The role of working memory in cardinality judgement

The role of working memory in cardinality judgement, ordinality judgement and calculation Helene Vos (KU Leuven), Wim Gevers (ULB) & Bert Reynvoet (KU Leuven)

Numerical symbol processing Calculation

Underlying numerical symbol processing • Domain-specific factors • Domain-general factors o o Cardinality Ordinality o Working Memory

Cardinality • Number of elements in a set (e. g. a football team consists of eleven football players)

Cardinality • Numerical Comparison Task: Standard Distance Effect (Ansari, 2008)

Ordinality • Position of an item in a sequence (the football team is ranked seventeenth in the leage and is ranked after the sixteenth and before the eighteenth)

Ordinality • Ordinal judgement task: Reversed distance effect (Lyons & Ansari, 2015)

Working memory models • Structural Models o Baddeley’s Model (Baddeley & Hitch, 1974) • Functional Models o Oberauer’s model (Oberauer, 2002)

Baddeley’s working memory model (CE) (VSSP) (PL)

Oberauer’s working memory model • Activated part of long-term memory • Region of direct access • Focus of attention LTM Direct access Focus of Attention

Oberauer’s working memory model Focus of attention Active row: Direct Access Passive row: Long-term memory

Calculation • Retrieval is the dominant strategy for multiplication (96%) • Subtraction relies more on alternate strategies such as counting and inversion and less on retrieval (only 58%) (Campbell & Xue, 2001)

Research questions 1. What is the relation between cardinality and ordinality and calculation? 2. Is the relation between cardinality/ordinality and calculation still there when controlled for working memory?

Methods Individual session • Working memory tasks o Baddeley’s model: Digit Span & Corsi Blocks Ø Digit Span Forward: Phonological loop Ø Digit Span Backward: Phonological loop* Central Excutive Ø Corsi Blocks Forward: Visuo-spatial sketchpad Ø Corsi Blocks Backward: Visuo-spatial sketchpad * Central Executive o Oberauer’s model: Memory Updating task Ø Active row: Direct acces Ø Passive row: Long-term memory

Methods Collective session • Basic number processing tasks o o Cardinality: Numerical comparison task Ordinality: Ordinal judgement Task • Calculation tasks o o o Single Digit Multiplication Multiple Digit Subtraction Tempo Test Arithmetic

Methods • 89 participants o o Mean age= 19, 80 (SD= 5, 48) ♀ N= 71 ♂ N=18 • 21 participants removed o o 6 missing data: -3 no-shows in one of the sessions -3 participants one of the tasks logged incorrectly 15 outliers: performance >3 SD above group mean on one of the experimental tasks

Basic number processing tasks • Numerical comparison task Combined ACC/RT-score (RT/ACC) o Distance: distance 1 and distance 3 o 64 trials

Basic number processing tasks • Ordinal judgment task Combined ACC/RT-score (RT/ACC) o 84 trials Small Distance Large Distance Ascending (e. g. 123) (e. g. 159) Descending (e. g. 321) (e. g. 951) Non-order (e. g. 312) (e. g. 195) (Lyons & Beilock, 2009; Rubinsten, 2013)

Working memory tasks Baddeley • Digit Span ACC o o Forward: range span: 2 -9 Backward: range span 2 -8 Two items per span Task is terminated when 2 items of one span are recalled incorrectly WAIS-IV-NL (Wechsler, 2008)

Working memory tasks Baddeley • Corsi Blocks ACC o o Forward: range span 2 -9 Backward: range span 2 -8 Two items per span Task is terminated when 2 items of one span are recalled incorrectly (Kessels et al. 2000; 2008)

Working Memory tasks Oberauer • Memory Updating Median RT • 24 trials • 2 to 6 values (Oberauer, 2002)

Calculation tasks • Multiple Digit Subtraction Combined ACC/RT-score (RT/ACC) o 96 trials (16 trials in each condition) o Verification Task o Borrowing & Non-borrowing Borrowing (e. g. 32 -15) Non-borrowing (e. g. 69 -31) 17 38 Incorrect 27 48 Incorrect 13 32 Correct (Linsen, Verschaffel, Reynvoet & De Smedt, 2014)

Calculation tasks • Single Digit Multiplication Combined ACC/RT-score (RT/ACC) o 96 trials (32 trials in each conditions) o Verification task o Table related and table unrelated errors e. g. 7*7 Correct 49 Incorrect Related 56 Incorrect Unrelated 48 (Romero, Rickard & Bourne, 2006)

TTR • Time-limited test: One minute for each column ACC (De Vos, 1992) • Five columns of forty items o o o Addition Substraction Multiplication Division Mixed

Processing speed task • Control Measure: General processing speed • Participant has to react as fast as possible to an appearing square in the middle of the screen (Reigosa-Crespo et al. , 2012)

Analyses • Correlations o o o Basic number processing skills (Cardinality & Ordinality) Calculation tasks (Subtraction, Multiplication, TTR) Controlling for Processing Speed • Regression o o o Independent variables: Calculation tasks (Subtraction, Multiplication, TTR) Predictors: Working memory, Cardinality and Ordinality Controlling for Processing Speed

Results • Partial Correlations controlling for processing speed 1 2 3 4 Cardinality 1 Ordinality . 406** 1 Subtraction . 383** . 626** 1 Multiplication . 448** . 511** . 642** 1 TTR -. 369** -. 463** -. 544** -. 751** *p<. 05; **p <. 01 5 1

