Learning to see pipes mathematically: preapprentices’mathematical activity in pipe trades training

Lionel LaCroix

Published online: 6 February 2014# Springer Science+Business Media Dordrecht 2014

Abstract This study examines adult students enrolled in pipe trades preapprenticeshiptraining to identify distinguishing features of the mathematical activity within their programand sources of their mathematics-related difficulties. Two closely related sociocultural per-spectives, specifically cultural historical activity theory and the theory of knowledge objec-tification, frame this investigation. Like workplace mathematics practices reported elsewhere,mathematics within this preapprenticeship program was inextricably tied to the efficientproduction of objects of workplace activity, in this case, the design and fabrication of alimited number of well-defined objects of pipe trades production. This form of mathematicalactivity was also mediated intensely by semiotic tools and norms of practice specific to thepipe trades. The contribution of this study is the finding that the great majority of themathematics difficulties encountered by students can be attributed to their novice levels ofawareness of the objects of pipe trades production and related ways of working rather thanthe mathematics understandings that they brought to this endeavor. Implications are identifiedfor the teaching of mathematics in skilled trades training generally, developing students’mathematical subjectivities within trades training, and the mathematics preparation of sec-ondary school students who will be entering skilled trades. Questions are also raised forfurther research.

Keywords Workplacemathematics .Mathematics education . Skilled trades .Cultural historicalactivity theory . Objectification . Subjectification

1 Introduction

Preparing students to use mathematics effectively in the workplace is an important andcomplex challenge in mathematics education. While the utility of mathematics is commonlyaccepted as justification for its central place in the school curriculum, research conducted in avariety of contexts shows that workplace mathematics practices can differ in significant ways

Educ Stud Math (2014) 86:157–176DOI 10.1007/s10649-014-9534-6

L. LaCroix (*)Peel District School Board, Mississauga, ON, Canadae-mail: lionel.lacroix@cogeco.ca

from traditional school mathematics (e.g., Jurdak & Shahin, 2001; Masingila, 1993; Noss,Hoyles, & Pozzi, 2000; Triantafillou & Potari, 2010; Williams & Wake, 2007b; Zevenbergen& Zevenbergen, 2009). A number of reports also indicate that many students leave schoolunprepared to face the numeracy demands of the workplace and those of workplace training(e.g., Fownes, Thompson, & Evetts, 2002; Hoyles, Noss, Kent, & Bakker, 2010; Zevenbergen& Zevenbergen, 2004, 2009). Moreover, school mathematics can even hinder workers’abilities to perceive the mathematics that is used in the workplace (Forman & Steen, 2000).Thus, a deeper understanding of the transition that students face as they are initiated intoworkplace mathematics practices is essential for informing workplace mathematics trainingand school mathematics.

The work on situated cognition by Lave (1988), for example, and others has informedmuch of the workplace mathematics research that has been done over the past two decadesby drawing attention to the central role of structuring resources provided by the sociocul-tural contexts where mathematics is used. More recently, many researchers have adoptedcultural historical activity theory (CHAT) (Leont’ev, 1978) as a theoretical perspective foranalyzing workplace mathematics. While addressing contextual dimensions of cognition,this sociocultural orientation provides a comprehensive set of analytic tools for analyzingthe socioculturally, historically situated, and dynamically inter-related elements of culturalpractices including mathematics (Williams & Wake, 2007a). To date, a diverse range ofworkplace mathematics practices have been examined in the research literature includingthat of carpet installation (Masingila, 1994); investment banking, pediatric nursing, andaviation (Noss et al., 2000); Arabic plumbing (sheet metal work in North American skilled-trade parlance) (Jurdak & Shahin, 2001); fish hatching (Roth, 2005); an industrial chemistrylaboratory, automated (CNC) machine operation in a metal workshop, and fuel managementwithin the utilities department of an industrial chemical plant (Williams & Wake, 2007a);telecommunications system repair (Triantafillou & Potari, 2010); and boat building(Zevenbergen & Zevenbergen, 2009). At present, it is recognized that mathematics practicesin the workplace are shaped by the immediate and practical requirements of workplaceproduction; that is, getting particular tasks accomplished with ease and efficiency usingresources at hand rather than the goals and norms of generality, formality, and internalconsistency commonly associated with school mathematics (Noss et al., 2000; Williams &Wake, 2007b). Furthermore, it is also recognized that mathematics is often obscured by theproduction goals, technology, artifacts, and established routines of workplace activity(Williams & Wake, 2007a).

In comparison to the range and depth of research on workplace mathematics practices, thebody of empirical research on workplace mathematics training and learning, and newcomers’encounters with workplace mathematics practices in situ, is considerably less developed.Researching this area for the purpose of making generalizations to improve workplacemathematics training, and mathematics in school for students going on to workplace training,is complicated by the diversity of workplace production practices across different sectors and,in turn, the associated workplace mathematics practices. It is complicated further by the varietyof forms that workplace training takes across different sectors, as well as the way thatmathematics on-the-job is inextricably bound up within, and obscured by, workplace produc-tion practices that are often the primary focus of workplace training and workers’ attentionduring training.

A number of research reports on workplace mathematics training activity have focused on asmall group of trainees or an individual, often having difficulty with a single mathematics-related task. Eberhard (2000), for example, provided a brief examination of students in atechnical secondary school program working on the task of laying out measurements from

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specifications for the foundation of a garage, provided on a technical drawing. Williams,Wake, and Boreham (2001) examined the challenges faced by a college mathematics andchemistry student reading the idiosyncratic graphical output of a materials testing machine inan industrial laboratory during a 1-week work placement that was part of her college program.From an earlier session of the training program that provides context for the present study,Martin, LaCroix, and Fownes (2005) examined the difficulties of a pipe trades preapprenticewith fractions of an inch while working on a pipe assembly construction exercise. Taking adifferent tack, Martin and LaCroix (2008) examined the diversity of mathematics-relatedknowledge that was apparent as three ironwork apprentices successfully completed an authen-tic workplace problem-solving task during the year-two college component of their appren-ticeship accreditation training.

Broader perspectives of training and learning in workplace mathematics training areprovided by Ridgway (2000) and FitzSimons, Mlcek, Hull, and Wright (2005). In an effortto make recommendations regarding the mathematics preparation of students at school prior toentering apprenticeship training, Ridgway surveyed the mathematics used by engineeringapprentices (precision machinists in North American trade parlance) and the predictors ofsuccess within their formal training, as indicated by various measures of educational attain-ment. FitzSimons et al. surveyed the mathematics-related practices related to chemicalspraying applications and the training of new workers within a variety of horticultural,agricultural, and outdoor recreational workplaces. Their findings on numeracy training withintheir sector were of a general nature, describing how learning takes place—most often throughhands-on experience on the job under the direction of more experienced employees. Severalother significant workplace mathematics training research studies have also been undertaken.Hoyles et al. (2010) conducted a comprehensive study to explore techno-mathematicalliteracies—the coordination of mathematical, information technology, and workplace-specific competencies—within the workplace practices of companies in a variety of industries.They then created software applications for developing this form of literacy with intermediatelevel managers in these companies. Bakker and Akkerman (2013, this issue) used a targetedpedagogical intervention during students’ internships within a medical testing laboratory toexamine the integration of school-taught statistics concepts with workplace-related knowledge.Hahn (2011, this issue) developed and analyzed a targeted intervention to help business studiesapprentices enrolled in a university Masters course apply statistics concepts within workplacecontexts. Finally, Bakker, Groenveld, Wijers, Akkerman, and Gravemeijer (2012) conducted atargeted intervention to help senior secondary school vocational students develop proportionalreasoning in relation to the concentration and dilution of liquids within the context of theirscience laboratory training.

