constructing and deriving reciprocal trigonometric ... · pdf fileconstructing and deriving...

CONSTRUCTING AND DERIVING RECIPROCAL TRIGONOMETRICRELATIONS: A FUNCTIONAL ANALYTIC APPROACH

CHRIS NINNESS

STEPHEN F. AUSTIN STATE UNIVERSITY

MARK DIXON

SOUTHERN ILLINOIS UNIVERSITY

DERMOT BARNES-HOLMES

NATIONAL UNIVERSITY OF IRELAND, MAYNOOTH

RUTH ANNE REHFELDT

SOUTHERN ILLINOIS UNIVERSITY

AND

ROBIN RUMPH, GLEN MCCULLER, JAMES HOLLAND, RONALD SMITH,SHARON K. NINNESS, AND JENNIFER MCGINTY

STEPHEN F. AUSTIN STATE UNIVERSITY

Participants were pretrained and tested on mutually entailed trigonometric relations andcombinatorially entailed relations as they pertained to positive and negative forms of sine, cosine,secant, and cosecant. Experiment 1 focused on training and testing transformations of thesemathematical 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 framesof opposition (reciprocal of ) followed by more tests of novel relations. All assessments of derivedand novel formula-to-graph relations, including reciprocal functions with diversified amplitudeand frequency transformations, indicated that all 4 participants demonstrated substantial im-provement in their ability to identify increasingly complex trigonometric formula-to-graph rela-tions pertaining to same as and reciprocal of to establish mathematically complex repertoires.

DESCRIPTORS: combinatorial entailment, construction-based training, mathematicalrelations, mutual entailment, matching to sample, trigonometry, relational frame theory

_______________________________________________________________________________

International indicators of mathematicalperformance suggest that the mathematicalskills of high-school and entry-level collegestudents in the U.S. are causes for concern. U.S.

high-school students have consistently rankedlower on math literacy than many industrializedand nonindustrialized countries (24 of 29countries) in the Organization for EconomicCooperation and Development (OECD). TheProgramme for International Student Assess-ment (PISA), a worldwide appraisal of high-school student performance, makes it obviousthat the performance levels of U.S. high-schoolstudents can only be described as discouraging,with nearly every developed nation surpassingU.S. 10th-grade students in all quantitative skillareas. The most troubling areas in the 2003evaluation were math literacy and problem

Portions of this paper were presented at the 33rd annualconference of the Association for Behavior Analysis. Earlierversions of the Web-based training protocols used inExperiment 1 are described in Ninness et al. (2009).

Correspondence concerning this article should beaddressed to Chris Ninness, School and BehavioralPsychology, P.O. Box 13019 SFA Station, Stephen F.Austin State University, Nacogdoches, Texas 75962 (e-mail: [email protected]).

doi: 10.1901/jaba.2009.42-191

JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2009, 42, 191–208 NUMBER 2 (SUMMER 2009)

191

solving, with U.S. students proving to beincapable of responding to items that requirefundamental algebra and the most rudimentarylevels of basic computations (PISA, 2003). Thesemathematical difficulties were also reflected inthe most recent PISA outcomes on scienceliteracy, which indicate that ‘‘when older U.S.students are asked to apply what they havelearned in mathematics, they demonstrate lessability than most of their peers in other highlyindustrialized countries’’ (PISA, 2006, as cited inU.S. Department of Education, 2006, p. 24).

Insufficient exposure to mathematics instruc-tion may account for some of these findings. Inthe highest achieving Asian and Europeancountries, eighth-grade mathematics curriculainclude the basics of three-dimensional geom-etry, proportionality problems, and transforma-tion of geometric functions (see Schmidt,Houang, & Cogan, 2002, for a detaileddiscussion). Japanese high-school students ac-quire more advanced math skills (e.g., quadraticfunctions, permutations and combinations,trigonometric functions, limits, derivatives,and the applications of integrals) than almostall U.S. students encounter in their requiredcollege course work (Conway & Sloane, 2005).Despite recent legislative attempts such as NoChild Left Behind (2001), student performancehas failed to show significant improvement.

Notwithstanding, innovative behavioral re-search conducted by Mayfield and Chase(2002) confirmed the beneficial effects ofcumulative practice on the acquisition of mathskills among low-performing college students.Also, Lynch and Cuvo (1995) illustrated thatteaching fifth and sixth graders to match fractionratios to their corresponding graphical represen-tations and graphs to the corresponding decimalvalues resulted in the emergence of nontrainedskills to match decimal to fraction and fraction todecimal without direct training. The emergenceof untrained relations, termed stimulus equiva-lence (Sidman, 1994) or relational framing whenthe stimuli are not directly equivalent (e.g., more

than, less than; Hayes, Barnes-Holmes, & Roche,2001), represents a possible advantage forteachers with limited time for math instruction.

Relational frame theory (RFT) extends theequivalence paradigm by suggesting that ver-bally competent participants can learn torespond in accordance with sameness and othermore complex relations among stimuli (Hayeset al., 2001). As in equivalence terminology, aframe of coordination refers to stimuli being thesame as or equivalent to other stimuli; a frame ofcomparison involves relating stimuli along somedimension of quantity or quality (e.g., largerthan, higher than). A frame of opposition refersto stimuli that can be contrasted along somedimension in which objects or events have someorder (e.g., opposite of, inverse of, reciprocalof ).

Ninness and colleagues have developedseveral computer-interactive protocols directedat establishing advanced math skills via derivedstimulus relations. The protocols have em-ployed matching-to-sample (MTS) strategiesto teach formula-to-graph relations for mathe-matical transformations about the coordinateaxes (Ninness, Rumph, McCuller, Vasquez, etal., 2005) and to teach formula-to-factored-formula and factored-formula-to-graph rela-tions for vertical and horizontal shifts on thecoordinate axes (Ninness, Rumph, McCuller,Harrison, et al., 2005). Participants were able todemonstrate derived relations by identifyingstandard formulas and their graphical represen-tations and vice versa, even though theexperimental preparations specifically precludedany direct training of these relations.

