Mining Mathematics in Textbook Lessons Erlina Ronda & Jill Adler

20 Pages·2016·0.88 MB·English

Mining Mathematics in Textbook Lessons
Erlina Ronda & Jill Adler
International Journal of Science and
Mathematics Education

ISSN 1571-0068

Int J of Sci and Math Educ
DOI 10.1007/s10763-016-9738-6
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IntJofSciandMathEduc
DOI10.1007/s10763-016-9738-6
Mining Mathematics in Textbook Lessons
ErlinaRonda1,2 &JillAdler2
Received:9May2015/Accepted:2March2016
#MinistryofScienceandTechnology,Taiwan2016
Abstract In this paper, we propose an analytic tool for describing the mathematics
made available to learn in a ‘textbook lesson’. The tool is an adaptation of the
MathematicsDiscourseinInstruction(MDI)analytictoolthatwedevelopedtoanalyze
whatismadeavailabletolearninteachers’lessons.Ourmotivationtoadapttheuseof
the MDI analytic framework to textbooks is to test the relative robustness of the
frameworkinmovingacrossdifferentpedagogictexts(e.g.videoofalesson,atextbook
lesson). Our initial findings suggest it has applicability across pedagogic texts, thus
opening possibilities for using a commonframework and languageinresearchand in
professionaldevelopmentactivitiesinvolvingthewrittenandenactedcurricula.
Keywords Analyticframework.Curriculumstudies.Mathematicsdiscourse.
Opportunitiestolearn.Socio-culturaltheory.Textbooksstudies
Introduction
Textbooksareinstructionaltextswrittentoteachtheirusers.Assuch,textbookshavesome
similaritytoclassroomlessons—ofcourse,withoutpossibilitiesforactual(asopposedto
imagined)contributionsfromlearnersinthedevelopmentofthetexts.TIMSS1textbook
study defines a textbook lesson as Ba segment of text material devoted to a single main
mathematicalorscientifictopicintendedtocorrespondtoateacher’sclassroomlessonon
that topic taught over one to three instructional periods^ (Valverde, Bianchi, Wolfe,
Schmidt&Houang,2002,p.139).Itthuscorrespondstoaunitofachapterintextbooks.
Textbook lessons, like classroom lessons, use a range of examples and tasks and
1TrendsinMathematicsandScienceStudy
* ErlinaRonda
[email protected]
1 NationalInstituteforScienceandMathematicsEducationDevelopment,Universityofthe
Philippines,Quezon,Philippines
2 UniversityoftheWitwatersrand,Johannesburg,SouthAfrica
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E.Ronda,J.Adler
accompanyingexplanatorytextintheformofanalogies,illustrations,definitions,etc.,to
mediatethemathematics.Startingfromthesesimilaritiesinpurposeandoftheelementsin
ateacher’sandatextbooklesson,weargueinthispaperthattheanalytictoolwedeveloped
toinvestigatewhatismadepossibletolearninteachers’lessons(Adler&Ronda,2015)
maybeadaptedtoinvestigatesuchaffordancesintextbooklessons.
OurtextbookstudyispartofWitsMathsConnect-Secondary(WMCS)Project—a
researchandprofessionaldevelopmentprojectsupportingmathematicsteachersinten
disadvantaged schoolsinSouthAfrica.2Arelatedstudyinthe projectonthe teacher-
textbookrelationshiphasfoundthatthisrelationshipis‘weak’,andthatwhiletextbook
useisevidentinteachers’lessons,selectionsarenotdeliberateandsuggestlimitations
in teachers’ awareness of the affordances of the textbooks (Leshota, 2015). Leshota’s
(2015)studyfocusedonasmallsampleofteachersinsomeoftheWMCSschools.Her
findings nevertheless suggest the need for a tool or framework that could be used to
makevisibletheseaffordances.
Visibilisingwhatismadepossibletolearninatextbooklessonhasimportbeyondits
useandrelationshipwiththeteacher,asitisalsoatextthatsupportslearners’learning.
