Towards a hypothetical learning trajectory for rational number

ORIGINAL ARTICLE

Towards a hypothetical learning trajectoryfor rational number

Vince Wright

Received: 1 August 2012 /Revised: 26 September 2013 /Accepted: 9 December 2013# Mathematics Education Research Group of Australasia, Inc. 2014

Abstract A hypothetical learning trajectory Simon (Journal for Research inMathematics Education, 26(2), 114–145, 1995) based on Kieren’s (1980, 1988,1993, 1995) sub-constructs for rational number was developed and used as a frame-work in a year-long design experiment in a New Zealand Middle School. Instructionalsequences were designed using the trajectory. Case studies of six 12–13-year-oldstudents provided evidence of their progression through phases of the hypotheticallearning trajectory. The patterns of progress for individual students were displayedusing visual maps which revealed considerable variability across the sub-constructs andpredicted growth within sub-constructs. The findings supported the usefulness of thehypothetical learning trajectory as an instructional tool supported by other forms ofpedagogical-content-knowledge. Developmental connections between the sub-constructs were also suggested.

Keywords Hypothetical learning trajectory . Rational number . Pedagogical-contentknowledge

Introduction

In this article, I propose a Hypothetical Learning Trajectory (HLT) that is based onKieren’s (1980, 1988, 1993, 1995) sub-constructs for rational number. Firstly, I discussSimon’s (1995) original meaning of a HLT and subsequent work of other researchers.Secondly, I lay out theoretical and research based evidence in support of the proposedHLT for rational number. Thirdly, I report on the results of a design experiment inwhich the HLT was used as a tool for guiding instruction. The progress of six studentsduring the course of one academic year is mapped against the schemes of the HLT toinvestigate the research question:

Math Ed Res JDOI 10.1007/s13394-014-0117-8

V. Wright (*)Mathematics Education/School of Education/Faculty of Education, Australian Catholic University,Locked Bag 4115, St Patrick’s Campus, Level 4, 115 Victoria Parade, Fitzroy MDC VIC 3065, Australiae-mail: [email protected]: www.acu.edu.au

To what extent is a learners’ progression through phases of the HLT consistentand predictable across learners and to what extent is it variable?

In conclusion, I discuss the implications of the research for learners’ progressionin rational number and the HLT construct.

Hypothetical learning trajectories

The construct of a hypothetical learning trajectory (HLT) was first proposed by Simon(1995). He saw the teacher as a field researcher who modifies activities upon reflectionon their students’ learning. HLTs have three features: learning goals, predicted concep-tual growth paths in a specific domain, and aligned activities. These features have beenkey elements of Japanese lesson study and realistic mathematics education in theNetherlands since the 1970’s (Gravemeijer 2001; Isoda et al. 2007; Van den Heuvel-Panhuizen 2001). While the concept of an HLT is not new, (Baroody et al. 2004)Simon’s work provided a common language that initiated productive research anddebate about support structures for teaching and learning.

Researchers develop HTLs using different methods, including design researchsituated in a small number of classrooms (Bowers et al. 1999; Cobb 2009), in-depthcase studies of individual learners (Olive 1999; Steffe 2003, 2004), and large-scalestudies across many students, schools and classes (Clements et al. 2004; Confrey andMaloney 2010). The deterministic view of conceptual growth suggested in somestudies has been criticised as unrealistic given the considerable variation inconcept development exhibited between students and in the responses of indi-vidual students to different situations (Lesh and Yoon 2004; Watson and Mason2006). Yet, there is evidence associating teachers’ knowledge of progressionframeworks for number with strong gains in student achievement within profes-sional development programmes (Bobis et al. 2005). The place of HLTs as asubset of the knowledge needed for teaching is rarely made explicit (Ball et al.2008; Ball and Bass 2000; Chick et al. 2006; Chick 2007). There is irony in thisnon-connection, given that the value of a HLT is largely in its utility forcurriculum design and teaching (Baroody, et al. 2004).

Issues of structure and implementation

Three related issues about the structure of HLTs are significant models for conceptualdevelopment, the validity of frameworks for progression, and the predictability ofstudents’ learning. Two metaphors dominate the literature on conceptual development,hierarchies and networks (Hiebert and Carpenter 1992). HLTs can present as an ordinalmodel, for example, Steffe and Cobb’s (1988) trajectory for counting types, as anetwork, or as combination of order and connection, for example, Confrey’s (2008)synthesis of rational number progression. The challenge is creating a model of pro-gression that balances the simplicity of linearity with the complexity of key knowledgethat is connected, blended and compressed in the development of concepts. As Baroodyet al. (2004) suggested models that are simplistic can still be informative to teachers.

