ERIC EJ750832: Transformations of Mathematical and Stimulus Functions

2006·1.6 MB·English

by  ERIC

JOURNALOFAPPLIEDBEHAVIORANALYSIS 2006, 39, 299–321 NUMBER3 (FALL2006)
TRANSFORMATIONS OF MATHEMATICAL
AND STIMULUS FUNCTIONS
CHRIS NINNESS
STEPHENF.AUSTINSTATEUNIVERSITY
DERMOT BARNES-HOLMES
NATIONALUNIVERSITYOFIRELAND,MAYNOOTH
AND
ROBIN RUMPH, GLEN MCCULLER, ANGELA M. FORD, ROBERT PAYNE,
SHARON K.NINNESS,RONALD J.SMITH,TODDA.WARD,AND MARC P.ELLIOTT
STEPHENF.AUSTINSTATEUNIVERSITY
Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric
functions received a brief presentation on the rectangular coordinate system. Next, they
participated in a computer-interactive matching-to-sample procedure that trained formula-to-
formulaandformula-to-graphrelations.Then,theywereexposedto40novelformula-to-graph
tests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantial
improvementinidentifyingformula-to-graphrelations;however,inthetestofnovelgraph-to-
formula relations, participants tended to select equations in their factored form. Next, we
manipulatedcontextualcuesintheformofrulesregardingmathematicalpreferences. First,we
informed participants that standard forms of equations were preferred over factored forms. In
a subsequent test of 10 additional novel graph-to-formula relations, participants shifted their
selections to favor equations in their standard form. This preference reversed during 10 more
testswhenfinancialrewardwasmadecontingentoncorrectidentificationofformulasinfactored
form.Formulapreferencesandtransformationofnovelmathematicalandstimulusfunctionsare
discussed.
DESCRIPTORS: value, preference, mutual entailment, combinatorial entailment, trans-
formation offunction
_______________________________________________________________________________
Many mathematical functions have graphs ticalrelations.Inotherwords,allowingstudents
thataretransformationsofalibraryoffunctions to‘‘see’’thetransformationsthatoccurwhenan
on Descarte’s rectangular coordinate system. equation is modified in its particular character-
Historically, transformation of graphs of func- istics (Larson & Hostetler, 2001) helps the
tions has been a major component of many learner to understand families of functions and
levels of algebra as well as of more advanced their relations to each other. This can be
courses in mathematics. Showing students how particularly beneficial in showing how transfor-
a variable changes in defined stages has been mations are applied to mathematical functions
a dynamic learning tool in training mathema- onthecoordinateaxis,andhowtheequationof
a function can be systematically modified to
Portionsofthispaperwerepresentedatthe31stannual transform the graph of any function (Sullivan,
convention of the Association for Behavior Analysis, 2002).
Chicago,May2005.
Nevertheless,instructionalstrategiesaimedat
Correspondence concerning this article should be
directed to Chris Ninness, School & Behavioral Psycho- training transformation of mathematical func-
logy Program, P.O. Box 13019, SFA Station, Stephen tions have been circumvented in many high
F. Austin State University, Nacogdoches, Texas 75962
school and college intermediate algebra classes
(e-mail: [email protected]).
doi:10.1901/jaba.2006.139-05 (R. Huettenmueller, personal communication,
299
300 CHRIS NINNESS et al.
June 25, 2005). This is somewhat surprising in to be needed, and indeed recent research has
that, traditionally, many state assessment in- begun to address this issue (Ninness, Rumph,
struments of algebra have included questions McCuller, Harrison, et al., 2005; Ninness,
regarding transformation of functions (e.g., Rumph, McCuller, Vasquez, et al., 2005). This
Texas Education Agency, 2002), and most work has drawn directly on some of the
Algebra I textbooks (e.g., Kennedy, McGowan, procedures and concepts used in various types
Schultz, Hollowell, & Jovell, 1998; Saxon, of research on stimulus relations (e.g., Lane,
1997) and intermediate algebra textbooks Clow, Innis, & Critchfield, 1998; Leader &
(e.g., Larson & Hostetler, 2001) emphasize Barnes-Holmes, 2001; Lynch & Cuvo, 1995)
the importance of exposure to these concepts. and in particular on relational frame theory
Moreover,thissubject matteris often addressed (RFT;Hayes, Barnes-Holmes, & Roche, 2001).
in varying degrees of complexity within college One of the core postulates of RFT is that much
algebra (e.g., Larson & Hostetler, 1997), of human relational responding, including
trigonometry (e.g., Smith, 1998), precalculus mathematical reasoning, is established in the
(e.g., Sullivan, 2002), and calculus (e.g., form of generalized relational operants through
Finney, Weir, & Giordano, 2001) textbooks. appropriatehistoriesofmultiple-exemplartrain-
Concurrently, math performance in the United ing (e.g., Y. Barnes-Holmes, Barnes-Holmes,
States has been in a state of continuous lag Smeets,Strand,&Friman,2004).Infact,awide
behind most of the countries in the industrial- range of new training protocols using stimulus
ized world. Findings from the Programme for relations are beginning to appear in the applied
International Student Assessment (PISA, 2003) literature. For example, Rehfeldt and Root
study indicate that 15-year-old students in the (2005) confirmed that a history of relational
United States are performing at a disappointing responding generated requesting skills among 3
level in fundamental math concepts when adults with severe developmental disabilities. In
contrastedwiththeircounterpartsinotherparts addition, 1 of the participants demonstrated
of the industrialized world. For example, the tacting and textual behavior. Rehfeldt and Root
United States ranked 24th of 29 countries with speculate that establishing derived relations has
regard to mathematics literacy. Moreover, the the potential togenerate novelforms ofrequest-
PISA study indicated that 25% of American ing and perhaps a wide range of complex verbal
students performed below the lowest possible skills. According to RFT, there are many such
level of competence in mathematics. This lack relational operants (or relational frames), but
of fluency in the fundamentals of basic they all possess three defining behavioral prop-
mathematical operations has migrated into erties: mutual entailment, combinatorial entail-
postsecondary education, where more than ment, and the transformation of stimulus
one in three college students must enroll in functions. It is important that the latter should
a remedial math program prior to taking not be confused with the mathematical variety,
college-level courses (Steen, 2003). transformations of graphs of functions. Both
From the perspective of behavior analysis, it types of transformations are detailed below.
seems likely that the omission of component The concept of mutual entailment refers to
math skills would contribute to a cumulative the derived relations that may obtain between
dysfluency in prerequisite and related problem- two stimuli or events. For example, if a given
solving skills (see Binder, 1996, for a discussion stimulusisrelatedtoanothersuchthatStimulus
of cumulative dysfluency). Consequently, ap- A is the same as Stimulus B, then the derived
plied behavior-analytic methods for training relation—B is the same as A—is mutually
transformationofmathematicalfunctionsappear entailed. The concept of combinatorial entail-
TRANSFORMATIONS OF FUNCTIONS 301
ment refers to derived relations among three or relation between ‘‘stuiver’’ and a ‘‘dubbeltje’’
more stimuli. For example, given that A is the wasthe‘‘opposite’’ofthatofsimilarcoinsinher
opposite of B and B is the opposite of C, C nativeland,shemightwellderivethata‘‘stuiver’’
same as A and A same as C are defined as is twice the value of a ‘‘dubbeltje.’’ Moreover,
combinatorially entailed relations. Among whilestilloperatingunderthecontrolofsuchan
many other possibilities (e.g., greater than or inaccurate rule, if given an opportunity to select
less than), it might be that A is the same as B oneofthetwocoins,thissameindividualislikely
andBisthesameasC,inwhichcaseAremains to show a preference for the physically larger
the same as C (and vice versa) as combinato- ‘‘stuiver.’’Here,thespecificvaluefunctionofthe
rially entailed. There are virtually an unlimited coins transforms in accordance with their re-
number of ways in which stimuli or events spective contextual cues (verbal descriptions of
might be mutually or combinatorially entailed, their relative value) and independent of any
and many of these may result in comparisons direct training or reinforcement addressing the
other than sameness. Nevertheless, in all cases, monetary systems involved.
