Year

2013

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

This study explores two challenges faced by tertiary ESL students learning mathematics, the complexity of learning the mathematics discipline together with the challenge of learning in a second language. It draws together the theoretical knowledge as to how bi-lingual students learn, using strategies such as codeswitching, borrowing, multimodal approach, an understanding of the problems experienced by students learning mathematics with the Cognitive Load Theory (CLT) providing a basis for understanding the function of working memory.

Drawing on the theories regarding cognitive load, the use of different approaches to teaching to reduce the cognitive load of students learning tertiary mathematics in English as a second language is examined. To facilitate generalisation of results, the effectiveness of teaching strategies is compared through two cases studies wherein mathematics is taught to ESL students in two vastly different contexts. The first case study at the King Abdul-Aziz University (KAU), Saudi Arabia, involves 198 male students taught in their home country in English, which is not their first language. The second case study at University of Wollongong College (UOWC), Australia, involves a mix of 74 students comprising both domestic students and international students taught in a second language, English.

In terms of design, the second case study replicates the first. In each case study data was collected over three iterations involving different groups of students undertaking a tertiary mathematics subject, each with a curriculum covering the topics, Functions, Exponents, Quadratic Equations, Logarithms, Geometry, and an Introduction to Statistics. In the first iteration baseline data was gathered using student questionnaires, lecturer interviews and examination of teaching materials regarding students’ experiences of learning via the methods, worked examples and problemssolving and student results achieved on topic tests. Following baseline data gathering, in the second iteration, a different cohort of students were taught the six topics wherein the teaching methods alternated between worked examples and problem-solving techniques, resulting in for three topics taught with each method. For the third iteration which involved a third cohort of students, the teaching strategies implemented in the second iteration were swapped for four of the topics and then faded worked examples were introduced as the method of teaching for the remaining two topics, one previously taught with problem-solving and one with worked examples.

The principal finding from both case studies was that worked examples which direct the attention of a learner to the problem stated, and show the steps required in solving a particular type of problem, facilitated learning. For both case studies, the performance of ESL students was improved by the use of worked examples. In the KAU case study, over the three phases a greater proportion of students indicated that having worked examples (80%) improved their study than did problem-solving (20%). At UOWC, over the three phases a greater proportion of students indicated that having worked examples (72%) improved their work than did problem-solving (26%).

This improvement in learning is consistent with cognitive load theory that suggests a reduction in cognitive load should generally make learning easier. Seventy percent of KAU students surveyed and fifty-six percent of international students in the Australian case study indicated that learning mathematics was preferable through the use of worked examples.

In terms of perceived learning outcomes it was found that for both cases studies there is an improved attitude toward studying mathematics, ‘increases my confidence about solving more problems’, ‘liking mathematics more’ and ‘reduces anxiety’. In the KAU case study, worked examples was found to enhance Quicker to study, Improved my review of mathematics notes and lab work, Easier to learn mathematics, Requires less mental effort, Makes mathematics learning more interesting. In UOWC case study, worked examples were found to enhance mathematics understanding, Increases my confidence about solving more problems, liking mathematics more and reduces anxiety. Students like to learn mathematics with worked examples more so than problem-solving even though they agreed that problem-solving increases their confidence in learning mathematics. Also, students have positive experiences in terms of learning outcomes with worked examples.

With respect to the use of faded worked examples, for both case studies, marks were significantly higher for the topic Geometry when taught with faded worked examples rather than worked examples. One could have expected that students in 2012 should have experienced higher cognitive load for this topic, however, faded worked examples increased their confidence which resulted in an increase in their marks (mean difference FWE-WE=-4.96, p=.000) for KAU. Marks were significantly higher for the topic Introduction to Statistics taught with faded worked examples rather than problem-solving. Students in 2012 should have experienced lower cognitive load for this topic. This was confirmed (mean difference FWE-PS=-4.58, p

Moreover, language of teaching mathematics has an impact on students learning if they learn in their second language. At KAU students’ ability to learn mathematics in English was seen to be lower than their ability to learn in mathematics and was also seen to decline with (67%) of students perceiving their ability to be fair/very good in 2010, declining to two percent of students in 2012. As for the ability to learn mathematics when it comes to learning mathematics in English, UOWC students’ perceived ability is relatively constant in each cohort, with (68%) of students perceiving their ability to be fair/very good in 2010, and a comparable (69%) of students in 2012.

Therefore, worked examples would be preferable to problem-solving and faded worked examples in terms of lowering the cognitive load which results from a language barrier and the difficulty of learning mathematics. This may explain that over the three phases a greater proportion of students indicated that having worked examples improved their mathematics studying than did problem-solving and faded worked examples.

In conclusion, it is important for teachers to find ways to teach mathematics effectively for their students to learn and understand their subjects. These findings support an increase in the use of worked examples for students who are learning mathematics in a second language. Implementation of the worked example pedagogy in teaching mathematics should facilitate learning for those students learning mathematics in a second language. Further examination of the use of faded worked examples as a scaffold to problem-solving is recommended as performance in both case studies improved in the topics Geometry and Introduction to Statistics. So, it is important for teachers to find ways to teach mathematics effectively. This remains a realistic challenge for the Saudi and Australian governments and indeed other governments to give teachers the required training in hybrid pedagogy.

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.