10 December 2025

What decades of tutoring, classroom teaching, and neuroscience taught me about how students truly learn

After more than 25 years working one-on-one with learners one truth shapes everything I do:

Students don’t struggle with “hard” topics — they struggle with the weak structures beneath them.

From my brief time in a plasticity lab, one principle has stayed with me:
the brain doesn’t learn by stacking rules. It learns by strengthening networks.

When foundational ideas are shaky, anything built on them wobbles.

This is true everywhere in mathematics, but nowhere is it more obvious than in the way students learn exponents and logarithms. 

These topics make a perfect case study — not because they are difficult, but because they expose every structural weakness beneath them.

Many students first encounter exponents as a mysterious shorthand and logarithms as an even more mysterious calculator button. They memorise rules, repeat procedures, and never build the conceptual structures that make the ideas simple.

So the way I teach — not just exponents and logs, but everything — is built around strengthening those deeper structures first.

Rules don’t create understanding. Meaning does.

Students instantly relax when they see that:

• exponents are just repeated multiplication, and 
• a logarithm is simply an exponent — the power that produces a given number.
   

Once meaning is clear, the “rules” stop being rules.
They become common sense consequences.

Most mistakes in exponents and logs aren’t actually exponent or log mistakes —
they’re algebra mistakes resurfacing in a new context.

So I never treat algebra as something students “should already know.”
The brain doesn’t compartmentalise. Knowledge strengthens when it is reused across situations.

Students don’t need to slow down; they need clarity strong enough to support speed.

So I briefly show:

• what exponents actually do, 
• how logarithms undo that action, 
• why each algebraic step is justified.
   

Once the logic is visible, confidence increases — and speed follows naturally because the cognitive load has been decreased.

Misconceptions don’t disappear when corrected once.
They fade with repeated, varied, low-stress exposure.

So my method returns—quietly—to the same core ideas:

• exponents grow by repeated multiplication, 
• logs are exponents, 
• rules work both forwards and backwards, 
• and everything rests on algebraic structure.
   

Not drill.
Reinforcement.

Students don’t notice the repetition; they just feel themselves getting stronger.

Tutors see the exact moment the structure fails:

• the step where hesitation begins, 
• the misconception that appears every year, 
• the algebra habit that quietly undermines everything else.
   

So I teach proactively.
I strengthen the structure before it collapses.

And it works — not just for exponents and logarithms but for every topic that depends on solid conceptual foundations.

Because students deserve clarity, structure, and confidence rooted in real understanding.

The real work is building understanding that lasts — understanding students can rely on when the maths gets harder, not shakier.

That’s the kind of learning I care about. And it’s the standard I hold my teaching to.