The Structural Grammar of Mathematical Tasks

A Pre-Solve Grammar for Problem Structure

© 2026 Channing Cornell Powers and Axiron LLC. All rights reserved.
6QRC™ and Universal 11™ are protected intellectual property.

Scope: Mathematical Tasks (all solvable problem-types)

Claim Statement

6QRC introduces the layer mathematics leaves unformalized: the human layer. It brings context, intention, interpretation, and human understanding into the mathematical act, before symbolic manipulation begins. It stabilizes the learner’s cognitive frame: eliminating the human reasoning failures that destabilize traditional math learning.

6QRC is not a pedagogy.
It is a schema: a formal layer that stabilizes the human entry into any math situation.

Intended Readers

This document is written for education professionals who work directly with mathematical tasks: teachers, instructional leaders, curriculum designers, interventionists, researchers, assessment developers, and tool builders.

Problem Structure is Finite and Coachable

WHY 6QRC EXISTS

WHAT 6QRC IS 

Part 0: The Structural Layer

Every math problem has a hidden structure: a small set of information types (expanded in the element section below) that determine what kind of thinking action is required from the student. 

Until now, teachers have been coaching students through problems with no unified language for that structure.  

Teachers coach on the fly.
Students take a swing and hope it lands.
Both sides work harder than they should.

6QRC changes that. 

6QRC is a discovered system. It reveals the six information elements present in every solvable math problem, across every grade and domain, and binds each element to the action it requires. The result is a shared, load‑bearing language for problem structure, and a reliable way to act on it. 

The Six Information Elements 

Part 1: The Six Structural Elements

Every math problem contains these six elements: 

• Quantities — the measurable or countable values 
• Relationships — how things compare, connect, or change 
• Conditions — the situational facts that define what is true 
• Constraints — the limits or boundaries that restrict what is allowed 
• Objective — the directive, what the problem is asking for 
• Representations — the forms such as tables, diagrams, equations, that express structure 

These elements are not optional.
They are inherent properties of mathematical tasks.
Each element activates a specific cognitive posture in the learner.

The Bound Cognitive Actions 

Part 2: The Cognitive Verbs (Cognimits)

6QRC identifies five cognitive actions that arise directly from the elemental properties of the information: 

• CAPTURE / EXTRACT — notice and record what is explicitly stated 
• INTERPRET — make meaning from semantic or situational information 
• ITEMIZE — organize information into trackable groups 
• ASSEMBLE / TRANSLATE — build or convert representations 
• SOLVE — construct visible reasoning 

Each information element binds to the following cognitive action it requires:

• Quantities → EXTRACT → INTERPRET → ITEMIZE
• Objective → EXTRACT → INTERPRET → ITEMIZE
• Relationships → EXTRACT → INTERPRET → ITEMIZE
• Conditions → EXTRACT → INTERPRET → ITEMIZE
• Constraints → EXTRACT → INTERPRET → ITEMIZE
• Representations → TRANSLATE (cognitively) / ASSEMBLE (instructionally) 
  

Teachers can guide students to adopt the posture each element requires:

• the extractor for what is stated
• the interpreter for what carries meaning
• the itemizer for what must be organized
• the assembler for what must be built

*Universal 11 = The parent system

*6QRC = Math descendant

*The six elements + the five Cognimits = the complete pre‑solve reasoning system powering 6QRC (Universal 11).

This introduction is not the system manual. Large‑scale implementation requires the 6QRC Central Schema and System Manual to avoid structural errors.

WHY 6QRC MATTERS

For Students

6QRC gives students a stable way to see the structure of any math problem.
Before solving, they can name:

• what to extract
• what to interpret
• what to itemize
• what they are solving for
• what they can assemble to reach it

This reduces cognitive-action uncertainty and increases precision.

For Teachers

6QRC creates a structurally precise language between teacher and student.
Instead of vague coaching such as “read it again,” “slow down,” or “think carefully,” teachers can coach the exact structure of the task.

For Curriculum Designers and Assessment Writers

6QRC provides a rubric for prompt demand and structural consistency.

For Researchers and Reviewers

6QRC offers a substrate‑level model that can be validated through rater agreement and the predictive accuracy of student moves.

This introduction is not the system manual. Large‑scale implementation requires the 6QRC Central Schema and System Manual to avoid structural errors.

HOW EDUCATORS CAN TRY 6QRC NOW

A simple five‑step entry point: 

1. Annotate a task using the six elements. 
2. Ask students to identify what they can EXTRACT and what they must INTERPRET. 
3. Have students ITEMIZE the extracted and interpreted information. 
4. Require a visible representation: students ASSEMBLE a diagram, table, or equation. 
5. Use that representation to SOLVE, then diagnose breakdowns by element. 

This gives students a transferable structure for navigating any problem.

This introduction is not the system manual. Large‑scale implementation requires the 6QRC Central Schema and System Manual to avoid structural errors.

HOW TO USE THIS DOCUMENT

Different readers will use 6QRC differently. To make this usable across classrooms, homes, tutoring sessions, and research settings, this introduction is written in layers:

Parents and Students
Use 6QRC as a “What is this problem made of” checklist before choosing a formula or computing.

Teachers
Start with the Six Elements, then move to “How Educators Can Try 6QRC Now.”

Tutors
Focus on element‑by‑element diagnosis and the action sequence
EXTRACT → INTERPRET → ITEMIZE → ASSEMBLE → SOLVE.

Designers
Use 6QRC to audit prompt demand and unintended ambiguity.

Researchers
Treat 6QRC as a candidate labeling scheme for problem structure and student moves.

You will see technical language in places because precision matters. Classroom implementation becomes straightforward only after terminology is exact. The goal is everyday usability: adults and students should be able to point to the same features of a prompt and say, “This is the kind of thinking this problem needs.”

This introduction is not the system manual. Large‑scale implementation requires the 6QRC Central Schema and System Manual to avoid structural errors.

About Access to System Materials 

Axiron makes the 6QRC introduction and system overview public, and shares implementation materials under agreement. 

This document is public. Implementation materials are shared under agreement.