Modali K–12
by LiquidVisuals
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Mathematical & Clinical Derivation Architecture

Psychometric Methodology & Strategy Derivation

How 100 raw diagnostic indicators translate into UDL Strategies, Accommodations, Gap Analyses, and Visual Profile Graphs.

Explore the mathematical formulas, clinical decision trees, and Universal Design for Learning (UDL) engines that power Modali’s automated learner profiling.

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Diagnostic Visualization Engine

1. How the Visual Profile Graphs are Computed & Plotted

Modali generates two mathematically rigorous visual charts to diagnose student cognitive-sensory balance and isolate outlier strengths and support needs.

Multidimensional Radar Chart
Scale: 1.00 – 5.00
Mathematical Formula:
\[S_k = \frac{1}{|I_k|} \sum_{i \in I_k} x_i\]

Where \(x_i \in \{1, 2, 3, 4, 5\}\) is the Likert rating and \(I_k\) is the set of items mapped to subscale \(k\).

How It Is Plotted & Used:
  • 13 Radial Axes: Evaluates VIS, AUD, KIN, VER, LOG, SPA, APP, INTER, INTRA, ENV, GSTI, SMMR, and STSR.
  • Purple Solid Polygon: Plots the student's actual subscale scores (\(S_k\)). A bulge outward signifies sensory dominance or cognitive assets.
  • Dashed Teal Benchmark (\(\mu_k\)): Plots the normative cohort mean. Comparing the purple polygon against the dashed line immediately identifies whether a student is performing above, at, or below baseline across each axis.
Cohort Standardized Z-Scores
\(Z_k \in [-2.5\sigma, +2.5\sigma]\)
Mathematical Formula:
\[Z_k = \frac{S_k - \mu_k}{\sigma_k}\]

Where \(\mu_k\) is the cohort benchmark mean and \(\sigma_k\) is the population standard deviation.

Threshold Color-Coding & Interpretation:
  • 🟢 Green Bars (\(Z_k \ge +0.85\sigma\)): Dominant Strengths (\(\alpha\)) — Statistically significant assets to anchor learning strategies.
  • ⚪ Slate Bars (\(-0.84\sigma \le Z_k \le +0.84\sigma\)): Universal Tier 1 — Normative development covered by core instruction.
  • 🔴 Red Bars (\(Z_k \le -0.85\sigma\)): Support Need Areas (\(\delta\)) — Statistically acute vulnerabilities triggering Tier 2/3 scaffolds.
Clinical Translation Engine

2. How Accommodations & UDL Strategies Are Derived

Rather than generic advice, Modali maps the student’s dominant sensory intake (\(VIS, AUD, KIN\)), cognitive typology, and \(Z_k\) outlier deviations directly to evidence-based Universal Design for Learning (UDL) guidelines.

UDL Principles (1, 2 & 3)

Derivation Logic: Triggered by the student's dominant sensory modality and cognitive reasoning strengths (\(\alpha\)):

  • Representation (Principle 1): High \(VIS\) triggers dual-coded graphic organizers and visual mind maps. High \(AUD\) triggers oral lecture recordings and discussion.
  • Action & Expression (Principle 2): High \(KIN\) or \(APP\) triggers tactile lab modeling, project-based demonstrations, and authentic artifacts instead of pure written exams.
  • Engagement (Principle 3): High \(INTER\) triggers collaborative peer learning; high \(INTRA\) triggers independent study contracts.

Targeted Accommodations

Derivation Logic: Automatically prescribed when support need areas (\(Z_k \le -0.85\sigma\)) or executive vulnerabilities are detected:

  • Low \(VER\) (\(Z_{VER} \le -0.85\sigma\)): Provides sentence frames, paragraph structural templates, and guided rubrics.
  • Low \(GSTI\) or \(SMMR\): Visual step checklists, chunked assignments, and structured start cues.
  • High \(ENV\) Sensitivity: Reduced ambient clutter, preferred quiet seating, and dimmed lighting options.

Assistive Technologies

Derivation Logic: Matched to compensate for specific mechanical or perceptual processing bottlenecks:

  • Speech-to-Text / Dictation: When handwriting stamina or fine-motor mechanics (\(Pillar\ 3\)) lag behind verbal intelligence.
  • Text-to-Speech & Screen Masking: When reading decoding speed (\(Pillar\ 2\)) creates a barrier to academic content.
  • Visual Timers (Time Timer): For executive pacing and task completion pacing (\(GSTI\)).
Psychometric Gap Analysis Rule Engine

3. How Gap Analysis & Instructional Trajectory Are Derived

A critical innovation in Modali is distinguishing between English Language Acquisition Barriers (ELL) and true Cognitive Processing Deficits.

Linguistic vs. Cognitive Discrepancy Rule

In traditional testing, English Language Learners (ELL) or students with oral communication delays are frequently misclassified with learning disabilities because test instructions demand high linguistic proficiency.

Modali Gap Analysis Diagnostic Formula:
\[\Delta_{\text{Lang-Cog}} = \frac{S_{LOG} + S_{SPA} + S_{APP}}{3} - \frac{S_{VER} + S_{REC} + S_{EXP}}{3}\]

If \(\Delta_{\text{Lang-Cog}} \ge 1.20\), Modali automatically diagnoses an English Language Acquisition Barrier rather than a cognitive deficit, instructing the teacher to provide visual dual-coding rather than remedial math/logic.

Tiered Intervention Index (TII) Trajectory

The student’s overall instructional placement is computed via the Tiered Intervention Index (\(TII\)), formulated by combining executive function, self-monitoring, and strategy receptivity:

Tier Level Determination:
\[TII = 5.00 - \frac{S_{GSTI} + S_{SMMR} + S_{STSR}}{3}\]
\(TII \le 2.00\)
Tier 1 Universal
\(2.00 \lt TII \le 3.00\)
Tier 2 Targeted
\(TII \gt 3.00\)
Tier 3 Intensive

Experience the Diagnostic Protocol Firsthand

Take the 100-indicator assessment to generate your personalized Multidimensional Radar, Standardized Z-Score Graph, and Appendix B Profile.

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