6QRC™ Structural System Manual: Public Research Edition (Limited Release)

Scope: K-12 Math/numeracy

Intended Audience

Researchers and reviewers evaluating 6QRC. Teams running reliability, validity, or implementation studies.

This document provides the schema‑level definitions of 6QRC™ for research and review. It includes the core operators and raw information elements needed to understand the structure of the system.

The full 6QRC™ Structural System Manual—including the complete decomposition protocol, implementation rules, diagnostic procedures, and operational guidance, is proprietary and available only by agreement with Axiron LLC.

Researchers may request access to the full manual for study use, schema‑level mapping work, or participation in pilot‑site evaluations.

For access inquiries: channing@axironsystems.com

Ultimate Purpose (Learner Courtesy): Make the schema level of a Prompt (mapping to Task) explicit before solving. Disentangle Prompt (surface text) from Task (cognitive event). Specify the Objective and the Cognitive Verbs. Specify the six raw elements: Quantities, Relationships, Conditions, Constraints, Objective, and Representations (Clarification: In 6QRC, “Objective” here is not a learning objective, instructional goal, or outcome statement. It is the Task’s governing completion‑criterion as a raw information element, because it is the mentally represented target the learner must hold. This is a technical construct unique to 6QRC, and distinct from how ‘objective’ is used in curriculum or instructional terminology.). And establish a compact, load-bearing language for math problem structure to tighten teacher and student communication.

Primary use cases:

Structure: In a given Prompt, specify the element set the Task depends on, and what’s not there.
Sequence: Specify the expected verb progression, plus permissible variation.
Communication: Provide a load-bearing language for naming what a problem requires before solving.
Diagnosis: Localize breakdowns to a specific 6QRC element.
Design: Make required elements guaranteed at Task-entry.
Assessment: Eliminate false equivalence: same words ≠ same Task.

Terminology (heads up): Terms are ordered by conceptual dependency, not alphabetically.

Introduction:

6QRC draws a hard boundary between the Prompt and the Task: it supplies a precise vocabulary for describing the cognitive event (Task) a student performs when solving a math problem (Prompt). In this manual, 6QRC is presented as a schema-level analytic framework: a decomposition model for mapping surface Prompts to their underlying cognitive structure and information elements. It targets the cognitive event, not just the words on the page. In many classrooms, the Prompt becomes the default stand-in for a problem’s meaning, because it is the only layer made explicit. 6QRC aims to make the schema level of a Prompt (mapping to Task) explicit and speakable: the information elements that are present (6QRC: Quantities, Relationships, Conditions, Constraints, Objective, and Representations), how these elements interact, and how the governing Objective structures the cognitive event, so that learners can see through surface variation to the invariant and enter a Task with a clear cognitive survey before reasoning, planning, or solving.

At a Glance (core terminology used throughout):

The Five Schema‑Level Operators

Prompt
Task
Tasklet
Objective
Cognitive Verb

The Six 6QRC Elements

Quantities
Relationships
Conditions
Constraints
Objective
Representations

Terminology (for accuracy): In this manual, a component is any defined part of the 6QRC system.

There are two kinds of components:

Schema‑level operators (which govern the cognitive event)
Raw information elements (the information a learner must handle within that event).

In this manual, 6QRC is presented as a set of component definitions (operators and raw information elements): components that govern numeracy Tasks, not as pedagogical moves or instructional strategies, but as an invariant vocabulary and decomposition protocol for mapping the information a Prompt makes available.

6QRC formalizes the schema level of a Prompt: the information elements it carries, and what Cognitive Verbs the learner must execute on those elements to satisfy the Objective.

It does this by defining:

(1) the schema‑level operators that structure a Task and,

(2) the raw information elements those operators act on.

* Worked Example: How 6QRC tells you whether two items are actually the “same problem”.

* Why this Example Exists: In 6QRC, the Objective is the invariant structural element that defines a Task’s identity.

Do not compare difficulty, alignment, or student performance until it is specified what output the learner must produce and what information the Prompt must supply (including representation demands). Expert intuition is valid only when it is rule-governed: the criterion must be stated and consistently applied.

