Elaborative Interrogation

There's a difference between knowing that something is true and understanding why it's true.

You can memorise that the angles in a triangle add up to 180°. But do you know why? Could you convince someone who didn't believe you?

Elaborative interrogation is the practice of constantly asking "why?" and "how?" — and it transforms surface-level memorisation into genuine understanding.

What Is Elaborative Interrogation?

Elaborative interrogation means asking yourself questions like:

• Why does this work?
• How does this connect to what I already know?
• What would happen if I changed this?
• When would I use this instead of something else?
• Why does this make sense?

Instead of just accepting facts and formulas, you interrogate them. You demand explanations. You make connections.

This does two powerful things:

1. It helps you understand why things work, not just that they work
2. It creates links between new information and what you already know — making both more memorable

Why "Why?" Works

When you ask "why?", you're forced to:

• Think more deeply about the concept
• Connect it to your existing knowledge
• Identify gaps in your understanding
• Create a richer mental model

Research shows that students who ask themselves "why?" while studying remember significantly more than students who just read the same material.

It works because understanding why something is true gives you multiple ways to reconstruct it. If you forget a formula but understand the principle behind it, you can often figure it out again.

The Self-Explanation Effect

A related technique is self-explanation: explaining your reasoning to yourself as you work through problems.

Instead of just doing the steps, you narrate why you're doing each step:

Without self-explanation: "Okay, I'll expand these brackets... then collect terms... then solve for x..."

With self-explanation: "I need to expand these brackets because I can't solve the equation while x is trapped inside them. I'll use FOIL because I have two binomials... now I'll collect like terms on each side because I want all the x terms together... I'm subtracting 3x from both sides because I want x only on the left..."

The second approach takes longer. But it builds much deeper understanding — and when you encounter an unfamiliar problem, that understanding helps you adapt.

Maths Examples: Asking "Why?"

Let's apply this to some GCSE maths content:

Example 1: The Quadratic Formula

You know that $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves $ax^2 + bx + c = 0$.

But why does this work?

The quadratic formula comes from completing the square on the general quadratic equation. If you complete the square on $ax^2 + bx + c = 0$, you eventually arrive at the formula.

Why does this matter?

If you understand this, you can:

• Derive the formula if you forget it
• See why the discriminant ($b^2 - 4ac$) tells you about the number of solutions
• Connect completing the square and the quadratic formula as related techniques

Example 2: Angles in a Triangle

You know angles in a triangle sum to 180°.

But why?

1. Draw a triangle.
2. Extend one side to create a straight line.
3. The angle on the straight line plus the interior angle equals 180° (angles on a straight line).
4. The other two angles of the triangle are equal to the alternate angles formed by a parallel line through the top vertex.
5. So the three angles of the triangle can be "moved" onto the straight line — showing they sum to 180°.

Why does this matter?

This connects triangle angles to parallel lines and angles on a straight line. It shows these aren't separate facts to memorise — they're connected parts of the same system.

Example 3: Dividing Fractions

You know that $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ (flip and multiply).

But why do we flip the second fraction?

Division asks "how many of the second thing fit into the first?" When you flip the fraction, you're finding the reciprocal — the number that multiplies to give 1. Dividing by something is the same as multiplying by its reciprocal.

Think: $6 \div 2 = 3$ and $6 \times \frac{1}{2} = 3$. Dividing by 2 is the same as multiplying by $\frac{1}{2}$.

Why does this matter?

You stop seeing "flip and multiply" as an arbitrary rule. It becomes a logical consequence of what division means.

Example 4: Pythagoras' Theorem

You know $a^2 + b^2 = c^2$ for right-angled triangles.

But why?

Imagine squares built on each side of a right-angled triangle. The area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. There are many visual proofs of this — rearranging the smaller squares to fill the larger one.

Why does this matter?

You understand why it only works for right-angled triangles (the proof relies on the right angle). You can connect it to the distance formula: the distance between $(x_1, y_1)$ and $(x_2, y_2)$ uses Pythagoras on the horizontal and vertical differences.

Making Connections

Elaborative interrogation isn't just about asking "why?" — it's also about asking "how does this connect?"

Look for Patterns

Many maths topics are connected:

Topic A	Topic B	Connection
Gradient of a line	Rate of change	Gradient is rate of change
Pythagoras	Distance formula	Distance formula is Pythagoras in coordinates
Multiplying by 0.5	Dividing by 2	They're the same operation
$x^{-1}$	$\frac{1}{x}$	Negative indices mean reciprocals
Solving equations	Rearranging formulas	Same skills, different context
Probability	Fractions	Probability often is fractions

When you learn something new, ask: "What does this remind me of? What's similar?"

