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* Time Estimation Utilities
*
* Provides sophisticated time estimation for practice sessions based on:
* - Historical response times from completed problems
* - Problem complexity (number of terms, digits, skill difficulty)
* - Part type modifiers (abacus vs visualization vs linear)
*
* ## Key Concepts
*
* ### Seconds per Term (SPT)
* The average time it takes to process one term in a problem.
* A problem like "45 + 27 + 13" has 3 terms, so:
* - Total time = SPT × 3 + base overhead
*
* ### Complexity Units
* A more advanced metric that accounts for skill difficulty:
* - Simple direct operations: 1 unit
* - Five complement: 1.5 units
* - Ten complement: 2 units
* - Multi-digit carries: additional units per digit
*
* ### Part Type Modifiers
* Different part types have different time characteristics:
* - Abacus: 1.0x (baseline - student uses physical abacus)
* - Visualization: 1.3x (mental imagery takes longer)
* - Linear: 0.8x (no visual processing, just mental math)
*
* ## Usage
*
* ```typescript
* import {
* calculateSecondsPerTerm,
* estimateProblemTime,
* estimateSessionProblemCount,
* } from '@/lib/curriculum/time-estimation'
*
* // Get avg seconds per term from historical data
* const spt = calculateSecondsPerTerm(sessionResults)
*
* // Estimate time for a specific problem
* const timeMs = estimateProblemTime(problem, spt)
*
* // Estimate how many problems fit in a duration
* const count = estimateSessionProblemCount(durationMinutes, avgTermsPerProblem, spt)
* ```
*/
import type { SessionPart, SlotResult } from '@/db/schema/session-plans'
// ============================================================================
// Constants and Configuration
// ============================================================================
/**
* Default time estimation values (in seconds)
*/
export const TIME_ESTIMATION_DEFAULTS = {
/** Default seconds per term when no historical data is available */
secondsPerTerm: 8,
/** Minimum reasonable seconds per term (prevents unrealistic estimates) */
minSecondsPerTerm: 3,
/** Maximum reasonable seconds per term (caps outliers) */
maxSecondsPerTerm: 30,
/** Base overhead per problem (thinking time, reading, etc.) in seconds */
problemOverheadSeconds: 2,
/** Minimum problems per part to ensure meaningful practice */
minProblemsPerPart: 2,
} as const
/**
* Part type time multipliers
*
* These adjust time estimates based on the cognitive load of each part type:
* - Abacus: Physical manipulation is relatively fast once learned
* - Visualization: Mental imagery requires more processing time
* - Linear: No visual processing, fastest for practiced students
*/
export const PART_TYPE_MULTIPLIERS: Record<SessionPart['type'], number> = {
abacus: 1.0,
visualization: 1.3,
linear: 0.85,
}
/**
* Skill complexity weights
*
* Used to weight time estimates based on which skills a problem exercises.
* Higher weights = more complex = takes longer.
*/
export const SKILL_COMPLEXITY_WEIGHTS: Record<string, number> = {
// Direct operations (simplest)
'add.direct': 1.0,
'sub.direct': 1.0,
// Five complement operations
'add.five': 1.5,
'sub.five': 1.5,
// Ten complement operations (most complex single-digit)
'add.ten': 2.0,
'sub.ten': 2.0,
// Multi-digit operations add complexity
carry: 1.5,
borrow: 1.5,
}
// ============================================================================
// Core Calculation Functions
// ============================================================================
/**
* Calculate average seconds per term from historical session results
*
* This is the core metric for time estimation. By measuring time per term
* rather than per problem, we can more accurately estimate time for problems
* of varying complexity.
