Exploratory Case Study on Solving Word Problems Involving Triangles by Pre-Service Mathematics Teachers in a Regional University in Australia

Abstract

Studies have shown that solving real-world problems involving triangles is one of the most difficult topics for the pre-service secondary mathematics teachers engaging study and training in metropolitan institutions. We have known little about performances of the pre-service mathematics teachers from regional, rural and remote (RRR) areas engaging study and training in regional universities. This case study intends to explore whether solving word problems involving triangles would also be challenging for the RRR pre-service mathematics teachers, and what unique factors may negatively affect the RRR student teachers’ performances in solving word problems involving triangles. This study compared the works of two groups of the first-year pre-service mathematics teachers who enrolled in an undergraduate education program in a regional university in Australia. The two chosen word questions for comparison were parts of the assignments involving solving triangles to the students. Through statistical analysis, this study indicates that the considerable proportion of no attempts among the enrolled RRR students is the distinctive difference between the metropolitan and RRR pre-service mathematics teachers. Although still challenging, the RRR student teachers who attempted the word problems seemed performed better than the metropolitan students in solving word problems involving triangles.

1. Introduction

Triangles are not only closely associated with geometry and trigonometry in elementary and secondary mathematics, but also commonly connected to solving numerous scientific and engineering problems in the real world. Hence, understanding properties of triangles and solving problems involving triangles by various means have been featured in mathematics teaching and learning from elementary schools to secondary schools in all countries. For example, elementary students begin with familiarizing different types of triangles and basic properties [1,2]. Junior secondary students continue with learning Pythagorean theorem and basic trigonometry defined as ratios of sides and using these basic rules to solve problems associated with right triangles [3,4,5]. Senior secondary students progress further with trigonometric identities, functions and relationships, law of sines, law of cosines and law of tangent to solve oblique triangles in scientific and engineering problems [6,7,8,9]. As a result, triangles and their applications have been an important part in mathematics curriculums for pre-service mathematics teachers in tertiary institutions in the world [10,11,12,13].

However, studies have shown that trigonometry and solving word problems involving triangles are among the most difficult topics for the pre-service secondary mathematics teachers. For example, Ubah [12] reported that many graduating pre-service mathematics teachers were struggling with semiotic transformation of similar triangles. Based on a designated study for more than 100 pre-service mathematics teachers, Nabir et al. [13] found that about 80% of the participants thought that trigonometry was difficult and too rigid and abstract. This study also revealed that only 40% of the participants were able to fully understand the origin of laws of sines and cosines from the knowledge of right triangles. Walsh et al. [14] reported that among the fifty final-year pre-service secondary mathematics teachers, about 20% of the participants could not satisfactorily understand and use the properties of right triangles and basic trigonometric rules as the ratios of sizes; more than 80% of the participants had difficulties associated with solving obliques triangles; more than 90% were not able to apply laws of sines and cosines to solve a scientific problem described in words. The similar findings in pre-service mathematics teachers’ difficulties in dealing with word problems describing real-life scenarios were also reported by Fyhn [15] and Dundar [16]. Tallman [17] found that the secondary teacher’s knowledge of sine and cosine was only on the behavioral proficiencies, not the mental actions and conceptual operations. Gholami et al. [18] even reported that trigonometry was the least preferred reaching subject rated by many mathematics teachers.

In regional universities in Australia, students enrolled in many tertiary programs are diverse in age, mathematical background, study mode, time availability, and level of commitment to their learning due to various personal, family, and professional issues [19,20,21]. A typical education mathematics course offered in a semester or term at a regional university would have about 15–30 students enrolled from different geographical locations, predominantly the regional, rural and remote (RRR) areas where the regional university resides in and serves for, with a few from major cities or other regions nationwide. It is interesting to see how the pre-service teachers in the RRR areas would approach solving word problems involving triangles in such a heterogeneous learning environment, and hence to explore the commonalities shared by them and the differences with respect to age band and gender in solving the same problem assigned to them. Specifically, the goals of this case study are to explore

• Whether solving word problems involving triangles would also be challenging for the RRR pre-service mathematics teachers?
• Whether age and gender of the students would affect the performances in solving word problems involving triangles?
• What factors may negatively affect the RRR student mathematics teachers’ performances in solving word problems involving triangles?
• What changes would be needed to better assist the RRR pre-service teachers’ mathematics learning?