The present study examines the mathematics-related activity of preapprenticeship studentswithin a pipe trades training program conducted at a trade union run school in BritishColumbia, Canada, with particular emphasis on the students’ mathematics-related difficultiesas well as sources of these difficulties. The particular session of this workplace trainingprogram is unlike other training programs described in the research literature in that it focusedexplicitly and intensely on a variety of workplace mathematics competencies required forgaining accreditation as a tradesperson: it addressed workplace mathematics through highlyintegrated classroom and practical workshop learning experiences, and it led directly to paidemployment as an entry-level apprentice for most, if not all, of the participants. This studymakes a unique contribution to workplace mathematics research by providing a comprehen-sive examination of the specific features of pipe trades mathematics within this trainingcontext, as well as the sustained efforts of individual students in resolving the full range ofmathematics-related difficulties that they encountered over the duration of the course.

Mathematical activity in pipe trades training 159

2 Research questions

Two research questions guided this inquiry:

& What is the nature of mathematical activity in pipe trades training?& What are sources of the preapprentices’ difficulties when doing mathematics within this

workplace training program?

The term pipe trades refers here to the closely related skilled trades of plumbing, steamfitting, sprinkler fitting, and gas fitting.

3 Theoretical perspective

Cultural–historical activity theory and Radford’s (2008b) closely related cultural–semiotictheory of knowledge objectification (TO) are used for this analysis. CHAT frames humanthinking and behavior as co-constituent elements of collectively organized, culturally mediat-ed, historically evolving, and practical object-oriented activity systems, the fundamental unit ofanalysis (Roth & Lee, 2007). It thus serves to distinguish the unique and situated dimensionsof different forms of workplace mathematics. Basic to CHAT is a commitment to dialectical-materialist ontology (Roth, Radford, & LaCroix, 2012). From this perspective, all of theconstituent elements of workplace production activity are recognized as structuring resourcesthat give it mathematical and vocational meaning for participants (Pozzi, Noss, & Hoyles,1998). Radford’s theory, specific to mathematics, provides analytic tools for understanding ingreater detail the historical, cultural, and semiotic dimensions of mathematical thinking andlearning, as well as the dimension of subjectivity as individuals become participants withinparticular forms of mathematical activity.

The analytic categories of objects, goals, tools or artifacts, and rules or norms from CHATdraw attention to ways that mathematics within particular forms of workplace activity aredistinct forms of mathematics practice, inseparable from their corresponding context ofworkplace production (see Roth, 2005; Roth et al., 2012). Analytic tools used from the TOare: (a) knowledge objectification (or simply objectification), (b) subjectification, (c) theterritory of artifactual thinking, and (d) semiotic systems of cultural signification. Together,they highlight the socially and semiotically mediated and dialectical nature of thinking withinthis activity and further distinguish this form of mathematical activity. As part of a broaderCHAT perspective, Radford (2008b) conceptualizes mathematics learning as an interactive andcreative encounter with historically constituted forms of thinking—a process of objectification.Drawing on Hegelian phenomenology, objectification refers to a dialectical process of becom-ing consciously aware of, making sense of, and becoming critically conversant with culturalobjects and systems of thought as one comes to participate within and contribute to culturalforms of activity, such as mathematics (see also Radford, 2008c, 2009). In this light, studentsfacing irresolvable mathematics difficulties or breakdowns reflect inabilities, under particularcircumstances, to mobilize sufficient resources to achieve a sufficient level of objectification tocomplete the tasks at hand. Learning, however, involves much more than objectification; it is“a process where knowing and being are mutually constitutive” (Radford, 2008b, p. 215); thatis, a reflexive process through which the learner also becomes someone—a process ofsubjectification.

An important aspect that makes Radford’s view of objectification distinctive from otherviews of learning is the close relationship that it bears with the Vygotskian concept of

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consciousness and its mediated nature (Vygotsky, 1979; see also Leont’ev, 1978). From thisperspective, consciousness is formed through encounters with other subjects and the historicalintelligence embodied in artifacts and signs that mediate our actions and reflections. This ideais reflected in Radford’s (2008a) concept of territory of artifactual thought or technology ofsemiotic mediation. This foregrounds the role of semiotic tools and other artifacts as constit-uents of what we think and feel, and our very being within cultural historical activity, ratherthan as mere technical instruments used to get things done. Closely related to this is Radford’s(2003) concept of semiotic systems of cultural significations; that is, cultural symbolic systemsthat “make available varied sources for meaning-making through specific social signifyingpractices” (p. 60). This relates to notions of mathematical truth or reality, its assumed nature,evidence, methods of enquiry, the manner in which mathematical objects are considered to beknowable, and the way that mathematical knowledge can be represented.

4 Method

This interpretive study is part of a larger project that examined mathematics learning within anumber of construction trades training courses in British Columbia, Canada. I served as thefield researcher for this project, visiting and collecting data within various training programsover a 21-month period. This included intensive interaction with students and data collectionwithin a previous session of the pipe trades preapprenticeship course that is the focus here, aswell as a fourth year plumbing apprenticeship course at a technical college.

4.1 Setting and participants

The 8-week pipe trades preapprenticeship program was taught by a qualified and experiencedsteamfitter. The classroom component of this course addressed: (a) workplace safety; (b) firstaid; (c) piping materials, related hardware, and pipe trades terminology; (d) reading andpreparation of isometric and orthographic technical drawings; (e) some of the physicalscience related to the production methods; (f) mathematics required for the design andfabrication of various types of pipe assemblies; and (g) the use of trade-specific referencedocuments. The practical component of the course, conducted in the school workshop,addressed tool and equipment use and the physical construction of different types of pipeassemblies that incorporated the mathematics done in the classroom. Mathematics-relatedtopics were typically introduced in the classroom and related pencil-and-paper exercisescompleted in advance of the students applying their learning to complete fabrication tasks inthe workshop, and a typical day included time spent in both places. While not part of theformal apprenticeship credentialing process, this course was designed to provide studentswith theoretical and practical knowledge and skills so that they would be able to workproductively and safely as entry-level apprentices. A second goal was to provide studentswith a head start on some of the course content addressed subsequently in the collegecomponents of their formal apprenticeship training. A typical apprenticeship in the pipetrades in British Columbia takes 4 years, with 6 to 8 weeks per year spent at a technicalcollege, and the remainder of the time spent as a paid employee in the workplace. At the endof this period, apprentices must pass a final credentialing exam set by the provincial trainingauthority.

All but one of these select groups of 15 students were secondary school graduates. Theirbackgrounds reflected a variety of secondary and post-secondary experiences; the greatmajority had five or more years of work experience in unskilled occupations, and all were

Mathematical activity in pipe trades training 161

highly motivated because, upon successful completion of this course, they would becomeeligible for placement within entry-level apprenticeship positions in companies associated withthe trade union running the program. Last, but not least, it should be noted that all of thesestudents successfully completed the preapprenticeship course.

4.2 Procedure and data collection

Throughout the course, I attended the class regularly and served as a mathematics tutor at adesignated table at the rear of the classroom for any preapprentices who wanted to work withme or with fellow students. At other times, I observed and interacted with students about theirmathematics-related coursework as they completed pencil-and-paper classroom activities andpractical workshop tasks. My goal as researcher was to reveal and document (a) themathematics-related coursework in the preapprenticeship program; (b) the students’ thinkingwhile completing this work, with a minimum of direction from me in the process; and (c) thestudents’ interactions while resolving their difficulties. My modus operandi was to attendpatiently and carefully to the students’ comments, questions, and written work while theycompleted their assigned tasks; to pay particular attention to situations when they experienceddifficulties or breakdowns; and to allow them to lead their conversations with me or oneanother as much as possible. The focus on students’ difficulties was similar to the approachused by Pozzi et al. (1998) and Williams and Wake (2007a), who used breakdowns in themathematical activity of nurses on the job and college students investigating mathematicspractices in the workplace, respectively, as a means to identify salient mathematics. Whendifficulties were encountered, I prompted the preapprentices to articulate their thinking to helpthem to be clear and explicit about this, both for themselves and for research purposes, beforeoffering direction. Then, when offering help, I endeavored to keep my directions to a minimumso that students were encouraged to resolve their difficulties for themselves as much aspossible.

