Transforming students’ approaches to learning and applying mathematics

D. Moran and H. Owen,

Higher Diploma Foundations,Dubai Men’s College, HigherColleges of Education, Dubai, United Arab Emirates

Abstract

The increasing scope of curriculum content, time restraints and gateway assessment require mathematics teachers to actively explore alternative learning strategies in order to effectively address students’ learning needs. In addition to meetingcurriculum-specific objectives, teachers are often expected to directly address graduate outcomes at each stage of the learning experience. A cooperative learning approach is one effective strategy to meet required objectives, while also addressing the requisite holistic outcomes to help students be competitive in today’s work environment.

This paper discusses an approach utilised to integrate mathematics within the interdisciplinary Foundations program: Computing, Research Skills and Projects (CRSP), at Dubai Men’s College (DMC). The CRSP course, originally developed to address English language and research skills objectives, has been designed with the intent of transforming students’ attitudes and approaches to learning. CRSP is a blended learning course delivered through fifty percent face-to-face sessions, and fifty percent through the learning management system, WebCT. The latter effectively incorporates the meaningful application of learning technologies such as audio, video, animations, applets and online quizzes. Mathematical tasks have been integrated in such a way as to exploit the course’s instructional design strategies, as well as its inherent adherence to adult learning principles and learning technologies. Mathematic objectives are dynamically related to other disciplines by providing relevant tasks within authentic contexts. Students develop new mathematical understandings, critical thinking and decision making skills that build on and enrich their knowledge, application and appreciation of mathematics. These active learning experiences then provide the opportunity for students to effectively transfer skills, knowledge and attitudes to Math-specific Foundations courses, as well as in subsequent years of study.

This paper will refer to research findings which indicate that students who complete the CRSP course have positive attitudes to the learning of mathematics and are better able to solve problems requiring mathematical skills outside the classroom.

Introduction

Mathematics is a discipline that might be defined as the “deductive study of numbers, geometry, and various abstract constructs, or structures” (Britannica Concise Encyclopaedia, 2006). However, this definition omits the applied nature of mathematics necessary to solve and explain the problems of everyday existence and fails to recognise the dynamic nature of mathematics (Hamming, 1980). The learning and teaching of mathematics has been affected by the discipline’s increasing complexity along with the requirement for “complex reasoning and performance on problem-solving tasks” especially when applied to situations outside the learning environment (Bransford, Brown, & Cocking, 2000, p. 2). Additionally, Corr (2007) indicates that frequently graduates do not have the ability to take abstract mathematical concepts and apply them, which “reflects the rigidity of education systems”. In response to the identification of issues such as these, the Foundations’ faculty at Dubai Men’s College (DMC) adopted an approach that integrated content areas (including mathematics) with the Computer, Research Skills and Projects (CRSP) course.

This paper describes the theory behind designing and implementing such an approach, and reports the results to date. Findings suggest that students who complete the CRSP course have positive attitudes to the learning of mathematics and are better able to solve problems requiring mathematical skills outside the classroom.

Alternative learning strategies in the UAE and Gulf Region

Education is currently experiencing a paradigm shift that acknowledges learners as cultural and social entities who are part of a wider community (Jaworski, 1996), and who need to be actively engaged in order to enable them to acquire knowledge and skills. Learning in a more traditional sense is underpinned by the notion that the world is ultimately ‘knowable’, and that students canlearn proven facts. In contrast, student-centered learning involves working collectively to solve problems (to which there may not be a ‘correct’ answer), hone and apply skills, and produce artifacts that are relevant outside of the classroom environment (Scardamalia & Bereiter, 2002), thereby preparing students for a ‘knowledge society’.

For the Arab world in general, and the UAE and Gulf region in particular,The Arab Human Development Report (United Nations Development Programme,2003) found that “students can do little but memorise, recite and perfect rote learning” (p. 54). They also indicated that the pedagogical methodology is largely didactic, teacher-centered and passive,with assessments that require memorisation of formulae and superficial identification of set problems. Conversely, an active“learning style is characterized by field-independent and analytical thinking” (Al-Harthi, 2005, p. 3), in a student-centered environment with an active teaching approach (Seels & Glasgow, 1998). These findings bleakly expose the gap between the preconceptions, skills and expectationswith whichmany students enter tertiary education and the magnitude of the required transformation of their approaches to learning. As such, education in the Arab world needs to shift its focus from “providing instruction to producing learning” (Barr & Tagg, 1995) equipping learners with the “wherewithal to think, understand, create,…and adapt within many different material and cultural contexts”(Hall, 1996, p. 27).

When considering such a dramatic shift in pedagogical methodology, affective factors need to be considered if effective learning is going to be achieved (Henderleiter and Pringle, 1999). For example, student anxiety and resistance to a shift in theapproach to learning and teaching must not be underestimated. Hofstede’s (1991) uncertainty avoidance index reveals that Arab countries have stronger uncertainty avoidance (score = 68) than, for example, the United States (score = 46)(Al-Harthi, 2005). However, these reactions can, in part, be addressed by initially providing strongly-scaffolded cooperative learning (CL) tasks, with clear instruction and expectations, and then gradually removing the scaffolding in subsequent tasks.

Alternate learning strategies in Mathematics

“The failure of research and curriculum reform during the past fifty years to significantly affect the mathematics classroom calls for alternative research paradigms that may bridge the gap between research concerns and problems from the reality of the classroom” (Atweh, 2002, p. 1).

The Higher Colleges of Technology’s graduate outcomes include demonstrable requirements of mathematical literacy as follows:

  1. Applying relevant numerical analytical tools to solve problems in authentic contexts.
  2. Analysing and communicating mathematical concepts with confidence in authentic contexts (Higher Colleges of Technology, 2006).

To support these outcomes, HCT has developed a Numeracy Learning Process Model, the core of which is the effective application of Numeracy,as shown in figure 1. Whilst not prescribing one specific methodology, the model does recommend activities applied to realistic situations.

Figure 1: HCT Numeracy Process Model, (UAE Ministry of Education 2005)

The HCT system requirement, along with the need to equip students with the range cognitive and metacognitive skills which enable students to take control of their own learning and to monitor their own progress (Bransford, Brown, & Cocking, 2000), alerted Foundations faculty at DMC to the necessity of moving from a text-book focussed math course, to an actively applied skills focus.

Implementation Issues

It can be difficult for Mathematics teachers to include applied tasks in real world contexts when faced with curriculum content volume and inflexible gateway assessments.Teachers ofLimited English Proficiency (LEP) students are doubly challenged in a content course. In addition to the intellectual obstacles their students must overcome to understand the content, they are faced with the difficulty of doing so through the filter of their limited English (Orthman, 2005). Orthmann (2005) suggests that, in general, adult LEP learners have the cognitive ability to comprehend even complex content once the language barriers have been breached. Whilst this ability in students supports teaching the effective application of Numeracy in relevant contexts, the obstacles to overcoming the language barrier should not be underestimated. When determining a strategy to teach a specific content objective, a mathematics teacher must consider whether s/he has the time not only to teach the content vocabulary, but also the ‘context’ vocabulary required by the student to understand the problem/situation presented.

Integrated Approach

Historically, the Foundations programme at DMC comprised four discrete courses: mathematics, computing, Arabic and English. Study, research and academic skills were taught as part of each course and supported by Learning Centre input sessions. The main issues included sparse communication between the courses and ‘projects’ being completed in isolation and viewed as an ‘add-on’ to the English course. Consequently, students were not graduating from Foundations with the required level of study, research and higher-order thinking skills. In response to this recognized learning deficit, Foundations faculty designed and implemented an integrated programme that was informed by adult learning theory, and which exploited the benefits of CL (see Figure 1).


Figure 2: The CRSP course and its integration with the Foundations’ content courses (Martin, 2005).

The resulting interdisciplinary CRSP Foundations program is a blended learning course, fifty percent of which is delivered through face-to-face sessions, and fifty percent by means of the learning management system, WebCT. The latter effectively incorporates meaningful application of learning technologies such as audio, video, animations, applets and online quizzes. The course is designed to foster students’ research and study skills, while simultaneouslysupporting English language learning strategies and providing authentic opportunities to apply mathematical and computing skills. Furthermore, it has been designed with the intent of transforming students’ attitudes and approaches to learning.

Mathematical tasks have been integrated in such a way as to exploit the course’s instructional design features, as well as its inherent adherence to adult learning principles and learning technologies. Mathematic objectives are dynamically related to other disciplines by providing relevant tasks within ‘real world’ contexts. As a result of this approach, students develop new mathematical understandings, critical thinking skills and decision making skills that build on and enrich their knowledge, application and appreciation of mathematics. These active learning experiences equip students with transferable skills, knowledge and attitudes to the Math-specific Foundations course that they then apply during subsequent years of study.

The Foundations’ integrated approach aimed to assist students in their acquisition ofthe cognitive and metacognitive skills required to overcome their sense of disorientation when progressing from a passive to an active learning environment,as well as enabling them to study effectively after graduating from Foundations. An iterative, experiential process has been developedaround four main projects which were evenly distributed over the forty-week academic year. This approach within CRSP course enables students to encounter concrete experiences, conceptualise, reflect and actively experiment (Kolb, 1985)with the skills and concepts from the respective integrated content courses. Students, through a cumulative process produce one main artifact per project which consists of a variety of elements, including associated mathematics tasks.

Role of Information and Communication Technology: Learning strategiestransformation

The versatility of information and communication technology (ICT), when used as a tool to enhance the learning process, has the potential to support learners by, for instance, enabling educators to provide multi-media learning objects that suit a wide range of learning styles. Furthermore, it provides the opportunity to make available pedagogically sound active learning experiences that “allow for multi-disciplinary projects, CL groups, flexible-scheduling, and authentic assessments”(Cheaney & Ingebritsen, 2005, p. 3), especially in a blended-learning setting.