*
* @param results - Array of slot results with timing and problem data
* @param options - Configuration options
* @returns Seconds per term, or null if insufficient data
*/
export function calculateSecondsPerTerm(
results: SlotResult[],
options: {
/** Minimum number of results needed for reliable estimate */
minResults?: number
/** Whether to exclude outliers (>3 std dev from mean) */
excludeOutliers?: boolean
} = {}
): number | null {
const { minResults = 5, excludeOutliers = true } = options
// Filter to results with valid timing and problem data
const validResults = results.filter((r) => r.responseTimeMs > 0 && r.problem?.terms?.length > 0)
if (validResults.length < minResults) {
return null
}
// Calculate seconds per term for each result
let sptsRaw = validResults.map((r) => {
const termCount = r.problem.terms.length
const seconds = r.responseTimeMs / 1000
return seconds / termCount
})
// Optionally exclude outliers
if (excludeOutliers && sptsRaw.length >= 10) {
const mean = sptsRaw.reduce((a, b) => a + b, 0) / sptsRaw.length
const stdDev = Math.sqrt(
sptsRaw.reduce((sum, spt) => sum + (spt - mean) ** 2, 0) / sptsRaw.length
)
const threshold = 3 * stdDev
sptsRaw = sptsRaw.filter((spt) => Math.abs(spt - mean) <= threshold)
}
if (sptsRaw.length === 0) {
return null
}
// Return weighted average (more recent results weighted slightly higher)
// For simplicity, we just use arithmetic mean for now
const avgSpt = sptsRaw.reduce((a, b) => a + b, 0) / sptsRaw.length
// Clamp to reasonable bounds
return Math.max(
TIME_ESTIMATION_DEFAULTS.minSecondsPerTerm,
Math.min(TIME_ESTIMATION_DEFAULTS.maxSecondsPerTerm, avgSpt)
)
}
/**
* Calculate average seconds per term from session plan data
*
* Convenience function that extracts results from session plans.
*
* @param sessions - Array of session plans with results
* @returns Seconds per term, or null if insufficient data
*/
export function calculateSecondsPerTermFromSessions(
sessions: Array<{ results: SlotResult[] }>
): number | null {
const allResults = sessions.flatMap((s) => s.results)
return calculateSecondsPerTerm(allResults)
}
/**
* Estimate time in milliseconds for a specific problem
*
* @param problem - Problem with terms array
* @param secondsPerTerm - Average seconds per term
* @param partType - Part type for modifier (optional)
* @returns Estimated time in milliseconds
*/
export function estimateProblemTimeMs(
problem: { terms: number[] },
secondsPerTerm: number,
partType?: SessionPart['type']
): number {
const termCount = problem.terms.length
const baseSeconds = termCount * secondsPerTerm + TIME_ESTIMATION_DEFAULTS.problemOverheadSeconds
const modifier = partType ? PART_TYPE_MULTIPLIERS[partType] : 1.0
return Math.round(baseSeconds * modifier * 1000)
}
/**
* Estimate time in seconds for a specific problem
*
* @param problem - Problem with terms array
* @param secondsPerTerm - Average seconds per term
* @param partType - Part type for modifier (optional)
* @returns Estimated time in seconds
*/
export function estimateProblemTimeSeconds(
problem: { terms: number[] },
secondsPerTerm: number,
partType?: SessionPart['type']
): number {
return estimateProblemTimeMs(problem, secondsPerTerm, partType) / 1000
}
// ============================================================================
// Session Planning Functions
// ============================================================================
/**
* Estimate the number of problems that can fit in a given duration
*
* @param durationMinutes - Target session duration in minutes
* @param avgTermsPerProblem - Average number of terms per problem (default: 3)
* @param secondsPerTerm - Seconds per term (default: from constants)
* @param partType - Part type for modifier (optional)
* @returns Estimated problem count
*/
export function estimateSessionProblemCount(
durationMinutes: number,
avgTermsPerProblem: number = 3,
secondsPerTerm: number = TIME_ESTIMATION_DEFAULTS.secondsPerTerm,
partType?: SessionPart['type']
): number {
const totalSeconds = durationMinutes * 60
const modifier = partType ? PART_TYPE_MULTIPLIERS[partType] : 1.0
const secondsPerProblem =
(avgTermsPerProblem * secondsPerTerm + TIME_ESTIMATION_DEFAULTS.problemOverheadSeconds) *
modifier
const count = Math.floor(totalSeconds / secondsPerProblem)
return Math.max(TIME_ESTIMATION_DEFAULTS.minProblemsPerPart, count)
}
/**
* Estimate total session duration given problem count and complexity
*
* @param problemCount - Number of problems
* @param avgTermsPerProblem - Average terms per problem
* @param secondsPerTerm - Seconds per term
* @returns Estimated duration in minutes
*/
export function estimateSessionDurationMinutes(
problemCount: number,
avgTermsPerProblem: number = 3,
secondsPerTerm: number = TIME_ESTIMATION_DEFAULTS.secondsPerTerm
): number {
const secondsPerProblem =
avgTermsPerProblem * secondsPerTerm + TIME_ESTIMATION_DEFAULTS.problemOverheadSeconds
return Math.round((problemCount * secondsPerProblem) / 60)
}
/**
* Convert seconds per term to approximate seconds per problem
*
* Useful for UI display and backwards compatibility with existing code
* that expects "seconds per problem" as input.
