SOURCES

1. Illustrative Mathematics — Principles for Organizing Mathematical Content. https://illustrativemathematics.org/principles-for-organizing-mathematical-content/
2. Illustrative Mathematics — Research. https://illustrativemathematics.org/research/
3. Illustrative Mathematics — Key Structures, K–5. https://accessim.org/k-5/grade-1/course-guide/key-structures?a=teacher&lang=en
4. Illustrative Mathematics — Exploring the Lesson Synthesis. https://illustrativemathematics.blog/2023/03/02/exploring-the-lesson-synthesis-when-do-i-actually-teach/