Inourinitialprofessionaldevelopmentworkwithteachers,weobserveddisconnections
andincoherenceinsomeoftheteachers’lessons(Venkat&Adler,2012),afunctionat
timesofcolloquiallanguageusedandnon-mathematicalsubstantiationsformathemat-
ical concepts and procedures (Adler & Venkat, 2014; Adler & Ronda, 2016). We
arguedthatsuchmathematicaldiscourseininstructionlimitspossibilitiesforlearnersto
reproducecorrectandvaluedmathematicsontheirown.Whatthenofthemathematics
teaching in textbook lessons? In what ways does the mathematics discourse in text-
booksopenorcloseopportunitiestolearnmathematics?Whatkindoflensmaybeused
tovisibilisetheseaffordancesorconstraints?
Ourmotivation to putour MathematicalDiscourse inInstruction (MDI) analytic tool
developed for classroom lesson analysis to work on textbook lessons is, as suggested
earlier,totesttherelativerobustnessoftheMDItoolinmovingacrossdifferentpedagogic
texts (i.e. from videotape of a lesson to a textbook lesson). This feeds into our goal of
developing a common language and framework for use across multiple contexts of
mathematicsteachingandteacherlearningintheprojectandmorespecificallyforuseas
adiscursiveresourceorartifactinprofessionaldevelopmentactivities.Whilewehavenot
testedtheanalytictoolwereporthereonawiderangeoftextbooklessonsandonlyonsome
ofthetopicsthatcorrespondstoourteachers’lessons,ourinitialanalysissuggeststhatthe
MDI analytic tool has applicability across pedagogic texts, albeit with some adaptations
following from the engagement with the empirical. Our analysis also suggests that the
adaptedMDIanalytictoolhasthepotentialtoserveasatoolforcomparingtextbooksand
asapracticaltoolforteacherstoexaminewhatthetextbookoffers—potential,thereforeasa
resourceforteacherstousetheirtextbooksmoredeliberatelyintheirinstruction.Weofferit
hereforfurtherdevelopmentanduseacrosscommunitiesofresearchersandteachers.
We begin with a review of research on mathematics textbooks, so as to locate our
textbook work in this wider field. Following a brief description of MDI, we then
presenthowwehaveadaptedittodescribethemathematicsmadeavailabletolearnina
textbook.Finally,weillustratehowwehaveusedtheadaptedframeworkontextbook
lessonsonquadraticinequalityand,throughthis,argueforitssalience.
2FurtherdetailonWMCSisavailableatwww.wits.ac.za/WitsMathsConnect.
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MiningMathematicsinTextbookLessons
Studies in Textbook Analysis
Asweknowfrompreviousresearch,textbooksareubiquitousinmathematicsteaching
and learning (Askew, Hodgen, Hossain & Bretscher, 2010). It remains the most
available resource particularly in developing country contexts (Nagao, Rogan &
Magno,2007)includingSouthAfrica,whereteachingandlearningresourceprovision
isconstrained (e.g. Adler, 2000).A few studies related totextbooks and mathematics
teachingandlearninghavebeenconductedinSouthAfrica(e.g.Bowie,2013;Fleisch,
Taylor, Herholdt& Sapire, 2011; Leshota,2015) and contributetoa growing poolof
international research in this field. In their recent review of mathematics textbook
research,Fan,ZhuandMiao(2013)showapredominanceofcontentanalysisstudies,
focusedtypicallyonaparticulartopicandgearedtowardscomparisonacrosstextbooks
withinandacrosscountrycontexts.Aswithothermathematicscontentanalysisstudies,
the particulartopic has influencedthe analyticframework developed for the study. In
Bowie’s (2013) study, for example,the distinction between differentkinds of geome-
tries(deductiveandvisual)wascentraltotheanalysis.Similarly,manyoftheinterna-
tional studies that involve analysis ofmathematical content are topic specificand use
topic-specificanalytictools(e.g.Dole&Shield,2008;Reys,Reys&Koyama,1996;
Shield&Dole,2013).
There are content analysisstudies thatare not topic specific, but ratherfocused on
mathematical practices, such as opportunities for abstraction (Yang, 2013), justifica-
tions and explanations (Dolev & Even, 2013) and proofs and proving (Stacey &
Vincent, 2009; Stylianides, 2009), which need to be factored in when describing the
mathematics made available to learn in textbooks. The analytic frames developed for
these,whilenottopicspecific,are‘practice’specific.Eachandallofthesestudieshave
enhanced our understanding of different analytic approaches to various topics and
practices in textbooks and what these enable and constrain. Our interest and need,
however, are for a more generic tool, one that may be used to illuminate the mathe-
maticsmadepossibletolearnintextbooks,whateverthetopic,andincludingparticular
practices.