Most of the activity of individuals and groups of preapprentices interacting with me orothers at the help table, some of the whole class instruction provided by the course instructor,and some of the students’ mathematics-related practical work in the workshop were docu-mented on 26 h of video tape using a portable camera. Over the course duration, 12 of the 15apprentices in the class were captured on video at the help table working and/or seeking help.The sample of students’ efforts to resolve mathematics-related difficulties captured on videowas, therefore, a substantial sample of difficulties experienced by the students over the entirecourse. It should be noted, however, that not all of the students would have found it necessaryor preferable to come to the help table to resolve their mathematics-related difficulties and, attimes, with up to eight individuals sorting things out among themselves on camera at the helptable, it was not possible to attend to everything that was said from the video recordings.Throughout this, and the previous session of the preapprenticeship course, I had regularconversations with the course instructor to ensure that I understood the important ideas behindthe work assigned to students from his perspective as a pipe trades insider, and kept field notesfrom our encounters. Print materials used by students in the course, selected copies ofpreapprentices’ written work, and background information on individual students were alsocollected for analysis. Print materials used in all four levels of the plumbing program at a localtechnical college (one of the institutions receiving students from this preapprenticeshipprogram), field notes from discussions with the pipe trades instructors and fourth yearplumbing apprentices there, and the formal provincial plumbing apprenticeship programsyllabus published by the Industry Training Authority of British Columbia (2009) alsoinformed this analysis.

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4.3 Data analysis

The identification and characterization of the different kinds of tasks that comprised thepreapprentices’ mathematical activity was based on analysis of the complete set ofmathematics-related print materials that were used during the course (i.e., instructional mate-rials, all assigned classroom exercises and practical fabrication tasks, technical referencematerials, and student evaluation tasks), the video record of the students using these materials,copies of students’ written work retained from class, as well as my field notes taken during thecourse. The first step in the analysis involved categorizing the tasks completed by thepreapprentices and the mathematics associated with each category. This was a straightforwardprocess in that the great majority of the tasks assigned involved finding the measurements orspecifications for one of a small number of objects common to the pipe trades. There were onlytwo types of tasks assigned where this was not the case. In these cases, the focus was onpracticing particular calculations in isolation for subsequent use in practical applications. Oncethe set of task categories was established, I identified the following attributes for each: (a) thephysical or mathematical object involved, (b) the specific kinds of the mathematics calcula-tions (actions) involved, (c) the goals or objective of the calculations, (d) the tools or artifactsused, and (e) any pipe trades-specific norms related to the calculations. Once the initial phaseof analysis was complete, the next step involved analysis of the preapprentices’ mathematicalactivity across tasks to identify and characterize the semiotic systems of cultural significationand territory of artifactual thinking that were inherent to the preapprentices’ mathematicalactivity as a whole. The final step was the construction of the description of the nature of thepreapprentices’ mathematics activity within their training program.

Identification of the sources of the preapprentices’ mathematics-related difficulties firstrequired the identification of instances from the video data when individuals encountered amathematics-related difficulty while completing their assigned coursework and which weresubsequently resolved. In each case, analysis of the discourse involving the preapprenticesserved as the primary basis for identifying each difficulty and its source(s) or, at least, asignificant part of the source(s), recognizing that all elements of the preapprentices’ activityserved to mediate their thinking. From the perspective of the theory of knowledge objectifi-cation, efforts to resolve difficulties such as these are regarded as a process of objectification—the social and semiotically mediated process through which individuals achieve a level ofawareness of essential aspects of their activity that enables them to achieve their goals inculturally appropriate ways (Radford, 2008b)—in this case, for the preapprentices to completetheir assigned coursework successfully.

Given that experiencing and overcoming difficulties is a normal and expected part oflearning mathematics, I focused on the more serious kinds of difficulties that, under differentcircumstances (i.e., without the availability of a mathematics tutor, or with less resourcefulstudents), would be ones more likely to impede the preapprentices’ progress and success intheir training if left unaddressed. In operational terms, I observed and identified the occurrenceof these kinds of mathematics difficulties when students (a) encountered irresolvable break-downs requiring the help of other individuals, (b) made unnoticed errors, (c) completedmathematics tasks correctly but by methods that did not reflect the cultural norms of reliabilityand efficiency associated with pipe trades workplace production (as conveyed to the class bythe course instructor), or (d) experienced other kinds of difficulties that interfered with theirprogress. Difficulties were identified (a) from students’ self reports of having difficulty, (b)when students sought assistance with an assigned task from me, another student, or thespecialist workplace educator who visited the class for a few hours each week, or (c) fromerrors evident in students’ discourse relating to something that they were expected to have

Mathematical activity in pipe trades training 163

known or figured out from their prior schooling, life experience, or course informationprovided. The following were not considered difficulties in this analysis: (a) when additionalinformation was needed to complete a task that students weren’t expected to know or be ableto figure out on their own, (b) when students sought confirmation that their results werecorrect, or (c) when direction was provided to students but the root of their difficulty was notclear from the available data. In other words, instances where efforts were made to provideassistance, by themselves, were not considered. This definition is open to interpretation and Idrew upon my own industrial and education sector experience to make these determinations.

The video recording of the discourse and copies of the print artifacts used in each episodewhere a preapprentice experienced a mathematics-related difficulty were then analyzed toidentify the source(s) of the individual’s difficulty in each case. Categories provided byCHAT and TO served to orient this analysis. Initially, these categories included: (a) students’understanding or level of objectification of the piping object in the task they were working on,(b) their ability to make sense of and use the pipe trades tools and artifacts associated with thetask, and (c) their ability to employ norms of pipe trades practice in their work. As the analysisproceeded, a distinction was made between students’ level of awareness of the physical objectsof their activity and their awareness of the mathematical thinking or mathematization behind thedesign and fabrication of these objects. In addition, I examined students’ spontaneouslyexpressed feelings about mathematics as they completed their work. These comments revealedpreapprentices’ ways of being, or subjectification, in relation to the mathematics they weredoing. Throughout the analysis, the precise definitions of the CHAT- and TO-based categoriesof the sources of the preapprentices’ difficulties within this activity were reviewed and refinedto ground them in the empirical data. The outcome is a set of six distinct categories of sources ofstudents’ difficulties that accommodates all of the instances identified.

5 The nature of mathematical activity in pipe trades training

In this section, I describe the essential features of preapprentices’ pipe trades mathematicalactivity, along with the instructor’s means of introducing constituent kinds of tasks and formsof mathematics practice to them. Together, these comprise the mathematical activity within thepreapprenticeship course. Categories from CHAT and the TO serve to foreground the distinc-tive features of this mathematical activity.

5.1 Objects, actions, and goals

The preapprentices’ classroom mathematical activity was centered on the completion ofpencil-and-paper exercises related to a limited number of well-defined and common pipetrades production tasks. These exercises reflected the calculations or actions needed for varioustypes of piping assemblies and related equipment—the ideal and material objects (in Russian–ob’ekt, German–Objekt) of pipe trades production activity that, in turn, reflect the object/motive (in Russian–predmet, German-Gegenstand) of pipe trades production activity as awhole (see Roth, 2014). Specifically, the preapprentices’ work involved mathematics used inthe production of (a) screwed pipe, bent pipe, and welded pipe assemblies; and (b) rolling andparallel offset pipe assembly configurations (see, e.g., Fig. 1), as well as (c) calculationsrelating to the specifications of various kinds of pipes and storage tanks. These exercises werevery similar to those that the preapprentices would encounter on their written tests in thecourse. A review of the print materials from all four levels of the plumbing program at the localtechnical college and discussions with the pipe trades instructors there, along with a review of

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the provincial plumbing apprenticeship program syllabus, confirmed that the mathematicalactivity within the preapprenticeship program reflected that of formal apprenticeship trainingin the pipe trades that followed. Throughout their training, an important, although notexplicitly articulated, goal was for the students to objectify common piping objects in a pipetrades mathematical way; that is, to see these objects in ways similar to experiencedtradespersons.