Students solving a situational word problem should be not only about providing the correct solution to the problem, but also about demonstrating their competency in choosing the most confident approach for individuals to deal with the process of problem solving if such options are available. The chosen approach by a student may not be the best possible way to solve the problem by means of the mathematical procedure, but it is the most confident way the student can use for solving the problem. This has been recently reported by Duris et al. [22] for student mathematics teachers. Hence, it would also be interesting to explore how the RRR student mathematics teachers chose different mathematical methods in solving word problems involving triangles in this case study.

To achieve these goals, this study compared the works of two groups of first-year pre-service mathematics teachers (or student mathematics teachers) who enrolled in an undergraduate education program with a specialization in secondary mathematics at Central Queensland University (CQU), a regional university whose headquarter is located in Rockhampton, a regional city in the northeast of Australia (Figure 1). These two groups of students undertook the same foundation mathematics course in the same year but in separate terms (equivalent to semesters) instructed by the same teacher before the COVID-19 pandemic. The two chosen word questions for comparison were parts of the assignments for the students and involving solving triangles. This foundation mathematics course is the prerequisite for the intermediate mathematics courses scheduled in the subsequent year, followed by the advanced mathematics courses in the third year. The main objective of this foundation mathematics course is to streamline the basic topics in algebra, geometry, and trigonometry that students had learnt in secondary schools through systematic reviews with further conceptual reasoning, logical articulation, and real-world applications. This also provides those students who left the school more than five years with an opportunity to refresh their previously obtained mathematical knowledge and/or bridge the gaps in their original mathematics learning in secondary schools.

In the rest of this paper, Section 2 introduces the background of the RRR areas in Australia and characteristics of the RRR students, from which the research method is chosen for this study. The two cases of word problems involving solving triangles along with students’ performances are presented in Section 3 and Section 4, respectively. Section 5 discuses students’ results and implications through statistical analysis. Section 6 summarized this study.

2. Background and Research Method

2.1. The Background Information

Australia has a large territory with relatively a small population just over 25 million people, among whom two-thirds live in less than 20 major cities mainly along the coastal zones (Figure 1). By the classification of Rural, Remote and Metropolitan Areas (RRMA), Australian regions are classified into five categories of Major Cities, Inner Regional, Outer Regional, Remote, and Very Remote areas [23]. Except Major Cities, the other four regions are commonly referred to as the regional, rural and remote (RRR) areas which cover more than 95% of the Australian territory [23,24].

Compared to the richness level of educational resources and the standard of academic achievements in schools located in or near the major cities, schools located in the most RRR areas are far behind the marks of the metropolitan schools in almost all aspects due to various issues, for example, the shortage of experienced STEM teachers, outdated infrastructure, and a low rate of participation in tertiary education which led to the shortage of teachers in the RRR schools. Studies have shown that once the students from the RRR communities enter to a tertiary program with a regional university, most graduates are likely to choose a career in the RRR communities after graduation [25,26,27].

The low rate of participation of school leavers in tertiary education has a profound impact on the curriculum design, pedagogy, quality assurance, and sustainability of tertiary programs in regional universities, particularly undergraduate degree programs. Running classes with a small number of students is always costly and consequently, a university can only run programs as long as they are viable, which is especially important for regional universities. To make a tertiary program financially viable, regional universities have both to encourage as many eligible school leavers as possible to engage in tertiary education and to offer preparatory courses for those people (both new school leavers and adults) who are academically ineligible to enroll in the tertiary program that they really want to take part. Hence, in a class at a regional university, student’s backgrounds are typically diverse in age, academic preparation, study mode and type, time commitment to learning, and so forth, which is particularly common in many STEM programs.