More recently, Ninness et al. (2006) em-ployed similar computer-interactive protocolsin the analysis of transformation of stimulusfunctions and found that participant preferencesfor either standard (the more difficult) orfactored formulas alternated with the prevailingrules and independent of prevailing contingen-cies. As discussed by Ninness et al. (2006,pp. 315–316), derived stimulus relations ap-

192 CHRIS NINNESS et al.

pears to be well suited to facilitate the efficientacquisition of a wide array of complex mathe-matical relations and mathematical reasoning(e.g., Ensley & Crawley, 2005; Ensley &Kaskosz, 2008), especially as it applies tolearning abstract concepts employed in algebra,trigonometry, precalculus, and calculus (e.g.,Sullivan, 2002). Within mathematics, trigono-metric relations address the sides and the anglesof triangles and produce repeating values on thecoordinate axes over a specific period identifiedby their formulas. Learning how the basictrigonometric relations operate on the coordi-nate axes is prerequisite to the acquisition ofprogressively more elaborate mathematicaltransformations employed in calculus andhigher level mathematics (Sullivan, 2002).

The current study extends and differs fromthe previous studies by Ninness et al. (2005,2006) in that more complex math conceptsentailing same as relations and opposite ofrelations were trained. Specifically, duringExperiment 1, we trained eight two-membertrigonometric classes that address transforma-tion of amplitude and frequency functions. Theinstruction was unique in that online construc-tion-based responding was required in additionto traditional MTS selection procedures. InExperiment 2, we used offline MTS proceduresto develop two four-member relational net-works that are especially relevant to basictrigonometry because these functions are absentfrom menus of graphing calculators (i.e.,students must learn these basic reciprocalidentifiers to become fluent in all trigonometricrelations).

EXPERIMENT 1

METHOD

Participants and Setting

After informed consent had been obtained, apretest was administered to determine partici-pants’ skill levels with regard to identification ofvarious types of trigonometric relations. Indi-viduals who were able to identify more than

four of 15 pretest formula-to-graph items wereexcused from the experiment. Four students, allwomen who ranged in age from 23 to 28 years,participated in pretraining and the two exper-iments. Although 1 participant had an academichistory that included a precalculus class duringhigh school, she correctly identified only threeof 15 items on the pretest. Thus, she waspermitted to participate. Interestingly, at thestart of the experiment, this participant wasunable to pronounce the names of several of thetrigonometric functions used in the study. Theother participants had no recollection ofexposure to the subject matter of this study.

Participants were recruited from variousacademic disciplines by way of agreements withprofessors to provide extra credit for taking partin university research projects. Participantsreceived five points on their final examinationsfor their involvement in the study. Also, eachparticipant could earn $0.10 per correctresponse during the assessment of novel rela-tions (maximum $6.00, which included aminimum of $2.00 for participation). Aftercompleting the study, all participants weredebriefed and reimbursed according to thenumber of correct responses emitted duringthe assessment of novel trigonometric relations.

Apparatus and Software

Training, MTS procedures, and the record-ing of responses were controlled by thecomputer programs, which were written bythe first author in Visual Basic and FlashActionScript 2.0. Error patterns were identifiedwith software written in the C++ programminglanguage. These software systems providedinteractive math instructions, displayed formu-las and graphs, and recorded the accuracy ofresponses during both experiments. The accu-racy of the computer’s data compilation wasconfirmed prior to initiating MTS segments.Before each participant began, the computerprogram’s data collection was compared againsthand tallies of correct and incorrect responses.These were found to match precisely.

DERIVING TRIGONOMETRIC RELATIONS 193

Design and Procedure

Following a brief presentation on the detailsof the Cartesian coordinate axes and concisedefinition of reciprocal relations, participantswere pretrained regarding positive and negativeforms of sine, cosine, secant, and cosecantfunctions. Then, during baseline, participantswere tested on a series of novel formula-to-graph relations. This was followed by onlinetraining and testing with regard to how graphstransform in amplitude and frequency inaccordance with particular types of formula-to-graph and graph-to-formula relations. Wethen provided another assessment of novelrelations regarding these functions. Figure 1illustrates the sequence of training and testingfor Experiments 1 and 2. Note that Experiment1 was conducted in an attempt to lay thefoundation skills needed for Experiment 2.

Stage 1: Pretraining. During Step 1, partici-pants were given a 4-min preliminary trainingpresentation in which they were exposed to anoverview of the fundamentals of the rectangularcoordinate system. In Step 2, a succinct definitionof reciprocal relations was provided, and in Step3, participants were pretrained regarding positiveand negative forms of sine, cosine, secant, andcosecant functions. All the pretraining proceduresin Step 3 were adapted from training strategiesemployed by Ninness, Rumph, McCuller, Har-rison, et al. (2005) and Ninness et al. (2006).During this stage, participants were pretrainedand tested on A-B and B-C trigonometricrelations, mutually entailed (B-A and C-B)relations, and combinatorially entailed (A-Cand C-A) relations as they pertain to how thepositive and negative forms of sine, cosine, secant,and cosecant transform on the coordinate axis.

Baseline. During our nonconcurrent multiplebaseline, Participants 1 and 2 attempted toidentify five novel MTS relations, and Partic-ipants 3 and 4 attempted to identify 10 novelMTS relations (i.e., five additional items andthe same five items as Participants 1 and 2 onItems 6 through 10). All remaining items in

Experiment 1 were the same for all participants.The assessment of novel relations addressedamplitude and frequency transformations. Dur-ing each assessment item, participants attempt-ed to match a novel sample formula with acomparison graph from an array of six graphsthat had not been used during any previoustraining or testing condition (see Figure 2).

