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Dermot Barnes-holmes - One of the best experts on this subject based on the ideXlab platform.
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The impact of high versus low levels of derivation for Mutually and combinatorially entailed relations on persistent rule-following.
Behavioural processes, 2018Co-Authors: Colin Harte, Dermot Barnes-holmes, Yvonne Barnes-holmes, Ciara McenteggartAbstract:The effects of rules on human behaviour have long been identified as important in the psychological literature. The increasing importance of the dynamics of arbitrarily applicable relational responding (AARR), with regards to rules, has come to be of particular interest within Relational Frame Theory (RFT). One feature of AARR that previous research has suggested may differentially impact persistent rule-following is level of derivation. However, no published research to date has systematically explored this suggestion. Across two experiments, the impact of levels of derivation was examined on persistent rule-following at two stages of relational development: Mutual Entailment (Exp. 1) and combinatorial Entailment (Exp. 2). A Training IRAP was used to establish a Mutually entailed relational network in Experiment 1 and a combinatorially entailed network in Experiment 2, and to train these networks to different levels of derivation. This was followed by a contingency switching Match-to-Sample (MTS) task to assess rule persistence. Results from both experiments were generally consistent with the suggestion that lower levels of derivation produce more persistent rule-following. Unexpectedly, however, the findings from Experiment 1 also indicated that persistence was moderated by the type of novel word employed. Variations in results across both experiments and their implications for future research are discussed.
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Deriving Trigonometric Relations 1 Running head: DERIVING TRIGONOMETRIC RELATIONS Constructing and Deriving Reciprocal Trigonometric Relations: A Functional Analytic Approach
2015Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Stephen F. Austin, Dermot Barnes-holmes, Mark Dixon, Ruth Anne Rehfeldt, James Holl, Ronald Smith, Sharon K. NinnessAbstract:Participants were pretrained and tested on Mutually entailed, trigonometric, relations, and combinatorially entailed relations as they pertain to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same-as) and frames of opposition (reciprocal-of) followed by more tests of novel relations. All assessments of derived and novel formula-to-graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their facility to identify increasingly complex trigonometric formula-to-graph relations pertaining to same-as and reciprocal-of to establish mathematically complex repertoires. DESCRIPTORS: combinatorial Entailment, construction-based training, four-member relations, mathematical relations, Mutual Entailment, matching-to-sample, reciprocal
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Relational Frame Theory and Analogical Reasoning: Empirical Investigations
International journal of psychology and psychological therapy, 2004Co-Authors: Ian Stewart, Dermot Barnes-holmesAbstract:A central assumption of Relational Frame Theory (RFT; Hayes, Barnes-Holmes & Roche, 2002) is that language and cognition may be explained in terms of derived relational responding. Furthermore, RFT research has by now modelled a number of arguably important areas of linguistic-cognitive functioning based on controlled laboratory demonstrations of this phenomenon. The main purpose of the present article is to consider one particular strand of RFT-based research that has focused on providing a derived relations-based model of analogical language.RELATIONAL FRAME THEORY, DERIVED RELATIONS AND LANGUAGEThere is now an appreciable quantity of evidence to suggest that verbal/cognitive and derived relational phenomena are closely linked. Relational Frame Theory explains this link by arguing that the former are in fact examples of the latter. The theory explains all derived relational phenomena as generalized contextually controlled relational responding which, in RFT terminology, is referred to as arbitrarily applicable relational responding. Naming is perhaps the simplest example. When a child is taught to name, he or she is taught explicitly, using numerous exemplars, that if a name is a certain object (A is B) then the object is also the name (B is A). Eventually, aspects of the context in which this bi-directional relationship is learned, such as the word 'is' for example, come to control the application of the relation. Hence, when the child learns that a wholly novel name X 'is' a wholly novel object Y, he or she will derive that the object Y is also the name X. This particular example of derived relational responding constitutes what linguists refer to as reference (name = object; object = name) and what behavioural researchers refer to as symmetry (A = B; B = A). RFT sees this type of derived or arbitrarily applicable relational responding as the initial and most basic form. Continued exposure to human socio-verbal interactions produces many more complex patterns than this, including, for example, responding in accordance with relations of equivalence, opposition, difference and comparison.Despite the variety of forms of derived relational responding, RFT argues that all forms involve the following three core properties:(i) Mutual Entailment describes the relations between two stimuli or events. For example, if an experimental participant is trained that A goes with B then s/he will derive the Mutually entailed relation of B goes with A. This is the symmetrical relation demonstrated in equivalence research. If the relation between A and B is not one of sameness or equality, however, then the Mutually entailed relation may differ from the trained relation, in which case the relational pattern is non-symmetrical. For example, if a participant is taught that A is greater than B then they will derive that B is less than or samaller than A.(ii) Combinatorial Entailment refers to a derived relation in which two or more stimulus relations combine. For example, if a participant is trained that A is the same as B and B is the same as C then s/he will derive the combinatorially entailed relations of A same as C and C same as A. In this case, the trained relations are of sameness, and thus the resulting combinatorially entailed relations are those of transitivity and equivalence, respectively. However, if training involves non-sameness relations then a different derived relational pattern may emerge. For example, if the participant is taught that A is opposite to B and B is opposite to C then the resulting combinatorial relational pattern of A same as C and C same as A is non-transitive and non-equivalent (i.e., two opposite relations combine to produce same, not opposite).(iii) Transformation of function is perhaps the most important of the three properties from an applied psychological perspective. For example, if a participant is trained such that s/he derives the Mutually entailed relation A less than B, and a punishing function is trained to A, then the previously neutral function of B may be transformed such that B now acquires a more aversive function than that directly trained to A. …
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Evidence-Based Educational Methods - Teaching the Generic Skills of Language and Cognition
Evidence-Based Educational Methods, 2004Co-Authors: Yvonne Barnes-holmes, Dermot Barnes-holmes, Carol MurphyAbstract:Publisher Summary This chapter discusses how to teach the generic skills of language and cognition, focusing on the contributions from relational frame theory (RFT). From the perspective of RFT, arbitrarily applicable relational responding has three defining properties: Mutual Entailment, combinatorial Entailment, and the transformation of stimulus functions. RFT employs the generic term “relational frame” to describe particular patterns of arbitrarily applicable relational responding. Furthermore, RFT suggests that the frame of opposition will emerge later than coordination because the combinatorially entailed relations within frames of opposition are frames of coordination. From the perspective of RFT, over-arching relational skills can be taught, and subsequent improvement in relational responding should lead to improved abilities in areas of cognition and language, as well as in intelligence in general. Establishing a manding repertoire is very important for children with language deficits, because it provides immediate control of the social and non-social environment and it facilitates the development of speaker and listener repertoires.
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Derived Relational Responding As Generalized Operant Behavior
2000Co-Authors: Olive Healy Dermot, Dermot Barnes-holmes, M. SmeetsAbstract:relational responding emerged and stabilized, response patterns on novel stimulus sets were controlled by the feedback delivered for previous stimulus sets. Experiment 2 replicated Experiment 1, except that during Conditions 3 and 4 four comparison stimuli were employed during training and testing. Experiment 3 was similar to Condition 1 of Experiment 1, except that after the mastery criterion was reached for class-consistent responding, feedback alternated from accurate to inaccurate across each successive stimulus set. Experiment 4 involved two types of feedback, one type following tests for Mutual Entailment and the other type following tests for combinatorial Entailment. Results from this experiment demonstrated that Mutual and combinatorial Entailment may be controlled independently by accurate and inaccurate feedback. Overall, the data support the suggestion, made by relational frame theory, that derived relational responding is a form of generalized ope
Christian Barker - One of the best experts on this subject based on the ideXlab platform.