Newton(2012),workingwithSfard’s(2008)notionofmathematicsasaspecialtypeof
discourse,usedagenericframeworktoanalysemathematicsinaparticulartextbook.Her
study engaged with the issue of how to compare the mathematics offered in written
curriculummaterials(likeatextbook)withthemathematicsofferedintheenactmentof
those materials in classroom practice. She argued that an orientation to both texts as
discoursesenabledonetoexamineandcomparethemathematicsdiscourseproducedin
each.Newton(2012)compared howmathematicalobjectsandtheirsignifiersarecom-
municatedinthesecurricularcontextsusingfractions/rationalnumbersassamplecontent.
OurstudyofmathematicsintextbooksalignswithNewton’sworkandSfard’sview
ofmathematicsasaspecialisedformofdiscourse.ForSfard(2008):
[mathematical discourses] are made distinct by their tools, that is, words and
visualmeans,andbytheformandoutcomesoftheirprocesses,thatis,theroutines
andendorsednarrativesthattheyproduce(emphasisintheoriginal).(p.161)
Itfollowsfromthisviewofmathematicsasadiscourse,withitsspecialisedtoolsand
processes, that to learn mathematics means to participate in and with these tools and
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processes.Weconcurandarguethattoteachmathematicsistocreateopportunitiesfor
learners to participate in this discourse. How then are these opportunities created in
mathematics instruction? The MDI analytic framework has developed out of the
question:Whatarekeytoolsandprocessesofinstructionaldiscourse?Withourbroader
aim in the project to develop a common language and tool to support teaching and
studyinstruction,ourtoolisbuiltfromcommonelementsofmathematicsinstruction.
Teachersdesignmathematicslessonsintermsofexamplesandtasksandhowtheywill
explainandsonameandlegitimatewhattheysayordointeachingmathematics.We
described these features as cultural tools that mediate mathematics in classroom
instruction,andweusetheminthispapertomineformathematicsintextbookslessons.
Each of them creates particular opportunities for engaging the different aspects and
featuresofmathematicaldiscourse.
The MDI Framework
TheMDIFrameworkisasocio-culturalframeworkthatarosefromourresearch-linked
professional development project focused on developing teachers’ mathematical dis-
courseininstruction(Adler&Ronda,2015).Theframeworkcharacterisestheteaching
of mathematics as about mediating an object of learning (Marton & Tsui, 2004) via
exemplificationandtheaccompanyingexplanatorytalk—twocommonplacepractices
that work together with the opportunities provided for learners to participate in
mathematicsdiscourse.
One ofthe waysweuse theMDI framework istodescribeteachers’ mathematical
discourse in instruction with respect to the degree to which it makes possible the
development of scientific concepts (Vygotsky, 1978) which aligns with Sfard’ (2008)
notionofanBobjectifiedfull-fledgemathematicaldiscourse^(p.289).Tothisend,we
developedtheMDIanalytictool(Adler&Ronda,2015)whichenablesustodescribe:
(a) whether and how the examples in a lesson and (b) the tasks in which they are
embeddedaccumulatetowardsgenerality,(c)theformaland/orinformalnamingofthe
mathematicalcontent,(d)whetherandhowthecriteriausedtolegitimatewhatcounts
as mathematics enables the mediation of mathematics as coherent and systematic
knowledgeand(e)thenatureoflearners’participationinthediscourse.
Inthenextsection,weexplaintheseconstituentconstructsoftheMDIAnalytictool
(excludinglearnerparticipation)anddescribehowwearrivedatindicatorsandadapted
themfortextbookanalysis.
Adapting the MDI Analytic Tool for Textbooks Analysis
To analyse the mathematics made possible to learn in a textbook lesson, we initially
usedtheMDIanalytictool,backgroundingthelearnerparticipationcomponentsinceit
is not visible in a written curriculum. What is visible is what learners are invited to
participatein,andthisisseeninthetaskssetandaccompanyingtextsorauthors’talk,
andthesearepartoftheframeworkwepresent.