Stage 2: Online training and testing ofamplitude and frequency transformations by wayof construction-based responding. Construction ofgraphs of trigonometric formulas requires theparticipant to respond in accordance with thedetails of each function. During this stage,participants were trained and assessed ongraphical transformations of trigonometricfunctions built on the skills acquired duringpretraining of positive and negative forms of thesine, cosine, secant, and cosecant functions.Throughout this stage, we trained and sequen-tially assessed eight two-member trigonometricclasses that addressed transformation of ampli-tude and frequency. At the beginning of thisstage, the experimenter provided training withonline software (http://www.faculty.sfasu.edu/ninnessherbe/graphCalcCN07.html). Table 1illustrates these trained and derived relations.

Our procedure included aspects of modeling,direct instruction, clear antecedent instructions,multiple-exemplar training, feedback, and rulesfor responding (Berens & Hayes, 2007). Todepict a given function prior to its transforma-tion, we first provided the basic (nontrans-formed) graph of each trig function on thecoordinate axes. For example, in training theamplitude transformation of y 5 cos(x), thescreen displayed a solid line representing thecosine function on the coordinate axes as itwould appear prior to transformation by achange in frequency or amplitude. Participantswere trained to construct a transformation ofthis function by adjusting the graphing anchorsof the green line to draw (construct) a change inamplitude in accordance with the details of thesample formula. At the completion of each


Figure 1. The sequence of training and testing procedures in Experiments 1 and 2.


graph constructed by a participant, the exper-imenter produced a computer-generated graphof the same function by typing the formula intoa text box, and the precise function then wasdisplayed on-screen in conjunction with theparticipant’s constructed graph. Accuracy of theparticipant’s graph was determined by visuallycomparing it to the computer-generated graphof the same function. The experimenter and atrained observer examined each graph andformula construction. If both the experimenterand observer agreed the participant’s graph wasa reasonable approximation of the computer-generated graph, the participant advanced to thenext assessment of graph-to-formula relations.Similarly, accuracy of the participant’s formulaconstruction was determined by comparing theparticipant’s formula to the experimenter’sformula.

During Step 1, the experimenter explainedthat multiplying the cosine function by a numbergreater than 1 stretches each point vertically, tothree times its original distance along the y axis.

The easiest points to observe were the high andlow points. The high points stretch from 1 to 3,and the low points stretch from 21 to 23. Usingthe text box in the lower left field of the graphingscreen, the experimenter changed the formulafrom y 5 cos(x) to y 5 3*cos(x), where theasterisk was used as a multiplication operator.After the graph button above the formula wasclicked, the amplitude of the graph stretched to23 and +3 along the y axis. Next, theexperimenter used the graphing anchors toconstruct a superimposed graph of this functiondirectly over the existing graph. The experimenterthen asked the participant to perform the sameexercise using her mouse to manipulate thegraphing anchors. Using numbers greater than1 as multipliers, this process was repeated untilthe participant was able to arrange the graphsuccessfully and independently. (Note that con-struction of a graph with this software simplyrequires mouse dragging each of the five red

Figure 2. Sample function of a negative sine functiontransformed in amplitude and frequency.

Table 1

Training and Testing of Amplitude and Frequency

Cosine amplitude transformations with multipliers greater thanand less than 1

y 5 3*cos(x) y 5 0.5*cos(x)

Train/test Test Train/test Test

A1-B1 B1-A1 A2-B2 B2-A2

Cosine frequency transformations with multipliers greater thanand less than 1

y 5 cos(2*x) y 5 cos(0.5*x)


A3-B3 B3-A3 A4-B4 B4-A4

Secant amplitude transformations with multipliers greater thanand less than 1

y 5 3*sec(x) y 5 0.5*sec(x)


A5-B5 B5-A5 A6-B6 B6-A6

Secant frequency transformations with multipliers greater thanand less than 1

y 5 sec(2*x) y 5 sec(0.5*x)


A7-B7 B7-A7 A8-B8 B8-A8


graphing anchors from the top of the screen tolocations on the coordinate axis such that thedesired shape of the graph is fashioned. Thesoftware system also allows users to identify wherea constructed point on the graph falls on thecoordinate axis by observing a floating text boxthat moves in conjunction with the mouse arrowor cursor; see top left side of Figure 3.)

To test an A-B vertical stretch (amplitudeincrease) of the cosine function, the experimentertyped a formula within a text box [e.g., y 5

3*cos(x)] and stated, ‘‘Using the red graphinganchors, construct the graph of this formula.’’ Atthis step in testing, the graph of the basic cosinefunction, y 5 cos(x), was displayed on-screen. Torespond, the participant moved the graphinganchors to the screen locations necessary toconstruct a transformation of the graph inaccordance with the newly displayed formula[i.e., y 5 3*cos(x)]. Then, to assess B-A mutuallyentailed relations, the experimenter typed thesame or a similar formula (multipliers alwaysranging between 2 and 5) into the lower right textbox. This text box (outlined in red), was labeled‘‘input values are hidden,’’ and did not permit ascreen display of the formula typed into the textbox. When the experimenter clicked the graphbutton, a graph of this hidden formula wasdisplayed on-screen. Thus, the participant wasunable to see the specific formula responsible forproducing the graph when the experimenterstated, ‘‘Please type the formula needed toproduce this graph in the far left text box.’’

After typing the formula into the text box,the participant clicked the graph button toverify that her response matched the existingon-screen graph. In the event that a participanterred during the assessment of either A-B or B-A relations, the A-B training protocol for cosinewas repeated immediately, and another assess-ment of A-B or B-A relations was conducted. Acorrect response to this test is illustrated on thetop left side of Figure 3. If the participant hademitted more than three consecutive errors, shewould have been reimbursed for her time,

debriefed, and excused; however, all participantseasily attained these criteria (see Table 2).

During Step 2, the experimenter informedthe participant that multiplying the cosinefunction by a number less than 1 causes itsgraph to compress along the y axis, consistentwith the value of the multiplier. The experi-menter provided the following rule, ‘‘Multiply-ing the cosine function by a number less than 1(e.g., 0.5) compresses each point vertically tohalf its original distance along the y axis. Again,the easiest points to watch are the high and lowpoints. You will see that the high points willcompress from 1 to 0.5, and the low points willcompress from 21 to 20.5.’’ To demonstratethe operation of this rule, the experimenterchanged the cosine formula by multiplying it by0.5 in the text box in the lower left text field ofthe screen [y 5 0.5*cos(x)].

