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Journal of Vocational Education & Training
ISSN: 1363-6820 (Print) 1747-5090 (Online) Journal homepage: https://www.tandfonline.com/loi/rjve20
Is the mathematics classroom a suitable learning
space for making workplace mathematics visible?
– An analysis of a subject integrated
team-teaching approach applied in different learning
spaces
P. Frejd & K. Muhrman
To cite this article: P. Frejd & K. Muhrman (2020): Is the mathematics classroom a suitable learning space for making workplace mathematics visible? – An analysis of a subject integrated team-teaching approach applied in different learning spaces, Journal of Vocational Education & Training, DOI: 10.1080/13636820.2020.1760337
To link to this article: https://doi.org/10.1080/13636820.2020.1760337
© 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
Published online: 06 May 2020.
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ARTICLE
Is the mathematics classroom a suitable learning space
for making workplace mathematics visible?
– An analysis
of a subject integrated team-teaching approach applied
in di
fferent learning spaces
P. Frejdaand K. Muhrmanb
aDepartment of Mathematics, Linköping University, Linköping, Sweden;bDepartment of Behavioural Sciences and Learning, Linköping University, Linköping, Sweden
ABSTRACT
This article presents an analysis of a team-teaching approach, applied in two learning spaces: a regular mathematics class-room; and a hairdressing salon at an upper secondary voca-tional education and training (VET) school. A mathematics teacher and a VET teacher jointly developed, planned and carried out the teaching activities in these two learning spaces. The overall goal was to prepare their 15 students for professional life. Observations made from the two lessons were analysed with the aim of identifying the extent to which the outcome of the team-teaching approach is dependent on the choice of the learning space. Drawing on Engeström’s activity theory and research literature on learning spaces, our results indicate that tools, norms, division of labour and community differ significantly within the two learning spaces. The environment of the salon appeared to be more effective in promoting discussion, and encouraging self-confidence and identity-making, in comparison with the mathematics classroom, and there was a more visible inter-action between mathematics and vocational subjects. This raises questions about the optimal design of the applied team-teaching approach and about whether the mathe-matics classroom can be considered a suitable learning space for facilitating students’ learning of workplace mathematics.
ARTICLE HISTORY
Received 15 April 2019 Accepted 4 April 2020
KEYWORDS
Activity theory; subject integration; team-teaching; vocational education; workplace mathematics; learning space
Introduction
According to OECD (2010), a well-educated and skilled labour force is an indispensable condition for developing a knowledge-based economy in the globalised world. To satisfy the labour market’s demand for highly skilled work-ers, efficient and adequate education and training are required. One approach with a focus on education for the workplace is vocational education and training (VET), and one part of VET relates to the teaching and learning of mathematics.
CONTACTP. Frejd peter.frejd@liu.se
https://doi.org/10.1080/13636820.2020.1760337
© 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduc-tion in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.
The main goal of vocational mathematics education is to either prepare indivi-duals for future work or to increase the competence of already qualified workers (FitzSimons2014). Another goal is to provide students with formal mathema-tical knowledge in order to prepare them for higher education.
However, there seems to be a clear tendency in vocational mathematics education at upper secondary school in Sweden to focus primarily on the latter aim (e.g. Muhrman 2016; Skolinspektionen 2014). One consequence of this focus is that students risk missing opportunities for employment, because they lack mathematical training and, consequently, the mathematical compe-tences that the labour market demands (Muhrman2016). Mathematics in the workplace is often characterised as complex, situation-dependent and hidden in workplace-specific tools (e.g. Noss, Hoyles, and Pozzi 2000; Wedege 2010; Yasukawa, Brown, and Black2013). The specific technologies and social, political and cultural dimensions found in the workplace are rarely present in educa-tional settings (ibid.). Therefore, other teaching approaches that take the mathe-matics of the workplace into account are needed.
There is evidence that teaching approaches focusing on connecting math-ematical and vocational learning lead to more positive attitudes among students towards learning mathematics (e.g. Casey et al. 2006; Muhrman
2016). There is a wide range of teaching approaches that support an integra-tion between mathematical and vocaintegra-tional learning. One approach is that mathematics and VET teachers work collaboratively in or outside of mathe-matics classrooms with projects that are directly related to the students’ vocational course (Dalby and Noyes 2015). Research literature indicates that teaching approaches that include authentic real-life workplace contexts moti-vate students to learn mathematics and provide them with accessible insight for their future vocational profession (Casey et al.2006; Dalby and Noyes2015,
2016). However, the learning spaces, how the classrooms are organised and situated, and where the teaching approach is going to be applied, will impact on the teaching approach to be used and on students’ learning outcomes (Martin2002; Stadler-Altmann2015).