As the textbook is a different empirical text (from a classroom lesson), we have
adapted our description of some of the indicators while retaining the characteristic
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featuresofthe tool, inparticular, indicatorsthatposit a trajectory towards developing
scientificdiscourseinthetextbook.Asweexplainbelow,eachofthekeyelementsof
instruction((a)–(d)above)foregroundsdifferentelementsofmathematicsdiscoursein
instructionandwethusdealwiththemasseparateanalyticconstructs.
‘Object’ofLearning
As we have argued in our earlier work, learning is always about something and
bringing to the learner what this is, the ‘object’ of learning, is central to the work of
teaching(Adler&Ronda,2015).Thisobjectoflearning,ideallywhichisthefocusof
thelesson,hasbothcontent(bethisconcept,method,relationshipandprocedure)anda
capabilitycomponent(Lo,2012).Inatextbooklesson,theintendedobjectoflearning
maybedeterminedfromthesectiontitlelike‘SolvingQuadraticInequalities’.Here,the
concept in focus in the lesson is quadratic inequalities and the capability that is
expectedofthelearnertodevelop issolvingthese inequalitieswhich entails attention
tothemethodsofsolvingandexpressingthefinalsolutionoranswer.
While an announcement of the object of learning in the title foregrounds to the
textbookreaderthe‘name’ofthemathematicscontentandtherelatedcapabilitythatis
infocusortalkedabout(ideally)inthelesson,thatisallitis—aname.Themeaningsto
beassociatedwiththatnamehaveyettobemediated.Wearguethatthewaytheauthor
uses examples, tasks, words and legitimations affords or constrains opportunities for
learningmathematics.
Examples
Ininstruction,the mathematicalcontent isannounced orvisibilised through examples
orthroughitsname.Inthissection,wefocusourattentionfirstonexamples.Wedefine
example, following Zodik & Zaslavsky’s (2008) definition, as Ba particular case of a
largerclass,fromwhichonecanreasonandgeneralise^(p.165),asaninstantiationof
thecontentinfocus(asin9>2x−3isanexampleofalinearinequality).Examplesare
usuallypresentedinsymbolicformorinvisualformlikedrawingsofparallelograms,
graphsoffunction,etc.Theycanhighlightfeaturesoftheconceptthatisexemplifiedin
alesson.
Our analysis of how examples mediate the object of learning draws more directly
from Variation Theory which posits that the key to better learning involves bringing
attentiontopatternsofvariationamidstinvariance(Marton&Pang,2006).Thus,ifwe
wantlearnerstoattendtoaparticularfeaturecrucialtotheobjectoflearning,weneed
togiveasetofexamplesthatwillforegroundthisfeatureinthelesson.Inalessonon
quadraticinequality,forinstance,anexamplespace(meaning,asetofexamples)may
consist of x2 > 4, x2 < 4 and x2 = 4. Another example space might consist of the
expressions x2 > 4, x2 − 4 > 0, (x − 2)(x + 2) > 0 together with their graphical
representations. These example spaces highlight different features of a quadratic
inequality. The first example space provides opportunities for seeing contrast—for
comparing and contrasting the solutions of the three different relationships between
two numbers or function. The second example space shows how the same inequality
x2>4appearsindifferentrepresentationalsystemsandthushighlightsdifferentways
ofsolvingthem.Thisseeingofsimilaritythusprovidesopportunityforgeneralizingthe
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features of x2 > 4 that remain invariant (in this case, its solution) in the different
representations. If the second example space was expanded to include further
inequalities such as x2 < 6x, x2 + 3x + 1 > 0, then we consider this expanded
example space as providing opportunities for discerning simultaneous dimensions of
variation (in this case, the dimensions that vary are the functions or expressions
involved, the inequality relationships, as well as the corresponding form of
solutions). This would signal a move to higher level of generality for the quadratic
inequality. The three examples spaces just described show variation that involves
contrast (C) (through noticing of difference), generalization (G) (through noticing of
similarity)andfusion(F)respectively(Lo,2012).
As with our analysis of teachers’ lessons (Adler & Ronda, 2015), we found it
necessary toform and describea set ofprogressive indicators for the example spaces
in the textbook lessons as follows: Level 1 if only one pattern of variation is used
throughoutthelesson,Level2iftwodifferentpatternsofvariationareusedandLevel3
ifallthreepatternsofvariationareused.Iftherearenopatternsofvariationthatcanbe
detectedintheexamplespace,thenwecodeitasNONE.NONEdoesnotmeanthatthe
author did not provide any examples. It means that the author did not provide
opportunitiesforlearnerstodiscernkeyfeaturesofthecontent.
Tasks
Wedefinetaskstorefertowhatlearnersareaskedtodowiththeexamples(e.g.graph
y=2x−3;writeafunctionparalleltoy=2x−3,etc.).Asamediationalmeans,atask
is linked with, but different from, an example. Examples are selected to mediate the
object of learning by making visible the feature(s) of the content that are key in
mediating meanings of the object of learning while tasks are designed to mediate the
capabilitieswithrespecttothecontent.
Workingwithvarioustasksrelatedtotheobjectoflearningcanincreaseopportunity
to learn through different experiences of the content. Thus, in the analysis, we
considered not only whether the tasks addressed the capability stated in the object of
learning but also whether the tasks have the potential to engage the learners to make
connections among features of mathematical content. We coded a task as known
procedure/fact(KPF)ifitonlyinvolvesapreviouslylearnedknowledgeand/orproce-
dure associated with the object of learning. For example, in a topic on quadratic
inequality,ataskthatasksthelearnerstowritethevaluesofxwherex2>4isaKPF
task.Itinvolvesinterpretinganexpressionthatisalreadyknownbylearnerswhenthey
reach this point in the curriculum. However, if the task involves the current content
topicorrequireslearnerstoapplytheprocedurethatisbeingintroducedinthecurrent
lesson, then we coded it as Current Topic/Procedure (CTP). For example, a task that
asksthelearnerstosolveaquadraticinequality(giveninalgebraicfrom)usinggraphs
would beclassifiedas CTPtask asthisrequires the learner touse a ‘new’ method of
solving introduced in the lesson. We would also classify the task solve the inequality
x2>4x[oranyinequalitydifferentinform/structurefromtheworkedexamplesection]
as CTP. Tasks that involve making a decision as to the procedure and concepts that
need to be called upon to answer the task or requiring connections between
concepts were coded as application/making connectionstasks(AMC).Anexample
ofanAMC task from Textbook B (Fig. 2) analysed belowisTosolvetheequation
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(x+4)(x−1)=0,weusethefollowingpropertyofzero:ifab=0,thena=0orb=0.
Canweusethispropertyinaninequality?AnotherexamplealsofromTextbookBofan
AMCtaskwouldbeifab>0isitalwaystruethata>0orb>0?
As with the example spaces, after categorizing the tasks, we further developed
indicatorsfordescribingmovementtowardscientificconcepts:Level1—thetextbook
lessonprovidedKPFtasksonly;Level2—thetextbooklessonprovidedCTPtasks,but
noAMCtasks;andLevel3—includesCTPandAMCtasks.
Tasks in the worked example3 section of the textbook lesson require further
discussion, as they are not previously known though individual steps in them might
be, but they also do not require application. As worked examples are provided, and
steps typically illustrated and explained, we code these as KPF. The students’ partic-
ipation inworkingwithexamplesistointerpretthe author’s solution, followingsteps
drawingonpreviouslyknownmathematics.However,theremightbeadditionaltasks
in this section that the author leaves for the learner to work on, and so some form of
application.Inthiscase,theywouldbecodedaccordingly.
Naming/WordUse
We have argued in our earlier paper (Adler & Ronda, 2015) that how we name
mathematical concepts (i.e. the specific words we use) and the way we name
procedures/actions carried out on them focuses learners’ attention in particular ways.
According to Wagner (2015), pointing things out and naming them draws one’s
attention to something in particular and gives that thing a signifier to facilitate com-
munication about it. In our analysis of teachers’ lessons, we examine the use of
mathematical and non-mathematical words to refer to mathematical concepts, the
relationships between them and to the procedures carried out on them. From the
empirical data, we were able to identify types of word use during instruction and
assignedcorrespondinglevelstoshowthedegreeofappropriateandmoreformaluses
of mathematical words and phrases. In our analysis of textbook lessons, as would be
expected,wefoundappropriateuseofmathematicalwordsandsoitwasnecessaryto
further analyze the texts or the author’s talk in terms of how the use of mathematical
wordssupportsthemovetowardformalmathematicaltalk.
Inwrittenmathematicaldiscourse(suchasintextbooks),Bactionsandprocessesare
beingturnedintonouns^(Pimm&Wagner,2003,p.163).Whilenoun-basedtalkisthe
privileged talk in mathematics (Schleppegrell, 2007), this could invite concern about
accessibilityespeciallyfor learnerswhoseexperience with mathematics may be more
about‘action’than‘entities’.InthestudybydeFreitas,Wagner,Esmonde,Knipping,
Lunney Borden and Reid (2012), for example, teachers who noticed the nominalised
talk in textbooks during their workshop proposed to turn the nouns into verbs as
nominalisation is not typical of how they speak in their class and because learners
could find this way of talking difficult. This nominalisation-verbification (de Freitas
et al., 2012) dilemma4 points to the need for noticing how the tension between these
3Noticethatthewordexampleshereisnotthesameastheywaywedefineexampleintheanalytictool
describedintheprevioussection.
4WeusethisterminthesenseofAdler(1999,2001)tomeanthatwhileanalyticallydistinct,andappearasa
dichotomy,therearenoteither-orsintheworkofteaching.Dilemmashavetobe‘managed’.
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different types of ‘talk’ plays out in written curricula, and so whether and how
nominalisation might function in a textbook to make it less accessible to learners, or
on the other hand, whether there is a lack of nominalisation and so limiting opportu-
nities for learner to have exposure to and access to valued ways of talking
mathematically.
The context in which mathematical words are used also matters (Schleppegrell,
2007). As we engaged with the texts, we observed that mathematical words are
sometimes used as labels. For example, a label or stand-alone use of mathe-
matical word would be ‘number line solution’ to label a particular way of
writing the solution of inequality. In our textbook analysis, we coded this type
of words use as L for label.
Mathematical words are also embedded in statements about mathematical proce-
dures (foundintasks and inworked examples) or about the propertiesormeaning of
theconcept.Suchstatementsdiffer.Forexample,(1)Tosolvex2−9>0,lookforall
possible values of x which make the expression x2-9 positive; or Solve the inequality
x2−9>0arestatementsaboutaprocedure,andthewayitisnamedimpliesactionon
the inequality which is to solve. For our purposes here, we call this kind of narrative
aboutproceduresasactiontalk.Anotherbutdifferentwayofsayingthiswouldbe(2)
Tofindthesolutiontox2−9>0,lookforallthepossiblevaluesofxwhichmakethe
expressionx2-9positive.Thisstatement,whilealsoaboutprocedures,focusesattention
onthesolution,anoun,andsoaconceptinitself.Inouranalysis,statementssuchasin
(1) are coded PA (procedure-action) while statements such as in (2) are coded PN
(procedure-noun)
The author’s talk may also be about the meaning of the mathematical concept
directly. For example, the narrative (3) An inequality gives the relationship between
two expressions using the signs > and < is talk about the meaning of inequality.
Anotherexample would be(4)BThesolutiontox2 −9> 0are the valuesofx which
maketheinequalitytrue^isalsotalkaboutthemeaningofthesolutionoftheinequality
andnothowtofindthesolution.Wecallthiskindoftalkoruseofwordasobjecttalk.
However, iftheauthor’s statementissomethinglike(5)The solutionofaninequality
consistsofarangeofvaluesofx,thenwhilethisisalsoobjecttalk,itreferssimplytoa
featureofthesolutionandnotthemeaningofasolution.Wecodedwordusesuchas
thoseinnarratives(3)and(4)asOM(object-meaning)whilethoseinnarrativessuchas
(5)asOF(object-feature).
Sincethemeaningofthewordisderivedfromitsuse(WittgensteininSfard,2008,
p. 73), we privilege the use of mathematical names that are embedded in various
narrativesoverthosewordsthatweresimplyusedaslabels.Furthermore,wearealso
privileging the noun form over the verb form or action-based names as the former is
ultimately required in mathematics. Lastly, we privilege talk about a concept that
defines what it is, over talk focused only on its features. Thus, for level categories,
weusedthefollowingrules:Level1—anyonetypeofworduseispresent(L,PA,PN,
OF,orOM),Level2—atleastanytwoofthecodesarepresentandLevel3—atleast
threeofthecodes(PA,PN,OF,OM)arepresentinthelesson.
Word use shows the manner and extent of formal mathematics talk in the
text particularly the talk on procedures and talk of concepts. What remains to
be investigated that is not ‘seen’ in the discussion above is how concepts and
procedures are substantiated.