To test an A-B relation, the experimentertyped the above formula into the text box [i.e., y5 0.5*cos(x)] and stated, ‘‘Using the redgraphing anchors, construct the graph of thisformula.’’ In response, the participant moved thegraphing anchors to the screen locations necessaryto construct a transformation of the graph inaccordance with the newly displayed formula[i.e., y 5 0.5*cos(x)]. A correct response to thistest is illustrated on the top right side of Figure 3.

To assess B-A mutually entailed relations, theexperimenter typed this formula into the lowerright text box. As in the assessment of verticalcosine stretches described above, the participantwas unable to see the formula responsible forgenerating the particular on-screen graph.Pointing to the text box with his mouse arrow,the experimenter asked the participant to typethe formula needed to generate the graphdisplayed on-screen.

To test A-B and B-A relations, the multiplierwas set at 0.5. If an error was emitted by aparticipant during the assessment of A-B and B-A relations, she was immediately reexposed tothe A-B training of cosine vertical compressionand an additional test of A-B or B-A relations.


Reexposure was limited to training only A-B(formula-to-graph) relations in terms of thevertical cosine compression function. At notime were participants given any trainingaddressing B-A (graph-to-formula) relations.

This type of correction procedure was employedthroughout assessments conducted during Ex-periment 1.

During Step 3, the experimenter informedparticipants that multiplying the argument (the

Figure 3. Sample online construction-based responding.


variable inside the parentheses) of the cosinefunction by a number greater than 1 causes itsgraph to compress along the x axis consistent withthat number. For example, if the argument of acosine function were multiplied by 2, the graphcompressed such that it became twice as frequentbut half as wide within one period of the function(a period for all functions in this study consistedof 2p). In this Step, the experimenter changed theformula of the graph by multiplying it by 2 in thelower left text field of the screen where y 5

cos(2*x). Clicking the graph button resulted inthe frequency of the graph being compressed, andthe graph became twice as frequent but half aswide along the x axis.

In the assessment of A-B relations thatfollowed, the experimenter typed the formula,y 5 cos(2*x), into the text box, pointed to theformula with his mouse arrow, and asked theparticipant to construct a graph of the formula.In response, the participant moved the graphinganchors to the screen locations necessary toconstruct a transformation of the graph inaccordance with the newly displayed formula.The bottom left panel of Figure 3 shows anaccurate construction-based response to the testof A-B relations.

To assess B-A mutually entailed relations, theexperimenter typed a formula into the lowerright text box. As in previous assessments, theparticipant was unable to see the formularesponsible for generating a horizontally com-pressed graph on-screen. Pointing to a far lefttext box with his mouse arrow, the experimenterasked the participant to type the formulaneeded to generate the graph displayed on-screen. After typing a formula into the text box,

the participant was requested to click the graphbutton and reveal whether her typed formulaproduced the correct graph. In the event that aparticipant erred during the assessment of eitherA-B or B-A relations, she was reexposed to theA-B relations training and another assessment ofA-B or B-A relations.

During Step 4, the experimenter informedparticipants that multiplying the argument of acosine function by a number less than 1 causes thegraph to stretch horizontally along the x axis. Forexample, if the argument of the cosine function ismultiplied by 0.5 [i.e., y 5 cos(0.5*x)], its graphon the coordinate appears half as frequent buttwice as wide relative to the basic cosine function,y 5 cos(x). The experimenter changed theformula of the graph by multiplying theargument by 0.5 in the text box in the lowerleft text field of the screen, where y 5 cos(0.5*x).When the graph button above the formula wasclicked, the frequency of the graph stretchedhorizontally, and the graph became half asfrequent but twice as wide along the y axis. Inthe assessment of A-B relations that followed, theexperimenter typed the formula, y 5 cos(0.5*x)into the text box, pointed to the formula with hismouse arrow, and asked the participant toconstruct a graph of the formula using the redanchors. To respond, the participant adjusted thegraphing anchors to the screen locations requiredto construct a transformation of the graph inaccordance with the newly displayed formula.The bottom right side of Figure 3 illustrates acorrect response to this item.

To assess B-A mutually entailed relations, theexperimenter typed a formula [e.g., y 5

cos(0.5*x)] into the lower right text box and

Table 2

Number of Exposures Required to Attain Mastery on Construction of Cosine and Secant Amplitude and

Frequency Functions

Participant A1-B1 A2-B2 A3-B3 A4-B4 A5-B5 A6-A6 A7-B7 A8-B8 Total

1 2 2 2 2 1 1 1 1 122 3 3 3 2 2 1 1 1 163 2 1 1 1 1 1 1 1 94 2 1 1 1 1 1 1 2 10


clicked the graph button. Pointing to the far lefttext box with his mouse arrow, the experimenterasked the participant to type the formulaneeded to produce the graph displayed on-screen. In the event that a participant erredduring the assessment of either A-B or B-Arelations, she was reexposed to A-B relationstraining, and another assessment of A-B or B-Arelations was immediately conducted.

Participants were exposed to the sametraining as described above for transformationsaddressing the secant function. They were nottrained regarding the transformation of sine andcosecant functions; instead they were simplyinformed that with regard to amplitude andfrequency, the sine and cosecant functionstransform in the same manner as cosine andsecant. Because secant transformations weretrained with the same rules as the cosinetransformations, the specific training graphsthat addressed secant are not shown.

Interobserver agreement. Each graph and for-mula construction were assessed by an observerand then by the experimenter. In total, there wereonly six of 94 occasions (i.e., 93.6% agreement)during which the observer and experimenterdisagreed with respect to the adequacy of aparticipant’s graph or formula construction.