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Scopability and sluicing
Linguistics and Philosophy, 2013Co-Authors: Christian BarkerAbstract:This paper analyzes sluicing as anaphora to an anti-constituent (a continuation), that is, to the semantic remnant of a clause from which a subconstituent has been removed. For instance, in Mary said that [John saw someone yesterday], but she didn’t say who, the antecedent clause is John saw someone yesterday, the subconstituent targeted for removal is someone, and the ellipsis site following who is anaphoric to the scope remnant John saw ___ yesterday. I provide a compositional syntax and semantics on which the relationship between the targeted subconstituent and the rest of the antecedent clause is one of scopability, not movement or binding. This correctly predicts that sluicing should be sensitive to scope islands, but not to syntactic islands. Unlike the currently dominant approaches to sluicing, there is no need to posit syntactic structure internal to the ellipsis site, nor is there any need for a semantic Mutual-Entailment requirement. Nevertheless, the fragment handles phenomena usually taken to suggest a close syntactic correspondence between the antecedent and the sluice, including case matching, voice matching, and verbal argument structure matching. In addition, the analysis handles phenomena exhibiting antecedent/sluice mismatches, including examples such as John remembers meeting someone, but he doesn’t remember who, Open image in new window , and especially so-called sprouting examples such as John left, but I don’t know when, in which there is no overt subconstituent to target for removal. In Sect. 5, I show how the analysis accounts for Andrews Amalgams such as Sally ate [I don’t know what] today, in which the antecedent surrounds the sluiced clause. Finally, in Sect. 6, I propose a new semantic constraint on sluicing: the Answer Ban, which says that the antecedent clause must not resolve, or even partially resolve, the issue raised by the sluiced interrogative.
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DOI 10.1007/s10988-013-9137-1 RESEARCH ARTICLE Scopability and sluicing
2013Co-Authors: Chris Barker, Christian BarkerAbstract:Abstract This paper analyzes sluicing as anaphora to an anti-constituent (a continuation), that is, to the semantic remnant of a clause from which a subconstitu-ent has been removed. For instance, in Mary said that [John saw someone yesterday], but she didn’t say who, the antecedent clause is John saw someone yesterday, the subconstituent targeted for removal is someone, and the ellipsis site following who is anaphoric to the scope remnant John saw _ _ yesterday. I provide a compositional syn-tax and semantics on which the relationship between the targeted subconstituent and the rest of the antecedent clause is one of scopability, not movement or binding. This correctly predicts that sluicing should be sensitive to scope islands, but not to syntactic islands. Unlike the currently dominant approaches to sluicing, there is no need to posit syntactic structure internal to the ellipsis site, nor is there any need for a semantic Mutual-Entailment requirement. Nevertheless, the fragment handles phenomena usu-ally taken to suggest a close syntactic correspondence between the antecedent and the sluice, including case matching, voice matching, and verbal argument structure matching. In addition, the analysis handles phenomena exhibiting antecedent/sluice mismatches, including examples such as John remembers meeting someone, but he doesn’t remember who he met, and especially so-called sprouting examples such as John left, but I don’t know when, in which there is no overt subconstituent to target for removal. In Sect. 5, I show how the analysis accounts for Andrews Amalgams such as Sally ate [I don’t know what] today, in which the antecedent surrounds the sluiced clause. Finally, in Sect. 6, I propose a new semantic constraint on sluicing: the Answer Ban, which says that the antecedent clause must not resolve, or even partially resolve, the issue raised by the sluiced interrogative
Justin Klocksiem - One of the best experts on this subject based on the ideXlab platform.
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Against reductive ethical naturalism
Philosophical Studies, 2019Co-Authors: Justin KlocksiemAbstract:This paper raises an objection to two important arguments for reductive ethical naturalism. Reductive ethical naturalism is the view that ethical properties reduce to the properties countenanced by the natural and social sciences. The main arguments for reductionism in the literature hold that ethical properties reduce to natural properties by supervening on them, either because supervenience is alleged to guarantee identity via Mutual Entailment, or because non-reductive supervenience relations render the supervenient properties superfluous. After carefully characterizing naturalism and reductionism, we will present, explain, and raise objections against each of the main reductionist arguments: (a) that supervenience does not support the claim that ethical properties and their subvenient natural properties are Mutually entailing; (b) that reductive views undermine the claim that ethical properties yield resemblance; and (c) that supervenience does not entail that non-descriptive ethical properties are superfluous in the most fundamental sense.
