The Role of Naming in the Formation of Equivalence Classes

1. The Role of Naming in the Formation of Equivalence Classes R.W. Maguire, M.J Persson, L. Nunes, L. Von Thaden & M.J. Cameron        Simmons College Simmons Department of Behavior Analysis Annual Conference 2010

2. Acknowledgements • This talk is based on two studies: • The Instruction of Multiplication and Division Skills via Equivalence Class Formation (2009). M.J. Persson, R.W. Maguire & M.J. Cameron  • The Formation of Equivalence Classes Following Name (Tact) Training (2010) L. Nunes, R.W. Maguire, L. Von Thaden & M.J. Cameron 

3. Equivalence and Naming • Sidman (1971): Subject: 17-year-old, severe mental retardation “Cat” “Cat” (Spoken by (Spoken by Experimenter) Subject) C – A - T Solid Lines = Established performances; Dashed Lines = Trained Relations Dashed,Red Lines = Emergent relations without direct training.

4. Equivalence and Naming • Equivalence relations “are a built in effect of reinforcement contingencies” (Sidman, 1994, p. 554) • Typically, relating a minimum of three stimuli via two conditional discriminations is followed by the emergence of novel stimulus-stimulus relations. Train: 8 x 3 = 24 Train: 4 x 6 = 24 Emergent: 24 = 8 x 3 24 = 4 x 6 8 x 3 = 4 x 6 4 x 6 = 8 x 3 And perhaps: 8 x 3 = “24” 4 x 6 = “24”

5. Equivalence and Naming Train: 8 x 3 = 24 24 Train 8 x 3

6. Equivalence and Naming Train: 8 x 3 = 24 Train: 4 x 6 = 24 24 Train 8 x 3 Train 4 x 6

7. Equivalence and Naming 24 Train: 8 x 3 = 24 Train: 4 x 6 = 24 Emergent: 24 = 8 x 3 24 = 4 x 6 8 x 3 = 4 x 6 4 x 6 = 8 x 3 Train 8 x 3 Train 4 x 6

8. Equivalence and Naming Train: 8 x 3 = 24 Train: 4 x 6 = 24 Emergent: 24 = 8 x 3 24 = 4 x 6 8 x 3 = 4 x 6 4 x 6 = 8 x 3 And perhaps: 8 x 3 = “24” 4 x 6 = “24” 24 Train 8 x 3 Train “24” 4 x 6 Naming

9. Equivalence and Naming • “Naming is a bi-directional speaker–listener relation that establishes category relations between a set of stimuli, given only that each particular stimulus in the set occasions the same name. Indeed, so defined, naming is categorization” (e.g., stimulus class formation). • Horne, Hughes, & Lowe (2006, p. 247) Speaker Relation Listener Relations “Cat” Spoken by Experimenter “Cat” Spoken by Participant C – A - T

10. Equivalence and Naming • “Naming is a bi-directional speaker–listener relation that establishes category relations between a set of stimuli, given only that each particular stimulus in the set occasions the same name. Indeed, so defined, naming is categorization” (e.g., stimulus class formation). • Horne, Hughes, & Lowe (2006, p. 247) Speaker Relation Listener Relations “Cat” Spoken by Experimenter “Cat” Spoken by Participant C – A - T

11. Equivalence and Naming ▒ Train: ▒ - “Cug” ۩ - “Cug” ¶ - “Cug” ۩ “Cug” ¶

12. Equivalence and Naming ▒ Train: ▒ - “Cug” ۩ - “Cug” ¶ - “Cug” Emergent: ▒ - ۩ ۩ - ¶ ¶ - ▒ ۩ “Cug” ¶

13. Equivalence and Naming Train: ▒ - “Cug” ۩ - “Cug” ¶ - “Cug” Emergent: ▒ - ۩ ۩ - ¶ ¶ - ▒ “Cug” - ▒ “Cug” - ۩ “Cug” - ¶ ▒ ۩ “Cug” “Cug” ¶

14. Equivalence and Naming • “When naming is acquired, the presence of one member of the class evokes a tact…[which] in turn…evokes selecting other members that are part of the same relation” • “It follows that to establish arbitrary stimulus classes, one would need only to learn how to name each member of the designated class” • Miguel, Petursdottir, Carr & Michael (2008, p.384)

15. Equivalence and Naming • Miguel, etal ‘s (2008) argument that naming alone should prove a sufficient prerequisite to equivalence class formation has not always proven accurate • It would seem that for a tact to exert control over selection of other stimuli it must be more than a tact….perhaps a contingency specifying stimulus (Skinner, 1969) or an example of joint echoic-tact control (Lowenkron, 1997) or an intraverbal (Commins, 2010). • This means that a tact that is also a name (as opposed to merely a label) must also be a discriminative stimulus O’Hora, Dermot Barnes-Holmes, and Roche (2001).

