Mathematics: Prerequisite diagnosis
Find out that the thing you can't do isn't the thing you can't do, it's something from two years ago that never quite landed.
What you'll be able to do: Find out that the thing you can't do isn't the thing you can't do, it's something from two years ago that never quite landed.
Why mathematics needs this more than any other subject
Mathematics has the deepest dependency chains in the curriculum. Integration by parts rests on the product rule, which rests on the chain rule, which rests on function composition, which rests on understanding what function notation means.
That last one is taught in a week, early, and assumed forever.
So a student who cannot do integration by parts may have a perfectly good grasp of integration by parts and a broken idea of what f(g(x)) denotes. Everything above the break fails unpredictably, and every failure looks like difficulty with the topic on top. They work harder on integration. It cannot help.
This is the subject where "I'm just bad at maths" is most often a prerequisite gap wearing an identity, and the identity is expensive, because it ends study rather than redirecting it.
The chains worth knowing
These recur constantly, and knowing them lets you guess where to look:
| Can't do | Usually broken at |
|---|---|
| Integration by parts | Product rule → chain rule → function composition |
| Solving trig equations | Periodicity → unit circle → what a function's range means |
| Anything with logs | That log is an inverse function → what an inverse is |
| Partial fractions | Polynomial division → factorisation → factor theorem |
| Related rates | Implicit differentiation → chain rule → dependent variables |
| Eigenvalues | Determinants → matrix as a transformation → linear independence |
| Hypothesis testing | Sampling distribution → distribution → random variable |
| Vector problems | Components → trigonometry → similar triangles |
| Series convergence | Limits → what "approaches" means → inequalities |
Notice how many bottom out in function notation, inequalities, or fractions, Those three are the load-bearing floor of school mathematics and are taught early, briefly, and never revisited.
How to run it
I can't do [SPECIFIC THING, not the topic, the move]. Don't teach it to me,
Instead:
1. list what it depends on, and what those depend on, three levels down;
2. give me ONE short diagnostic question for each, starting from the bottom;
3. wait for my answers;
4. tell me the lowest link I failed, that's what I actually need to fix.Two rules that matter more here than elsewhere:
Answer the easy ones honestly. The temptation to skip "what does f(g(x)) mean?" is enormous and the gap is usually there. It survived precisely because it's beneath you to check.
Check below the first failure. Gaps have gaps. A student who fails at the chain rule may be fine with the chain rule and broken at composition.
What a real diagnosis looks like
A student says: "I can't do implicit differentiation."
The chain: implicit differentiation → chain rule → function composition → dependent variables → what it means for y to be a function of x.
The diagnostic questions, bottom up:
- If y depends on x, and x changes, what happens to y?. Fine.
- What does d/dx mean, in words?: "Differentiate with respect to x." Fine, but recited.
- Differentiate y³ with respect to y.:
3y². Fine. - Now differentiate y³ with respect to x.. Stall.
The gap is at 4, and it's not implicit differentiation. It's that y is a function of x and therefore d/dx of anything containing y needs the chain rule. Ten minutes on that, and implicit differentiation becomes routine, because it was never the problem.
Scaling
- School: two levels, run by an adult. Admitting you can't do the easy one is socially costly at this age and needs making safe.
- A-level: run it at the start of every module, not when stuck. The dependency map for the module, tested at the bottom, saves a term.
- Undergraduate: chains get longer and cross into other subjects, analysis rests on logic and set notation that nobody teaches explicitly.
- Returning after a break: the highest-value use. Your gaps will be unusual and specific; general revision will waste weeks on things you retained.
Pitfalls
- Skipping the easy questions. Where the gap almost always is.
- Accepting a recitation. "Differentiate with respect to x" said fluently is not evidence of understanding. Ask for a case.
- Stopping at the first failure. Check one level below it.
- Taking the teaching instead of the diagnosis. If it explains at step 2, you'll feel better and learn nothing about where you're broken.
- The tell: you diagnosed a gap and went back to grinding the top-level topic anyway.
Try this today
Name the one thing in maths you consistently can't do, the specific move, not the chapter.
Ask for its chain three levels down with one diagnostic question per level, bottom-up. Answer the one you're tempted to skip.