mutually and combinatorially entailed relations Inthesamewaythataparticulardiscriminative
constitute a relational network, and when such stimulus may control a class of unique responses
a network has particular stimulus functions, the in alternating contexts, a particular rule that
functions of other events in that network may describesacontextmayalteraclassofresponses,
transform or alter in accordance with the but only in the context defined by the rule.
derived relations. Informingachildthatwhensheisinaparticular
country, coins are the same as or the opposite of
Transformation of Stimulus Functions similar coins in her native land may control her
The concept of transformation of stimulus preference for those coins, but only for the
function refers to the changes that occur in the duration of her stay in that country. The rules
behavioral functions of stimuli based on their function as a form of instructional control, but
participation in a particular relational network. onlyconditionally.Althoughtheaboveaccountis
In the present study, transformation of stimulus only a theoretical example, it has practical
function addresses how networks of newly implications for training several types of mathe-
learned relations may become preferred over matical relations. The current study focused on
others. As a practical matter, consider the the transformation of preference functions for
following brief example from Stewart, Barnes- specific relational or mathematical networks.
Holmes, Hayes, and Lipkens (2001). Perhaps
Transformation of Mathematical Functions
an English-speaking child has learned that
a nickel has a particular type of financial value When graphs of functions change by the
addition (or subtraction) of terms within
in relation to a dime,
formulas for those functions, they are said to
and furthermore has learned the arbitrary compar-
transform. Collectively, these techniques are
ativerelationofsize,suchthatwhenofferedanickel
or a dime she will avoid the physically larger nickel often described as transformations of graphs of
topickthearbitrarily‘‘larger’’dime.Ifwhenvisiting functions(Larson&Hostetler,2001).Critically,
theNetherlandsthechildisprovidedcontextualcues
the concept of relating relations may be
for the derivation of a relation between a ‘‘stuiver’’
important in developing a behavior-analytic
anda‘‘dubbeltje,’’thatisanalogoustothatbetween
anickelandadime(e.g.,‘‘stuiveristodubbeltjeas understanding of the transformation of math-
nickel is to dime’’) then she may now derive that ematical functions. When a student successfully
a‘‘stuiver’’ishalfthevalueofa‘‘dubbeltje.’’(p.77)
learns to relate a particular graph to a particular
Conversely, if in the above example, the child formula, for example, this involves more than
were to be informed (incorrectly) that the simply relating two discrete stimulus events.
302 CHRIS NINNESS et al.
Both the graph and the formula are composed (Ninness, Rumph, McCuller, Harrison, et al.,
of multiple stimuli that relate to each other in 2005), participants took part in an MTS
specific ways, so relating graph to formula thus procedure in which they received training on
involves relating one set of relations to another particular standard formula-to-factored formula
set of relations. andfactoredformula-to-graphrelationsasthese
For instance, many functions have graphs formulas pertain to reflections and vertical and
that interrelate in their various types of vertical horizontal shifts. Participants demonstrated
and horizontal shifts, reflections, compressions, combinatorial entailment by identifying stan-
andstretches.Asoneexampleofamathematical dard formula-to-graph relations and showed
transformation, the addition of a positive high levels of accuracy in identifying 40 novel
constant inside of the argument shifts the transformations of graphs of mathematical
functiontotheleft(negativedirection),whereas functions. The fact that the participants were
the subtraction of a constant inside of the capable of identifying novel transformations
argument produces a shift to the right (positive indicated that they had not simply learned to
direction). For many students, horizontal relate formula and graphs as discrete stimulus
transformations run contrary to expectation, events. Instead, they had learned to relate the
and many students have difficulty learning to relations contained within the graphs and
identify graphed representations of such trans- formulas to each other, and this relating of
formations when displayed individually or in relations was then applied to the novel trans-
multiple combinations of horizontal and verti- formation problems. The present study sought
cal shifts. Pilot research in our laboratory to extend this research.
suggests that horizontal shifts are especially In most mathematical textbooks, equations
difficult for many students when the argument of functions are illustrated in standard form,
entails a negative xpcoefficient and a negative starting with the highest degree term and
constant (e.g., y ~ ffi{ffiffiffiffixffiffiffiffi{ffiffiffiffiffiffi4ffiffiffi). However, our continuing with terms in descending order.
previous investigations have demonstrated that
(Dependingonthefunction,sometextsreferto
training students to factor via matching-to-
this as the general form of the equation.) In
sample (MTS) procedures allowed them to
contrast, factored equations are written as the
learn horizontal shifts more efficiently and to
products of lower degree expressions, and they
derive more complex transformations for
are most often used for explanatory purposes or
a much wider range of formulas and graphed
to demonstrate the derivation of various
analogues.
mathematical relations (Sullivan, 2002).