Assessment of novel relations before andafter training. During baseline assessments,Participants 1 and 2 were tested on five novelformula-to-graph relations, and Participants 3and 4 were assessed on 10 novel formula-to-graphrelations. After completing online training andtesting of amplitude and frequency transforma-tions, each participant was assessed with 10 novelformula-to-graph relations of the same type asthose employed during the baseline assessment ofnovel relations. These items were novel in thesense that they involved positive and negativeforms of sine, cosine, secant, and cosecant as theytransformed in amplitude and frequency whenmultiplying the function, the argument of thefunction, or both, by a number greater or lessthan 1 (see Figure 2).

RESULTS

All exposures to training A-B (formula-to-graph) relations, including the initial exposureplus any reexposures required to demonstrate acorrect construction-based response, are provid-ed in Table 2. All 4 participants failed toconstruct the graph of y 5 3*cos(x) on their firstattempts and were reexposed to graph construc-tion training for the amplitude transformation(A1-B1) at least once (see Column 2 ofTable 2). Following reexposure to training,each successfully constructed the graph given aformula as a sample stimulus, and correctlytyped the formula given a graph of y 5 3*cos(x)as a sample stimulus. All participants masteredsubsequent functions with similar or fewerexposures to training (follow each participantacross all columns). If errors occurred duringthe assessment of A-B and B-A relations,participants immediately returned to the A-Btraining (formula to graph), and an additionaltest of A-B and B-A (untrained) relations.

Figure 4 shows the multiple baseline designacross paired participants. Participant 1 failed toidentify any of the baseline novel relations itemsbut identified eight of 10 novel formula-to-graph assessment items after training. Partici-pant 2 correctly responded to one of five novelassessment items during baseline and respondedcorrectly to all 10 items after training. Partic-ipant 3 failed nine of the 10 baseline items andonly one of the 10 items after training.Similarly, Participant 4 identified only two of10 baseline items but was able to identify all 10novel formula-to-graph assessments followingtraining.

EXPERIMENT 2

Pretraining and Experiment 1 were conductedin an attempt to lay the critical foundation forExperiment 2. Indeed, pilot research in ourlaboratory has indicated that learning the math-ematical relations in Experiment 2 depends onthe skills addressed in the first part of this study.


METHOD

Participants and Setting

Experiment 2 was an extension of Experi-ment 1, the same participants served in thesame setting, and this experiment was conduct-ed within the same approximate 2- to 2.5-hrtime period.

Design and Procedure

Immediately following Experiment 1, all 4participants were exposed to MTS protocolsthat afforded training in accordance with framesof coordination (same as) and frames ofopposition (reciprocal of ) as they entail trigo-nometric operations. Targeting two four-mem-ber relational networks, participants weretrained and tested on A-B, B-C, and C-Drelations and assessed on mutually entailed (D-C, C-B, and B-A) and combinatorially entailed(B-D, D-B, A-D, D-A, A-C, and C-A)relations. This was followed by a posttreatmentassessment of novel formula-to-graph relations,including complex reciprocal functions anddiversified amplitude and frequency transfor-mations. In this nonconcurrent multiple base-line design across paired participants, Partici-pants 1 and 2 were assessed on 15 novelformula-to-graph relations (following training),and Participants 3 and 4 were assessed on 10novel formula-to-graph relations. (Items 26through 35 presented to Participants 1 and 2

were the same as Items 31 through 40 presentedto Participants 3 and 4.)

Baseline. At the beginning of Experiment 2,participants returned to their previous computerscreens and keyboards facing away from theexperimenter, such that they were not in directvisual contact with the experimenter throughoutthe remainder of the experiment. Baseline tests ofnovel formula-to-graph relations involved identi-fication of graphical transformations pertainingto positive and negative forms of sine, cosine,secant, and cosecant functions in conjunctionwith transformation pertaining to changes inamplitude and frequency. Moreover, the first 10MTS items for all participants addressed formu-la-to-graph relations in the form of reciprocals.Participants had not been exposed to any form oftraining that would have prepared them torespond to these items (e.g., Figure 5).

Stage 1: Training and testing of cosine andsecant reciprocals. We trained three fundamentalrelations (A reciprocal of B, B same as C, and Csame as D) to see if participants could derivetheir mutually entailed relations (D same as C,C same as B, B reciprocal of A), as well as theircombinatorially entailed relations (B same as D,D same as B, A reciprocal of D, D reciprocal ofA, A reciprocal of C, and C reciprocal of A) asfour-member relational networks (see the top ofFigure 6 as an illustration of trained andderived relations). Thus, we attempted to have

Figure 4. Correct and incorrect responses displayed in a multiple baseline design across paired participants. The bolddouble line demarcates the locations at which treatment was implemented and terminated. Errors pertaining to tests ofnovel relations are identified as shaded blocks containing ones, and correct responses are depicted as zeros.


participants acquire two four-member relationalnetworks pertaining to cosine reciprocal ofsecant and sine reciprocal of cosecant.

Having been exposed to the basic formula-to-graph relations addressing the positive andnegative forms of the sine, cosine, secant, andcosecant functions during pretraining, partici-pants were well positioned to identify thegraphical representation of the secant functionas being incorrect in the presence of the cosineformula as a sample stimulus. Having beenexposed to pretraining and Experiment 1training, participants were able to discriminatethe graphical representation of cosine as beingincorrect in the presence of the secant formulaas a sample stimulus. Thus, during theassessment of trained and derived relations, wewere able to rotate the sample and comparisonstimuli across trials such that the placement oftargets and distracters were counterbalanced. In

the event that a participant erred while beingprobed on any of the tested relations, she wasimmediately reexposed to the training of allthree A-B, B-C, and C-D primary relations andwas reassessed on all derived relations within thefour-member relational network.

Correction strategies and MTS fluency criteria.Fluency (in the form of rate of accurate problemsolving) is a consistent and reliable indicator ofthe probability of occurrence of a newly learnedbehavior (Binder, 1996). Some form of fluencyseemed to be particularly important to addresswhen gauging math progress; thus, our softwareplaced a limited hold on the time allocated torespond to each MTS item. Specifically, anyresponse that required more than 30 s wasidentified as an error, and if such a delay tookplace, the programmed contingencies requiredthe participant to engage in reexposure training.If a participant had required more than fourexposures, the program would have automati-cally ended and that participant would havebeen compensated, debriefed, and excused fromthe study; however, all participants easilyachieved these criteria (see Tables 3 and 4).