16. Equivalence and Naming • Questions: • How does a tact acquire contingency specifying properties in the absence of direct reinforcement? • Can the Joint Control hypothesis (that a name exerts simultaneous echoic and tact control) account for this phenomenon? (Lowenkron, 1997) • Why doesn’t naming ALWAYS result in class formation and why doesn’t naming ALWAYS emerge following class formation?

17. Equivalence and Naming • Ms. M.J Persson will present an applied study in which class formation was established between multiplication sentences and their related products. Interesting, naming did emerge despite the fact that auditory stimuli were not used in the study. • Ms. L. Nunes will present an experimental study in which class formation was established between nonsense forms following tact training. This training provided a sufficient history for the emergence of equivalence classes, including auditory-to-visual matching-to-sample performances.

18. The Instruction of Multiplication and Division Skills via Equivalence Class Formation Joyce PerssonRussell W. MaguireMichael J. CameronSimmons College

19. Purpose • The purpose of this study was to extend prior studies in mathematics and to determine whether matching-to-sample procedures involving multiplication would be effective for students with limited math skills and error histories in multiplication.

20. General Procedure • The goal of this study was to establish three, 3-member equivalence classes of multiplication families. • Training and testing of the 24-family occurred first. Participants were taught to match the visual multiplication sentence 4 X 6 to a visual sample 24 (relation A-B). • Then they were taught to match the visual multiplication sentence 3 X 8 to the same visual sample 24 (relation A-C). B S+:4 X 6 D A 24 “24” S+:3 X 8 C Following this training all possible stimulus-stimulus relations, including naming of visual stimuli, were assessed.

21. General Procedure B S+:6 X 6 D A 36 “36” S+:4 X9 C Following training and testing with the 24-Family training and testing with the 36-Family occurred, following the same procedures for the 24-Family.

22. General Procedure B S+:6 X 8 D A 48 “48” S+:4 X 12 C Finally, following training and testing with the 36-Family training and testing with the 48-Family occurred, following the same procedures for the 24 and 36 Families. Note: In the interest of time only the procedures and results for the 24-Family will be presented.

23. Method: Participants • Jerry: 15-year old boy with Autism Spectrum Disorder • Kyle: 11-year old boy with Asperger’s Syndrome • Larry: 12-year old boy with Asperger’s Syndrome • All demonstrated deficits on formal testing, especially in the areas of multiplication and division…all were functioning at a 3rd grade level, mathematically

24. Setting • Jerry’s sessions were conducted in a public library in his home town • Kyle and Larry’s sessions were conducted in a classroom at the middle school they attended • Setting contained a table, chairs, and a lamp

25. Pretesting • Pretesting occurred in a quiet room with just the participant and the experimenter, unless reliability data were being collected. In these cases, a third person stood away from the table, behind and out of eye sight of the participant. • Participants were presented the multiplication worksheets first and were required to write answers to each problem until all were completed or the participant said they could not answer any more correctly. • Participants were then presented the division worksheets, as described above. • Finally, generalized identity matching-to-sample and visual-visual arbitrary matching-to-sample were verified using index cards containing addition facts. Testing occurred in a three choice array and sample-comparison combinations and comparison placement were counterbalanced. • Pretesting occurred in the absence of any teacher assistance and any form of reinforcement.

26. Multiplication Pretest

27. Division Pretest

28. Matching-to-Sample Pretest 4 2 + 2 1 + 1 1 + 0

29. Procedure • Correct responses were followed experimenter praise and incorrect responses were followed by the next scheduled trial. • Each session contained between 40 and 60 trials, conducted in blocks of 10, separated by breaks of approximately 3 minutes during which participants were able to read a magazine or converse with the experimenter. • Prior to testing to evaluate the emergence of untrained symmetrical and transitive relations training of the prerequisite relations for each potential equivalence class underwent six training phases.

30. Phase 1: AB1 Training • Initially, the S+ comparisons were unknown multiplication sentences and the two S- comparisons were known multiplication sentences* • (pretesting had revealed that all the participants were capable of accurately producing the correct products for all single digit X single digit multiplication problems involving the numbers 1, 2, 5 and 10). • Schedule of Reinforcement: FR1 B S+:4 X 6 S-: 2 X 1, 2 X 2 A 24

31. Training Example:(Initial AB1 Training) 24 24 4 X 6 2 X 1 2 X 2 S+ S- S-

32. Phase 2: AB1 Training • Once participants were errorless during a session of Phase 1 AB1 training they were exposed to Phase 2; • During Phase 2 the Phase 1 S- stimuli were replaced unknown multiplication sentences • Schedule of Reinforcement: FR1 B S+:4 X 6 S-: 6 X 6, 4 X 9 A 24

33. Phase 3: AC1 Training • As during Phase 1 training the S+ comparisons were unknown multiplication sentences and the two S- comparisons were known multiplication sentences • Schedule of Reinforcement: FR1 B S+:4 X 6 A 24 S+:3 X 8 S-: 2 X 1, 2 X 2 C