Chris Ninness - One of the best experts on this subject based on the ideXlab platform.
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Deriving Trigonometric Relations 1 Running head: DERIVING TRIGONOMETRIC RELATIONS Constructing and Deriving Reciprocal Trigonometric Relations: A Functional Analytic Approach
2015Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Stephen F. Austin, Dermot Barnes-holmes, Mark Dixon, Ruth Anne Rehfeldt, James Holl, Ronald Smith, Sharon K. NinnessAbstract:Participants were pretrained and tested on Mutually entailed, trigonometric, relations, and combinatorially entailed relations as they pertain to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same-as) and frames of opposition (reciprocal-of) followed by more tests of novel relations. All assessments of derived and novel formula-to-graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their facility to identify increasingly complex trigonometric formula-to-graph relations pertaining to same-as and reciprocal-of to establish mathematically complex repertoires. DESCRIPTORS: combinatorial Entailment, construction-based training, four-member relations, mathematical relations, Mutual Entailment, matching-to-sample, reciprocal
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A FUNCTIONAL ANALYTIC APPROACH TO
2014Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, K. Ninness, Stephen F. Austin, State UniversityAbstract:Following a pretest, 11 participants who were naive with regard to various algebraic and trigonometric transformations received an introductory lecture regarding the fundamentals of the rectangular coordinate system. Following the lecture, they took part in a computerinteractive matching-to-sample procedure in which they received training on particular formulato-formula and formula-to-graph relations as these formulas pertain to reflections and vertical and horizontal shifts. In training A-B, standard formulas served as samples and factored formulas served as comparisons. In training B-C, factored formulas served as samples and graphs served as comparisons. Subsequently, the program assessed for Mutually entailed B-A and C-B relations as well as combinatorially entailed C-A and A-C relations. After all participants demonstrated Mutual Entailment and combinatorial Entailment, we employed a test of novel relations to assess 40 different and complex variations of the original training formulas and their respective graphs. Six of 10 participants who completed training demonstrated perfect or near-perfect performance in identifying novel formula-to-graph relations. Three of the 4 participants who made more than three incorrect responses during the assessment of novel relations showed some commonality among their error patterns. Derived transfer of stimulus control using mathematical relations is discussed. DESCRIPTORS: Mutual Entailment, combinatorial Entailment, mathematical relations, stimulus equivalence, novel relations, matching to sample, relational frame theor
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A RELATIONAL FRAME AND ARTIFICIAL NEURAL NETWORK APPROACH TO COMPUTER-INTERACTIVE MATHEMATICS
2014Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, Stephen F. Austin, Sharon K. Ninness, Eleazar Vasquez, Ashley Capt, Anna BradfieldAbstract:Fifteen participants unfamiliar with mathematical operations relative to reflections and vertical and horizontal shifts were exposed to an introductory lecture regarding the fundamentals of the rectangular coordinate system and the relationship between formulas and their graphed analogues. The lecture was followed immediately by computer-assisted instructions and matching-tosample procedures in which participants were e)(posed to computerposted rules regarding the relationship between particular types of formulas and their respective graphs. After participants demonstrated Mutual Entailment on formula-to-graph and graph-toformula functions, they were assessed for 36 novel relations on complex variations of the original training formulas and graphs. In Experiment 1, 5 of 15 participants demonstrated perfect or near perfect performance on all novel relationships. Experiment 2 was directed at the remaining 10 participants who failed to correctly identify all mathematical relationships assessed in Experiment 1. The error patterns for these 10 participants were classified with the help of an artificial neural network self-organizing map (SOM). Training in Experiment 2 was directed exclusively at the types of errors identified by the SOM. Following remedial training, all participants demonstrated a substantial reduction in errors compared to their performance in Experiment 1. Derived transfer of stimulus control using mathematical relations is discussed. Many everyday occurrences entail two or more elements that are associated by some rule of correspondence. The mathematical term for such a correspondence is a relation. Within the stimulus equivalenc