*
* @param secondsPerTerm - Seconds per term
* @param avgTermsPerProblem - Average terms per problem (default: 3)
* @returns Approximate seconds per problem
*/
export function convertSptToSecondsPerProblem(
secondsPerTerm: number,
avgTermsPerProblem = 3
): number {
return Math.round(
avgTermsPerProblem * secondsPerTerm + TIME_ESTIMATION_DEFAULTS.problemOverheadSeconds
)
}
/**
* Convert seconds per problem to approximate seconds per term
*
* Useful for converting legacy data or user input to the new model.
*
* @param secondsPerProblem - Seconds per problem
* @param avgTermsPerProblem - Average terms per problem (default: 3)
* @returns Approximate seconds per term
*/
export function convertSecondsPerProblemToSpt(
secondsPerProblem: number,
avgTermsPerProblem = 3
): number {
const netTime = Math.max(0, secondsPerProblem - TIME_ESTIMATION_DEFAULTS.problemOverheadSeconds)
return netTime / avgTermsPerProblem
}
// ============================================================================
// Advanced: Complexity-Weighted Estimation
// ============================================================================
/**
* Calculate complexity units for a problem based on skills required
*
* This provides a more nuanced estimate than just counting terms.
*
* @param skillsRequired - Array of skill IDs the problem exercises
* @returns Total complexity units
*/
export function calculateProblemComplexityUnits(skillsRequired: string[]): number {
if (skillsRequired.length === 0) return 1 // Minimum complexity
let totalComplexity = 0
for (const skill of skillsRequired) {
// Check for exact match first
if (SKILL_COMPLEXITY_WEIGHTS[skill]) {
totalComplexity += SKILL_COMPLEXITY_WEIGHTS[skill]
continue
}
// Check for partial match (e.g., "add.+5.direct" matches "add.direct")
const parts = skill.split('.')
const operation = parts[0] // add, sub
const technique = parts[parts.length - 1] // direct, five, ten
const partialKey = `${operation}.${technique}`
if (SKILL_COMPLEXITY_WEIGHTS[partialKey]) {
totalComplexity += SKILL_COMPLEXITY_WEIGHTS[partialKey]
} else {
// Default complexity for unknown skills
totalComplexity += 1.0
}
}
return totalComplexity
}
/**
* Calculate seconds per complexity unit from historical results
*
* This is the most sophisticated estimation method, accounting for
* both problem structure and skill difficulty.
*
* @param results - Array of slot results
* @returns Seconds per complexity unit, or null if insufficient data
*/
export function calculateSecondsPerComplexityUnit(results: SlotResult[]): number | null {
const validResults = results.filter(
(r) => r.responseTimeMs > 0 && r.problem?.skillsRequired?.length > 0
)
if (validResults.length < 10) {
return null
}
const ratios = validResults.map((r) => {
const complexity = calculateProblemComplexityUnits(r.problem.skillsRequired)
const seconds = r.responseTimeMs / 1000
return seconds / complexity
})
const avg = ratios.reduce((a, b) => a + b, 0) / ratios.length
return Math.max(2, Math.min(15, avg)) // Clamp to reasonable bounds
}
// ============================================================================
// Utility Types
// ============================================================================
/**
* Time estimation summary for a student
*/
export interface TimeEstimationProfile {
/** Seconds per term (primary metric) */
secondsPerTerm: number
/** Seconds per complexity unit (advanced metric, if available) */
secondsPerComplexityUnit: number | null
/** Equivalent seconds per problem for UI display */
secondsPerProblem: number
/** Number of results used to calculate these estimates */
sampleSize: number
/** Whether these are default values (no historical data) */
isDefault: boolean
}
/**
* Get a complete time estimation profile for a student
*
* @param results - Historical slot results
* @returns Time estimation profile
*/
export function getTimeEstimationProfile(results: SlotResult[]): TimeEstimationProfile {
const spt = calculateSecondsPerTerm(results)
const spcu = calculateSecondsPerComplexityUnit(results)
if (spt === null) {
return {
secondsPerTerm: TIME_ESTIMATION_DEFAULTS.secondsPerTerm,
secondsPerComplexityUnit: null,
secondsPerProblem: convertSptToSecondsPerProblem(TIME_ESTIMATION_DEFAULTS.secondsPerTerm),
sampleSize: 0,
isDefault: true,
}
}
return {
secondsPerTerm: spt,
secondsPerComplexityUnit: spcu,
secondsPerProblem: convertSptToSecondsPerProblem(spt),
sampleSize: results.length,
isDefault: false,
}
}
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