For example, Ninness, Rumph, McCuller,
To delineate the present study from our
Vasquez, et al. (2005) employed a series of
previous work in this area, in the present
MTS procedures in training formula-to-graph
investigation we attempted to expand our
relations. Participants were tested on 36 novel analysis to include novel graph-to-formula
variations of the original equations, and most relations as well as the transformation of
demonstrated substantially improved perfor- stimulus functions of formulas. Unlike most
mance. Participant error patterns were identi- laboratory research in the area of derived
fiedwiththehelpofanartificialneuralnetwork stimulus relations (e.g., Dougher & Markham,
system called the self-organizing map (Koho- 1994; O’Hora, Roche, Barnes-Holmes, &
nen, 2001). Subsequently, revised software was Smeets, 2002), the current investigation at-
developed to remediate errors, and participants tempted to address contextual control by
showed improved performances in identifying manipulating verbal instructions that address
mathematical relations. In a follow-up study novel mathematical relations. First, as in our
TRANSFORMATIONS OF FUNCTIONS 303
previous research (Ninness, Rumph, McCuller, were undergraduates or graduate students at
Harrison, et al., 2005; Ninness, Rumph, Stephen F. Austin State University.Participants
McCuller, Vasquez, et al., 2005), participants were recruited from various academic programs
were given a brief lecture, followed by comput- by way of prearranged agreements with profes-
er-interactive training on the relations between sorstoofferthisopportunitytotheirclasses.All
standard formulas and more easily understood participants received five test points on their
factored formulas as well as factored formulas final examination for taking part in the
and their graphical representations. MTS pro- experiment. There were 70 tests of novel
cedures were used to assess mutual entailment relations in total. Each participant was paid
and combinatorial entailment, and participants a maximum of $8.00 for the entire experiment,
wereprobedoveranarrayofnovelandcomplex with 10 cents per correct response during the
formula-to-graph relations. However, unlike assessment of novel relations (i.e., the first 60
any of our previous studies in this area, items) and 20 cents per correct response during
participants also were probed on 10 novel the last10trials.Aftercompleting thestudy,all
graph-to-formula relations, in which two of the participants were debriefed and compensated
six comparison items were correct. One of the according to the number of correct responses
correct formulas was arranged in standard form they provided in the assessment of novel
and the other in factored form. Then, we formula-to-graph and graph-to-formula rela-
attempted to transform participants’ preference tions. All sessions were conducted in rooms of
for particular forms of correct answers by the university that remained free of external
informing them that mathematicians prefer to diversions throughout the course of the exper-
express equations in standard rather than iment.
factored form. Finally, we implemented a con-
trol condition in which the experimental Apparatus and Software
contingencies were altered by informing partic- Participantswereseatedindividuallyatatable
ipants that they would receive increased finan- in a small session room containing a Hewlett
cial reward for selecting correct equations Packard Pavilion ze5170 (Pentium 4 2-GHz
represented in the factored form rather than processorwith512MBRAM)laptopcomputer
the standard form. thatdisplayedbothmathematicalequationsand
graphicalrepresentationsofequationsonawhite
background. A Labtec AM 252 microphone
METHOD
adjacent to each computer was conspicuously
Participants and Setting
attached to a side port of the computer.
After obtaining informed consent and ad- Training and MTS procedures and the re-
ministering a pretest to determine individuals’ cording of responses were controlled by the
familiaritywithvariousmathematicalfunctions, computer program, which was written by the
the experimenters dismissed anyone who dem- first author in Visual Basic 6 and C++. The
onstrated any prior knowledge of transforma- software provided math instructional tutorials,
tion of mathematical functions. Correctly displayed graphs, and recorded the accuracy of
answering more than three of 15 pretest items responses throughout all phases of the study.
precluded an individual from participating in The experimental sessions were conducted on
the experiment. the laptop with an attached infrared mouse.