34. Phase 4: AB1 Training • Once participants were errorless during a session of Phase 3 AC1 training they were exposed to Phase 4; • During Phase 4 the Phase 3 S- stimuli were replaced unknown multiplication sentences • Schedule of Reinforcement: FR1 B S+:4 X 6 A 24 S+:3 X 8 S-: 6 X 6, 4 X 9 C

35. Phase 5: AB1 and AC1 Training • AB1 and AC1 trials were intermixed unsystematically • Errorless performance was required for one session (10 trials) to move to Phase 5 • Schedule of Reinforcement: FR1 B S+:4 X 6 A 24 S+:3 X 8 C

36. Phase 6: AB1 and AC1 Training • AB1 and AC1 trials were intermixed unsystematically • Errorless performance was required for one session (10 trials) on a VR 8 schedule to move into testing • Schedule of Reinforcement gradually decreased across two sessions to VR8 B S+:4 X 6 A 24 S+:3 X 8 C

37. Testing for the Emergence of Untrained Symmetrical, Transitive and Naming Relations • Following the six training phases testing was conducted to assess for the emergence of untrained stimulus-stimulus relations indicative of equivalence class formation. Testing was conducted within a baseline of known performances (AB and AC trials) and only correct responses to these trials were reinforced on a VR 8 schedule B S+:4 X 6 D A 24 “24” S+:3 X 8 C Solid lines indicate trained relations; dashed lines indicate emergent performances

38. Testing for the Emergence of Untrained Symmetrical, Transitive and Naming Relations • Testing occurred in three sessions: • First, the BC and CB relations were tested; • Second, the BA and CA relations were tested; • Third, naming of the B and C stimuli was assessed B S+:4 X 6 D A 24 “24” S+:3 X 8 C Solid lines indicate trained relations; dashed lines indicate emergent performances

39. Finally, participants ability to verbally produce answers to visual division problems, related to the potential classes already trained and tested, was assessed in extinction: Division Visual Correct Division Verbal Problem Response 24 ÷ 4 “6” 24 ÷ 6 “4” 24 ÷ 3 “8” 24 ÷ 8 “3” 36 ÷ 6 “6” 36 ÷ 4 “9” 36 ÷ 9 “4” 48 ÷ 6 “8” 48 ÷ 8 “6” 48 ÷ 4 “12” 48 ÷ 12 “4”

40. Training • Once all training and testing was completed for one potential stimulus class (e.g., 24, 3 X 8, 4 X 6) the next was trained (e.g., 36, 6 X 6, 4 X 9). This process continued for 48 (6 X 8, 12 X 4).

41. Training Phases: Summary

42. Results • After training matching-to-sample relations involving the visual stimuli 24, 4x6 and 3x8 participants were able to relate 4x6 to 3x8 and 3x8 to 4x6 (transitivity), 4x6 to 24 and 3x8 to 24 (symmetry). • Further, all participants were able to name all the visual stimuli. • Finally, related division facts emerged without direct training. • Although there were three participants in this study only data regarding a single participant’s performance will be presented. The other two participants demonstrated immediate emergence of trained multiplication, naming and division relations at 100% accuracies

43. B S+:4 X 6 D A 24 “24” S+:3 X 8 C

44. B S+:4 X 6 D A 24 “24” S+:3 X 8 C

46. Discussion • Training two simple discriminations within each potential multiplication class resulted in: • The emergence of four arbitrary matching-to-sample multiplication relations without direct training; • The emergence of two oral naming relations for multiplication sentences without direct training; • The emergence of four oral naming relations for division sentences without direct training

47. Discussion • These outcomes clearly documented the formation three member equivalence classes, in which each member of the class was mutually substitutable for one another…..that is, each member of the class exerted equivalent discriminative and conditional control even if these functions were not directly trained

48. Discussion • Why oral naming of multiplication sentences emerged remains a puzzle….. • Prior to training participants were unable to “name” multiplication sentences as the corresponding product (e.g., 8 x 3 = “24”). • After training, naming emerged, despite the fact that auditory and verbal stimuli were not used at any time during this study…… • It may be that because the participants were able to name the visual stimuli (e.g., 24, 26, 48) prior to the study that once the these stimuli became related to the multiplication sentences via training naming automatically transferred.

49. Discussion • Why oral naming of division sentences emerged remains a puzzle…..especially since the “name” was different for each division sentence (e.g., 24 ÷ 4 = “6” and 24 ÷ 6 = “4” • Did the operator (x versus ÷) serve as contextual control of four-term contingencies (second order control, Bush etal , 1989) • Did participants merely respond with the stimulus from the class that was missing?

50. Discussion • Regardless of the questions that remain the current procedure suggests an efficient methodology for remediating basic mathematical skills and even for their initial instruction. • Future research should investigate methods to promote stimulus equivalence methodologies in general educational settings.