The Questions That Showed Up This Week — And What They Really Reveal
Every Sunday, I go back through the week's sessions and look for patterns. Not just "what topics came up" but how students were thinking about them — where the logic broke down, where confidence collapsed, where one small reframe completely changed someone's understanding. This week was a particularly good one for that. Students from the UK, UAE, Singapore, Australia, Ireland, and the US all came in with questions that, on the surface, looked different — but underneath, shared the same root cause.
So here's a proper breakdown of what came up this week across subjects, with the kind of detail I'd give you if we were sitting together at a table working through it. Whether you're following Cambridge, Edexcel, IB, or AP, there's almost certainly something here that's relevant to you.
Mathematics: The Calculus Problem That Isn't Really About Calculus
What keeps going wrong with differentiation from first principles
I had three separate students this week — one on Cambridge A-Level, one on IB HL, one doing AP Calculus — all stuck on the same type of question. Differentiation from first principles. And every single one of them made the same mistake. Not an arithmetic error. A conceptual one.
They were expanding (x + h)² and getting x² + h². No middle term. Just... gone.
Here's the thing though — this isn't really a calculus error. It's an algebra error wearing a calculus disguise. The first principles formula f'(x) = lim(h→0) [f(x+h) - f(x)] / h only works if you expand f(x+h) correctly. If that step collapses, everything after it is wrong, and students often don't catch it because they're focused on the limit concept, not the expansion.
What I always tell my students: before you touch any calculus problem involving first principles, write out your expansion on a separate line and check it independently. Treat it like a sub-calculation. (x + h)² = x² + 2xh + h² — that middle term 2xh is the entire reason the derivative of x² is 2x. It's not a detail. It's the answer.
The worked example that clicked for one of my students this week: we took f(x) = x³ and I asked her to just expand (x + h)³ first, nothing else. She got it right immediately. Then we plugged it into the formula, subtracted f(x), and the x³ terms cancelled cleanly, leaving 3x²h + 3xh² + h³ in the numerator. Divide by h — you get 3x² + 3xh + h². As h approaches zero, the last two terms vanish. Derivative: 3x². Suddenly it wasn't mysterious anymore.
Common pitfall: rushing the algebra to get to "the calculus bit"
Students — especially strong students — often feel like the algebra is beneath them. They speed through it to get to what they perceive as the real work. This is backwards. In calculus, the algebra IS the real work. The limit is often the easy part.
Statistics and Further Maths: When Probability Gets Philosophical
A Further Maths student (Edexcel, Year 13) came to me genuinely frustrated this week. She understood conditional probability mechanically — she could use the formula P(A|B) = P(A∩B) / P(B) — but she kept getting tree diagram questions wrong and couldn't figure out why.
We went through her working together and I spotted it immediately. She was applying the correct formula to the wrong branch. In a two-stage probability tree, she was conditioning on the first event when the question was asking her to condition on the second. That's a reading problem, not a maths problem.
My approach here: read the conditional probability question and physically point to which event is given before you write anything. The phrase "given that" is your anchor. Whatever comes after "given that" — that's your B. Everything before it is your A. Force yourself to label them before you calculate anything.
If you're working through questions like these independently, try our free study tools in the Thula learning centre — there are structured probability walkthroughs that break down exactly this kind of reading comprehension issue in statistics problems.
Chemistry: Organic Reactions and the "Why" Nobody Teaches
Nucleophilic substitution — understanding mechanism, not memorising it
A-Level and IB Chemistry students often treat organic mechanisms like vocabulary lists. They memorise that a nucleophile attacks a carbon, that a leaving group leaves, that the arrow goes this way. But ask them why a nucleophile attacks that specific carbon, and you get silence.
Here's the non-obvious insight: organic chemistry mechanisms are all electron accounting. Electrons move from areas of high density to areas of low density. If you understand that, you don't need to memorise much. You can reason your way through most mechanisms.
In SN2 reactions, the nucleophile attacks the carbon bonded to the leaving group because that carbon is electron-deficient — it's bonded to something electronegative. That makes it an electrophilic site. The nucleophile, being electron-rich, is attracted to it. The leaving group departs with the electrons from the C–X bond because it's stable as an anion (iodide, bromide, etc.). When I explained it this way to a student this week — framing it as electron logic rather than a memorised sequence — she said "oh, so it's basically just electrons going where they want to go?" Exactly. That's exactly what it is.
Economics: The Diagram Students Draw Wrong Every Single Time
I genuinely cannot count how many times I've seen this. A student draws a perfect supply and demand diagram, labels everything correctly, then explains price elasticity completely backwards in their written answer.
This week a Cambridge A-Level Economics student made a claim in his essay that a price increase would lead to a proportionally larger fall in demand — and then labelled this as inelastic demand. His diagram was right. His words were wrong. And in an exam, the words are what get marked.
The fix that works: stop using the words "elastic" and "inelastic" as if they're arbitrary labels, and instead think about what they literally mean. Elastic = stretchy = quantity responds a lot to price changes. Inelastic = stiff = quantity barely responds. If you're struggling with microeconomics or macroeconomics concepts like elasticity, monopoly pricing, or fiscal policy, working with a dedicated economics tutor through one of our tutoring plans can make an enormous difference — not just for understanding, but for writing technically accurate answers under exam pressure.
English Literature: The Essay Feedback Nobody Likes to Hear
Why "analysing language" isn't the same as making an argument
I read three practice essays this week from English Literature students — one IB, two A-Level — and all three had the same structural problem. Every paragraph identified a technique, quoted it, and explained what it meant. That's PEE (Point, Evidence, Explain). It works at GCSE. At A-Level and IB, it's not enough.
What examiners want — and I've read enough mark schemes to say this with confidence — is an argument that uses textual evidence, not a textual tour that mentions an argument at the start. There's a difference. A tour says: "Shakespeare uses metaphor here. The metaphor is X. This shows Y." An argument says: "The corruption of nature imagery throughout Act II signals something more disturbing than guilt — it suggests Macbeth's ambition has made the world itself complicit." Then you bring in the evidence to prove that claim.
One concrete technique: write your topic sentence last. Draft the paragraph, find the most interesting idea that emerged, then write a topic sentence that makes that idea a claim. You'll be surprised how much sharper your paragraphs become when you stop forcing evidence to fit a pre-written point.
Computer Science and Programming: The Logic Error That Wastes Hours
A student doing AP Computer Science came in with a debugging problem that had cost him two hours. His loop wasn't terminating. We looked at it together and found the issue in about ninety seconds — an off-by-one error in the loop condition. He'd written < when he needed <=.
Off-by-one errors are so common in programming that experienced developers have a name for them (OBOE — Off By One Error). But here's the broader lesson: when a loop doesn't behave as expected, the first thing you check is the condition. Not the body. Not the variables inside. The condition. Is it using strict inequality or inclusive inequality? Does it start at 0 or 1? Is it checking against length or length-minus-one?
Build yourself a checklist for debugging loops specifically. It sounds over-engineered, but after you've wasted two hours on the same class of error twice, a ten-second checklist feels like a bargain.
One Final Thought on This Week
What struck me most looking back across all these sessions is how rarely the problem is what it appears to be. The calculus student had an algebra problem. The statistics student had a reading problem. The economics student had a vocabulary problem. The English student had a structural problem. The programmer had a logic problem hiding in a single character.
Good tutoring — the kind we try to do at Thula Academy whether you're in London, Dubai, Melbourne, or anywhere else — isn't about explaining topics. It's about finding the actual gap between what a student knows and what they're demonstrating, and building a precise bridge across it. That's harder than it sounds, and it's different for every person.
If any of what's above resonated with something you've been stuck on, it might be worth working through it properly rather than hoping it resolves itself before the exam. You can start your free 21-day trial and see what targeted, subject-specific tutoring actually feels like. No generics. No one-size-fits-all approaches. Just the kind of session where something finally clicks.
See you next Sunday.
Originally published on Thula Academy — Smart online tutoring for Cambridge, Edexcel, IB, AP & University students. Start your 21-day free trial →
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