The present paper is a part of a larger study of how mathematics and vocational subjects can be better integrated and of how interdisciplinary colla-borations can be encouraged between mathematic teachers and VET teachers, in order to achieve a smooth transition from school to workplace practice. In particular, we aim to explore the potential and limitations of collaborations and teaching approaches within different activity systems and learning spaces, which all support workplace learning. To address parts of this aim we will, in this paper, investigate a subject integrated team-teaching approach implemen-ted in two learning spaces– the hair dressing saloon and the regular classroom – as two different activity systems. Using the third generation of activity theory (CHAT), we will analyse the differences in tools, rules, community and division of labour that are visible in the different activity systems and how these differences
affect the extent to which students are prepared for using mathematics in their future workplace. The following research questions are explored:
● What differences in tools, rules, community and division of labour are visible in the activity systems that are generated by a subject integrated team-teaching approach and applied in the learning spaces of the mathematical and vocational classroom?
● What differences in outcomes of the two generated activity systems are visible in terms of the processes for developing students’ professional awareness?
To clarify and frame the research questions within the theoretical framework of CHAT, the following section will provide a language that is adequate to describe the key elements in our study: the activity system, learning space and a subject integrated team-teaching approach.
Activity theory, learning space and subject integrated team-teaching approaches
Activity theory
Activity theory has been extensively used as a theoretical framework for explor-ing real-world problems connectexplor-ing workplaces with mathematics education (see, e.g. Jurdak2016; Jurdak and Shahin2001; Nussbaumer2012; Williams and Wake2007; Yasukawa, Brown, and Black2013). Activity theory aims to describe an entire activity or work system (including work tasks, organisations, etc.) embedded in a community. The third generation of activity theory, Cultural Historical Activity Theory (CHAT; see Engeström1987,2001) builds on the basic subject-object-tools or mediator relation developed by Vygotsky (1978), where individuals (subject) try to fulfil the motive (object) to create an outcome by using physical objects, language, gestures, etc. (tools). However, the subject-object-tools relation is also influenced by the social, cultural and historical environments in terms of community, rules or norms, and division of labour (see the nodes of a triangle inFigure 1).
The rationale for using CHAT as a language for the description of the analysis in this project is that it offers a lens through which to view and analyse a teaching approach for developing students’ professional awareness in differ-ent activity systems. The third generation of CHAT is designed as a research tool for an understanding of multiple perspectives and networks of interacting activity systems with a common object. These common objects are described in the research literature (e.g. Jurdak 2016) as boundary objects and may facilitate students in overcoming the boundaries between mathematics in school and mathematics in the workplace.
The two activity systems we intend to explore are different activity sys-tems that share the common object of developing students’ professional awareness and, in particular, ‘teaching students to develop mathematical knowledge that is useful for their future vocation’ (see Figure 1). There are many similarities between the two activity systems at a more general level, such as the motives (teaching through subject integration with a team-teaching approach in order to educate students in mathematics and in vocational subjects according to the national curriculum), the subject (mathe-matics and VET teachers at the hairdressing programme) and the community (15 hairdressing students, one mathematics teacher, one VET teacher). However, although some parts of the activity systems are identical at this general level, findings from the research literature (e.g. Muhrman 2016; Yasukawa, Brown, and Black 2013) indicate that the historical and cultural setting nevertheless influences the outcomes of various team-teaching arrangements for integrating mathematics in vocational education training, which makes the two activity systems rather different from each other. Differences in cultural settings, for example, can influence how teachers perceive their role in the team-teaching approach and what the students are expected to learn in the different activity systems. While mathematics has a strong tradition of teacher-centred teaching, hairdressing has an apprentice tradition. Within the field of mathematics, there is also a strong tradition of what is considered to be valid mathematical knowledge. Similar traditions can be assumed to exist in the hairdressing profession. Differences in what count as legitimate teaching practice and knowledge within and between activity systems, may cause tensions or contradictions. However, within activ-ity theory tensions are not seen as conflicts, but rather as providing informa-tion which may enable organisainforma-tional change (Engeström 2001).
Figure 1.Model of the two Activity systems related to Engeström’s (2001) third generation of activity theory.