V. Wright

HTLs usually define progression with schemes that are action structures (Olive andSteffe 2002). Piaget (1985) used schemes extensively in his research to describe thestimulus-motor responses of children in situations. The scheme construct wasreconceptualised extensively by Von Glasersfeld (1989, 1995) as a triadic cognitivestructure connecting situation recognition, goal directed activity and anticipated result.Scheme theory underpins a volume of significant research into children’s numberdevelopment (for example. Steffe 1983; Steffe and Cobb 1998). The modern interpre-tation was captured by Vergnaud (2009, p.88) who described schemes as “the invariantorganisation of activity for a certain class of situations.” His definition implies theconstruction of an abstraction that facilitates transfer across situations and that thelearner knows-to apply their conceptual knowledge to relevant situations, howeverunfamiliar (Mason and Spence 1999). Use of schemes is advantageous in that itprovides a set of behaviours in conjunction with classes of problems, therebyrepresenting increased sophistication in an observable way. Much is assumed in servicein schemes, including goals, beliefs, possession and co-ordination of knowledge(DiSessa 2008), key conceptions (and misconceptions), control in the face of distractorsand reliable application to different contexts. Yet, consistent transfer of concepts bylearners between situations is illusive and unpredictable (Greeno 2006; Lobato 1997,2006; Royer et al. 2005; Siegler 2007). As Carraher et al. (2006) stated, “it would benaive to assume that the challenges are conquered once and for all. We never cease tostumble when confronted with variations of mathematical problems that we havealready encountered” (p. 111).

This evidence points to cognitive and situational variation confounding attempts todefine progression in fine-grained steps that every student moves through in precisesequence or generalisation of progression in particular situations to all other structurallysimilar situations. HLTs are situated in that they reflect the creator’s perspectives onconceptual development, how best to describe progression and their degree of ac-knowledgement of situational variation. Compromises are made by researchers in theinterest of science in an effort to balance complexity and utility. I will now describe therationale for my creation of a HLT for rational number before moving on to the resultsof a design experiment that tested its efficacy. In doing so, I acknowledge myperceptions that influenced the initial design of the HLT and the subsequent research.

Design considerations

In creating a hypothetical learning trajectory for rational number, I adopt a hierarchicalmetaphor while well aware of the rich connections between sub-constructs. The HLT isorganised in broad phases of progression. Schemes are used as the descriptors ofprogression. A functional growth path defined by fine-grained knowledge elementsseems prohibitive given the complexity and connectedness of these elements in rationalnumber. I assume that conceptions and misconceptions, and that task variables, prob-lem types and situations are implied in Vergnaud’s (2009) definition of “class ofsituations”. I also accept the consistency of progression in learners’ schemes in thelong-run, based on previous research (Young-Loveridge and Wright 2002a, b), whileaccepting short-term variation.

Two structural features of the HLT are supported by the literature: the connectionbetween multiplicative thinking with whole numbers, and the usefulness of Kieren’s

Towards a hypothetical learning trajectory for rational number

(1980, 1988, 1993, 1995) sub-constructs as a way to organise the conceptual field.First, the development of multiplicative thinking with whole numbers plays a criticalrole in learners’ understanding of rational number (Dole et al. 2012; Jacob and Willis2003; Kilpatrick et al. 2001; Siemon et al. 2006; Thompson and Saldanha 2003). Thisis particularly true of the ability to recognise division and multiplication relationshipsbetween sets of number pairs, such as y ¼ 1

3 x in the relation {(9,3),(24,8),(15,5),…}.To assume a strictly prerequisite relationship between the development of multiplica-tive thinking with whole numbers and facility with rational number is erroneous sincethe developments occur concurrently and are significantly influenced by opportunitiesto learn and access to productive discourse (Anthony and Walshaw 2007; Lamon2007).

Second, Kieren’s sub-constructs survive the test of scholarly scrutiny and recentadaptations involve applications to varying situations rather than the proposition of newconstructs (Adjiage and Pluvinage 2007; Alatorre 2002; Alatorre and Figueras 2004,2005). Initially, Kieren conceived of five sub-constructs: part-whole, measures, quo-tients, operators and rates/ratios for rational number, though he later integrated part-whole with the other four (Kieren 1995). The constructs might be thought of ascharacteristics of rational numbers activated by learners in applying their schemes.Rich relationships exist between the sub-constructs when they are applied in situations.For example, a well-developed concept of ratios involves regarding the part-wholerelations as measures. Kieren believed that the sub-constructs were built on threeintuitive constructs of equal partitioning, equivalencing and unit-forming. Partitioningrefers to splitting an object or setting it into smaller parts, either equal or non-equal.Equivalencing is making two objects or sets equal in respect of a chosen quantifiableattribute, like number or mass. Unit forming is creating objects or sets, and regardingthem as units that can be reasoned with. Increased sophistication in these contributoryconstructs is prevalent in the literature about progression in rational number (Confreyand Maloney 2010; Confrey and Smith 1994; Lamon 2002; Olive 1999; Resnick andSinger 1993; Steffe 2003, 2004). The proposed HTL for rational number uses Kieren’sfour sub-constructs and assumes the significance of increasingly sophisticated schemesfor reasoning with units, equal partitioning and equivalencing.

Hypothetical learning trajectory

Progression in reasoning with rates and ratios

A range of studies suggests at least four broad phases of progression in the develop-ment of schemes for solving problems with rates and ratios (Alatorre and Figueras2005; Ben-Chaim et al. 1998; Hart 1988; Hart et al. 1981; Kaput and West 1994; Loand Watanabe 1997; Noelting 1980; Steinthorsdottir 2005; Tourniaire and Pulos 1985).These phases are the following:

1. Inappropriate centrations characterised by no application of the ratio or rate, orinappropriate focus on one measure, the total parts (for ratios) or the differencebetween measures. For example, the colour ratio 2:3 (blue:yellow) is judged

V. Wright

incorrectly to be a lighter than 3:5 because it has less measures of blue.Comparison of rates or ratios is possible where one measure is equalised, forexample, 2:3 compared to 2:4.

2. Additive build-up characterised by treating the ratio or rate as a composite unit andreplicating it by repeated addition. For example, 2:3 is replicated to create 4:6 and6:9 which is compared to 6:10 created by replication of 3:5 twice. Comparison ispossible where one measure is the same or the total parts (for ratios) are equal.

3. Multiplicative build-up characterised by use of multiplication or division to abbre-viate the build-up or equi-partitioning of a composite ratio or rate to find the targetratio/rate. For example, 2:3 is equivalent to 16:24 (multiplying by 8) and comparedwith 15:25, created by multiplying each measure of 3:5 by 5 (both have a whole of40 parts).

4. Proportionality characterised by treatment of ratios/rates as iterable and equi-partitionable units, and the flexible use of integral and non-integral operatorswithin measures and between measures. For example, 2:3 is treated as 2

5 blue

which is greater than 38 blue (from 3:5).

Variation and consistency in progression for other sub-constructs

Any claim of uniform progression is tempered by research on situational variation(Adjiage and Pluvinage 2007; Alatorre and Figueras 2005; Panoutsos et al. 2009. Thetension between consistency and variation was captured by Case (1992), Kaput andWest (1994), and Behr et al. (1984) who described the development of proportionalreasoning as initially localised in context but eventually generic in nature.

The four broad phases of progression for rates and ratios, as described above, havesynergy with learners’ schemes documented in the literature about Kieren’s other three sub-constructs. Learners also apply inappropriate centrations on numerators and denominators,and additive relations between numerators and denominators in fraction ordering tasks(Clarke and Roche 2009; Gould 2006; Hart et al. 1981; Pearn and Stephens 2004;Streefland 1993). One view is that whole number thinking with fractions is evidence ofmisconception, another is that it is a sign of learners reorganising their whole numberschemes (Olive 1999; Post et al. 1986; Streefland 1993; Steffe 2003). I hypothesise thatprogression in learning about fractions as measures or quantities mirrors the phases evidentfor rates and ratios.

Similar shifts from additive tomultiplicative thinking are also documented in the literatureon learners’ strategies for quotients (sharing) (Confrey and Maloney 2010) and operators(Post et al. 1993). A phase of recognising co-variation but not being able to consistentlyrelate both dimensions is common to learners’ strategies with fractions (Mitchell and Horne2011) and with inverse rates in balance situations (Jansen and van der Mass 2002).

Proposed trajectory

A HLT for progression across all four sub-constructs is presented in Table 1. It consistsof a two-way table with broad phases of conceptual progression and Kieren’s constructsas the dimensions.


Table1

Hypotheticallearning

trajectory

forrationalnumber

Sub-

construct

Unitform

ing

Unitco-ordination

Equivalence

Com

parison

Measure

Halving-based

splittin

grecognisingmoreequal

partsresults

insm

aller

parts.

Directphysicalcomparison

ofequivalence.

Repeatedequi-partitioning

preserving

equiva-

lence.Non-unitfractio

nscomposedof

itera-

tions

ofunitfractio

ns.R

elatingim

proper

fractio

nsto

wholenumbers.

Equivalentfractio

nsdeterm

ined

byspatial

andmultiplicativerelationships

and

recognised

assameequivalent

quantities.

Com

bining,separating,

andfinding

differenceswith

fractions

(related

denominators).

Measuring

onewith

non-unitfractions.

Orderingof

fractions

usingequivalence,

benchm

arks

andqualitativ

erelationships

betweennumeratorsanddenominators.

Com

bining,separating,

andfinding

differenceswith

fractio

ns.

Measuring

fractio

nswith

fractions.

Operator

Unitfractio

nsof

quantities

byequalsharing,

anticipated

byadditiv

ebuild

-up.

Non-unitfractio

nsof

quantitiesas

iteratio

nsof

unit-fractio

nsusingadditiveandmultiplicative

relatio

nships.

Non-unitfractio

nsof

quantitiesby

multiplicationanddivision,including

fractio

nto

whole.

Recognitio

nof

both

operator

andam

ount

affectingproduct.

Propertiesof

operations

with

whole

numbersappliedto

fractions.

Findingunknow

nfractio

noperator.

Quotient

Practicalequalsharingby

halvingwith

namingof

shares

ascombinatio

nsof

unitfractio

ns.

Practicalequalsharing(openshares)anticipated

byiterationof

parts.

Shares

named

asfractionof

areferent

whole.

Shares

anticipated

andnamed

asfractions,

a�b¼

a b.

Equalandunequalshares

anticipated

where

dividend

ordivisorsareequal.

Com

parisonof

shares

usingequivalence,

benchm

arks

orqualitativerelatio

nships.

Rem

aindersfrom

division

expressedas

fractio

nsin

context.

Rateand

ratio

Practicalreplicationof

rates

andratiosby

doublin

gandadditiv

ebuild-up.

Equivalentratesandratio

santicipated

byreplication.

Part-w

holerelationships

inratiosexpressedas

fractio

ns.

Equivalentratesandratio

santicipated

byscalar

multip

licationanddivision,

includingunitratestrategies

andpartto

partrelatio

nships

(ratios).

Com

parisonof

rates(sam

eandconverted

measures)andratio

s(part:p

art,

part:whole,w

hole:whole)usingwith

inandbetweenrelatio

nships.

Findingequivalent

measuresfrom

inverse

rates.

V. Wright

The column headings of the trajectory are hypothesised by the researcher to conveypotential synergy of thinking reflected in the schemes listed below them. Unit formingrefers to the creation of unit fractions, equal shares, rates and ratios by practical means.At unit co-ordination, a learner anticipates the result of creating non-unit and improperfractions, equal sharing and equivalent ratios, using additive and simple multiplicativethinking, such as doubling and halving. At equivalence, multiplicative relationships areapplied to recognise equivalence and non-equivalence in fractions, equal shares, ratesand ratios, mainly by iteration, and to treat the units formed as composites of twomeasures. At the next phase, comparison, a learner applies equivalence to compare andoperate with fractions, equal shares, rates and ratios, partitioning the relevant units aswell as iterating them.

I acknowledge that the wording of the scheme descriptors in the HLT was alteredpost-study from that in the originally proposed trajectory for better clarity. There was nosignificant change to the meaning of the phases or schemes in the original trajectory.

The development of schemes across all four sub-constructs is unlikely to beconsistently synchronous. Fractions, ratios and rates are different types of quantities(Schwartz 1988; Post, et al. 1993). The HLT refers only to fractions. There areparticular issues associated with the learning of decimal place value (Roche andClarke 2006; Steinle and Stacey 2004), percentages (Lachance and Confrey 2002),and particular representations of proportional relationships such as gradients (Lobatoand Siebert 2002). Despite these limitations, this simple HLT is still potentially usefulin guiding instruction and curriculum design.

Research method

I next discuss the method used to address the research question:To what extent is a learners’ progression through phases of the HLT consistent and

predictable across learners and to what extent is it variable?

Context

A classroom based design experiment in 2007 provided the data for the discussionwhich follows. The study took place in a class of eight students (aged 11–13 years) overan academic year. The school was located in a large rural town in New Zealand and wasselected because I had established a working relationship with teachers at the school inmy role as a consultant. Students generally came from families of moderate to higheconomic status and from a mixture of urban and rural communities.

In keeping with some recent models for design experiments (Ball 2000), I co-taughtthe class alongside their usual teacher for a total of 15 weeks during the course of theyear. At the time, I was an experienced mathematics educator. The class was selectedbecause I had a strong previous working relationship with the teacher and there wasdaily use of digital technology. I planned most instruction by reviewing work samplesand discussions one lesson at a time in keeping with the iterative nature of designexperiments. For most lessons, I taught an introductory session to the whole classfollowed by a rotation in which the students either solved problems in achievement-based groups with either the teacher or me, or worked on tasks independently, in pairs


or threes. Lessons usually concluded with an opportunity for students to share andjustify their computational strategies with the whole class. Instruction involved consid-erable teacher to student and student to student discussion, extensive connected use ofmultiple representations (materials, diagrams, and symbols), use of computer technol-ogy, and a co-operative, investigative approach with considerable discussion. Planningand instruction were strongly influenced by students’ use of schemes in the HLT. So,the data on individual learners’ conceptual development reported here is heavilyinfluenced by the research-based instructional design.

Six students were identified for a case-study group (see Table 2). The group wasrepresentative of the diversity of gender, ethnicity, age and initial achievement levels ofstudents in the class. Initial achievement level was determined by use of standardisedassessment tools, Progressive Achievement Test (PAT) and Assessment tools forteaching and learning (asTTle) (Darr et al. 2006; Hattie et al. 2004; Ministry ofEducation 2003).

Data sources

Multiple data sources were synthesised to evaluate students’ possession of schemesfrom the HLT at the beginning of the year and at the end of terms one, three and four(terms are divisions of the school year of about 10 weeks duration). Due to ill health, Iwas unable to carry out interviews and evaluate student progress at the end of term two.The data sources were interviews, pencil and paper assessments, written work samples,item responses to standardised tests, conversations recorded in the teaching diary, andstudent and student recording in group modelling books.

All students in the class were interviewed at the beginning of term one using theGlobal Strategy Stage (GloSS) Interview (Ministry of Education 2003) to provide dataabout their mental calculation strategies. Four interviews using a teaching protocol(Hershkowitz et al. 2001) were conducted with students in the case-study group duringthe year. Two different interview types were used, “think aloud” in which a studenttalked to the interviewer as they solved the problem for the first time and “rehearseddiscussion” in which a student at first, attempted the problems independently and thenlater, explained their strategies to the interviewer. All interviews were video-taped andtranscribed.

Table 2 Case-study students

Studenta Gender Age (beginning of year, February) Ethnicity Initial achievement

Ben Male 11 years 1 month European NZ Above class average

Rachel Female 12 years 0 months European NZ Above class average

Odette Female 12 years 9 months Maori/European Average for class

Jason Male 12 years 8 months European NZ Average for class

Linda Female 13 years 2 months European NZ Below class average

Simon Male 12 years 9 months Maori/European Above class average

a Pseudonyms

V. Wright

Two norm-referenced standardised tests, PAT and asTTle (Darr et al. 2006; Hattieet al. 2004) were attempted by all students at the beginning and end of the year. Theitem responses for each case-study student were analysed to identify patterns in theunderstanding of the sub-constructs. All work samples that students produced frominstructional group and independent settings were examined to inform the next day ofteaching. Work samples from the students in the case-study group were copied anddated. Notes about case-study students’ scheme development were also written andadded to their individual files on a weekly basis. Daily, I added notes to a teachingdiary. These notes contained reflections on students’ understanding of concepts, eval-uations of instructional approaches and activities, and records of conversations withstudents. Passages pertaining to individual students were extracted and added to theirfiles.

All these data were collected in an individual paper and electronic file for each case-study student. Table 3 shows the sources of data and the times at which the data werecollected.

Cycles of data generation

The complexity and volume of the data necessitated several cycles of data generation.Data in the individual file of each case-study student were sorted by the sub-constructs,part-whole and measures, quotients, operators and rates/ratios. Additional categorieswere created for probability, decimals, percentages and graphs. Naturally, there wasintersection so some data appeared in more than one category. The next cycle requiredtracking the progression in schemes within the HLT over time. I created a two-waylearning trajectory table that correlated phases of the HLT with the sub-constructs andschemes for whole number multiplication and division (Wright 2011). The cell de-scriptors in the table were mostly schemes expressed as key understandings such as“equivalence (of fractions) as multiplicative relation” or “Quotient theorem a� b ¼ a

b ”though some knowledge elements such as “known division facts—common factors”were also included to provide additional detail. The data were analysed to create a tablefor each case-study student at four time points in the year, beginning of term one, end ofterm one, end of term three, and end of term four. Three types of shading were used inthe cells, white for no evidence of the scheme or knowledge, grey for evidence of someuse of the scheme and black for comprehensive evidence of a transferable scheme.

Hypothetical learning trajectory maps

In the final stage, data from the learning trajectory tables were synthesised into agraphical HLT map to represent progressive growth in each student’s schemes at thefour time points described above (see Fig. 1). The concentric hexagons representprogressive phases of the HLTwith progress to a later phase represented by movementto an outer hexagon. The corners of each hexagon represent a phase in one of the sub-constructs: measures, quotients, operators, rates and ratios. Three phases of schemes formultiplication and division schemes for whole numbers were also graphed (top left) toconsider relations between these schemes and the development of rational numberschemes. Schemes for multiplication and division broadly align with phases of skipcounting/repeated addition, knowing and deriving facts using properties of


Table3

Tim

esequencedsourcesof

data

Tim

epoints

Standardised

norm

-referenced

tests

Interviews

Worksamples

Teaching

diary

Modellingbooks

Term

one

Weektwo

PAT,

AsTTle(allstudents)

GloSS

(allstudents)

Interview

one(case-studystudents)

Term

one

Weeks

threeto

six

Case-studystudents

(allwrittenwork)

Researcherobservations

Discussions

with

classteacher

Allstudentteachinggroups

Term

one

Weekseven

Interview

two(case-studystudents)

Term

two

Weeks

threeto

four

Case-studystudents(all

writtenwork)


Discussions

with

classteacher


Term

three

Weekone

Allstudents,exceptB

en,

(writtentest)

Term

three

Weeks

oneto

four


writtenwork)


Discussions

with

classteacher


Term

three

Weekfive

Interview

three(case-study

students)

Term

four

Weeks

threeto

six


writtenwork)


Discussions

with

classteacher


Term

four

Weekseventoeight

PAT,

AsTTle(allstudents)

Interviewfour

(case-studystudents)

V. Wright

multiplication and division, and relational multiplicative thinking. These phases arebased on substantial research (Jacob and Willis 2003; Mulligan and Mitchelmore 1997;Sherwin and Fuson 2005; Young-Loveridge 2006; Young-Loveridge and Wright2002a, b). No further progressive phase is conjectured, so the third largest hexagonof the HLT map represents a limit for multiplication and division strategies with wholenumbers.

The rate and ratio sub-constructs were separated to test a conjecture that studentstreat these concepts differently despite their structural similarity. White, grey, and blackcircles represent students’ breadth and consistency of application for schemes, as takenfrom the HLT tables. So progression from grey (partial possession) to black circles(consistent possession) indicates improvement in breadth and consistency of schemeapplication to different situations. Lines connecting the circles give the profile shape,but do not indicate the state of connection between the sub-constructs.

The map provides a concise view of scheme progression and allows for analysis ofconceptual development of several students over time simultaneously. The representa-tion provides a window for considering the research question about the consistency ofstudent progression through the HLT. I acknowledge some limitations of graphingprogress in this way. There is considerable complexity and breadth to the schemescontained in each cell of the HLT. It was neither possible nor desirable to assess all ofcontributory knowledge elements and skills at given points in time. Some assumptions,based on the data from previous time points, were made about the location of circleswhere new data were not available. No movement of circles outwards occurred unlesssupported by data. Some conceptual growth occurred but was insufficient to warrant achange to darker shading or outward progression. This means that some improvement inbreadth and consistency was unrepresented at the time it first occurred. Similarly, a circlefor a given construct was only located at an earlier phase if evidence of reversion existed.

Results

Simultaneous views of the hypothetical learning trajectory maps

Data representation using the HLT maps revealed features of the progression ofstudents through the phases. Figure 2 provided a simultaneous view of the learningmaps for all six students for four time points during the year. While the progression of

MeasuresMultiplication and Division

Rates Operators

QuotientsRatios

Unit Forming Phase

Unit Co-ordination Phase

Equivalence Phase

Comparison Phase

Indicates partially consistent application

of equivalence in operator situations

Fig. 1 Example of learning trajectory map


Multiplication and Division

Rates

Ratios

Operators

Measures

Quotients

Simon

Ben

Rachel

Odette

Jason

Linda

Beginning End Term 1 End Term 3 End Term 4

Key

Nodata

Consistentapplication

Partialapplication

Fig. 2 Simultaneous views of students’ learning maps

V. Wright

students was uniformly to more sophisticated schemes over time, there was consider-able variation between students in their pattern of progression through phases, theirconnections between sub-constructs and their responses to different situations andrepresentations. Students appeared to move through a consistent sequence of progres-sively more complex schemes within any given sub-construct over time. There wasalso similarity of shape of the maps for the high achieving students, Ben, Rachel andSimon at the end of terms three and four. All three students operated at the comparisonphase for all four rational number sub-constructs, though partially consistent schemeapplication (grey dots) was more prevalent than reliable application (black dots). Twopossible explanations for the shape were teaching effect and the ceiling effect of theHLT in that there was no fifth phase of progression. All three students showed strongtendencies to connect between sub-constructs in their solutions to problems. A morelikely reason for the regularity was that proportional reasoning involves the integrationof the sub-constructs that allows students to draw on multiple views of rational numbersin solving problems in context.

The maps for the middle achievers, Odette and Jason, and the lower achiever,Linda, were different from each other in shape at all four time points. Thesestudents had individualised relative strengths in the sub-constructs initially andtheir progressions of development were variable. The three maps displayed irreg-ularity as some sub-constructs developed without corresponding advance in others.Odette began the year with relative strength in rates and quotients but made thegreatest gains in measures. Linda had poor understanding with all sub-constructsinitially and made considerable progress with rates and operators over the year.Jason began with relative strength in the quotient sub-construct but made thegreatest gains in his understanding of measures.

The evolution of the individual maps of the middle and low achieving studentscontrasted with the greater consistency in progression of the high achievers, Ben,Rachel and Simon. This supported the idea that development of the sub-constructswas variable and unconnected in the early stages then became more generalised andconnected, if only for the higher achieving students. Regularity of the hexagonsindicated that schemes in all sub-constructs were sufficiently developed to enableconnected application in situations, for example, the use of measures to describe theequal shares in quotient situations such as, “Four pizzas shared between five people.”Linda, the lowest achiever, had the most irregular map and had the only instance ofregression as previously learned knowledge of a sub-construct was absent later.Irregularity suggested the possibility that a different synchronicity of schemes was abetter fit to the data. There was no clear pattern in the emerging shapes of the maps forOdette, Jason and Linda to support this contention. A feature of all six maps was thatconsistent application of a scheme to all applicable situations was not a necessarycondition for the student developing more sophisticated schemes in the same sub-construct later. This showed as instances where outward movement was preceded bygrey dots at less sophisticated schemes. Consistent application of a scheme to allpresented situations (black dots) usually preceded progression to a more sophisticatedscheme shortly after, but, sometimes, students consolidated their schemes for a sub-construct for extended periods. Advancement in schemes seemed dependent on ade-quacy of knowledge possession and co-ordination of that knowledge rather thancompleteness. Students were able to progress to applying a more advanced scheme


without showing completely reliable application of the earlier scheme to all situationsthey encountered.

Connections between sub-constructs

The understanding and flexible application of the properties of multiplication anddivision through mental strategies on whole numbers seemed critical to progress inschemes for rational number. This was not attributable to teaching effect, as therelationship was independent of the order of instruction. For Jason, absence of multi-plication basic fact knowledge restricted his ability to make sense of instruction aboutrational numbers as measures and ratios during term one. Similarly, for Linda, herunreliable calculation and inability to see multiplicative relationships with wholenumbers restricted her understanding and use of fractions as equivalent measures.

All three high achieving students used divisibility, particularly through identifyingcommon factors, in determining equivalent fractions and ratios. Other examples oftemporal order in scheme development were noted. Knowing non-unit fractions asiterations of a unit fraction (measure) appeared requisite to applying non-unit fractionsas operators. For example, 3/8 equals 1/8 plus 1/8 plus 1/8, so 3/8 of 24 can be found byfinding 1/8 of 24 then multiplying by 3. Recognising equivalent fractions as the samemeasure, e.g. 3

8 ¼ 1540 , seemed necessary for comparing shares in complex quotient

situations, e.g. 3 shared among 8 gives a lesser share than 2 shared among 5.Understanding equivalence was also required for ordering ratios by some attributeusing part-whole comparisons. For example, considering mixtures of blue paint toyellow paint, 3:5 was a lighter green than 2:3 because 3/8 was less than 2/5.

Considerable variation occurred in the ways individual students transferred schemesfrom one sub-construct to meet the demands of a task principally involving anothersub-construct. In most cases, this transfer was enabling in that proficiency with onesub-construct informed growth in another. For example, Simon knew an algorithm forfinding equivalent fractions at the beginning of the year but did not recognise equiv-alent fractions as expressions of the same quantity in ordering tasks. Yet, he usedequivalence to compare correctly the equal shares for five boys with three pizzas andthree girls with two pizzas (see Fig. 3).

During terms one and two, Simon learned to apply equivalence to comparingfractions as measures and to comparing frequencies (part-whole relationships in ratios).For example, in a basketball shooting context he compared 32 successful shots out of40 with 39 successful shots out of 50. His co-ordination of sub-constructs improvedconsiderably over time but was susceptible to changes in situations. For example, headded on the same measure to side lengths when asked to produce a larger copy of agiven right-angled triangle and did not notice equivalent fractions as operators in

Fig. 3 Simon solved a quotientproblem by creating equivalentfractions (21 February)

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comparing the distance from home of two families on respective journeys, e.g. 2/3 of210 km compared to 4/12 of 96 km.

The process of co-ordination across sub-constructs sometimes caused cognitiveconflict that appeared as confusion for the student at the time but led to significantadvances in their understanding of rational number later. At the beginning of the year,Jason had some understanding of fraction symbols. He knew a

b can mean “a out of b”

and “a iterations of 1b ”. These different views were applicable at different times. Jason

struggled to co-ordinate these two processes in determining the size of improperfractions like 8

6 . In time, he came to understand ab as a number that embodies both

meanings. He accepted that ab ¼ nanb by a process of equal partitioning and, in turn, used

that to divide ab by other fractions.

Some attempts to connect across sub-constructs appeared to be problematic. Odettetended to apply an additive model to ratios and did not appreciate that the attributeassociated with a given ratio, such as colour or flavour, was conserved through iterationof that ratio, e.g. 2:3, 4:6, 6:9,… She established a trusted procedure for convertingfractions to percentages and used percentages to express the part-whole relationships inratios. Consequently, incompatible responses to ratio comparisons co-existed (seeFig. 4). After deciding that all three mixtures of apple and blueberry juice were ofthe same flavour, due to the same difference, Odette then correctly calculated percent-ages, oblivious to the contradiction. In time, she relied on percentages exclusively tocompare ratios. By investing in percentages, as a trusted measure, it is unlikely that she

Fig. 4 Odette compared ratios by difference of measures (26 October)


ever engaged with and understood conservation of attributes such as flavour, colour orprobability.

Variations in student responses

As anticipated from the literature, variations in student response within and amongsituations and representations were commonplace. For example, Jason, Odette, Ben,Rachel and Simon went through a lengthy phase of indecision where they appliedadditive differences to rates in some contexts but not others. They were inclined toapply multiplicative thinking to ratios where the part-whole relations were explicit(frequencies) and to rates where the unit rate was calculated by integral division, $6 for9 avocadoes equals $2 for 3. However, in situations where the part-whole relationrequired inference (comparison ratios) or the unit rate was harder to conceptualise, likescaling or balance tasks, students frequently chose additive models. Noticing similaritybetween different situations was sometimes momentous. Simon’s understanding ofrates was enhanced through noticing that adding the same measure to each side didnot preserve the shape of triangles and that speed involved two measures. In thefollowing situation, he compared the speed of three runners displayed as ordered pairson a graph.

I: Have you completely discounted Jee? Why can’t she be the fastest?

S: Because she hasn’t run for as many minutes or for as many laps.

I: But the question is not for long she has run it is how fast she’s run. How do youdecide how fast she has run?

S: Measure how many laps she’s done.

I: How do you measure how fast a car goes?

S: K’s per hour…oh so minutes per lap

Despite the transposition of minutes and laps, Simon’s final comment suggestsrecognition of rate as a relationship between measure spaces. Representations alsoelicited considerable variation of student response even in different situations thatinvolved the same sub-construct. Making connections between words, symbols andphysical or diagrammatic models was difficult for all six students at various times.For example, when working with decimats, a paper model for representing deci-mals, Ben modelled the answer to 7÷5= in a quotient context (see Fig. 5). Yet, hewrote 1.20 (incorrect) for his answer, reflecting his preference for expressingdecimals in money-like notation. Yet, he recorded 1÷4=0.4 and 3÷4=0.75 in thesame exercise.

At the beginning of term three, different pencil and paper tests were created to caterfor the levels of achievement in the class. The three high-achieving students, Ben,Rachel and Simon, attempted the following task: “The decimal for three-eighths 3

8

� �is

0.375. Use this knowledge to work out the decimals for the following: (a) 324 = How

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did you work it out? (b) 198 = How did you work it out?” While the task afforded

easy calculation of the decimals once the relationship between 38 and 3

24 wasrecognised, no students used the strategy of dividing 0.375 by 3. Simoncalculated 19

8 ¼ 19� 8 ¼ 2:375 through applying the quotient rule a� b ¼ ab

� �

but Rebecca wrote “2.”, finding the whole number part of the mixed number yet beingunable to provide the fractional part. Making connections between fraction, quotient anddecimal symbolic representations were extremely difficult for all students.

The data paint a picture of some elements of consistent progression through the HLTas well as considerable variation. Progression through phases within a given sub-construct holds but students vary considerably in the connections they make betweensub-constructs, and, therefore, the order of progression across sub-constructs. Variationassociated with different situations and representations, particularly symbolic andgraphic, was pronounced.

