Instructional Systems

Fewer topics per year, studied more deeply, with a documented progression from concrete work to general structure. Coherence across grades is the organising principle: each grade is written to be built on by the next.

Note: The public licence permits copying and displaying the standards but not revising or recasting them. Store codes and links; never present a paraphrase as the standard text. Algebra I lands in Grade 8 or 9 depending on the district pathway; the atlas assumes Grade 9 for the core track and Grade 8 for the accelerated track.

A small number of topics per year taken to mastery, sequenced concrete to pictorial to abstract, with the bar model as a single visual language that carries a learner from Grade 2 addition all the way to ratio and algebra.

Note: The MOE syllabus documents are personal/non-commercial use only. The bar-model *method* is an unprotectable pedagogical idea and may be implemented freely; specific published wording, artwork, and problem sets may not be copied.

Mathematics as a structured deductive system taught early. Problems, not exercises, are the unit of work: a lesson is organised around analysing the structure of a situation before any arithmetic happens, and non-routine problems are normal rather than enrichment.

Note: Covers both the school tradition and diaspora programs such as RSM. Verified 2026-08-30 against the TIMSS 2015 Encyclopedia (authoritative for the aggregate Classes 1-4 / 5-9 / 10-11 structure) and peer-reviewed sources on program families: Davydov is a genuine scope-and-sequence outlier, deriving "number" from continuous-quantity measurement before counting, from Class 1 (peer-reviewed, e.g. PMC7711087); Peterson ("School 2000") is meaningfully accelerated toward early algebraic reasoning relative to the traditional Moro baseline; Zankov keeps Moro-like sequencing but removes drill in favour of harder problems introduced earlier. No primary document gives a clean grade-by-grade comparison across all four families, so program-family placement stays qualitative rather than grade-exact, and RSM (the US diaspora franchise) is lower confidence still — its Parent Handbook was not machine-readable. Do not copy RSM notebooks, artwork, or problem sets; describe pedagogy only.

Automaticity through small, self-explanatory increments. Placement is by demonstrated speed and accuracy rather than by age, and students usually start below their school grade so that the work is always comfortably executable without instruction.

Note: Level content/ordering verified 2026-08-30 as MEDIUM confidence — five independent secondary sources converge closely, and Kumon's own franchise materials exist (e.g. official Levels E/F/G handout pages were located) but returned as unparseable binary/image PDFs rather than clean text, so no primary document could be fully confirmed. Grade-equivalence numbers stay LOW-to-medium confidence: Kumon deliberately does not publish a level-to-grade table, using KIS instead precisely to decouple level from school grade — the ASHR bronze/silver/gold/platinum tiers ("years ahead of grade") are the closest official proxy. Worksheets and level names are proprietary — never reproduce them.

One carefully chosen problem per lesson. Students attempt it before being taught a method, several student strategies are then compared publicly, and the lesson ends by consolidating toward the most general or efficient of them.

Note: MEXT Course of Study English translations have no clear open licence; treat as reference. The structured-problem-solving format is a described practice, not a protected asset.

Incremental development with continuous distributed review. A new idea is introduced in a small increment each lesson, and every problem set mixes that increment with a long tail of previously learned material so that nothing is ever finished and forgotten.

Note: Saxon is a proprietary commercial program: describe the method only. Ray's Arithmetic and other pre-1931 US texts are public domain and may be adapted freely (with OCR cleanup and unit modernisation).

Abstraction is earned through the hand. Each concept begins as a physical material whose structure embodies the mathematics, and the materials are deliberately sequenced so each is a slightly more abstract version of the last, until the material is no longer needed.

Note: The materials and their sequence are long-published public pedagogy. Specific modern publishers' albums and illustrations are not.

Discovery before algorithm, at a deliberately high ceiling. Problems are chosen so that a learner who has only the previous tools can still make progress, and the general method is extracted from the struggle rather than supplied in advance.

Note: Wholly proprietary. Describe the approach; never reproduce problems, characters, or artwork.

Whole-class teaching for mastery: the class moves through small steps together, with variation designed so that what changes between examples isolates exactly the idea being learned.

Note: The National Curriculum is Crown copyright under the Open Government Licence v3.0, which permits commercial reuse with attribution — the most permissive of the national standards documents mapped here.

Problem-based instruction built on a documented mathematical progression: students work on a purposeful task, a whole-class synthesis names the mathematics that emerged, and a short cool-down checks it before the next lesson.

Note: The 2019-2021 FIRST EDITION is CC BY 4.0 and may be adapted commercially with attribution; the 2024 v.360 edition is CC BY-NC and must not be ingested. Pin the edition on every artifact. IM trademarks are excluded from the licence — strip all branding.