Learning space
The physical environments around teachers and students in terms of school space and classroom space also constitute part of the cultural and historical dimensions of teaching and learning. The way in which classroom space is organised influences pedagogical methods (Martin2002). A traditional mathe-matics classroom, for example, is frequently organised as ‘teacher-centred’, with desks arranged in rows facing the blackboard, whereas many VET training halls are more ‘student-centred’, simulating a workplace situation. Student interactions with the learning space seem to influence their own motivation for learning, which may have an indirect effect on learning outcomes (Stadler-Altmann 2015). However, drawing on research literature, a recent article by McNeil and Borg (2018) asserts that the relationship between learning and spaces is complex and that understanding of this relationship is under-developed. They argue that there is insufficient evidence about this relation-ship, as well as an insufficient focus on pedagogy, and call for more research. This paper explores the relationship by analysing two learning spaces, using activity for a particular type of pedagogy, namely subject integrated team-teaching.
Subject integrated team-teaching approaches
Teaching as a means to connect mathematical and vocational learning has multiple descriptions in the research literature. These descriptions all draw on some form of collaboration between teachers (Dalby and Noyes2015; Savage
2011; Sriraman and Freiman2010).
In this paper, we adopt what is called a team-teaching approach (Cook and Friend
1995) for mathematics in vocational education. Team-teaching is a co-teaching approach where two or more teachers share ideas and the giving of instructions to students in jointly organised teaching practice (Cook and Friend1995).
Our definition also resonates with the third model (shared delivery) of subject integration as a team-teaching approach, identified by Black and Yasukawa (2013) in the context of VET education in Australia. The three models are: study skills, vocational socialisation, and shared delivery. The first model, study skills, refers to a scenario in which a mathematics teacher plays a largely secondary role to the vocational teacher, primarily assisting students. The vocational socialisation model proposes a close collaboration between the VET and the mathematics teacher, who plan their lessons jointly and teach whole groups of students, taking turns to focus on their area of expertise. The third model, shared delivery, is when the teachers work very closely together with the same group of students, aiming to prepare them for their future occupation by sharing the same pedagogical disposition and practices.
The empirical study
As already stated, this paper is a part of a larger research project that took place during the school year of 2016–2017. The project was initiated by a mathematics teacher working at an upper secondary VET school. He contacted one of the authors and enquired about the possibility of developing a project based around teaching approaches of subject integration, also including the other author in the project. The authors’ role was to participate in the mathematics teachers’ planning meetings, to observe lessons, and to later evaluate these, together with the mathematics teachers. The authors also interviewed teachers and students three times during the project.
In terms of background information, the VET school has about 2700 students, 350 teachers and nine different programmes. The school building includes both traditional classrooms used for subjects such as mathematics and English, and vocational classrooms, such as a hairdressing salon, a car workshop, a bakery, and a restaurant. As afirst attempt to apply a subject integrated team-teaching approach and connecting the two school subjects of mathematics and hair-dressing, the study took place over the course of two lessons one in the mathematics classroom and one in a hairdressing salon.
In Swedish upper secondary vocational education, students learn mathe-matics both in vocational courses and in their mathemathe-matics course(s). One mathematics course of 100 hours, that aims to both prepare students for future occupation and higher education, is mandatory in all VET programmes. This course includes contents such as measurements, calculations of volumes, use of algebraic expressions and formulas (Skolverket2019b). In addition, aspects of mathematics are found in the syllabus for the vocational courses more or less implicitly. However, while the syllabuses define what content to teach and the goals of teaching, they do not stipulate the teaching methods to be used. This may cause some tension, since the teaching methods applied may not always reflect the dual aim of preparing students for both a future occupation as well as for higher education.
The VET teacher who participated in this study has substantial experience of working both as a hairdresser and as a VET teacher. Her teaching activities are mostly carried out in the training hall for hairdressing.‘Hands-on’, service, colla-boration and communication skills are emphasised in her teaching as necessary attributes for building and maintaining a business. The mathematics teacher is also very experienced. He typically adopts traditional pedagogical techniques, but also enjoys trying new teaching approaches. The rules shaped by the community of traditional mathematics teaching has a long history and is the most frequent teaching method used in Sweden (Jablonka and Johansson 2010). This largely consists of a teacher presenting a method or a concept at the white board and the students then working individually with textbooks, exercises and tasks, while the teacher walks around the classroom offering assistance. Both teachers had little
experience of working with subject integrated team-teaching before participating in the study.
The group of 15 female students taking part in the study are all of the same age, have similar school backgrounds and are all interested in hairdressing. While they may not have a homogenous background in terms of academic achievements and other interests, this paper’s focus is on the students as a group and, therefore, the observation scheme described below does not consider students individual backgrounds.
The activities in Lesson 1 and Lesson 2 were jointly planned by the mathe-matics and VET teachers, with an overarching object to apply the team-teaching approach, facilitate learning for students applying mathematics and to explore a context related to hairdressing, useful for the students’ future occupation. The mathematics teacher, by his own account, took the initiative to the collabora-tion and was the principal driving actor in the planning phase.Table 1provides information about the activities implemented in terms of the nodes of an activity system.
The two lessons were observed by two researchers (the authors). The obser-vational method included the use of an observation scheme designed to include aspects related to the subject integrated team-teaching approach, such as; frame for the lesson, mathematical content, ways of teaching, engage-ment, motivation/understanding.
To seek answers to research question 1, we used the notes from the observa-tion scheme to describe the nodes of the activity systems (instrument, rules, community and division of labour), and to investigate how these nodes differed between the two lessons. The characterisations of the nodes were also used to identify and analyse the processes which impact of the development of the students’ professional awareness, which relates to research question 2.
Table 1.Background information of the activities to be implemented in Lesson 1 and 2.
Lesson 1, Mathematics and shampooing in the
hairdressing salon Lesson 2, Business plan for a hairdressing salonin the mathematics classroom The activity Teaching through subject integration with a team-teaching approach to educate students in
mathematics and in vocational subjects according to the national curriculum Subject Mathematics- and VET teachers at the hairdressing program
Object Prepare students for their future occupation, within the context of how to shampoo a client’s hair and estimate the service charge
Prepare students for their future occupation, within the context of using excel to make a budget
Tools Mathematical knowledge, Pen and whiteboard, a set of tasks developed to be handed out, and equipment and accessories in the salon
Mathematical knowledge, Spreadsheets, digital projector and a set of tasks developed for students to be solved
Rules Time limitations, the facilities in vocational classroom, the norms used in the vocational classroom, etc.
Time limitations, the facilities in mathematics classroom, the norms used in the mathematics classroom, etc Community 15 hairdressing students, 1 mathematics teacher, 1 VET teacher
Division of labour
The mathematics teacher and the VET teacher intend to work collaboratively in the hairdressing salon and share equal responsibility to deliver the task, students are expected to work in pairs or groups of three
The VET teacher is invited to the mathematics classroom to take part in a discussion about making a budget, the mathematics teacher delivers the tasks and students are expected to work individually
Lesson 1: Mathematics and shampooing (in the hairdressing salon)
As described in Table 1, the first intervention took place in the hairdressing salon at the upper secondary VET school. The salon was furnished with hair-dressing chairs, mirrors and wash areas. Connected to the salon was a hall with desks. The lesson started in the morning, lasted for 2 h and 15 students participated. The VET teacher and the mathematics teacher introduced the lesson by describing and explaining the tasks (seeFigure 2to the students.
The aim of the tasks in Figure 2 was that students should learn how to shampoo a client’s hair and estimate the service charge, taking into account working time and amount of shampoo used. The teachers described the task together, although the VET teacher focused on the practical work of shampoo-ing, and the mathematics teacher focused on the calculations. The VET teacher discussed practical issues regarding the shampooing of a client’s hair, such as the correct type and quantity to use, depending on hair type, length, thickness, texture and colour. The mathematics teacher addressed other issues, such as how to measure the amount of shampoo used in cubic centimetres. During the teachers’ presentation, the students listened attentively and seemed eager to begin the tasks.
Shampooing- consumption and costs?
You are going to wash a customer’s hair. Your tasks are to investigate how much shampoo to consume and how much the customer should pay. You shall present your estimations and calculations in a report.Investigation
• Describe the hair of your customer.
• How much shampoo do you consume? Measure the volume as accurately as possible.
• Measure the time to perform a hair wash?
Calculations
• Draw a sketch of the shampoo bottle in scale 1:10 • Calculate the volume of the shampoo bottle. Compare
with the volume displayed on the bottle. Is this information correct?
• How many shampooings does the bottle last?
• How much should you charge for a hair wash? Consider the cost of shampoo and your working time. • Estimate the number of customers you shampoo
during a day. How many bottles of shampoo will you consume during a month?
The students organised themselves in pairs or in groups of three and sham-pooed either each others’ or dolls’ hair. To determine the correct quantity of shampoo, the students poured the amount of shampoo in the hand to estimate its quantity and thereafter put it into the measuring cup. The teachers also instructed the students to estimate the time needed for washing the customer’s hair.
Almost all students acted in a service-minded way while washing each others’ hair and carried out their tasks in a dedicated manner. Both the mathematics and hairdressing teachers moved around the classroom, giving advice to students during this part of the lesson. Once the students had finished shampooing, they took their places either in the salon or in the hall with desks connected to the salon and started working on their mathematical tasks. They used their estimates of consumed shampoo and washing time for calculating how much a hairdresser should charge a client to make a profit. In addition, they were asked to calculate the volume of the shampoo bottles, to compare this with the volume displayed on the bottle, and to estimate the consumption and cost of shampoo over a month. The prices of the shampoo bottles were given without VAT, obliging the students to perform calculations to establish the full cost.
The dedication and engagement of the students during the shampooing were sustained while they did their calculations. Students discussed their find-ings enthusiastically with others in their group and compared these with the results from other groups. When the groups realised that their answers differed, they evaluated their own calculations step by step in order to understand the discrepancies. During these discussions, the students used mathematical rea-soning to draw conclusions, based on theirfindings.
Several unexpected learning opportunities were observed as an outcome of the activity, which were not anticipated by the mathematics teacher. One learning experience related to the shape of the shampoo bottle. The stu-dents anticipated that the shampoo bottle had the shape of a regular cylinder. However, the bottle had a slight curvature towards its smaller bottom section, causing a deviation from this form. Having performed their calculations assuming a cylindrical form and using the diameter of the bottom disc, the students were amazed by the significant deviation from the displayed volume (1000 ml), which seemed difficult to explain from measurement errors. They were not convinced that the displayed volume was correct until they had done the calculations again, this time measuring the circumference of a circle around the middle of the bottle. Another learning experience was identified when students realised that they had calculated the cost of shampooing in two different ways. Some groups based their estimation of cost on hourly salary, while other groups estimated the cost of running the hairdressing salon for 1 h, including the cost of renting the salon, etc.
Lesson 2: Business plan for a hairdressing salon (in the mathematics classroom)
The second lesson with the same group of students and teachers was con-ducted 1 month after the first lesson. This lesson was the students’ first of the day and took place in the mathematics classroom: 13 students attended (two were absent). The desks in the classroom were organised in three columns of two desks, each placed so that the students faced the whiteboard.
The word‘Moms’ (VAT) was projected on the whiteboard when the students entered the classroom. Due to a minor snow storm, the VET teacher was late. The mathematics teacher started the lesson by distributing instructions,‘Tasks for spreadsheets’ and a compendium ‘Budget for renting a chair/sole proprietor-ship’ and explained to the students that they were going to work on these tasks. When the VET teacher arrived, the mathematics teacher asked the students to close their computers and pay attention. The two teachers then sat in the front of the classroom and played the roles of a press reporter (the mathematics teacher) and a hairdresser (the VET teacher) to set the scene for the lesson. The observed dialogue is displayed inTable 2.
The students sat attentively during the dialogue. After the end of the dialogue, the VET teacher left the classroom and the students were instructed to begin working on ‘Tasks for spreadsheets’. However, some of the students were not familiar with spreadsheets. To assist the students, the mathematics teacher dis-played the programme on the whiteboard and asked the students to follow his procedures. The teacher gave instruction of how to type numbers and text into the
Table 2. A dialogue between the mathematics teacher (reporter) and the VET teacher (hairdresser).
Questions and responses
Reporter -As a hairdresser do you own a salon/rent a chair or are you an employee?
Hairdresser −50/50, It used to be common for hairdressers to rent a chair, but the standard of practices has changed and it is safer to be an employee than being a business owner.
Reporter -What is the difference?
Hairdresser - You may get ill and you get your salary even if there are no customers. However, if you have your own chair you may make a better profit and you are more independent
Reporter -If you have your own business, is it important that you have knowledge about economics? Hairdresser -Yes, if I’d want to have a monthly salary of 20 000 SEK I might have to earn 40 000 SEK because half
is reduced by income tax and social security contributions. Reporter -Are you paid for all your working time?
Hairdresser -No, as a chair owner there is some’downtime’ such as answering the phone, or waiting for the hair colour to set. An income must also cover this time.
Reporter -What is a budget?
Hairdresser - It is an estimation of costs of rent, electricity, water etc. and resources in stock etc. Reporter - Is it possible to get help to draw up a budget?
Hairdresser -Yes, you may hand over the task to an auditor, but it is expensive and after signing the budget you are still responsible for it.
Reporter -Is‘VAT’ a part of the budget?
Hairdresser -Yes, it is and it is difficult and that was the reason why I let an auditor draw up my budget. Reporter -Is it reasonable for a hairdresser to earn 400 SEK/h excluding tax?
Hairdresser -Yes, that means you have to charge your costumier 500 SEK/h. As an entrepreneur you must either charge more and have fewer customers or charge less and work more.
cells. The next task was to derive a formula for the procedure (i.e. Think of a number, etc.) seeFigure 3.
The procedure to solve this task in Figure 3 was also displayed, by the teacher, on the whiteboard, since the students were having difficulties. After the students completed the task‘Think of a number’, they started working in the compendium. They opened a blank workbook and gave it a name, copied a table of working hours from a page on internet, calculated the mean values for monthly working days and working hours, and created a cell for effective working time (downtime subtracted from the total working time).
During the lesson, the students did not question why they needed to learn to use a spreadsheet, but some students did seem to become bored and brought out their cell phones. The lesson ended with the mathematics teacher explicitly emphasising the goals in the curriculum and the value of using spreadsheets in their future working lives. He also said that they would continue the work in the compendium during the next lesson.
Analysis of the data and results
Lesson 1 and Lesson 2 constitute two attempts of using a subject integrated team-teaching approach for teaching mathematics in VET education. The approach creates two different versions of activity systems, focusing on differ-ent aspects of mathematics within hairdressing. Both lessons were presdiffer-ented by the same teachers (subject), were directed towards the same students, and shared the overarching aim of developing mathematical knowledge which would be useful for students’ future careers (object). They thus generated a shared activity system. However, aspects of division of labour, tools, rules or norms, and community differed between the two lessons and these had an impact on the outcome.
Think of a number. Multiply the chosen number by six. Add the product by 12. Divide the sum by three. At last, subtract the double amount of the original number from the new number. What number do you get?
Type the text, as displayed in the picture.
In cell B2 type =B1*6 In cell B3 type =B2+12 I think you can figure out what to type in cell B4 In B5 type =B4-2*B1
Type different numbers in cell B1. What do you get?
Division of labour
An analysis of the division of labour between the two teachers indicates that the subject integrated team-teaching approach differs from Lesson 1 to Lesson 2. In Lesson 1, in the hair-salon, the teachers appeared to work as a team and shared equal responsibilities for the entire lesson. Although it was observed that they were both committed to help the students, they each had their respective‘area of expertise’. In terms of Black and Yasukawa (2013), such a team-teaching approach would be called vocational socialisation rather than shared delivery. The VET teacher was responsible for explaining the more vocational-related parts of the task, such as how to choose the right shampoo in relation to different hair texture, and the mathematics teacher was responsible for the explanation of the mathematical parts. In Lesson 2, the team-teaching differed from Lesson 1 in that the teachers did not share equal responsibilities, causing some tensions in the activity system; the mathematics teacher carried the load of informing, distributing instruction, answering questions, organising the dia-log and so on, while the VET teacher was only required to play the role of being a hairdresser, visiting the classroom.
The division of labour was observed to have an influence on the process of professional awareness in terms of identity-making in either becoming a hairdresser or being a mathematics student. In the salon, both teachers took equal responsibilities: the hairdresser assisted with‘hands on’ activities, helping with the development of the students’ their vocational identity; and the mathe-matics teacher emphasised the mathematical aspects of the activities. In the mathematics classroom, the teachers’ dialogue provides a basis for the students to develop their vocational identity. However, during the dialogue the VET teacher mentioned that she thought calculations of VAT were too complex for her to do, and left after the introductory dialogue (seeTable 2).This may have given the students an impression that VAT calculations are not a genuine part of the hairdressing vocation.
Tools
In both activity systems, mathematical knowledge can be described as a mediated tool that must be applied in a hairdresser’s work. However, the two activity systems also include mediated tools that are culturally linked to mathematics and hairdressing, respectively (seeTable 1). In Lesson 1 paper, pen and calculator were used, as well as a number of job-specific tools, such as showerheads, protective aprons, shampoo bottles, and hairdryers. In Lesson 2 no job-specific tools were used, with the exception of computers. The use of the work-specific tools illustrates their key roles as mediators between the students and the object (to use mathematics in vocational contexts to prepare students for their future occupation). Without these tools, the connection between the
two subjects would be less explicit. In Lesson 1, the students recognised, and were familiar with using, the work-specific equipment and accessories in the salon and this seemed to have an impact on the outcome, at least in terms of engagement. The shampoo bottle in particular raised several mathematical discussions. In Lesson 2, the students were not all familiar with spreadsheets, and the use of them in the workplace seemed to be less necessary to the students.
The tasks designed by the teachers may also be described as mediating tools. In Lesson 1, the task to measure the volume would probably not be a practice in the workplace, but does makes the concept of volume explicit and its relation to units. The calculations regarding service charge could be seen as a mediating tool, even though a standard charge for washing and shampooing might be expected. The mathematics required in Lesson 1 relates to geometry and economic calculations, which is explicit in the syllabus of mathematics (Skolverket 2019b) and implicit in the syllabus of hairdressing (Skolverket
2019a). The mathematics required in Lesson 2 relates to formulas and the use of digital tools and spreadsheets, and is also found in the VET syllabus. However, as seen from Figure 3, the instructions and the compendium focused on procedures with academic characteristics, in terms of following instructions, causing a tension between learning goals for higher education and future occupation.
Rules
Teaching techniques and learning spaces over the course of the two lessons impacted on the prevailing norms. All norms have cultural and historical roots and some are more resistant to change than others. The norms of traditional mathematics teaching were visible in the regular classroom, while the hair salon invited students to acquire norms of working life. When the students worked on ‘hands on activities’ in the hair-salon, they clearly seemed familiar with the rules or norms that prevail there, and quickly set to work. During the shampooing, the students treated each other with the same respect that customers would expect. They demonstrated a spirit of service and a high level of energy, and frequently asked questions and helped each other, which is common practice in the salon. When they had completed the shampooing, they sat down at desks next to the salon to work with the calculations, with continued high engagement.
In the mathematics classroom in Lesson 2, completely different rules or norms prevailed, evident even in how the room was furnished. The students sat in desks in lines facing the teacher’s desk to imitate his work on the spreadsheets. Lesson 2 has a more traditional design of teaching mathematics, where the students work individually, waiting for assistance from the teacher rather than asking peers. Many students, therefore, spent most of the lesson waiting for the teacher, resulting in low engagement, in contrast to Lesson 1,
where the students moved around in the classroom, discussing and helping each other solving both mathematical tasks and‘hands on’ activities.
The fact that the norms in the vocational classroom differed from the norms in the mathematics classroom had an impact on ways of working and behaving in these classrooms. In particular, they helped to create an atmosphere which facilitated the discussion of mathematics and vocational content in the voca-tional classroom, which was not observable in the mathematics classroom.
Community
Within the Community, the roles of teacher-teacher and teacher-student varied. In Lesson 1, the teachers perceived that they had equal roles during the lesson and the role of student-teacher varied. In some parts of the lesson, the teachers had a leading role, for example, in the introduction, but in the part of the lesson where the students were working with the shampooing, the students took the leading role, explaining to the mathematics teacher how and why they acted in a certain way. This shift of roles implies a new type of situation, where the mathematics teacher and students become more equal, akin to working collea-gues. In Lesson 2, no shift in the role teacher-student was observed and the mathematics teacher had a leading role throughout the whole lesson, during both the introduction and the implementation of the task.
The shift of teacher-student roles within the community seemed to encou-rage the development of self-confidence in the vocational training hall. This self-confidence was also evident in the process of solving the mathematical tasks in Lesson 1, where the students discussed with peers rather than asking the teachers. When the traditional teacher-student roles were maintained in Lesson 2 in the mathematics classroom; however, students in difficulty directed their questions to the teacher rather than to their peers.
Conclusion and discussion
Based on our analysis we conclude that mathematical learning was shaped differently in Lesson 1 than in Lesson 2. Applied in the two classrooms, the approach generated two different activity systems that were shaped by the subject, object, tools, rules or norms, division of labour, and community. Although the teachers shared the overarching aim of the object, that is, prepar-ing students for their future occupation, in both lessons, the impact of the other components in the activity systems influenced the outcome.
The outcomes of the interventions may indicate that the students linked the mathematics education and the vocational education better in the salon, since the activity system developed in the salon included more workplace-related tools and a workplace community that has a culture of responsibility sharing and assistance between colleagues. Using open problems with ‘hands on’
activities in the VET classroom facilitated the development of processes which better encourage mediation and identity making, compared to the teaching practice in the mathematical classroom. The kind of mathematics used in the two lessons illustrates that the mathematics in the workplace is often implicit and hidden in objects (Noss, Hoyles, and Pozzi2000; Williams and Wake2007). However, mathematics education has the power and maybe the responsibility to make the importance and uses of mathematics more explicit to students. In Lesson 1, focusing on the quantity of shampoo used and the time required to wash someone’s hair illustrated that these are factors that may be quantified in terms of mathematics. In this particular lesson, the shape of the shampoo bottle raised several questions concerning estimation errors and the use of different methods to calculate the same volume with more or less accuracy. In addition, the two ways of calculating the cost of shampooing, hourly salary versus the cost of running the hairdressing salon for 1 h, highlighted different methods of drawing up a budget and calculating costs. Altogether, this illustrates that while the mathematics used in workplace-related tasks in vocational education may not be mathematically advanced, they are made complex by the context (Muhrman2016). The students’ discussions show the extent to which teachers have an important and complex task in intervening at the right times and moments.
The teachers did not seize the opportunity to discuss and compare the different ways of calculating the cost of shampooing. A discussion about how to calculate service charges as realistically as possible might also have been used as a starting point for the second lesson. Nevertheless, the second lesson in the mathematics classroom highlighted how spread-sheets utilise ‘hidden’ mathematics – including, for example, formulas, rounding, VAT calculations, effective working time, percentage change fac-tor, on so on This mathematics is a prerequisite for making a budget as well as a part of the syllabus of mathematics and hairdressing (Skolverket 2011ab). Hence, the students were given the opportunity to see spread-sheets as an explicit mathematical tool used in the workplace. However, they did not take this view, and did not connect this mathematical and typical workplace tool with their future vocation. The VET teacher herself did not stress the importance either, since she left her budget to an auditor, which may have contributed to the low engagement during the lesson. The particular knowledge valued by the mathematics teacher and the VET teacher impact on what and how to teach. Even though the overall goal of the activity was to prepare students for their future occupation, the cultural and historical experiences of teachers shaped the activity towards academic knowledge in the mathematics classroom, compared to more practice-based knowledge in the VET classroom, causing tensions in the activity system.
The extent to which the two lessons produced outcomes that would qualify as adequate mathematical abilities demanded by the labour market (Muhrman
2016) is a matter of speculation. The results of this study are based on observa-tional analysis, which is influenced/affected by the observation scheme used, the observers and the method of analysis. The observation scheme used was designed to observe a subject-integrated team-teaching approach in different learning spaces and, following each lesson, the two researchers discussed and analysed their notes from the observations in collaboration in order to enhance reliability and validity. The results from the observation were also presented to teachers and school leaders at a meeting, with the aim of further developing subject-integrated teaching at the local VET school.
Based on our results, we conclude that the learning space of vocational class-room with access to workplace-authentic tools and a culture of workplace norms led to greater student engagement in discussion and collaboration about the mathematical tasks, in comparison to working in the mathematics classroom with the prevailing norms of traditional mathematical teaching. This suggests that, under suitable conditions, working with activities that connecting mathematics with a vocational subject can increase student engagement (cf. Casey et al.2006; Dalby and Noyes2016). The students’ self-esteem also appeared to be positively influenced by the activity system of the learning space of the vocational classroom, in comparison with the activity system of the mathematical classroom. This was followed up in more detail in an interview study (Muhrman and Frejd2018). There were many more mathematical discussions and more practices of different calcula-tions and unexpected problem-solving in the learning space of the vocational classroom. Most of the students were also able to calculate VAT in Lesson 1, while they had problems doing so during Lesson 2. The design of Lesson 1, therefore, seemed to give the students an understanding of VAT calculations which appeared to be absent in Lesson 2. This leads us to the conclusion that the students’ under-standing of how to solve vocational-related mathematical tasks may be increased when they are in an activity system that resembles professional life.
In conclusion, the results from this study indicate, in line with other research results (e.g. Casey et al.2006; Black and Yasukawa2013), that the team-teaching approach where teachers share the responsibility for leading the lesson equally and integrate mathematics with vocational subjects may have positive effects on VET students learning of mathematics if the lessons are carried out in an activity system similar to a workplace. This effect does not seem to be the same if the subject-integrated team-teaching approach is applied and carried out in a regular classroom. Traditional mathematics teaching has a long history and is resistant to change, causing tensions between the community of mathematics teachers and VET teachers when working with subjected integrated teaching (Muhrman 2016). Nevertheless, highlighting and discussing these tensions within and between activity systems such as valued knowledge, traditional/ practical-based teaching methods, and goals in the curriculum, as reported on
in this study, in addition to moving the mathematics teaching to a learning space where the culture of VET profession is strong, could also provide a ffor-dances for organisational change.
However, the relationship between space and learning is complicated (McNeil and Borg 2018) and a possible follow-up study could be to explore if a similar result would be generated if Lesson 2, the spreadsheet, had taken place in the salon. It is also important to emphasise that we did not consider the differences in the students’ individual responses to the inte-grated team-teaching approach. In a follow-up study (Muhrman and Frejd
2018), the students were interviewed, confirming our conclusions about the benefits of organising some of the mathematics teaching in the learning space of vocational classroom. However, the interviews also showed that some students preferred traditional mathematics teaching with less voca-tional integration, since they also have addivoca-tional goals with their VET education, such as further education study. Further research is necessary to explore and validate the usefulness of team-teaching for integrating mathematics and VET-subjects in different learning spaces.
Disclosure statement
No potential conflict of interest was reported by the authors.
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