During Step 1, we trained A1-B1 [i.e., y 5

cos(x) reciprocal of y 5 1/cos(x)] relations, inwhich the program presented a trigonometric ruleon-screen, and the experimenter requested par-ticipants to read aloud, ‘‘Two numbers arereciprocal of one another when their product is1. The reciprocal of 2, for example, is K because2 * K 5 1. The reciprocal of the cosine functionis illustrated below.’’ Participants read the ruleson-screen twice, and each time the experimenterused his mouse arrow to point to the respectiveformulas. Participants clicked ‘‘next’’ to advanceto the screen that assessed A1-B1 performance. Ifthe correct comparison items were selected,participants moved to the next set of instructedrelations; otherwise, they were reexposed to A1-B1 (formula-to-graph) relations training.

In Step 2, the same MTS procedure was usedto train and test B1-C1 [i.e., y 5 1/cos(x) sameas y 5 sec(x)] relations. As the program

Figure 5. An example of a test of novel formula-to-graph relations that required identification of transforma-tions including negative coefficients and changes inamplitude and frequency.


Figure 6. The top panel is a four-member relational network, with solid lines representing trained relations anddashed lines indicating derived relations (e.g., same as, reciprocal of). The bottom left panel shows assessment of B1-A1relations in the context of reciprocal of. The bottom right panel shows an assessment of B1-D1 relations in the context ofsame as.


presented a trigonometric rule on the computerscreen, the experimenter requested that partic-ipants read the mathematical rule aloud.Participants read the following rule: ‘‘Thereciprocal of the cosine function is the same asthe secant function. This is referred to as afundamental identity.’’

The identity formulas were depicted imme-diately below the rule, and the experimenterused his mouse arrow to point from oneformula to the next. Participants then read thefollowing rule at the bottom of the screen: ‘‘Inother words, when you see 1/cos(x), you shouldsay ‘secant.’ ’’ After reading these rules aloudtwice, participants clicked ‘‘next’’ and advancedto the screen that assessed B1-C1 relations. Ifthe correct comparison item were selected in thecontext of same as, participants were exposed tothe next set of instructed relations; if not, thesoftware reexposed participants to trainingbeginning at Step 1 of Stage 1.

In Step 3, a similar procedure was employedto train and test C1-D1 [i.e., y 5 sec(x) same asthe graphed representation of the secantfunction] relations. Participants read the fol-lowing rule: ‘‘From formula to graph, the secantfunction is illustrated below.’’ The basic secantformula and its graph were depicted immedi-ately below the rule, and the experimenter usedhis mouse arrow to point to the formula andthen to its graph. Subsequent to reading theserules aloud twice, participants clicked ‘‘next’’and advanced to the assessment of C1-D1relations. (Note that the C1-D1 relations hadbeen assessed for combinatorial entailment aspart of the pretraining protocol.) If the correctcomparison item were identified in the context

of same as, the program moved participants to aseries of assessments addressing four-termrelations that pertained to combinatoriallyentailed frames of coordination (same as) andframes of reciprocity (opposite of). If, in anytest, the correct comparison item was notidentified, participants were reexposed to train-ing, beginning with Step 1 of Stage 1.

During Step 4, participants were assessed onmutually entailed (D1-C1, C1-B1, and B1-A1)relations, as well as all combinatorially entailed(B1-D1 and D1-B1) relations, frames of coordi-nation, and frames of reciprocity (A1-D1, D1-A1, A1-C1, and C1-A1; see bottom of Figure 6for examples of B1-A1 and B1-D1 test ques-tions). In conjunction with these primaryassessments, several probe items were provided(i.e., tests in which the secant function was anincorrect response) such that the placement oftargets and distracters was counterbalanced. Ateach assessment, if the correct comparison itemwas identified, participants advanced to the nextset of assessments. If the correct comparison itemwas not identified, participants were reexposed totraining beginning at Step 1 of Stage 1.

Stage 2: Training and testing of sine andcosecant reciprocals. After completing the train-ing and testing of cosine and secant reciprocalfunctions, participants were exposed to a seriesof parallel steps pertaining to the sine andcosecant reciprocal functions. Addressing thetraining and testing of sine and cosecantreciprocal relations, participants were trainedand tested on A2-B2 [i.e., y 5 sin(x) reciprocalof y 5 1/sin(x)], B2-C2 [i.e., y 5 1/sin(x) sameas y 5 csc(x)], C2-D2 [i.e., y 5 csc(x) same asthe graphed representation of the cosecant

Table 3

Stage 1: Number of Exposures Required to Attain Mastery

of the Cosine-Secant Four-Member Relational Network

Participant A1-B1 B1-C1 C1-D1 Total

1 2 2 2 62 2 2 2 63 1 1 1 34 1 1 1 3

Table 4

Stage 2: Number of Exposures Required to Attain Mastery

of Sine-Cosecant Four-Member Relational Network

Participant A1-B1 B1-C1 C1-D1 Total

1 1 1 1 32 2 2 2 63 1 1 1 34 1 1 1 3


function] relations and then assessed regardingthe mutually entailed (D2-C2, C2-B2, and B2-A2) and combinatorially entailed (B2-D2, D2-B2, A2-D2, D2-A2, A2-C2, and C2-A2)relations. Just as in Stage 1, the placement oftargets and distracters was counterbalanced withprobe items. Because training and testing of thesine and cosecant reciprocals employed the sameprocedures as those directed at the training ofcosine-secant relations, the details of theseprocedures are not included.