The design of the CRSP course includes a suite of ICT support tools to scaffold and enhance students’ learning including MSN chat, WebCT, Calendar, Camtasia, podcasts, online quizzes, and audio and video learning objects. The adaptability of the tools enables teachers to gradually reduce student reliance on a teacher-centered approach, although care is taken to provide enough support so that students remain motivated, but not so much that they do not have a reason to collaborate with peers and complete tasks in a self-directed manner (Erlendsson, 2001).

STRATEGY / TOOL / SCAFFOLDING / ENHANCEMENTS PROVIDED / OUTCOME / SKILLS ACQUIRED
WebCT (online learning management system) /
  • Provides a searchable, central location for communication tools, calendar, quizzes etc, and a place to upload homework and completed artifacts
  • Gives access to all the models, examples, videos, podcasts, materials, resources, instructions and rubrics used during the 40 week course
  • Enables students to consult and retrieve resources 24/7
  • Empowers students as they do not have to wait for the teacher to “reveal” what they are going to do next
  • More advanced / motivated students can access / complete work ahead of time
  • Enables students who have to travel or be unavoidably absent to keep up with course requirements
/
  • Project management
  • Empowerment
  • Self-directed learning

Calendar (in WebCT) /
  • Gives students an overview of the 40-week semester
  • Has reminders of homework required each week
  • Gives final project deadlines
  • Provides live links from the calendar directly to the task, rubric, instructions, tool, or explanation referred to in a calendar posting
/
  • Time management
  • Project management
  • Self-directed learning

Laptops /
  • Allow students to collect and collate original research data outside classrooms
  • Encourages students to be organised (e.g. file naming conventions and file management)
  • Raises global awareness (e.g. security issues such as viruses)
  • Assists students to take responsibility for their own learning (e.g. regular backups)
/
  • Data gathering
  • Organisation
  • Production and presentation of artifacts
  • Communication skills

Table 1: Scaffolded tools provided for learners participating in the CRSP blended learning course
(Owen, 2007, adapted from Krajcik, Blumenfeld, Marx, & Soloway, 1998)

Relevant Tasks and Authentic Contexts – Integration of Mathematics

For those who learn to think meaningfully, mathematics gets essentially simpler, while for those who learn isolated techniques, it becomes increasingly complicated. (Hall, 1997). Teachers need to incorporatea variety of appropriate teaching strategies in the intended learning experiences which can be enhanced by available technologies and other resources (AAMT 2006).

The following section of the paper describes two fully-integrated math projects that have been run as part of the CRSP course in the 2004-2005, and 2005-2006 academic years. Each year has seen enhancements and additions to the math tasks in response to research data and feedback from stakeholders. The description below is the most recent version of these tasks.

Project 1 Careers: decision analysis activities

In Semester 2 of their Foundations year at DMC students must selectthe specialisation they are going to study after they have graduated from Foundations. The following specialisations are offered: Business, Aviation, IT, Communications Technology and Engineering. Many students are unsure which course will best prepare them for the workforce and what employment opportunities are available to them following graduation from a course. Thougheverybody makes decisions on a daily basis, Saaty (1996) suggests that making a decision is not purely a question of selecting the best alternative. Instead, he advises that one should apply mathematics to decision making,wherebydissecting a problem into its constituent parts, while establishing importance or priority by ranking the alternatives, is a comprehensive and general way to look at the problem mathematically.

Foundations students are not ready to undertake a quantitative process of multi-criteria decision making, yet the CRSP team’s task to teach students an effective process of decision analysis remained. The scope of this objective included developing students’ ability to make informed decisions using key considerations whilst ensuring they recognise the role mathematics plays in sound decision making.

Task 1 - Podcast and Analysis Task: To assist students in identifying and prioritising components which would assist them in course decision making, the CRSP team developed a podcast episode and linked it to an analysis task embedded in WebCT. Episode one of Research Skills Voice Podcast was entitled ‘Career Considerations’ and included a recording of six Foundations students who appeared on a local Talkback radio station, Dubai Eye. Students were recorded discussing which job they would like to undertake after completing their studies at DMC. After listening to the Podcast, students were required to:

  1. identify the key reasonsgiven bythe students for wanting go into particular industries;
  2. list the things that they themselves will consider before choosing a specialisation to study;and
  3. prioritise this list of considerations.

These integrated tasks were used to ensure that students reflected on and internalised the importance of considering a number of relevant factors when making decisions. It also further developed their understanding of the importance of prioritising these factors within a decision analysis process.

Task 2 - Using Excel to organise and display data: Opportunities should be taken to involve students in mathematical activities beyond the scope of the classroom in contexts of interest and relevance to the students (AAMT 2006). Students undertook an‘off-campus’ task which required them to interview people in the workplaceand gather data on employees’ occupations, skills, training and education. Data gathered by students also included the respondents’ nationalities and whether their job was part of the public or private sector. This data was sorted in tables and represented in graphs using MS Excel software; graphs displayed, for example, the representation of nationalities working within the public and private sectors. In addition, students were asked to answer questions about the hypothetical consequencesof the collected responses in connection to themselves, their respective careers decisions and where their courses may lead them. As such, the task was specifically designed to encourage the use ofcritical thinking skills and to exploit the use of technology in displaying and formulating data.