The preapprentices’ mathematical actions involved (a) fractions, mixed numbers, decimalnumbers; (b) arithmetic and algebraic formulas to determine length, perimeter, circumference,surface area, volume or capacity, weight, force, and pressure; (c) applying primary trigono-metric relationships and the Pythagorean theorem to solve for unknown side lengths or acuteangles in right triangles; (d) reading technical drawings, reference tables, graphs, as well as theinscription pattern on an imperial measuring tape; and (e) converting measurements betweenimperial, US, and metric units using conversion factors on reference materials provided.Throughout the preapprenticeship program, students’ work on pencil-and-paper exercises inthe classroom preceded their practical work fabricating the corresponding types of pipingobjects in the workshop. In two cases, the preapprentices were assigned pencil-and-paperexercises that focused on isolated mathematics processes. One involved solving for anunknown side length or acute angle in right triangles like introductory trigonometry exercisesfound in traditional school textbooks (for subsequent use in offset calculations with bent pipe).The other involved conversions between (a) inches and fractional parts of an inch; (b) inchesand decimal parts of an inch; (c) feet, inches, and fractional parts of an inch; and (d) feet anddecimal parts of a foot for use in area and volume calculations.

Mathematics within the context of the preapprenticeship course served instrumental pur-poses; that is, to meet pipe trades production requirements in an efficient and reliable manner.The instructor told the class that he was not concerned with the methods used, provided theirresults were “close enough” (which usually meant to the nearest 16th of an inch) on aconsistent basis. Nor did the instructor spend class time explaining the reasoning behind anyof the mathematics needed. The goal of efficiency with the use of mathematics was reinforcedby the mantra, repeated on a number of occasions by the course instructor, “time is money.”

The course instructor’s approach to teaching was traditional, transmission based. Itconsisted of (a) describing briefly each new type of piping object to be worked on and the

Fig. 1 A parallel offset screwedpipe calculation task for a pipe di-ameter to be specified. The threedarker sets of lines represent pipes.Students used the information giv-en to solve for unknown pipelengths a–g, accounting for thedistances (take-offs) taken up byindustry standard 45° elbow fit-tings. Note: Single prime and dou-ble prime signs following numbersare used in imperial measurementto signify feet and inches,respectively

Mathematical activity in pipe trades training 165

purpose served by, or the goal of, the calculations in the assigned exercises; (b) providing aworked example of the required mathematical operations for the class; (c) answering students’questions about the example; and (d) assigning a set of similar practice exercises onworksheets. During his explanations, the instructor introduced various tools and ways ofworking that were specific to pipe trades mathematics, including (a) formulas, (b) algorithms,(c) terminology, and (d) print reference materials (all of these being forms of semiotic tools), aswell as (e) industry norms related to the particular requirements of the piping objects involvedin the exercises. The print materials handed out in class often contained additional examplesworked out in a highly prescriptive manner (e.g., using semiotic tools like that show in Fig. 2).Students were encouraged to work together to resolve difficulties on their own. The instructorvariously discussed the answers to the assigned tasks with the class, provided written solutionsfor students to refer to, or directed students to check their answers amongst themselves.Normally, the course instructor was available to provide some assistance to students whilethey completed their work. However, during the present study, he left this job principally tome. A specialist workplace educator who regularly visited the class also provided assistance tostudents, often with me at the table at the back of the classroom, during this study, and herinteractions with students were also captured on video. Students were given the option of usingsome of their shop time to finish their classroom work, and many of the regular participants atthe help table took advantage of this to get help with their mathematics exercises.

The course instructor explained that, while the students were responsible for knowing all ofthe content addressed in their apprenticeship training (and reflected in the provincial appren-ticeship training syllabi), they would not encounter all of this when they entered the workplaceas apprentices or, eventually, as credentialed tradespersons. This assertion reflects the widerange of roles available in industry to qualified tradespersons in the pipe trades, and thepreapprentices did not question this. A pipe trade instructor at the technical college put this intosharper perspective, specifically for plumbing, when he commented, “Only about a third of themath covered here in formal [plumbing] apprenticeship training will be dealt with in yourcareer. One half would indicate a rich and varied career as a plumber.”

5.2 Tools

A number of formulas, algorithms, technical terms, symbols, sign systems, print resources, andother artifacts served as tools within the preapprentices’ mathematical activity. Many of theseare specific to the pipe trades, while others are more commonly used. One example was aformula (a form of semiotic tool) for calculating the take-off for a bend in a pipe,take off ¼ tan θ

2� radius of elbow, where θ represents the angle of the bend (customarilymeasured in degrees as the acute angle of deviation in the path from one straight segment of

To findside

When you

know side

Multiplyside

For a 45º

ell by

For a 22ºell by

T O O 1.414 2.613O T T .707 .383R O O 1 2.414O R R 1 .414T R R 1.414 1.082R T T .707 .933

Fig. 2 A semiotic tool for performing offset pipe calculations

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pipe to the next). This formula gives the portion of a straight pipe length taken up by a bend ofa specified size. It is necessary because of the convention or norm on pipe drawings to leaveout any representation of the curvature of fittings and bends between straight sections of pipe,and to specify distances from the center of one pipe bend or elbow fitting to the next (as is thecase for lengths a, b, and c in Fig. 1), or from the center of a bend or fitting to the other end of apipe if it is straight (as is the case for lengths d, e, f, and g in Fig. 1). The formula pipe bendlength=bend radius×degrees of turn×0.0174 is then used to determine the arc length of thebend so that the straight pipe can be cut to length before it is bent. The preapprentices weredirected by their instructor to use the sine of 1° from their calculators for the constant, insteadof memorizing this value, without further explanation (i.e., the sine of 1° provided anacceptable approximation of the radian measure equivalent for 1°).

A set of algorithms and an accompanying sign system (both forms of semiotic tools) usedto calculate the unknown side of the triangles encountered when constructing pipe assembliesusing either 45 or 22 1

2° elbow fittings or ells can be seen in Fig. 2. The students were directedto use this print artifact to solve for the side lengths in 45–45–90 triangles in parallel offset pipecalculation exercises, like that shown in Fig. 1, as well. Other than 90° elbows, 45° elbowfittings are, by far, the most common fittings used with screwed pipe assemblies in the pipetrades, and most tradespersons in the pipe trades commit the corresponding multiplicationconstants to memory, as did many of the preapprentices. No explanation was given regardingthe derivation of this set of algorithms. Students were expected to know simply how to followthese directions.

Throughout the course, regular use was also made of piping industry standard referencetables like those in The Pipe Fitter’s and Pipe Welder’s Handbook (Frankland, 1984), a pocket-sized reference book (semiotic artifact) commonly carried by tradespersons on the job. Itcontains the specifications for different kinds of pipes and fittings used in pipe assemblyconstruction; for example, inside and outside diameters for standard pipes, (empty) iron pipeweights per lineal foot, and the fitting allowances and thread engagement values for varioustypes and sizes of screwed pipe fittings. It also contains formulas and step-by-step algorithmsfor performing a wide variety of pipe trades mathematics procedures, like those discussed here,as well as other useful technical information. Other semiotic artifacts commonly used by thepreapprentices included imperial measuring tapes, a ubiquitous tool in the workplace with acomplicated system of markings signifying various binary fractions (halves, quarters, eighths,etc.) of an inch intervals simultaneously on a single number line, and electronic calculators.The use of computers as part of pipe trades production work was conspicuously absent withinthis program, and the course instructors and fourth year apprentices at the technical collegeconfirmed that this was also their experience in the workplace as well.

While all of the mathematics done within the preapprenticeship program was part of thecredentialing requirements in the pipe trades, the course instructor explained that there weresometimes quicker and easier ways of doing some things on the job. One example, applicablewhen working with fittings for smaller sizes of screwed pipe, involved approximating the take-off values of fittings (i.e., the distance from the end of a pipe screwed into a fitting and thecenter of the fitting) by eye-balling it (i.e., approximating it visually) with a measuring tape onthe fitting, rather than looking up the precise fitting allowance and thread engagement valueson a reference table and calculating the required take-offs precisely from these. The smallerrors resulting from this approach would be within acceptable tolerances. Another examplewas applicable when bending smaller gauges of copper pipe using a vice-mounted hand tool(like the bending tool referred to by Roth, 2012, this issue). Here, the measurement markingsbuilt into this tool, and the ability to trim these pipes to their required lengths easily afterbending, alleviate the need to calculate the take-off and bend length values beforehand. The

Mathematical activity in pipe trades training 167

course instructor indicated, however, that the take-off and bend length formulas were stillneeded when working with a larger pipe using a hydraulic bending machine because, in thiscase, it was necessary to cut the pipe to length precisely prior to bending.

5.3 Rules and norms of practice

Despite Canada being an officially metric jurisdiction, it remains the norm in the pipe trades tomeasure distances in imperial measure (feet, inches, and fractions of an inch) much of the time.The main reason for this is that the equipment used in the Canadian pipe trades ismanufactured for both the Canadian and US markets. In the case of screwed pipe, for example,the pipe and factory fittings used worldwide are manufactured exclusively in Americanstandard (imperial) sizes. Another norm of pipe trades mathematics used by the preapprenticeswas that of measuring and calculating lengths using binary fractions to the nearest 16th of aninch, with the exception of welded pipe calculations where tolerances were calculated to thenearest 32nd of an inch. Other mathematics norms used in the preapprenticeship course weresimilar, if not identical, to those used in other contexts including school mathematics, such asthe practice of writing fractional values in simplest or lowest terms.

There were a number of workplace norms related to the graphic representation of pipeassemblies and related equipment, reflected in semiotic tools used in the preapprenticeshipcourse. While many of these norms are common to the technical drawings used throughout theconstruction industry, a number were specific to the pipe trades. One example, mentionedearlier, was that of using straight lines meeting at vertices to represent physical pipe assembliescomprised of straight segments of pipe with curved components between them.

5.4 Semiotic system of cultural signification

As mentioned earlier, a prominent mathematical object within the preapprentices’ activitywas the discrete set of binary fractions used to designate parts of an inch. This mathematicalobject was crystallized in the pattern of markings on the imperial measuring tapes thatstudents used throughout the course. For the preapprentices, this set of fractions served asthe basis of their numerical reality for length values, given that all values that they workedwith were rounded off to 16ths or 32nds of an inch. This, in turn, mediated their relatedmathematical activity by focusing attention exclusively on computation methods that servedthis system of measurement. Additional steps to achieve greater degrees of accuracy or otherfractional values served no practical purpose and were therefore not a consideration. The onlyexception to working in binary fractions occurred when calculating areas and volumes basedon length values given initially in binary fractions of an inch. These values were calculatedwith an electronic calculator using equivalent decimal values, although results were roundedoff to specified and discrete units such as the nearest square or cubic inch or foot or someother unit of volume.

Within the pipe trades, the notion of fit (of the pipes) provides the ultimate justification orproof of one’s measurement calculations, and thus served as the basis for mathematical truth orvalidity within the semiotic system of cultural signification of the preapprentices’ mathemat-ical activity as well. Here and in most, if not all, other forms of material production work,qualitative and empirical methods served as the customary means for establishing fit—do themeasurement calculations serve getting the job done? Adherence to this norm drew thestudents’ attention to the functionality of the material objects of workplace production forvalidation of measurement calculations, rather than to the logic of computational processes. Toput this into perspective, compare the empirical approach just described with that of formal

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school mathematics where mathematical truth or validity is often based upon the preciseapplication of rational methods and symbolic representations (more often than not as arbitratedby the teacher).

5.5 Territory of artifactual thought

As described earlier, various semiotic tools common to the pipe trades were integral to thepreapprentices’ mathematical activity and thinking. These included (a) the imperial measuringtape with its sophisticated set of markings signifying various binary fraction-of-an-inchintervals and, in some cases, its use as number line for visualizing addition and subtractioncalculations involving fraction-of-an-inch values in lieu of pencil-and-paper computations; (b)print artifacts containing formulas, algorithms, and various sign systems like those discussedearlier; as well as (c) technical drawings of various pipe assemblies. The students often referredback and forth between their pipe drawings and their handwritten calculations to organize andground their thinking. For example, the preapprentices used the pipe drawings to determineand keep track of next steps in their sequences of calculations and also related the physicalmeanings of the numbers that they were calculating to the corresponding measurements of thepiping objects represented on the drawings in order to make sense of these.

6 Sources of preapprentices’ difficulties doing pipe trades mathematics

Altogether, 116 episodes of preapprentices having difficulty with the mathematical aspects oftheir assigned course work were identified from the video data. A set of CHAT based andempirically grounded categories of preapprentices’ difficulties was constructed from theanalysis of these episodes to account for the sources of the students’ difficulties. In 103 ofthe episodes, a single source of difficulty was identified. In 13 of the episodes, two distinctsources of difficulty were identified making for 129 times in the data when a source ofdifficulty was seen as interfering with the students’ mathematical activity. Of the total numberof times that a difficulty was identified, 78 % of these difficulties were experienced by onlyfour of the preapprentices who found the pipe trades mathematics exercises they were assignedto be particularly difficult and who sought assistance regularly.

The most common preapprentices’ difficulties reflected their novice levels of objectificationof the material objects of pipe trades production and the pipe trades mathematical thinkingbehind the design and fabrication of these objects, and the semiotic tools or resources usedwithin pipe trades mathematical activity. Difficulties identified also reflected, but to a lesserdegree, the students’ novice levels of objectification of the norms of pipe trades mathematicspractice, their troubles with common mathematics objects and tools like those associated withschool mathematics, and their ways of being with, or subjectification, in relation to doingmathematics within the preapprenticeship course.

6.1 Awareness of the objects of pipe trades production

Almost half of the preapprentices’ mathematical difficulties identified (62 of 129) wereattributed to their novice levels of awareness of the material objects of pipe trades produc-tion—the pipe assemblies and related equipment—encountered in their course work. Each ofthese difficulties fits one of two subcategories: 18 were attributed to the preapprentices’ levelof awareness of the material piping objects that they encountered in their work, while theremaining 44 were attributed to the preapprentices’ level of awareness of the pipe trades

Mathematical activity in pipe trades training 169

mathematical thinking behind the design and fabrication of these objects. The latter subcate-gory presumes, of course, that students were aware of the object in each case!

There were three ways that preapprentices’ difficulties attributed to their levels of awarenessof the objects of pipe trades production became evident. The first was when they soughtclarification about the physical features of particular types of pipe assemblies. The second waswhen they sought help with or made an error in relating the component measurementsindicated on pipe assembly drawings to actual pipes. The third was when they made anincorrect generalization about a measurement on a type of pipe fitting. Examples are providedbelow when episodes of difficulty attributed to multiple sources are discussed.