Discussion and conclusion

Consistency of progression

I sought to answer the question, “To what extent is a learners’ progression throughphases of the HLT consistent and predictable across learners and to what extent is itvariable?” The data raise significant issues about HLTs, particularly in a conceptualfield of considerable complexity, such as rational number. Variation in progressionthrough phases between students suggests that trajectories need to be interpretedstochastically not deterministically. While the three highest achieving students,Simon, Rachel and Ben, finished the school year with similar schemes for the measure,operator, quotient and rate/ratio sub-constructs (Kieren 1980, 1988, 1993, 1995) theirroutes were different. The early development of rational number of the case-studystudents, as exemplified by the progressions of Jason, Odette and Linda, supports theidea that learning is highly localised to situation and that transfer is unreliable (Behret al. 1984; Case 1992; Kaput and West 1994). The three higher achievers connectedacross the sub-constructs with increasing fluency as the year progressed suggesting thatcurriculum design and assessment must reflect a balance of the sub-constructs ratherthan privileging of one or two. Nonetheless, the demands of new-to-learner situations,like balances, new representations, such as graphs, and new connections, such asconnecting decimals and fractions, present even the most able students with challenges.

Fig. 5 Ben used decimats to model a quotient, i.e. 7÷5=1.2 (14 March)


Connectedness of the sub-constructs

Acknowledgement of connectedness of the sub-constructs means that assumptions oflinear progression need to be tempered. The data support the integrity of the within sub-construct progressions in the HLT. However, students’ progress was informed byconnection to other sub-constructs. Comparison of shares in quotient situations appearsheavily dependent on defining the shares as measures and the use of equivalence tocompare these measures. For example, comparing the equal shares for 3 among 5 and 2among 3 requires the ability to compare 3/5 and 2/3 as measures. This is similarly truefor comparison of ratios using part-whole measures. Identifying and applying non-integral multiplicative relationships between measure spaces in rates and ratios appearheavily dependent on identifying the unknown operator between values, that is

a� ba ¼ b . For example, to solve 5litres

3minutes ¼ ½�litres10minutes it is helpful to know the unit rate

between the measures, 5� 3 ¼ 53 litres per minute, and/or the within measures

operator, 103 ¼ 103 (see Figs. 6 and 7).

The tabular form of the HLT means that progressions in the sub-constructs aretreated discretely. Therefore, critical connections between sub-constructs are not rep-resented. Further study is needed to establish and sequence blends of sub-constructs,that is, instances where the development of a scheme in one sub-construct is shown tobe an essential enabler for the development of schemes in other sub-constructs.

Usefulness of the hypothetical learning trajectory

The results of this research reflect a specific situation. As both a teacher and aresearcher, I found the HLT useful as a planning tool, as a framework to describestudents’ learning in the long-run and as a lens to view their responses to instructionalsituations. I readily acknowledge that I called on finer-grained mathematical knowledgefor teaching that was not and could not be provided by a succinct HLT. “In-the-moment” instructional decisions call on a complex register of other knowledge, such

x 35

Litres Minutes

5 3

? 10

x 35

Fig. 6 Using a between measuresoperator to solve a rate problem

Litres Minutes

5 3

? 10

x 310x 3

10

Fig. 7 Using a within measuresoperator to solve a rate problem

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as anticipation of the impact of task variables, choice of representations, and recogni-tion of common learner misconceptions. Research into teachers’ use of HLTs duringtheir interactions with students requires a broader focus that encompasses the range ofknowledge needed for teaching (Ball et al. 2008; Ball and Bass 2000; Chick et al. 2006;Chick 2007; Empson and Jacobs 2008) and the significant roles of goals and beliefs.

Schemes as descriptors of progression

This research suggests that the use of schemes to describe progression has significantadvantages over other possible formats. Schemes are observable in action in differentsituations so are appropriate for assessment. Teachers can infer learners’ growth inunderstanding from seeing what learners do. In contrast, describing progression interms of knowledge or success on tasks is problematic due to the sheer number andspecificity of knowledge used by students in their development of rational number andthe diversity of tasks that could be selected. As von Glasersfeld (1989) pointed out, theco-ordination of situation, action and result, can become conceptual knowledge, forexample, a learner can treat the action of sharing 20 objects among 4 people as a trusted‘fact’ encoded as 20÷4=5. However, learners’ possession of knowledge is no guaran-tee that they can productively apply it. It is what knowledge they are able to enact insituations that matters. Beginning with schemes then analysing what applicable knowl-edge students do or do not use seems a feasible approach to describing progression andbetter aligned to the way teachers respond supportively to the solutions and ideasproposed by their students.

Conclusion

In conclusion, I agree with Sztajn et al. (2012) that HLTs constitute a significantelement of the mathematical knowledge needed for teaching and provide necessarydirectionality for curriculum design and assessment. I contend that HLTs should beevaluated in two ways: their utility as predictive tools for mapping scheme progression,acknowledging natural variation, and their utility as tools for informing instruction. Theinevitable variation in students’ patterns of progression means that HLTs must beinterpreted and adapted by teachers in their localised situations.

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Towards a hypothetical learning trajectory for rational number

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