RESULTS

Complete outcomes for each participantacross Stages 1 and 2 of Experiment 2 areprovided in Table 3. Participant 1 failed toidentify the combinatorially entailed formula-to-graph relation of A1-D1 [y 5 cos(x)reciprocal of secant graph] and was reexposedto all A1-B1, B1-C1, and C1-D1 relationstraining in Stage 1. She then passed theassessment of all derived relations within thefour-member cosine-secant network. Partici-pant 2 failed to identify the formula-to-graphcombinatorially entailed relation of A1-C1 [y 5

cos(x) reciprocal of y 5 sec(x)] during Stage 1.She was reexposed to A1-B1, B1-C1, and C1-D1 relations training in Stage 1 and then passedthe assessments of all mutually entailed andcombinatorially entailed relations within thecosine-secant network. During Stage 2, thisparticipant erred while attempting to derive acombinatorially entailed formula-to-graph rela-tion of A2-D2 in the form y 5 sin(x) reciprocalof cosecant graph and was reexposed to Stage 2A2-B2, B2-C2, and C2-D2 relations training.Following reexposure, she passed all assessmentsin the sine-cosecant relational network. Partic-ipants 3 and 4 emitted no errors throughout theduration of the assessments that addressed A2-B2, B2-C2, and C2-D2 trained relations; themutually entailed D2-C2, C2-B2, and B2-A2relations; or the combinatorially entailed rela-tions including B2-D2, D2-B2, A2-D2, D2-A2, A2-C2, and C2-A2.

Posttreatment assessment of novel formula-to-graph relations. Figure 7 (top) shows a binarygraph depicting trial-by-trial responding withthe results of the novel trigonometric assessmentitems obtained in Experiment 1, followed bythose obtained in Experiment 2. All 4 partici-pants were assessed on 10 novel formula-to-graph relations prior to training. After comple-tion of training and testing of reciprocalidentities, each participant was assessed with 10novel formula-to-graph relations that were of thesame type employed prior to training. Theseitems were novel in the sense that they addressedpositive and negative forms of sine, cosine,secant, and cosecant as they transform whenmultiplying the function, the argument of thefunction, or both, by a number greater or lessthan 1. Prior to training in Experiment 2,Participant 1 identified two of 10 formula-to-graph assessments (Items 16 to 25 in Experiment2; Figure 7, top); after training, she identified 13of 15 novel formula-to-graph assessments cor-rectly (Items 26 to 40 in Experiment 2). Prior totraining in Experiment 2, Participant 2 correctlyresponded to one of 10 novel assessments (Items16 to 25); following training, she correctlyresponded to 13 of 15 assessments of novelrelations (Items 26 to 40). Before training,Participant 3 correctly identified only one of the10 pretraining formula-to-graph assessments(Items 21 to 30); after training, she identifiednine of the 10 novel formula-to-graph assess-ments (Items 31 to 40). Prior to training inExperiment 2, Participant 4 identified only oneof 10 baseline formula-to-graph assessments(Items 21 to 30); after training, she identifiedeight of 10 novel formula-to-graph assessments(Items 31 to 40). Moreover, these assessments ofnovel relations used sample formulas in recipro-cal format (Figure 7, bottom). Participants 1and 2 were exposed to five fewer baseline itemsduring Experiment 1 and received five additionaltest items following treatment during Experi-ment 2. Otherwise, all items in the assessment ofnovel relations were the same for all participants.


In conjunction with these novel assessments ofreciprocal relations, we provided one probe item[y 5 2 sin(2x)]. Participant 3 emitted anincorrect response to this test item. All otherparticipants responded correctly to the probeitem.

GENERAL DISCUSSION

Experiment 1 showed that online construc-tion-based training and testing of amplitude

and frequency relations were sufficient toprepare participants to address more complexreciprocal trigonometric relations. During Ex-periment 2, we trained two four-memberrelational networks addressing reciprocal ofand same as trigonometric relations in accor-dance with reciprocal identities. All participantsderived mutually entailed relations (D same asC, C same as B, B reciprocal of A), andcombinatorially entailed relations (B same as D,D same as B, A reciprocal of D, D reciprocal of

Figure 7. The top panel depicts trial-by-trial responding for novel trigonometric assessment items obtained inExperiment 1, followed by Experiment 2 with the transition between experiments indicated by the dotted lines. Withineach experiment, training is designated by the solid heavy double lines. Problem numbers are listed along the x axis.Accurate responses are represented by zeros, and errors are shaded and represented by ones. The bottom panel representsnovel relations test items.


A, A reciprocal of C, and C reciprocal of A) thatconstituted four-member relational networks(see Figure 6, top). Participants then performedwell on tests of novel complex formula-to-graphrelations, including positive and negative formsof sine, cosine, secant, and cosecant transformedby multiplying the trigonometric formulas, theargument of the formulas, or both, by variednumbers above or below 1. Moreover, theformula-to-graph relations identified during theassessment of novel relations included a largeand diverse network of relations that extendedbeyond the exemplars used during pretraining,construction-based training, and MTS training.These findings suggest that participants werenot constructing or relating samples andcomparisons as unitary stimuli, but werelearning to identify the relations among allformulas and graphs, consistent with outcomesfrom other relational frame studies of similarlearned relations (e.g., same as, more than) andspatial and mathematical relations (Dymond &Barnes, 1995; Ninness et al., 2006).

What makes relational networks in trigo-nometry and precalculus powerful is theirability to extend existing abstract relations sothat they contribute to the student’s ability toderive similar networks while the basic charac-teristics of the original are retained. Forexample, the four-term cosine-secant reciprocalrelations network contains all combinatoriallyentailed relations that form the basis of thefour-term sine-cosecant network. Althoughthese trigonometric functions transform graph-ically according to patterns that are analogous,the graphical transformations of the ‘‘even’’cosine-secant reciprocal functions are not thesame as those of the ‘‘odd’’ sine-cosecantreciprocal functions. Each of the trigonometricfunctions trained in this study has uniquetransformation characteristics, and our baselineoutcomes suggest that the patterns for each ofthese functions are not intuitively obvious.However, our data suggest that once studentshave acquired the pretraining skills that pertain

to positive and negative forms of the sine,cosine, secant, and cosecant functions; learnedthe mutually entailed relations pertaining toamplitude and frequency; and acquired thecombinatorially entailed relations addressingthe functional identities; they are well posi-tioned to respond to a wide range of complexand novel formula-to-graph relations.