The preapprentices’ difficulties attributed to their levels of awareness of the pipe tradesmathematics thinking behind the design and fabrication of the material objects of pipe tradesproduction were evidenced in a number of ways as well. These included difficulties in:

& Relating the measurement values given in mathematical calculations or operations to themeasurements of corresponding parts of piping objects, and vice versa;

& Seeing the geometric organization of piping objects in order to determine unknownmeasurement values: for example, knowing where to superimpose right triangles on anoffset pipe configuration (as shown in Fig. 1) in order to carry out necessary trigonometriccalculations; and

& Making sense of the reasoning behind particular pipe trades arithmetic computations afterthese had been introduced and explained in class;

as well as in students’ use of particularly inefficient mathematical procedures, inappropriateprocedures, or omission of essential steps in their calculations.

6.2 Awareness of the semiotic tools associated with pipe trades mathematics and production

Approximately one third of the preapprentices’ difficulties (47 of 129) were attributed to theirnovice levels of awareness of the semiotic tools used in pipe trades mathematics andproduction. These included difficulties with pipe trades symbols and terminology as well asdifficulties using the information provided in the industry standard reference handbook usedthroughout the course. Specific examples included (a) a student’s interpretation of 8′ 1

2 ″ aseight-and-a-half feet instead of 8 ft and ½ in.; (b) a student writing the measurement value 1′7 1116 ″ as 1′ 7″ 11

16 , not recognizing that 7 1116 signifies a single value; (c) students having

difficulties reading fractions of an inch on their measuring tapes; and (d) students beingunaware of information to simplify their calculations that was readily available for them intheir reference materials. An instance of the last type of example occurred when some studentscalculated the volume of a very large pipe, with the length of more than 100 ft, in cubic inchesusing the inside pipe diameter, read from a reference table in their handbook before theyconverted this to gallons; whereas they could have more easily used the gallons per foot valuefor this size of pipe provided in another column on the same reference table and simplymultiplied this amount by the given length.

In 12 of the 13 cases where two sources of students’ difficulty were implicated (included inthe above tallies) the sources identified were their novice levels of awareness of the semiotictools they were using as well as their novice levels of awareness of the piping objects theywere working on. Examples included (a) a student misinterpreting a dimension line on atechnical drawing as part of a pipe assembly object; (b) a student asking why it was necessaryto calculate “the length of the hypotenuse” on the drawing of an offset pipe assembly when thehypotenuse line represented one of the pipe component lengths that was required; and (c)

170 L. LaCroix

students being unaware that the value they had calculated was, indeed, reasonable—in thiscase the enormous weight of a 100-ft long, 12-in. diameter iron pipe, filled with water.

6.3 Other sources of difficulty

The remaining difficulties account for less than one sixth of the total number observed andwere attributed to one of three sources: pipe trades mathematical norms, awareness of commonmathematical objects, or previous school mathematics experiences. Difficulties attributed topreapprentices’ novice levels of awareness of the norms used in pipe trades mathematicsaccounted for 10 of these. These were evidenced when individuals sought clarification of, ordid not attend to (a) rounding off final lengths and angles in their calculations so that theirresults would be within required tolerances, (b) the standard bend radius of manufactured pipeelbows used in welded pipe assemblies (1.5 times the pipe diameter), or (c) the standard gap of332 ″ required between components before assembly in welded pipe assemblies.

Preapprentices’ difficulties attributed to their level of awareness or objectification ofcommon mathematical objects, including mathematical relationships and tools commonlyassociated with school mathematics, accounted for seven of the remaining instances ofdifficulty. These were evidenced when individuals required assistance to (a) identify propertiesof basic geometric shapes, (b) interpret the meaning of common algebraic notation when itappeared in pipe trades-specific measurement formulas, (c) operate a basic scientific calculator,or (d) solve for a missing side length in a right triangle, given the measures of one side and oneacute angle.

Finally, there were three instances, all in the first week of the course, when preapprenticesexpressed having difficulty being with, or relating to, the pipe trades mathematics work theywere doing. In all of these cases, their difficulties were attributed to previous negativeexperience of mathematics in school. While providing only a glimpse of these students’subjectivities in relationship to the mathematics at this early stage of their pipe trades training,these examples, nevertheless, provide valuable clues to this dimension of the preapprentices’experience as learners. One student, while finishing his calculations for a rolling offset with myhelp, explained, “It’s easier than I thought.” When prompted to explain what it was that wasdifficult initially about this exercise, this student confided, “It’s just, whenever I hearPythagorean, or squared and things like that, it sounds hard to me. So it [the task at hand]gets more difficult!” After working with me to visualize the right triangles necessary to carryout the trigonometry calculations to solve the pipe lengths for the parallel offset pipe assemblyshown in Fig. 1, another student explained, “I’m just used to doing the rote learning fromschool.” This sentiment was echoed on another day when, in the course of asking for help, thissame individual explained:

I can’t visualize. I’m good on formulas. So all I know is, I made up my formula table [aset of algorithms that this student had created as a guide to use for various screwed pipefitting calculations]. (…) I know it’s easy if you just look at it and see it. But I’m caughtup in going, okay, step one I have to do this, step two I do this, step three I have to dothis. But I’m not understanding why I’m doing it, I’m just doing it.... I’m not looking atthe damned picture, and I should be looking at the picture.

These comments suggest instances of conflicted or, at best, tenuous relationships with themathematical aspects of their preapprenticeship work. They had significant potential toinfluence the ways in which these preapprentices engaged with the mathematics in pipe tradestraining, as well as their sense of ease and their success in doing so.

Mathematical activity in pipe trades training 171

7 Discussion

This study provides a detailed analysis of the nature of mathematical activity and sources ofstudents’ difficulties when doing mathematics within a skilled trades training program—alargely unexplored area within mathematics education research. Socio–cultural perspectiveshighlight the unique and culturally situated dimensions of this form of workplace mathematicswithin the broader context of pipe trades production activity.

Within the target pipe trades preapprenticeship program, the pencil-and-paper exercises thatmade up the preapprentices’ classroom mathematical activity all served the requirements of alimited number of well-defined objects of workplace production. The results of the greatmajority of the mathematical operations undertaken corresponded to the physical measure-ments of these objects. This relationship was reinforced through the organization of thepreapprenticeship coursework: The fabrication of particular forms of piping objects in theworkshop followed immediately from the introduction of the related mathematics in theclassroom.

This preapprenticeship training program, like the college components of the formal ap-prenticeship programs in each of the individual pipe trades, targeted standard forms ofworkplace production practice including mathematics. The preapprentices’ mathematicalactivity involved the ubiquitous presence of pipe trades-specific objects of production; specificterminology, symbols, print resources, algebraic formulas, algorithms, and forms of represen-tation; other tools; norms of production; as well as the more general workplace goals ofworking reliably and efficiently. Furthermore, the predominant number system (the discrete setof binary fractions), the computation methodology, and the empirical basis of mathematicalvalidation used all reflected the practical demands and constraints of material workplaceproduction in the pipe trades. This thorough penetration of pipe trades-specific mediatingelements within the apprentices’ mathematical activity uniquely distinguishes this form ofworkplace mathematics practice and thinking from that found in formal school mathematicsclasses and in other forms of workplace activity.