Results • Independent variable: Subtraction Model Standardized β t p R² ΔR 1. Processing Speed . 083 . 673 . 504 0. 007 2. Processing Speed . 037 . 290 . 773 0. 062 0. 055 Digit Span Forward -. 223 -1. 637 . 107 Digit Span Backward -. 017 -. 122 . 903 Corsi Blocks Forward -. 048 -. 388 . 699 Corsi Blocks Backward -. 027 -. 217 . 829 3. Processing Speed . 015 . 148 . 883 0. 446** 0. 384** Digit Span Forward -. 133 -1. 238 . 221 Digit Span Backward . 035 . 327 . 745 Corsi Blocks Forward -. 015 -. 158 . 875 Corsi Blocks Backward . 117 1. 146 . 257 Cardinality . 181 1. 662 . 102 Ordinality . 556 5. 147 <. 001 *p<. 05; **p <. 01

Results • Independent variable: Multiplication Model Standardized β t p R² ΔR 1. Processing Speed . 094 . 766 . 447 0. 009 2. Processing Speed . 054 . 416 . 679 0. 036 0. 027 Digit Span Forward -. 137 -. 993 . 324 Digit Span Backward -. 025 -. 178 . 860 Corsi Blocks Forward . 079 . 623 . 536 Corsi Blocks Backward -. 047 -. 368 . 714 3. Processing Speed . 027 . 249 . 805 0. 363** 0. 327** Digit Span Forward -. 055 -. 476 . 635 Digit Span Backward . 006 . 049 . 961 Corsi Blocks Forward . 105 . 999 . 322 Corsi Blocks Backward . 110 1. 007 . 318 Cardinality . 316 2. 711 . 009 Ordinality . 402 3. 475 . 001 *p<. 05; **p <. 01

Results • Independent variable: TTR Model Standardized β t p R² ΔR 1. Processing Speed -. 072 -. 582 . 562 0. 005 2. Processing Speed -. 056 -. 427 . 671 0. 017 0. 012 Digit Span Forward . 033 . 235 . 815 Digit Span Backward . 028 . 200 . 842 Corsi Blocks Forward . 002 . 014 . 989 Corsi Blocks Backward . 097 . 757 . 452 3. Processing Speed -. 034 -. 299 . 766 0. 260** 0. 243** Digit Span Forward -. 039 -. 314 . 754 Digit Span Backward -. 004 -. 033 . 973 Corsi Blocks Forward -. 022 -. 197 . 845 Corsi Blocks Backward -. 031 -. 263 . 793 Cardinality -. 224 -1. 784 . 079 Ordinality -. 386 -3. 094 . 003 *p<. 05; **p <. 01

Results • Independent variable: Subtraction Model Standardized t p R² ΔR β 1. Processing Speed . 083 . 673 . 504 0. 007 2. Processing Speed . 058 . 466 0. 044 0. 038 MU Active -. 069 -. 478. MU Passive . 222 . 1. 546 3. Processing Speed . 016 . 165 . 869 0. 422** 0. 377** MU Active . 031 . 261 . 795 MU Passive . 062 . 513 . 610 Cardinality . 163 1. 491 . 141 Ordinality . 547 4. 828 <. 001 *p<. 05; **p <. 01

Results • Independent variable: Multiplication Model Standardized β t p R² ΔR 1. Processing Speed . 094 . 766 . 447 0. 009 2. Processing Speed . 075 . 597 . 552 0. 032 0. 023 MU Active -. 057 -. 393 . 696 MU Passive . 176 1. 217 . 228 3. Processing Speed . 030 . 287 . 775 0. 343** 0. 310** MU Active -. 009 -. 069 . 945 MU Passive . 091 . 699 . 487 Cardinality . 304 2. 606 . 011 Ordinality . 367 3. 035 . 004 *p<. 05; **p <. 01

Results • Independent variable: TTR Model Standardized t p R² ΔR β 1. Processing Speed -. 072 -. 582 . 562 0. 005 2. Processing Speed -. 066 -. 520 . 605 0. 006 0. 001 MU Active . 003 . 022 . 983 MU Passive -. 039 -. 268 . 790 3. Processing Speed -. 029 -. 256 . 798 0. 261** 0. 254** MU Active -. 061 -. 463 . 645 MU Passive . 068 . 495 . 623 Cardinality -. 200 -1. 617 . 111 Ordinality -. 399 -3. 112 . 003 *p<. 05; **p <. 01

Discussion • Basic number processing skills are related to calculation • When controlled for WM only order processing stands out as significant predictor for subtraction and arithmetic • When WM controlled for WM cardinal and order processing are significant predictors for multiplication Cardinal processing and multiplication might rely on retrieval strategies as they are both highly automatized

Discussion • A specific feature of order processing is important for mathematical ability, which cannot be explained by WM. • Future research: what underlies order processing? o o Serial scanning? Associations between numbers?

Discussion • Future research: what underlies order processing? o Serial scanning: Items are scanned in short-term memory with one item at a time

Discussion o Associations between successive items where each item acts as a retrieval cue for the next item Item #1 Item #2 Item #3 Item #4 Item #5

Thank you for your attention! Contact: helene. vos@kuleuven. be