Sequencing, spiraling, and pacing

Order concepts so each one lands on ground the learner already holds — then keep returning to it, and know what to defend when time runs short.

The idea

Three planning moves decide whether a unit sticks. Sequencing: teach a concept only after its prerequisites are secure. Spiraling: revisit earlier ideas on a spaced schedule instead of teaching-then-abandoning. Pacing triage: when the calendar loses to snow days and assemblies, choose deliberately what to compress, what to cut, and what to defend.

Below, three small tools — one per move. They share the same six-concept unit, so you can feel how a choice in one ripples into the others.

How it works

Each move is a small, concrete discipline:

SEQUENCE   build a prerequisite graph, then teach in an order where
           every topic comes after the topics it depends on.
           Graphing lines needs Solving equations AND Ratios first.

SPIRAL     don't teach-and-drop. Re-touch each concept on a spaced,
           expanding schedule. Retrieval practice (recall, not reread)
           resets forgetting and stretches how long it lasts:

             blocked (no review) :  90% -> ... -> ~16% by week 6
             spiraled (wk 2 & 4) :  90% -> ... -> ~74% by week 6

PACE       when days vanish, triage by the dependency graph:
             DEFEND   foundations many topics depend on
             COMPRESS robust topics that survive a lighter touch
             CUT      leaves nothing else depends on (safest to lose)

The through-line: a curriculum is a graph, not a list. Sequencing respects its edges, spiraling keeps its nodes alive in memory, and pacing triage protects the nodes with the most edges leaving them.

When to use it

Move	Reach for it when…	The trade-off
Sequence by prerequisite	Topics genuinely build (math, languages, most skills).	Some topics have several valid orders — the graph constrains, it doesn’t dictate one path.
Spiral with spaced retrieval	Retention past the unit test matters — which is almost always.	Review time competes with new content; you must budget it in, not bolt it on.
Pacing triage	The calendar slips and something has to give.	Every cut has a downstream cost; the skill is choosing the cheapest one.

Watch out for

• Teaching in textbook order without checking prerequisites. A chapter sequence isn’t a dependency graph. Verify each topic’s foundations are actually secured first.
• Cramming review into the last week. Massed review right before the test feels productive and fades fast. Spacing beats blocking for durable memory.
• Confusing re-reading with retrieval. Looking over old notes feels easy but does little. The gain comes from making learners recall — a quick do-now, a mixed problem set.
• Cutting a foundation to save time. Dropping the topic everything leans on is the most expensive cut, not the cheapest. Cut leaves, defend roots.
• Compressing the hardest concept. The topic learners struggle with is usually the one to defend; compress the resilient ones instead.

Worked example

An interviewer asks: “You planned twelve days for this unit and just lost three to testing and a snow day. Walk me through your cuts.” A weak answer trims the last few topics because they’re “what we didn’t get to.” A strong answer reaches for the dependency graph: “I’d defend Number line and Variables — five other topics rest on them. I’d compress Ratios and Equations, which survive a tighter pass, and if I still needed a day I’d cut Graphing, since nothing downstream depends on it. And I’d protect the spaced-review do-nows — losing those costs more in June than the extra day costs in March.” That answer names the roots to defend, the resilient middle to compress, and the safe leaf to cut — and it treats review as content, not slack.