Ten participants (9 women and 1 man),
ranging in age from 19 to 35 years, began the Design and Procedure
experiment, but 2 did not complete the entire After a brief pretraining presentation (Stage
training and testing sequence. All participants 1),theparticipantwastakentothecomputerin
304 CHRIS NINNESS et al.
anindividualtreatmentroomforStage2,where the experimenter read rules regarding square
computer-interactive training was conducted. root operations from the screen and answered
As in Ninness, Rumph, McCuller, Harrison, et questions, but only to the extent that the
al. (2005), participants were trained and tested questions pertained directly to sample-to-com-
onA-BandB-Crelationsandtheassessmentof parison items(A-B or B-C). The relationsof B-
mutually entailed (B-A and C-B) and combi- A, C-B, A-C, and C-A were not addressed.
natoriallyentailed(A-CandC-A)relations.The Step 1: Provide A-B rules. In pretraining A-B
computer program then assessed the participant rules,theexperimenterexplainedthatanegative
on 40 novel relations between formulas and the coefficient of x inside the argument of the
graphs of these functions. standard form of a formula often makes
At the conclusion of the assessment of novel graphing the function more complicated and
formula-to-graph relations, participants’ prefer- that the standard form of the formula can be
ences for factored and standard forms of the factored to remove any negative signs that
equations were assessed. In this phase, we preceded the x variable. It was explained that
attemptedtoproduce an experimentalanalogue factoring the standard formula makes it more
of the transformation of stimulus functions, or conducive to graphing techniques.
change in formula preference, by way of In this narrative account of A-B relations,
providing contextual cues in the form of verbal standard formulas were samples, and factored
rules that described a mathematical preference formulas served as comparisons. For instance,
for particular types of formulas over others. In participants were shown a basic square root
attempting to examine the transformation of function in its standard form and how to
stimulus functions, rules were used to establish express it when a negative one coefficient is
contextual control over the discrimination of factoredoutoftheargument.Inthepretraining
correct formulas from the comparison array of and computer training phases, we used the
incorrectformulas.Participantswereexposedto words ‘‘negative sign’’ rather than ‘‘21 co-
comparison items with four incorrect formulas efficient’’ because most of our participants had
and two correct formulas. One of the correct no familiarity with the latter term.
formulas was represented in standard form and Step 2: Provide B-C rules. Because partici-
the other in factored form. After participants pants were unfamiliar with the Cartesian
respondedto10graph-to-formulacomparisons, coordinate system, the experimenter noted that
they were given verbal rules indicating that the horizontal number line is identified as the x
equations of graphs are preferred when they are axis and that the vertical number line is called
represented in their standard form. Subsequent the y axis. Participants were told that various
to responding to a set of 10 graph-to-formula types of formulas were called functions and
comparisons, we implemented a control condi- could be employed to produce graphs. Using
tion in which we told participants that they PowerPointH slides, the experimenter read rules
would receive supplemental reinforcement explainingthatanegativesigninsidetheradical
(20 cents rather than 10 cents) for selecting or the parentheses reflects the graph over in
the correct comparison formula in factored (about) the y axis and that a negative sign
form rather than standard form. outsidetheradicalortheparenthesesreflectsthe
Stage 1: Pretraining on basic mathematical graph down in (about) the x axis. Likewise,
relations.Theexperimenterprovidedapretrain- aconstantvalueaddedorsubtractedoutsidethe
ing lecture to participants individually using radicalortheparenthesesmovesthegraphupor
PowerPointH illustrations of the rectangular down the y axis in the same direction as
coordinate system. As illustrated in Figure 1, indicatedbythesign.Theillustrationsandrules
TRANSFORMATIONS OF FUNCTIONS 305
Figure 1. Stage 1 included giving rules and rules with exemplars by way of a PowerPointH presentation. Stage 2
involvedcomputer-interactive conditional discrimination MTS training.