The mathematics specialty in the Bachelor of Education program at CQU is a typical example that serves the needs of training mathematics teachers for secondary schools, particularly the RRR schools. CQU is a regional university with multiple campuses in the RRR areas located in local centers along the north-eastern coast of Australia. Its headquarter is located in Rockhampton, a regional city with a population of 70,000 people (Figure 1). The mathematics specialty consists of one statistics course and five mathematics courses across multiple academic levels over three years of full-time study. The five mathematics courses have been gradually realigned to the Queensland Senior Mathematics Syllabus [28] since 2019 with one foundation mathematics course, two intermediate mathematics courses, and two advanced mathematics courses. The foundation course is the prerequisite for the second-level mathematics courses scheduled in the subsequent year, followed by the advanced mathematics courses in the third year for full-time students.

A class of the foundation mathematics has typically 15–30 students aged from 17/18 years old (new school leavers) to the 50s in age. Many new graduates in their early 20s may take apprenticeships for living as their families could not afford to support their territory education. Others in their late 20s or older would be likely to have more family commitment in addition to the responsibilities for work and study, compared with new school leavers and the younger ones in their early 20s (or the younger group). Most students in the elder group (the late 20s and older) are likely to take part-time study with concurrent commitments to work and/or family. In Australia, a load of 3 or more courses enrolled in a term is classified as full-time study whereas a load of 1 or 2 courses is classified as part-time study. Moreover, most of these 15–30 students in a class are likely to live in different RRR areas in Queensland, with a few in other states of the country. Effectively, online delivery is the most viable option for such a group of diverse students.

2.2. Research Method and Considerations

This setting of mathematics classes in a regional university is vastly different from that in most existing studies with hundreds of students from multiple places or tens of students sitting in the same venue with a metropolitan university. Hence, for the purposes of this study focusing on the performances of solving word problems involving triangles by the pre-service mathematics teachers from the RRR areas, the research method of comparative case study is adopted [29], supported by simple statistics. The comparisons are conducted at three levels, the intragroup level concentrating on finding facts within the same group of students, the mixed group level comparing the facts identified from all the students in the two groups, and the cross-group level involving comparisons between the facts found in this study and existing studies around the world.

It would be even more beneficial if qualitative measures, such as one-to-one interview with the pre-service mathematics teachers, could be incorporated into this case study, just like many other studies reported in [2,10,12]. Unfortunately, as most RRR student mathematics teachers could hardly attend the weekly live online classes regularly, voluntarily participating in such a formal and serious one-to-one interview would be their last preference for engagement. Fortunately, given the four fact-oriented research goals regarding student’s procedural and technical efficiency in mathematical problem solving, similar to the works done previously by others [9,22,30], the chosen method would be sufficient to achieve the goals of this exploratory case study.

3. The First Case Study

3.1. The First Word Problem and Background Information

The word problem shown in Box 1 was part of an assignment for student teachers enrolled in the foundation mathematics course. This word problem was related to the Pythagorean theorem and basic trigonometric functions associated with right triangles. The problem aimed at testing students’ understanding of the Pythagorean theorem and basic trigonometric functions, and their ability to use these basic rules to solve an authentic problem described in words with a clear strategy. By the teaching and learning plan, students should have completed these topics and all the required exercises at least three weeks before submitting their assignments.

Twenty-six students enrolled in this mathematics course at the time, with twelve male students and fourteen female students. The age range was from eighteen to fifty-six years with a rough split into two age groups, the younger group who graduated from schools in less than five years, and the elder group who graduated from schools at least five years ago and was twenty-three years of age or older. A few students in the elder group were the first time to reengage with learning mathematics over 20 years since graduated from high schools. There were 11 and 15 students in the younger and elder groups, respectively.

Four students were on campus and could attend the weekly live class on campus, which was simultaneously telecasted to other distance students through the pre-set zoom connection. Only two students out of the four on-campus students attended the physical classes regularly during the term. The weekly teaching sessions were recorded, and the edited instructional videos were made available on the course website for all the students usually within 24 h after the live class.

Box 1. The first word problem assigned to the pre-service mathematics teachers.

An observer was on top of a cliff 255 m above the sea level. The observer first saw a boat in the sea at point A with an inclination of 64°. Two minutes later, the observer saw the same boat at point B with an inclination of 33°. If the boat travelled straight and by ignoring the observer’s height, what was the average speed of the boat?

3.2. The Reference Solutions

There are multiple ways to solve this problem. In general, these approaches can be classified into two broad strategies: by right triangles alone or by mixing right and oblique triangles. No matter which strategy was chosen by a student to solve this word problem, the student should firstly draw a diagram to depict the described scenario with the known and derived figures as a guide to craft a plan for solving the problem. The diagrams representing these two strategies are shown in Figure 2. Point C indicates where the observer was standing during the observation.

3.2.1. By Right Triangles

Referring to the right triangles OAC and OBC in Figure 2a, the distance from A to B (AB) can be found through the following processes.

$$\begin{array}{l}OA=OC\tan 26°(=OC\cot 64°)\\ OB=OC\tan 57°(=OC\cot 33°)\\ AB=OB-OA=OC\tan 57°-OC\tan 26°=OC(\tan 57°-\tan 26°)\end{array}$$

The average speed of the boat over two minutes (or 120 s) is calculated by

$$v=\frac{AB}{t}=\frac{OC(\tan 57°-\tan 26°)}{t}=\frac{255(\tan 57°-\tan 26°)}{2\times 60}=2.236\mathrm{m}/\mathrm{s}.$$

3.2.2. By Mixing Right and Oblique Triangles

Referring to the right triangle OAC or OBC and the oblique triangle ABC in Figure 2b, the distance from A to B (AB) can be found through the following processes.

$$\begin{array}{l}AC=\frac{OC}{\sin 64°}(=\frac{OC}{\cos 26°})\mathrm{or}BC=\frac{OC}{\sin 33°}(=\frac{OC}{\cos 57°})\\ \frac{AB}{\sin 31°}=\frac{AC}{\sin 33°}=\frac{BC}{\sin 116°}\\ AB=\frac{AC\sin 31°}{\sin 33°}=\frac{OC\sin 31°}{\sin 33°\sin 64°}(=\frac{OC\sin 31°}{\sin 116°\sin 33°})\end{array}$$

The average speed of the boat can be calculated similarly as above.

3.3. The Pre-Service Mathematics Teachers’ Performances

3.3.1. The Overall Performance

Sixteen out of the twenty-six students solved this problem correctly, which was about 62% of the total. Two students presented partly correct solutions to this problem, which was mainly due to some errors in calculating the associated angles or trigonometric values with employing a correct strategy to solve the problem. Three students were completely wrong in solving this problem, which seemed more relevant to their inability to understand the scenario described in the word problem. Five students did not attempt this problem, which counted for 19% of the total. The overall performances of the student teachers are summarized in

Table 1. Overall performances of students in solving the first word problem.

Outcome	Number	Percentage (%)
Correct	16	61.54
Partly correct	2	7.69
Incorrect	3	11.54
No attempt	5	19.23
Total	26	100

.

Gender seemed affected the performance with different patterns for the gender groups (

Table 2. Overall performances of students with different genders in the first word problem.

Outcome	Male	Female
Correct	9 (75%)	7 (50%)
Partly correct	0 (0%)	2 (14.3%)
Incorrect	2 (16.7%)	1 (7.1%)
No attempt	1 (8.3%)	4 (28.6)
Total	12	14

). By excluding no attempts, male students exhibited an uneven bimodal pattern for Correct solution versus Incorrect solution with a ratio of 9 to 2, e.g., nine out of eleven male students (75%) correctly solved the problem and the remining two male students were completely wrong in their attempts, none in the middle. On the other hand, there were seven, two and one female students solved the problem correctly, partly correctly and incorrectly, respectively. Four out of the five students who did not attempt this question were females.

Coincidentally, less the no attempts, the same performance patterns were observed for the age groups (

Table 3. Overall performances of students with different ages in the first word problem.