Build a Mental Map

As you revise, try to see how topics link together:

Algebra cluster:

• Expanding brackets → leads to → Factorising (reverse process)
• Factorising → leads to → Solving quadratics by factorising
• Solving quadratics → connects to → Quadratic graphs (x-intercepts are solutions)

Geometry cluster:

• Pythagoras → extends to → Trigonometry (right-angled triangles)
• Trigonometry → extends to → Sine/cosine rule (any triangle)
• All of these → apply to → 3D problems

Understanding these connections helps you:

• Remember topics better (they're not isolated facts)
• Recognise what methods to use (you see the connections to what you know)
• Solve unfamiliar problems (you can draw on related concepts)

The Feynman Technique

Named after physicist Richard Feynman, this technique tests whether you truly understand something:

The Steps

1. Choose a concept

(e.g., "Completing the Square")

2. Explain it as if teaching someone who's never seen it

• Use simple language
• Don't skip steps
• Give an example

3. Identify gaps

Where did you get stuck? What couldn't you explain clearly?

4. Go back and learn

Fill in the gaps, then try explaining again

5. Simplify

Can you explain it even more simply? Can you use an analogy?

Why This Works

If you can't explain something simply, you don't fully understand it. The act of explaining forces you to confront gaps in your knowledge.

You can do this out loud to yourself, write it down, or actually teach someone (a friend, family member, or even your pet — they don't need to understand, you just need to explain).

Practical Elaboration Activities

While Learning a New Topic

• After reading a section, close the book and explain it to yourself
• Ask "why does this make sense?" for every new fact or formula
• Before accepting a method, try to understand why it works

While Doing Practice Problems

• Narrate your reasoning as you work (self-explanation)
• When you use a formula, remind yourself why that formula applies
• After solving, ask "could I have done this differently?"

While Reviewing

• Don't just check if your answer is right — check if your reasoning matches
• When you see a new method in a mark scheme, ask "why does this work?"
• Look for connections: "This is similar to [other topic] because..."

Create "Why?" Flashcards

Add elaboration to your flashcard deck:

Front	Back
Why does $a^0 = 1$?	Following the pattern: $a^3 = a^2 \times a$, $a^2 = a^1 \times a$, $a^1 = a^0 \times a$. For the pattern to work, $a^0$ must be 1.
Why multiply probabilities for "and"?	If A has 3 outcomes and B has 4 outcomes, then "A and B" has $3 \times 4 = 12$ combined outcomes. Probability follows the same multiplication.
How are gradient and rate of change connected?	Gradient IS rate of change. Rise/run = change in y per unit change in x = how fast y changes as x changes.

Common "Why?" Questions for GCSE Maths

Here are some good questions to ask yourself:

Number:

• Why do two negatives make a positive when multiplying?
• Why do you flip the second fraction when dividing fractions?
• Why does the order of operations exist?

Algebra:

• Why do you do the same thing to both sides of an equation?
• Why does completing the square work?
• Why do you need the ± symbol in the quadratic formula?

Geometry:

• Why do angles in a triangle sum to 180°?
• Why does Pythagoras only work for right-angled triangles?
• Why is the tangent perpendicular to the radius?

Statistics:

• Why might the median be better than the mean?
• Why does multiplying probabilities give P(A and B)?
• Why do we square differences when calculating standard deviation?

For each of these, can you give an explanation — not just state the fact?

When Understanding Takes Time

Some "why" questions have deep answers that take time to fully grasp. That's okay.

You don't need to understand everything at a philosophical level. But even partial understanding helps:

• Understanding part of why something works is better than accepting it blindly
• Some understanding now creates hooks for deeper understanding later
• The process of asking "why?" builds the habit of thinking deeply

If you ask "why?" and can't find a satisfying answer, note it down. Ask your teacher. Search online. The question itself is valuable, even if the answer takes time.

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The bottom line: Don't just learn that things are true. Demand to know why. Connect new knowledge to what you already understand. This transforms fragile memorisation into robust understanding.

Related strategies:

• 💪 Desirable Difficulty — Elaboration is effortful (and that's why it works)
• 🧪 Retrieval Practice — Test yourself on the "why" not just the "what"
• 📈 The Science of Learning — Understand why all these strategies work