provided on the slide presentation were identi- Here, B represented the factored formula, and
cal to those of the computer-interactive MTS C was one of six possible graphical representa-
procedures that followed this presentation. tions of B. Again, using the laser pointer, the
Step 3: Provide A-B exemplar. Participants experimenter identified the correct comparison
wereexposedtoaninstanceofanA-Btestscreen. itemdisplayedonthescreen.(TheA-BandB-C
In this illustration, A was represented as PowerPointH illustrations were not the same as
a standard square root formula and B as one of those used in Stage 2 for the actual computer-
six factored square root formulas, or compar- interactive training.)
isons. The experimenter told participants that As in Ninness, Rumph, McCuller, Harrison,
the same type of screen image would be used et al. (2005), all participants were given the
duringthecomputer-interactivetrainingsession. same15-minpretrainingpresentationregarding
Usingahandheldlaserpointer,theexperimenter the basics of the rectangular coordinate system
identified the correct comparison item. and the relation between the square root
Step 4: Provide B-C exemplar. Participants formula and its graphical representation. Fol-
were exposed to a slide of a B-C test screen. lowing the presentations, participants were
306 CHRIS NINNESS et al.
escorted to the session room, and the experi- posed the participant to the entire MTS
menter demonstrated a point-and-click re- protocol, starting from the beginning.
sponse on a sample screen and made sure that Step 1: Train A-B relations, test A-B. As in
the participants could perform this type of Ninness, Rumph, McCuller, Harrison, et al.
response independently. Participants were told (2005), computer-interactive training of A-B
that mathematical rules would be posted on included the following details. As the program
various screens and were directed to read these displayed a mathematical rule on each training
rules into the microphone adjacent to the screen, the computer instructed the participant
computer each time the program instructed to read this rule into the microphone. For
them to do so. example, the participant read a rule indicating
Stage 2: Training and testing of mathematical that the standard form of the square root
relations. Before beginning the experiment, all formula can be factored to remove any negative
participants were advised that at various points signs that precede the x variable. After reciting
duringthesession,thecomputerwouldinstruct each rule twice, the participant clicked ‘‘next’’
them to stop responding. At that time, they to advance to the screen that assessed A-B
were to contact the experimenter, who was performance (see Appendix A). Unlike the
located in an adjacent office. Stage 2 involved previous training procedures employed in
conditional discrimination MTS training; the Ninness, Rumph, McCuller, Harrison, et al.
program presented the participants with visual (2005),thecomputerprogramdidnotgenerate
displays in the form of mathematical rules and any audio output expressing the rules displayed
emphasized that these rules should be stated on the screen, nor did we attempt to record
aloud by reading them into a microphone. participants’ reading of the rules displayed on
Thus, participants read and practiced the the computer screen.
behaviors specified by mathematical rules. After Step 2: Train B-C relations, test B-C. The
training and assessing standard formula-to- same MTS procedure was used to train and test
factored-formula (A-B) relations and factored- B-Crelationsbutfocusedonrulespertainingto
formula-to-graph (B-C) relations, participants factored equations and their graphs. Rules
were tested for mutually entailed formula-to- regarding reflections and vertical andhorizontal
formula (B-A) relations and graph-to-formula shifts associated with various functions on the
(C-B) relations. Then, the combinatorially coordinate system were displayed and then read
entailed relations between the standard form aloud by participants. For example, the screen
of the formulas and their respective graphs (A- displayed solid blue lines to represent the basic
C), as well as the graphs of the functions and square root functions and dashed red lines to
the original standard formulas (C-A), were illustrate transformations of the square root
tested. functions that occurred when the formulas
For samples and comparisons, formula size changed. These rules described mathematical
was set at 24 using Times New Roman font. relations such as, ‘‘Negative sign inside the
When displayed as samples, graphs were radical is reflected over in the y axis. A positive
approximately 2 in. square on the computer constant inside the radical or the parentheses
screen;however,whendisplayedascomparisons moves the function in the opposite direction
in groups of six, they were reduced to along the x axis.’’ (The program used the words
approximately 1.5 in. square each. On any ‘‘over in’’ rather than the conventional ‘‘about’’
occasion during which a participant emitted an based on pilot testing feedback that indicated
incorrect response, the program randomized all thisphrasingwaseasierformanyparticipantsto
comparison elements on all screens and reex- understand.SeeAppendixBforanillustration.)