The preapprenticeship program provided students with a quick overview of pipe tradesmethods with high stakes evaluation, given that the students had to achieve specified levels ofperformance on course requirements as a means of securing positions of employment withinthe pipe trades at a time when they had no pipe trades workplace experience to draw upon.This, together with the particularized form of work—including the mathematics—undertakenwithin this program, called for the preapprentices to conform to the established ways ofworking that they were being taught. In other words, these circumstances called for studentsto fit in and not be critical. While not an explicit focus of the empirical analysis here, thestudents’ subjectification as dutiful subjects under these circumstances, including their ways ofbeing with the pipe trades mathematics they were doing, was, undeniably, a significant part oftheir enculturation into the pipe trades at this point in their training. This is similar, in someways, to Roth’s (2012, this issue) account of the subjectification of electrician apprentices,although there was no sense from the preapprentices, their instructor, or the experienced fourthyear plumbing apprentices that I engaged with as part of my research that any parts within thistraining activity amounted to “putting in time,” as Roth reported. All of the preapprenticecoursework was perceived as targeted towards their being able to work productively on the job.

The vast majority of the mathematics-related difficulties that the preapprentices encountered(119 of 129) were attributed to their novice levels of awareness of pipe trades mathematics andpipe trades production activity including (a) mathematical objects of pipe trades work, (b)semiotic tools specific to these trades, and (c) norms of pipe trades practice. This is notsurprising, given that mathematics in the workplace is inextricably tied to particular objects

172 L. LaCroix

and goals of workplace production, and that workers’ understanding of specific forms ofworkplace mathematics co-emerges with their understanding of the objects of workplaceproduction (Roth et al., 2012). Also, the frequency of students’ difficulties related to theirawareness of commonly used or school mathematics skills and relationships was comparativelysmall (7 of 129 difficulties) and, clearly, not a major problem for them in the bigger picture.Together, these findings suggest that the source of the preapprentices’ difficulties was rarely alack of school mathematics skills. Their ready access to pocket calculators and their ability touse these effectively to carry out frequent computations certainly contributed to this result.

8 Implications and considerations for trades training

The training format of the preapprenticeship program—a relatively short but intensive stand-alone program, taught by an experienced tradesperson, and which integrated classroom andpractical components as a prelude to entering the workplace—appeared to be quite successful.All of the students were highly motivated, actively and meaningfully engaged with theircoursework, and all were successful in meeting the program exit requirements, including thoserequirements pertaining to pipe trades mathematics. This success suggests that a similarapproach to workplace training could be useful in other contexts, particularly for developingstudents’ workplace mathematics skills. However, it must be emphasized that the mathematicscomponent of this preapprenticeship course focused on skills required for becomingcredentialed as a tradesperson in the pipe trades. No claims can be made directly from thefindings of this study about the fit between the mathematics addressed (which reflected that ofthe formal provincial syllabi for pipe trades apprenticeship accreditation in Canada) and thefull range of mathematics skills needed on the job in these trades. It should be noted, however,that provincial apprenticeship curricula were written by industry experts with intimate aware-ness of workplace requirements. Further research to investigate ways that apprentices’ math-ematical practices and thinking develop in the workplace and the alignment of these with theprogression of mathematics addressed within formal apprenticeship coursework has potentialto provide useful insights for informing the design of workplace mathematics training.

The findings from this study exemplify the inseparability of workplace mathematicspractices and workplace production activity. Paraphrasing Wedege (2000), workplace mathe-matics competence involves the integration of mathematical knowledge with particular work-place practices and forms of work organization, and also includes a readiness to act. Thissuggests that it would be beneficial for students in workplace training to have their attentiondrawn explicitly to the mathematically pertinent details of the objects, tools, norms, and goalsof workplace production prior to, or coincident with, focusing on specific workplace mathe-matics practices. This aligns with Akkerman’s (2011) finding, in her analysis of boundarycrossing between cultural–historical activities, that the distinctive features of different activitiescan easily remain implicit for individuals moving between them, and that drawing explicitattention to and contrasting these features can trigger productive dialogical engagement. In thepresent study, this pertains to preapprentices’ transition from their prior workplace mathemat-ics practices and school mathematics to pipe trades preapprenticeship mathematics.

The mediating influence of negative expectations about doing mathematics that somepreapprentices carried with them from their school mathematics experience was identified asa source of difficulty. This raises questions about how best to support the continued develop-ment of students’ productive mathematical subjectivities in workplace training generally, andwhat goals would be most culturally appropriate in this regard. Further exploration of students’developing mathematical subjectivities in workplace training could serve to determine what

Mathematical activity in pipe trades training 173

effects existing training programs contribute in this regard, and provide a basis for consider-ation of alternative possibilities if warranted.

The findings from this study also suggest that mathematics instruction in trades trainingcould benefit from close collaboration between trades specialists and mathematics educators.Throughout the preapprenticeship course, mathematics was presented to students in aninstrumental fashion, much as other aspects of workplace production were taught. A mathe-matics education perspective, with students’ objectification of important workplace mathemat-ical practices and conceptual connections as its goal, could inform the design and implemen-tation of teaching materials and classroom teaching approaches. Of course, this would requirethat the mathematics educators involved become deeply aware, themselves, of the workplaceproduction activities and constituent forms of workplace mathematics to be addressed. Whilethis point has been made elsewhere (e.g., Gillespie, 2000; Hoyles et al. 2010), the presentstudy provides some indication of the scope of the workplace mathematics–production activityconnections that are needed in trades training to help students to become effective andconfident with workplace mathematics.

The findings from this study reveal the multi-dimensionality of workplace mathematicscompetence. As mentioned earlier, all but one of the preapprentices were secondary schoolgraduates (which involved a minimum of three mathematics courses in most, if not all cases),and problems with school mathematics skills were implicated in only a few instances as theycompleted their assigned mathematics-related coursework. With this in mind, implicationsfrom this study for secondary school mathematics programs intended for students entering aparticular trade or trade training program will be similar to those outlined above for workplacetraining programs. For students in general or nonspecific workplace mathematics courses, theimplications of this study and others (e.g., Wake, 2014; Williams &Wake, 2007a) suggest thatstudents would benefit from becoming aware of the kinds of practical and situated dimensionsof workplace mathematics practices generally. Some suggestions include:

& Making workplace semiotic tools and technology from select trades mathematics practices(e.g., formulas, algorithms, reference tables, measurement tools) objects of investigation tofamiliarize students with these kinds of resources and to provide them with experience inmaking sense of and using them;

& Introducing case studies of mathematics practices and problems from various skilledtrades, along with some of the practical considerations and norms of correspondingworkplace mathematics practices such as required tolerances for measurements and theneed to attend to efficiencies of time and the cost of materials during production; and

& Using shop (technical training) courses at the secondary school level, where students areoften immersed within contexts of material production, as a venue for introducingworkplace mathematics methods, related semiotic tools and norms, and the utility ofparticular ways of doing workplace mathematics, as well as the relation of all of these toworkplace production practices.

Finally, this study suggests other avenues for research to inform our understanding ofworkplace mathematics practices and learning. Among these are (a) further study of mathe-matics practices and learning in other workplace training programs, and with other populationsof students, to determine how the nature of mathematics activities and students’ experiences ofdoing mathematics vary across diverse programs and settings; (b) further study of students’subjective experiences of doing mathematics in workplace training as means for developingmore detailed and nuanced understandings of the sources of their mathematics-related diffi-culties; and (c) examination of the tensions and contradictions that students experience

174 L. LaCroix

between the mathematics in their trades training coursework and in the on-the-job componentsof their training given, as Roth (2012, this issue) shows, the potential for significant misalign-ment between these two contexts.

Acknowledgement This article is a result of research funded by The Social Sciences and Humanities ResearchCouncil of Canada (SSHRC). I wish to thank the pipe trades preapprentices and course instructor who were thesubjects of this study for their willingness and openness to engage in this research endeavourwithme, SueGrecki forher valuable assistance and insights during fieldwork, and Bill Evans and Mick Bryant for generously sharing theirtechnical expertise during the preparation of the manuscript. Finally, I would like to thank the reviewers for theirhelpful comments and, especially, Gail FitzSimons and Arthur Bakker for their valuable feedback on earlier drafts.

References

Akkerman, S. F. (2011). Learning at boundaries. International Journal of Educational Research, 50, 21–25.Bakker, A., & Akkerman, S. (2013). A boundary-crossing approach to support students’ integration of statistical

and work-related knowledge. Educational Studies in Mathematics. doi: 10.1007/s10649-013-9517-zBakker, A., Groenveld, D., Wijers, M., Akkerman, S., & Gravemeijer, K. (2012). Proportional reasoning in the

laboratory: An intervention study in vocational education. Educational Studies in Mathematics. doi: 10.1007/s10649-012-9393-y

Eberhard, M. (2000). Forms of mathematical knowledge relating to measurement in vocational training for thebuilding industry. In A. Bessot & J. Ridgway (Eds.), Education for mathematics in the workplace (pp. 37–51). Dordrecht, The Netherlands: Kluwer.

FitzSimons, G., Mlcek, S., Hull, O., & Wright, C. (2005). Learning numeracy on the job: A case study ofchemical handling and spraying. Adelaide, Australia: National Centre for Vocational Education Research.

Forman, S. L., & Steen, L. A. (2000). Bringing school and workplace together. In A. Bessot & J. Ridgway (Eds.),Education for mathematics in the workplace (pp. 83–86). Dordrecht, The Netherlands: Kluwer.

Fownes, L., Thompson, E., & Evetts, J. (2002). Numeracy at work. Burnaby, BC, Canada: SkillPlan-BCConstruction Industry Skills Improvement Council.

Frankland, T. W. (1984). The pipe fitter’s and pipe welder’s handbook. New York, NY: Glencoe/McGraw-Hill.Gillespie, J. (2000). The integration of mathematics into vocational courses. In A. Bessot & J. Ridgway (Eds.),

Education for mathematics in the workplace (pp. 53–64). Dordrecht, The Netherlands: Kluwer.Hahn, C. (2011). Linking academic knowledge and professional experience in using statistics: A design

experiment for business school students. Educational Studies in Mathematics. doi: 10.1007/s10649-011-9363-9

Hoyles, C., Noss, R., Kent, P., & Bakker, A. (2010). Improving mathematics at work: The need for techno-mathematical literacies. New York, NY: Routledge.

Industry Training Authority of British Columbia. (2009). Plumber program outline. Victoria, BC, Canada:Queen’s Printer of British Columbia.

Jurdak, M., & Shahin, I. (2001). Problem solving activity in the workplace and the school: The case ofconstructing solids. Educational Studies in Mathematics, 47, 297–315.

Lave, J. (1988). Cognition in practice. New York, NY: Cambridge University.Leont’ev, A. N. (1978). Activity, consciousness and personality (M. J. Hall, Trans.). Englewood Cliffs, NJ:

Prentice Hall.Martin, L. C., & LaCroix, L. N. (2008). Images and the growth of understanding of mathematics-for-working.

Canadian Journal for Mathematics, Science and Technology Education, 8(2), 121–139.Martin, L. C., LaCroix, L. N., & Fownes, L. (2005). Folding back and the growth of mathematical understanding

in workplace training. Adults Learning Mathematics, 1(1), 19–35. Retrieved from http://www.alm-online.net/images/ALM/journals/almij_volume1_1_june2005.pdf. Accessed 1 Sep 2013

Masingila, J. O. (1993). Learning from mathematics practice in out-of-school situations. For the Learning ofMathematics, 13(2), 18–22.

Masingila, J. O. (1994). Mathematics practice in carpet laying. Anthropology and Education Quarterly, 25, 430–462.Noss, R., Hoyles, C., & Pozzi, S. (2000). Working knowledge: Mathematics in use. In A. Bessot & D. Ridgway

(Eds.), Education for mathematics in the workplace (pp. 17–36). Dordrecht, The Netherlands: Kluwer.Pozzi, S., Noss, R., & Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in

Mathematics, 36, 105–343.

Mathematical activity in pipe trades training 175

Radford, L. (2003). On culture and mind. A post-Vygotskian semiotic perspective, with an example from Greekmathematical thought. In M. Anderson, A. Sáenz-Ludlow, S. Zellweger, & V. Cifarelli (Eds.), Educationalperspectives on mathematics as semiosis: From thinking to interpreting to knowing (pp. 49–79). Ottawa:Legas.

Radford, L. (2008a). Culture and cognition: Towards an anthropology of mathematical thinking. In L. English(Ed.),Handbook of international research in mathematics education (2nd ed., pp. 439–464). New York, NY:Routledge, Taylor, and Francis.

Radford, L. (2008b). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G.Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom andculture (pp. 215–234). Rotterdam, The Netherlands: Sense.

Radford, L. (2008c). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations ofpatterns in different contexts. ZDM–The International Journal on Mathematics Education, 40(1), 83–96.

Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings.Educational Studies in Mathematics, 70(2), 111–126.

Ridgway, J. (2000). The mathematical needs of engineering apprentices. In A. Bessot & D. Ridgway (Eds.),Education for mathematics in the workplace (pp. 189–197). Dordrecht, The Netherlands: Kluwer.

Roth, W.-M. (2005). Mathematical inscriptions and the reflexive elaboration of understanding: An ethnographyof graphing and numeracy in a fish hatchery. Mathematical Thinking and Learning, 7, 75–109.

Roth, W.-M. (2012). Rules of bending, bending rules: The geometry of conduit bending in college andworkplace. Educational Studies in Mathematics. doi: 10.1007/s10649-011-9376-4

Roth, W.-M. (2014). Reading activity, consciousness, personality dialectically: Cultural-historical activity theoryand the centrality of society. Mind, Culture, and Activity, 21(1), 4–20. doi:10.1080/10749039.2013.771368.

Roth, W.-M., & Lee, Y. J. (2007). “Vygotsky’s neglected legacy”: Cultural-historical activity theory. Review ofEducational Research, 77, 186–232.

Roth, W.-M., Radford, L., & LaCroix, L. (2012). Working with cultural-historical activity theory. ForumQualitative Sozialforschung/Forum: Qualitative Social Research, 13(2), Art. 23. Retrieved from http://nbn-resolving.de/urn:nbn:de:0114-fqs1202232. Accessed 15 Jun 2012.

Triantafillou, C., & Potari, D. (2010). Mathematics practices in a technological workplace: The role of tools.Educational Studies in Mathematics, 74, 275–294.

Vygotsky, L. (1979). Consciousness as a problem in the psychology of behavior. Soviet Psychology, 17(4), 3–35.Wake, G. (2014). Making sense of and with mathematics: The interface between academic mathematics and

mathematics in practice. Educational Studies in Mathematics. doi:10.1007/s10649-014-9540-8.Wedege, T. (2000). Mathematics knowledge as a vocational qualification. In A. Bessot & J. Ridgway (Eds.),

Education for mathematics in the workplace (pp. 127–36). Dordrecht, The Netherlands: Kluwer.Williams, J., & Wake, G. (2007a). Black boxes in workplace mathematics. Educational Studies in Mathematics,

64, 317–343.Williams, J., & Wake, G. (2007b). Metaphors and modes in translation between college and workplace

mathematics. Educational Studies in Mathematics, 64, 345–371.Williams, J., Wake, G., & Boreham, N. (2001). College mathematics and workplace practice: An activity theory

perspective. Research in Mathematics Education, 3, 69–84.Zevenbergen, R., & Zevenbergen, K. (2004). Numeracy practices of young workers. In M. J. Høines & A. B.

Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology ofMathematics Education, 4 (pp. 505–512). Norway: Bergen University College.

Zevenbergen, R., & Zevenbergen, K. (2009). The numeracies of boatbuilding: New numeracies shaped byworkplace technologies. International Journal of Science and Mathematics Education, 7, 183–206.

176 L. LaCroix