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Training and Deriving Precalculus Relations: A Small-Group, Web-Interactive Approach
2014Co-Authors: Jenny Mcginty, Chris Ninness, Robin Rumph, Glen Mcculler, Andrea Goodwin, Ginger Kelso, Angie Lopez, Elizabeth Kelly, Stephen F. AustinAbstract:A small-group, web-interactive approach to teaching precalculus concepts was investigated. Following an online pretest, 3 participants were given a brief (15 min) presentation on the details of reciprocal math relations and how they operate on the coordinate axes. During baseline, participants were tested regarding their ability to construct formulas for a diversified series of graphs. This was followed by online, construction-based, small-group training procedures focusing on the construction of mathematical functions and a test of novel relations. Participants then received group training in accordance with frames of coordination (same as) and frames of opposition (reciprocal of) formula-to-graph relations. Online assessment indicated that participants showed substantial improvement over baseline and pretest performances. This was true even though, during the tests of novel relations, graphs were displayed with scattered data points instead of solid lines on the coordinate axes. Although one participant was unable to complete the second half of the experiment, we were able to train this small group employing approximately the same number of exposures needed for individual training in previous research. Key words: group training, reciprocal, precalculus, Mutual Entailment, combinatorial Entailment, mathematical relations, four-member relations, construction-based training, matching to sample, relational frame theory Over the past decade, high school students in the United States have performed significantly below the mathematical achievement scores of their international peers. For example, outcomes from the 2006 Program for International Student Assessment (PISA) confirmed that U.S. 15-year-olds performed at levels below cohorts from 23 of 30 industrialized nations (Baldi, Jin, Skemer, Green, & Herget, 2007). Math achievement measure
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A Relational Frame and Artificial Neural Network Approach to Computer-Interactive Mathematics
The Psychological Record, 2005Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, Sharon K. Ninness, Eleazar Vasquez, Ashley Capt, Anna BradfieldAbstract:Fifteen participants unfamiliar with mathematical operations relative to reflections and vertical and horizontal shifts were exposed to an introductory lecture regarding the fundamentals of the rectangular coordinate system and the relationship between formulas and their graphed analogues. The lecture was followed immediately by computer-assisted instructions and matching-tosample procedures in which participants were e)(posed to computerposted rules regarding the relationship between particular types of formulas and their respective graphs. After participants demonstrated Mutual Entailment on formula-to-graph and graph-toformula functions, they were assessed for 36 novel relations on complex variations of the original training formulas and graphs. In Experiment 1, 5 of 15 participants demonstrated perfect or near perfect performance on all novel relationships. Experiment 2 was directed at the remaining 10 participants who failed to correctly identify all mathematical relationships assessed in Experiment 1. The error patterns for these 10 participants were classified with the help of an artificial neural network self-organizing map (SOM). Training in Experiment 2 was directed exclusively at the types of errors identified by the SOM. Following remedial training, all participants demonstrated a substantial reduction in errors compared to their performance in Experiment 1. Derived transfer of stimulus control using mathematical relations is discussed.
Sharon K. Ninness - One of the best experts on this subject based on the ideXlab platform.
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Deriving Trigonometric Relations 1 Running head: DERIVING TRIGONOMETRIC RELATIONS Constructing and Deriving Reciprocal Trigonometric Relations: A Functional Analytic Approach
2015Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Stephen F. Austin, Dermot Barnes-holmes, Mark Dixon, Ruth Anne Rehfeldt, James Holl, Ronald Smith, Sharon K. NinnessAbstract:Participants were pretrained and tested on Mutually entailed, trigonometric, relations, and combinatorially entailed relations as they pertain to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same-as) and frames of opposition (reciprocal-of) followed by more tests of novel relations. All assessments of derived and novel formula-to-graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their facility to identify increasingly complex trigonometric formula-to-graph relations pertaining to same-as and reciprocal-of to establish mathematically complex repertoires. DESCRIPTORS: combinatorial Entailment, construction-based training, four-member relations, mathematical relations, Mutual Entailment, matching-to-sample, reciprocal
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A RELATIONAL FRAME AND ARTIFICIAL NEURAL NETWORK APPROACH TO COMPUTER-INTERACTIVE MATHEMATICS
2014Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, Stephen F. Austin, Sharon K. Ninness, Eleazar Vasquez, Ashley Capt, Anna BradfieldAbstract:Fifteen participants unfamiliar with mathematical operations relative to reflections and vertical and horizontal shifts were exposed to an introductory lecture regarding the fundamentals of the rectangular coordinate system and the relationship between formulas and their graphed analogues. The lecture was followed immediately by computer-assisted instructions and matching-tosample procedures in which participants were e)(posed to computerposted rules regarding the relationship between particular types of formulas and their respective graphs. After participants demonstrated Mutual Entailment on formula-to-graph and graph-toformula functions, they were assessed for 36 novel relations on complex variations of the original training formulas and graphs. In Experiment 1, 5 of 15 participants demonstrated perfect or near perfect performance on all novel relationships. Experiment 2 was directed at the remaining 10 participants who failed to correctly identify all mathematical relationships assessed in Experiment 1. The error patterns for these 10 participants were classified with the help of an artificial neural network self-organizing map (SOM). Training in Experiment 2 was directed exclusively at the types of errors identified by the SOM. Following remedial training, all participants demonstrated a substantial reduction in errors compared to their performance in Experiment 1. Derived transfer of stimulus control using mathematical relations is discussed. Many everyday occurrences entail two or more elements that are associated by some rule of correspondence. The mathematical term for such a correspondence is a relation. Within the stimulus equivalenc
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A Relational Frame and Artificial Neural Network Approach to Computer-Interactive Mathematics
The Psychological Record, 2005Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, Sharon K. Ninness, Eleazar Vasquez, Ashley Capt, Anna BradfieldAbstract:Fifteen participants unfamiliar with mathematical operations relative to reflections and vertical and horizontal shifts were exposed to an introductory lecture regarding the fundamentals of the rectangular coordinate system and the relationship between formulas and their graphed analogues. The lecture was followed immediately by computer-assisted instructions and matching-tosample procedures in which participants were e)(posed to computerposted rules regarding the relationship between particular types of formulas and their respective graphs. After participants demonstrated Mutual Entailment on formula-to-graph and graph-toformula functions, they were assessed for 36 novel relations on complex variations of the original training formulas and graphs. In Experiment 1, 5 of 15 participants demonstrated perfect or near perfect performance on all novel relationships. Experiment 2 was directed at the remaining 10 participants who failed to correctly identify all mathematical relationships assessed in Experiment 1. The error patterns for these 10 participants were classified with the help of an artificial neural network self-organizing map (SOM). Training in Experiment 2 was directed exclusively at the types of errors identified by the SOM. Following remedial training, all participants demonstrated a substantial reduction in errors compared to their performance in Experiment 1. Derived transfer of stimulus control using mathematical relations is discussed.
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A Functional Analytic Approach To Computer-Interactive Mathematics
Journal of applied behavior analysis, 2005Co-Authors: Chris Ninness, Robin Rumph, Glen Mcculler, Carol Harrison, Angela M. Ford, Sharon K. NinnessAbstract:Following a pretest, 11 participants who were naive with regard to various algebraic and trigonometric transformations received an introductory lecture regarding the fundamentals of the rectangular coordinate system. Following the lecture, they took part in a computer-interactive matching-to-sample procedure in which they received training on particular formula-to-formula and formula-to-graph relations as these formulas pertain to reflections and vertical and horizontal shifts. In training A-B, standard formulas served as samples and factored formulas served as comparisons. In training B-C, factored formulas served as samples and graphs served as comparisons. Subsequently, the program assessed for Mutually entailed B-A and C-B relations as well as combinatorially entailed C-A and A-C relations. After all participants demonstrated Mutual Entailment and combinatorial Entailment, we employed a test of novel relations to assess 40 different and complex variations of the original training formulas and their respective graphs. Six of 10 participants who completed training demonstrated perfect or near-perfect performance in identifying novel formula-to-graph relations. Three of the 4 participants who made more than three incorrect responses during the assessment of novel relations showed some commonality among their error patterns. Derived transfer of stimulus control using mathematical relations is discussed.
