Translate between a rule that builds on the previous term and one that jumps directly to term n.
FUNC.RECURSIVE_EXPLICIT“I can convert between recursive and explicit forms of a sequence.”
Given a_1 = 5, a_n = a_(n−1) + 4, write the explicit rule and verify at n = 6.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Algebra II (Grade 10-11) | Sequences and series | intro | introduce | ✓ |
A fillable grid where a recursive rule is dragged down a column to iterate a model.
Recursion and modelling without programming syntax.
engine: editable grid with formula evaluation
Paired columns of inputs and outputs with a difference column exposing the rule.
Constant differences vs constant ratios distinguishes model families.
engine: table widget with computed delta column
Scene archetype: pattern_table
FUNC.REC.NO_SEEDWrites a recursive rule without stating the first term.
Repair: Require the initial condition every time.
One row per instructional system. Rows marked cluster carry authored treatment for this family of topics; rows marked template are derived from the system’s general pattern and are not topic-specific research. Confidence is recorded on every row.
Common Core places recursive and explicit definitions at a specific grade and expects it to be built on a named prerequisite from the year before rather than re-taught from scratch. The standard is usually phrased as understanding plus application, so the expectation is both a correct procedure and an explanation of why it works, developed through a documented progression of representations.
Singapore introduces recursive and explicit definitions concretely with objects the learner can move, then moves to a drawn model — usually a bar or a number bond — that keeps the structure visible, and only then to symbols. The drawn model is retained as the tool for word problems rather than discarded once the algorithm appears, which is why the same bar picture reappears years later for ratio and algebra.
The Russian tradition treats recursive and explicit definitions as structure to be analysed rather than a procedure to be executed. A typical lesson opens with a problem whose condition is schematised — a labelled segment drawing or a quantity table — so the relationships are visible before arithmetic begins, and closes with variations that break any pattern-matching the learner may have adopted. Non-routine and multi-step versions appear early rather than as extension.
Kumon reaches recursive and explicit definitions as a numbered worksheet level rather than a grade, and teaches it by graded repetition: the first pages of the level are barely harder than the last pages of the level before, so the method is inferred rather than explained. There is no manipulative and usually no context — the learner meets the bare computation, repeated until it is both accurate and fast, and repeats the level if the standard completion time is not met.
A Japanese lesson on recursive and explicit definitions usually opens with one problem the class has not been shown how to do, worked independently for a stretch, after which several student approaches are put on the board side by side and ordered from the concrete to the general. The teaching happens in that comparison rather than before it, and the board keeps the whole argument visible so the class can see why the efficient method is efficient.
The classical American approach introduces recursive and explicit definitions as a small increment inside a lesson that also reviews a dozen earlier ideas: the teacher shows a worked example, the learner imitates it, and the topic then reappears in mixed practice for months afterwards. Retention comes from distributed review rather than from a single deep unit, and fact fluency is drilled to automaticity on a timer before the topic is used elsewhere.
Montessori presents recursive and explicit definitions first as a material the child manipulates, chosen so that the structure of the mathematics is physically present in the object rather than explained about it. The child works with that material until the answer becomes predictable, then moves to a deliberately more abstract material representing the same idea, and only reaches written notation when the material has become redundant.
AoPS approaches recursive and explicit definitions by handing the learner a problem that the standard method would solve easily but that they have not yet been given the standard method for, and letting the method be reconstructed from the attempt. The treatment goes materially deeper than a grade-level curriculum, favours a structural argument over a computation, and follows up with non-routine variations chosen to defeat pattern-matching.
The English mastery approach teaches recursive and explicit definitions to the whole class in small steps, using a carefully varied sequence of examples in which one feature changes at a time so the learner can see what the idea does and does not depend on. Concrete and pictorial representations — usually a bar model or a part-whole diagram — sit alongside the symbols rather than before them, and pupils who finish early get deeper problems on the same topic rather than the next one.
IM builds recursive and explicit definitions out of a task students can start with what they already have, then names the mathematics in a whole-class synthesis once several approaches are on the table. Representations are introduced in a planned order within the unit — commonly a diagram, then a table, then the symbolic form — and a short cool-down at the end of the lesson checks whether the idea landed before the sequence moves on.