This study employed a novel strategy ofonline construction-based responding ratherthan selection-based MTS procedures. Theconstruction procedure is somewhat similar toSidman’s (1994) pretraining strategies usedprior to the implementation of MTS training.Although construction-based responding issomewhat more time consuming than MTSprocedures, the additional instructional timemay be well worth the effort. Perhaps, some-what analogous to the way in which multiple-choice test items allow faster responding thanshort essay test items, selecting correct compar-ison graphs via MTS procedures is moreefficient but less demanding than constructinggraphs. The more demanding task of graphconstruction may produce more robust re-sponding during training of abstract concepts,although this statement is speculative at thispoint and requires empirical verification.

Presently, we are pursuing variations ofconstruction-based responding in conjunctionwith MTS procedures with the ambition ofdeveloping more relational networks that estab-lish trigonometric identities, inverse trigono-metric functions, and conversion of polarcoordinates to rectangular coordinates and viceversa. Although these topics may appear to bebeyond the grasp of students who lack well-established mathematical repertoires, the pre-sent study and our ongoing investigationssuggest that mathematically inexperienced butverbally competent adolescents and adults arecapable of mastering complex and multifacetedabstract concepts efficiently when proceduresbased on derived relational responding are used.Future research might contrast protocols based


on stimulus relations training with existingbehavioral strategies that address higher levelmath operations.

Considering that the math literacy levels ofU.S. high-school students are inferior to almostall peers from OECD countries and alsoconsidering the potential benefits of blendingcomponents of high-school and college-levelmath curricula with strategies based on derivedstimulus relations, our ambition has been to laysome of the groundwork for more sophisticatedcomputer-interactive procedures. An instruc-tional technology of mathematics based onderived stimulus relations might provide onemethod that contributes to addressing thechallenges that face many of this nation’s highschools and colleges in the area of basicmathematics. We believe the emerging researchin derived stimulus relations may add animportant dimension to the technology ofteaching described so elegantly in the writingsof Skinner (1968).

REFERENCES

Berens, N. M., & Hayes, S. C. (2007). Arbitrarilyapplicable comparative relations: Experimental evi-dence for a relational operant. Journal of AppliedBehavior Analysis, 40, 45–71.

Binder, C. (1996). Behavioral fluency: Evolution of a newparadigm. The Behavior Analyst, 19, 163–197.

Conway, P. F., & Sloane, F. C. (2005). Internationaltrends in post-primary mathematics education. Reportto the National Council for Curriculum andAssessment, Dublin, Ireland.

Dymond, S., & Barnes, D. (1995). A transformation ofself-discrimination response functions in accordancewith the arbitrarily applicable relations of sameness,more than, and less than. Journal of the ExperimentalAnalysis of Behavior, 64, 163–184.

Ensley, D., & Crawley, W. (2005). Discrete mathematics:Mathematical reasoning with puzzles, patterns andgames. New York: Wiley.

Ensley, D., & Kaskosz, B. (2008). Flash and math appletslearn by example: Introduction to ActionScript program-ming for mathematics and science teaching and learning.Charleston, SC: BookSurge.

Hayes, S. C., Barnes-Holmes, D. & Roche, B. (Eds.).(2001). Relational frame theory: A post-Skinnerianaccount of human language and cognition. New York:Kluwer Academic/Plenum.

Lynch, D. C., & Cuvo, A. J. (1995). Stimulus equivalenceinstruction of fraction-decimal relations. Journal ofApplied Behavior Analysis, 28, 115–126.

Mayfield, K. H., & Chase, P. N. (2002). The effects ofcumulative practice on problem solving. Journal ofApplied Behavior Analysis, 35, 105–123.

Ninness, C., Barnes-Holmes, D., Rumph, R., McCuller,G., Ford, A., Payne, R., et al. (2006). Transformationof mathematical and stimulus functions. Journal ofApplied Behavior Analysis, 39, 299–321.

Ninness, C., Holland, J., McCuller, G., Rumph, R.,Ninness, S., McGinty, J., et al. (2009). Mathematicalreasoning. In R. A. Rehfeldt & Y. Barnes-Holmes(Eds.), Derived relational responding: Applications forlearners with autism and other developmental disabilities(pp. 313–334). Oakland, CA: New Harbinger.

Ninness, C., Rumph, R., McCuller, G., Harrison, C.,Ford, A. M., & Ninness, S. (2005). A functionalanalytic approach to computer-interactive mathemat-ics. Journal of Applied Behavior Analysis, 38, 1–22.

Ninness, C., Rumph, R., McCuller, G., Vasquez, E.,Harrison, C., Ford, A. M., et al. (2005). A relationalframe and artificial neural network approach tocomputer-interactive mathematics. The PsychologicalRecord, 51, 561–570.

No child left behind. (2001). 34 CFR Part 200.Programme for International Student Assessment. (2003).

Learning for tomorrow’s world: First results from PISA2003. Retrieved July 8, 2007, from http://www.pisa.oecd.org/dataoecd/1/60/34002216.pdf.

Schmidt, W., Houang, R., & Cogan, L. (2002). Acoherent curriculum: The case of mathematics.American Educator, 26, 10–26, 47–48.

Sidman, M. (1994). Equivalence relations and behavior: Aresearch story. Boston: Authors Cooperative.

Skinner, B. F. (1968). The technology of teaching. NewYork: Appleton-Century-Crofts.

Sullivan, M. (2002). Precalculus (6th ed.). Upper SaddleRiver, NJ: Prentice Hall.

U.S. Department of Education, National Center forEducation Statistics. (2006). U.S. students and adultperformance on international assessments of educationalachievement: Findings from the condition of education2006. Retrieved July 8, 2007, from http://www.nces.ed.gov/pubs2006/2006073.pdf.

Received July 11, 2007Final acceptance April 7, 2008Action Editor, Linda LeBlanc


constructing and deriving reciprocal trigonometric ... · pdf fileconstructing and deriving...

Documents