Pair 1: Different wording—same Task.

Prompt A: Solve for x: 2(x + 3) = 18

Prompt B: A number, increased by 3 and then doubled, equals 18. What is the number?

Claim: Prompt A and Prompt B have the same Task Identity because they demand the same output and require the same 6QRC elements.
Objective (invariant): determine the value of the unknown number x such that 2(x + 3) = 18.
Quantities: 2, 3, 18.
Relationships: “doubled after adding 3” corresponds to multiplication distributed over addition: 2(x + 3) = 18.
Conditions: the unknown represents a single number that satisfies the equality.
Constraints: none stated (unless a domain such as integers is specified).
Representations (required/allowed): either an equation form (Prompt A) or a verbal description convertible into an equation (Prompt B). The representational entry point differs, but the representational target is the same.

Likely Cognitive Verbs: (proprietary IP; available in the full manual by agreement)

* Why this matters: In a study, you can treat these as the same type of item only if you can show that they ask for the same output and require the same information. Here, both Prompts ask the student to find the same unknown and rely on the same quantities and relationships. If you do not make that explicit, an item bank can accidentally group together problems that look similar but demand different outputs—or split apart problems that look different but are the same Task.

Pair 2: Same equation—different Task

Prompt C: Solve for x: 2(x + 3) = 18

Prompt D: Solve for x and write a justification that explains why your steps preserve equality: 2(x + 3) = 18

Surface similarity: Prompt C and Prompt D share the same equation and many of the same Quantities/Relationships.
Task Identity check (6QRC): Prompt C and Prompt D do not have the same Task Identity because the required output changes.
Objective (Prompt C): determine x.
Objective (Prompt D): determine x and produce a justification (a second output type).
Representations: Prompt D requires an explanation/argument representation in addition to the algebraic representation; Prompt C does not.
Implication for studies: treating C and D as interchangeable “solve for x” items would confound math accuracy with explanation skill, language production, or argumentation norms. 6QRC prevents this by making the Objective and representation demands explicit before grouping items.

Likely Cognitive Verbs: (proprietary IP; available in the full manual by agreement)

* Why this matters: If you treat Prompt C and Prompt D as the same “solve for x” item, you contaminate every downstream inference: you are no longer measuring math solving alone—you are also measuring justification, language production, and explanation norms. That confound can inflate or depress apparent difficulty, distort alignment claims, and misattribute performance differences to mathematics when the real driver is an added output and representation demand. 6QRC prevents this by forcing you to state (and hold fixed) the required outputs before you compare, group, or interpret results.

By naming these operators and elements, 6QRC establishes a new conceptual layer in mathematics education: the schema that governs the internal logic a Task has, the information it carries, the cognitive event it demands (i.e., the verb sequence), and therefore the meaning the Task actually possesses—anchored by the Objective as governing success condition.

While 6QRC defines the schema itself, its operational environments, 6QRC Element Activation (SEA) and the Numeracy Recovery Notebook (NRN), are where this system becomes fully usable, teachable, and repeatable for classroom uptake across instruction.

With 6QRC made explicit, the next step is to name the schemalevel operators that form the schema’s internal structure. These operators are not instructional moves or prompt-level features; they govern how Tasks obtain their meaning in learner cognition. Each operator carries a specific identity, a set of boundaries, a functional role, and a defined relationship to the others. Together, they form the governing schema that all interdependent layers, Prompts, Tasks, Objectives, Cognitive Verbs, and the 6QRC raw information elements, draw their meaning from.

Each operator is defined through the following components:

Identity Statement: What this operator is.
Boundaries: What it is not.
Function: What it does in the system.
Relationships: How it interacts with other operators.
Examples / Notes (optional): Clarifying only.

Together, these definitions form the system’s schema, the governing structure that makes the interdependent layers nameable and comparable across Tasks.

The Five Schema‑Level Operators

(Schema‑Level Operators)

Numbered 1.–5.

End of Public Research Edition.

The full 6QRC™ Structural System Manual is available by agreement with Axiron LLC.