From “I Understand It” to “I Can Do It”

Why mathematics has to be done, not merely watched

There is a familiar moment in almost every mathematics classroom. A teacher works through an example on the board, carefully explaining each step. The student follows the solution and everything seems perfectly logical.

“Do you understand?”

“Yes.”

Then the student turns to the next question, which is only slightly different, and suddenly does not know where to begin.

What happened?

The student probably did understand the teacher's solution. The problem is that recognising and following someone else's mathematical thinking is not the same as being able to produce that thinking independently.

That distinction is one of the most important aspects of learning mathematics.

Watching mathematics can feel easier than doing it

When we watch someone solve a problem, many of the difficult decisions have already been made for us.

Consider a simple algebra problem:

Solve 3x + 5 = 20.

A teacher might explain:

3x + 5 = 20

Subtract 5 from both sides:

3x = 15

Divide both sides by 3:

x = 5

For a student watching this, every step may make sense. There is nothing particularly mysterious about the solution.

But give the student:

4x − 7 = 21

and something different is required. The student must now decide:

What should I do first?

That decision is part of the mathematics.

When watching a worked example, the student mainly has to recognise why each step is reasonable. When solving a question independently, the student must decide which mathematical ideas to use, in what order to use them and how to carry them out accurately.

That requires a much deeper level of understanding.

Worked examples are important - but they are the beginning

Worked examples are an extremely useful part of learning mathematics. They demonstrate techniques, show mathematical notation and provide students with a model they can follow.

But a worked example should be the starting point, not the finishing point.

A useful learning sequence is:

Understand → Practise → Apply

First, students need to understand the mathematical idea and see how it works.

Next, they need to practise it themselves.

Finally, they need to apply that knowledge to questions where the method may not be immediately obvious.

It is the middle step - practice - that helps turn “I understand it” into “I can do it.”

The example needs to disappear

One simple test of mathematical understanding is to take the worked example away.

Can the student still solve a similar problem?

It is tempting for students to keep referring back to an example while completing an exercise. Initially, that can be useful. However, there comes a point when they need to attempt questions without constantly checking what someone else did.

A student may progress through something like:

Step 1: Follow a worked example.

Step 2: Complete a similar question while referring to the example.

Step 3: Complete another question without looking at the example.

Step 4: Complete a question that looks different but requires the same underlying mathematics.

By the fourth stage, the student is no longer simply copying a procedure. They have to recognise the mathematics for themselves.

Why practice questions should change

Suppose a student has just learnt how to calculate 15% of $80.

If the next ten questions all say “Find 15% of...”, the student can become quite good at following the same procedure.

But what happens when the question becomes:

A jacket normally costs $80 and is reduced by 15%. What is the sale price?

Or:

A jacket is reduced from $80 to $68. What percentage discount has been applied?

The mathematics is related, but now the student has to decide what to do.

This is why good mathematics practice should not consist entirely of identical questions with different numbers. Questions should gradually become more varied and require students to make more decisions for themselves.

Struggling a little can be useful

Students naturally like the feeling of getting questions right. But if every question can be answered immediately by copying the example above it, the student may not be learning as much as they think.

Having to stop and think is not necessarily a sign that something has gone wrong.

Questions such as:

What information have I been given?

What am I trying to find?

Which method could I use?

Have I seen something similar before?

Does my answer make sense?

are all part of doing mathematics.

The aim is not to make mathematics unnecessarily difficult. It is to provide enough support for students to get started while gradually reducing that support as their confidence and ability develop.

Mathematics is a practical skill

There are many things we cannot learn simply by watching someone else.

We would not expect to become a good pianist by watching someone play the piano. We would not expect to become a good swimmer by watching swimming videos.

Mathematics has something in common with these activities.

Explanation is important. Demonstration is important. But eventually the learner has to do it.

They have to pick up the pen, attempt the question, make decisions, make occasional mistakes, correct those mistakes and try again.

Every time students retrieve a method from memory and successfully use it themselves, they strengthen their ability to use that mathematics again.

This is why well-structured practice matters

At NuLake, we believe mathematics resources should do more than present information.

A good mathematics workbook should provide clear explanations and worked examples, followed by carefully structured questions that allow students to practise what they have learnt. As confidence develops, questions can become less routine and require greater independence.

The objective is not simply for students to look at a solution and think:

“Yes, I understand that.”

The real goal is for them to turn the page, face a new problem and be able to say: