The Structural Grammar of K–12 Mathematical Tasks

Math isn’t the barrier. Invisibility is.
6QRC clears the path.

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

Problem Structure is Finite and Coachable

Every K–12 math problem can be fully stabilized using six information types and navigated using five repeatable cognitive actions. 

Intended Readers

This document is written for K–12 education professionals who work directly with mathematical tasks and their structure: teachers, instructional leaders, curriculum designers, interventionists, researchers, assessment developers, and tool builders.

Why 6QRC Exists

6QRC exists to make the internal structure of K–12 math tasks visible and coachable. When students hesitate at the start of a problem, the issue is rarely intelligence. It’s that the task's structure is invisible. 6QRC closes that gap by giving teachers and students a unified structural language for reading a math task before strategy selection or solving begins.

• Make the Task visible: expose the exact information elements a prompt contains (and doesn’t), so students know what they’re entering.
• Make the thinking predictable: reveal the required cognitive action sequence and permissible variations.
• Name the requirement precisely: give teachers and students a shared structural language for what the prompt demands prior to strategy selection.
• Locate breakdowns accurately: diagnose errors at the element level instead of attributing them to “carelessness” or "just not a math kid."
• Guarantee alignment: design and assess tasks by structure, eliminating false equivalence between prompts that look similar but demand different work.

Ultimate Purpose (Learner Courtesy) 

6QRC exists to make math feel human... by giving students a clear lens for seeing the full structure of any K–12 math problem before solving. And to establish a shared, load‑bearing language between every teacher and student, nationwide.

WHAT 6QRC IS 

A Cognitive Action System for Math Problem Structure

Every math problem has a hidden structure: a small set of information types that determine what kind of thinking the student must do. 

Until now, teachers have been coaching students through problems with no unified language of problem 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, not an invented one. It reveals the six information elements present in every K–12 math problem, across every grade and domain, and binds each element to the cognitive action it triggers. The result is a shared, load‑bearing language for problem structure, and a reliable way to act on it. 

The Six Information 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’s allowed 
• Objective — the directive: what the problem is asking for 
• Representations — the forms (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 

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

• CAPTURE — noticing and recording what is explicitly stated 
• INTERPRET — making meaning from semantic or situational information 
• ITEMIZE — organizing information into trackable groups 
• ASSEMBLE / TRANSLATE — building or converting representations 
• SOLVE — construct visible reasoning 

Each information element binds to a specific action: 

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

This binding is stable. It does not vary by grade, domain, or teacher preference. 

The Sequencing Principle 

The actions themselves are finite.
What changes is their firing order, and that order is determined by the element you’re handling.

Quantities / Objective:
CAPTURE → ITEMIZE

Relationships / Conditions / Constraints:
INTERPRET → ITEMIZE

Representations:
TRANSLATE → ITEMIZE (ASSEMBLE → ITEMIZE in classroom practice)

When students use the wrong sequence, their reasoning misfires.
When they use the right one, the structure snaps into place.

WHY 6QRC MATTERS

For Students 

6QRC gives students a stable lens for seeing the full structure of any math problem. 

 Before solving, they can name: 

• what they can capture 
• what they must interpret 
• what they need to itemize 
• what they must assemble 
• what they are ultimately solving for 

This reduces Cognitive Action Uncertainty and increases precision. 

For Teachers 

6QRC creates a shared, load-bearing language between teacher and learner. 

 Instead of vague coaching (“read it again,” “slow down,” “think carefully”), teachers can coach the exact structure of the task.

For Curriculum Designers and Assessment Writers 

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

For Researchers and Reviewers 

6QRC offers a substrate‑level model that can be independently validated through rater agreement and 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 CAPTURE and what they must INTERPRET. 
3. Have students ITEMIZE the captured/interpreted information. 
4. Require a visible representation: students ASSEMBLE a diagram, table, or equation. 
5. Use the representation to SOLVE, then diagnose breakdowns by element. 

This gives students a transferable structure for navigating any problem.

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: 

• If you’re a parent or student: treat 6QRC like a “what is this problem made of?” checklist before you compute. 
• If you teach: start with the Six Elements, then jump to “How Educators Can Use 6QRC Immediately.” 
• If you tutor: focus on element-by-element diagnosis and the action sequence (CAPTURE → INTERPRET → ITEMIZE → ASSEMBLE/TRANSLATE → SOLVE). 
• If you design curriculum or assessments: use 6QRC to audit clarity, cognitive demand, and unintended ambiguity. 
• If you research learning: treat 6QRC as a candidate labeling scheme for prompt structure and student moves. 

You’ll see technical language in places (because precision matters), but the goal is everyday usability: a way for adults and students to point to the same features of a task and say, “Ah, this is the kind of thinking this problem needs.” 

Closing 

6QRC is a cognitive‑action map for K–12 math problems, a way to make the invisible structure of numeracy tasks visible, nameable, and coachable. 

It gives students stability.
It gives teachers precision.
It gives the field a shared language.

And it opens the door to a more coherent, less mysterious experience of mathematics.

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. 

6QRC’s Central Schema, System Manuals, and operational demonstrations make up the system’s operational core. These materials are shared with implementing schools and districts, as well as research partners, under agreement.