TRANSFORMATIONS OF FUNCTIONS 307
After these rules were displayed twice, the
participant clicked ‘‘next’’ to advance to the
screenthatassessedB-Cperformance.Although
the assessment screens varied the specific values
oftheconstantswithintheformulas,thecorrect
answers were always obtainable by performing
in accordance with the previously displayed
rules.
Step 3: Test B-A, test C-B, test A-C, and test
C-A. This step assessed the mutually entailed
B-A and C-B relations and the combinatorially
entailed relations between the graphs of the
functions and the original standard equations
Figure 2. A square root function where A2 is in
(A-C).Also,thecombinatorialrelationsbetween standard form, B2 is in factored form, and C2 is the
the standard forms of the formulas and their graphical representation of thisfunction.
graphs (C-A) were tested (see Appendix C).
All mathematical rules, formulas, and graph- ended, and the participant was compensated,
ical representations for horizontal and vertical debriefed, and excused from the study.
shiftsaswellasforreflectionsinthexaxisandy Assessment of novel formula-to-graph relations.
axis were trained according to this protocol. In After demonstrating mutual entailment and
total, the program trained and tested two combinatorial entailment on both versions of
versions of the square root function and two the square root functions and both versions of
versions of the common logarithmic function. thecommonlogarithmicfunctions,participants
Figure 2 illustrates the trained and assessed weretestedonnovelrelationsbetweenformulas
relations addressing the log function A2, B2, and their graphs—specifically, 40 complex
and C2. Following the trained (A-B and B-C) variations of the trained mathematical formu-
relations, the program tested the mutually la-to-graph relations. For each test item, the
entailed (B-A and C-B) relations and combina- participant attempted to match a new formula
torially entailed (A-C and C-A) relations. with a graph from a selection of six graphs not
Correction procedures and mastery criteria. previously used during any training and
Mastery of the basic mathematical relations assessment screens. (Within any MTS test, the
required the participant to complete one error- comparison graphs were all the same type of
less sequence of all four classes of related mathematical function; for instance, a sine
mathematical functions. This included six formula was only compared against variations
MTS tests (A-B, B-C, B-A, C-B, A-C, and of sine graphs.) No accuracy feedback or
C-A) within each class of functions. Therefore, reexposure to training was given during assess-
mastery required 24 consecutive correct identi- ment of novel formula-to-graph relations. This
fications of the formula-to-graph and graph-to- task required participants to identify graphs
formula relations. Participants who made MTS relating to formulas with new combinations of
errors during the assessment of relations were positive and negative constants inside and
returned to the beginning of the program; the outside the arguments of 40 new mathematical
program randomized all comparison elements functions. These novel formula-to-graph assess-
before reexposing participants to the MTS ments included multiple combinations of
protocol. If a participant required more than reflections and vertical and horizontal shifts
five reexposures to the protocol, the program for complex logarithmic, square root, exponen-
308 CHRIS NINNESS et al.
Figure3. Oneofthe40testsofnovelformula-to-graphrelations.Thesolidlinesrepresentthebasiccubefunction(y
5x3),andthedashedlinesindicatethepossibletransformationwhentheformulabecomesmorecomplex.Aparticipant
whoidentifiedanovelvariationoftheformulathatincludedanegativeconstant4withintheargumentandanegative
constant 4 followingthe argument would select Casthe correct comparison item.
tial, square, cubic, tangent, and sine functions. Figure 3 illustrates one of the 40 novel
PerformingtheseMTStasksdidnotrequirethe formula-to-graph tests. The solid line on each
participant to become familiar with the dynam- of the six comparison graphs represents the
ics of each type of mathematical function. basic function prior to transformation. The
However, successful performance did require dashed line shows the graph produced by the
the participant to identify new and complex novel formula. When given the formula for
combinations of reflections and shifts as they a cube function with a negative constant 4
occurred among a wide array of diversified inside the argument (shifting right) and a neg-
functions. In effect, this task involved relating ative 4 outside the argument (shifting down),
specific relations expressed within the formula participants who answered correctly chose C as
to specific relations illustrated within the the mathematical transformation of this func-
graphs. tion.