Age	Correct	Partly Correct	Incorrect	No Attempt	Subtotal
18–22	7 (63.6%)	2 (18.2%)	1 (9.1%)	1 (9.1%)	11
≥23	9 (60%)	0 (0%)	2 (13.3%)	4 (26.7%)	15

The percentage inside the parentheses is the relative value against the subtotal of an age group.

). The younger group showed the same pattern as the female group whereas the elder group shared the same bimodal pattern as the male group. This was reasonable as eight out of the eleven students in the elder group were male, in which six out of the eight presented correct solutions so did the three females in the elder group. Although the elder group seemed outperformed the younger group in this case, four out of the five students who did not attempt this question were also from the elder group.

3.3.2. The Effect of Different Strategies on the Performance

Overweeningly, eighteen out of the twenty-one students (90%) used right triangles to solve this problem and only three used the strategy of mixed triangles. Among the sixteen students who solved the problem correctly (

Table 4. Effect of different strategies on the performances in solving the first word problem.

Method	Correct	Partly Correct	Incorrect
Right triangles	13 (81%)	2 (100%)	3 (100%)
Mixed triangles	3 (19%)	0 (0%)	0 (0%)
Subtotal	16	2	3

The percentage inside the parentheses is the relative value against the corresponding subtotal.

), thirteen (81%) used the method of right triangles perfectly, as shown in the example in Figure 3.

However, two partly correct solutions and the three incorrect solutions were also resulted from using the strategy of right triangles. The incorrect solutions were primarily resulted from a misunderstanding of the described scenario and hence drawing a wrong diagram for the situation. The partly correct solutions were resulted from applying the correct strategy of right triangles but using incorrect trigonometric relationships with the right triangles.

The strategy of mixed triangles was adopted by three students only, but all three solved the problem correctly with confidence in applying both the Pythagorean theorem and the law of sines, which is demonstrated by one student’s work shown in Figure 4.

4. The Second Case Study

4.1. The Second Word Problem and Background Information

The second word problem shown in Box 2 was part of an assignment to 19 student teachers enrolled in the foundation mathematics course in a different term. This word problem was primarily related to solving oblique triangles. The problem aimed at testing students’ understanding of the law of cosines and their ability to use this rule to solve an authentic problem described in words with a clear strategy. By the teaching and learning plan, students should have completed all the required exercises in solving oblique triangles at least two weeks before submitting their assignments.

Box 2. The second word problem assigned to the pre-service teachers.

A radar speed camera is placed at point C offside of a road. It takes two readings of time in seconds at Points A and B, respectively, for a moving vehicle (See figure below). If the time difference between A and B for vehicle X is 0.5 s whereas the time difference for vehicle Y is 0.4 s, determine the speed of X and Y, respectively.

Among the nineteen students enrolled in this mathematics course at the time, the numbers of male and female students were seven and twelve, respectively. The age range was from seventeen to fifty-three years old, which can be roughly split into two groups, being similar to the first case study, the younger group of students aged 17–22 who graduated from schools within five years, and the elder group of students aged 23 or older who graduated from schools five years or earlier, a few being 20–30 years since graduated from high schools.

One student was on campus and could attend the weekly live class in person whereas other eighteen were studying in distance mode. The weekly teaching sessions were recorded, and the edited instructional videos were made available on the course website for all the students usually within 24 h after the live class.

4.2. The Reference Solutions

There are multiple ways to solve this word problem too. In general, the law of cosines is the most direct approach to solve the oblique triangle ABC because two sides and the included angle are already known. Of course, the problem can also be solved by dividing the oblique triangle into two right triangles and then solving the two right triangles to determine the distance AB. If the latter approach is used, it would be equivalent to using the law of cosines by separate steps.

4.2.1. By Oblique Triangle

Referring to the oblique triangle ABC in the second problem, the distance from A to B (AB) can be found by using the law of cosines directly as follows.

$$AB=\sqrt{A{C}^2+B{C}^2-2AC\cdot BC\cos C}=\sqrt{{60}^2+{50}^2-2\cdot 60\cdot 50\cos 5°}=11.0830\mathrm{m}$$

The average speed for vehicles X and Y can be calculated, respectively, by

$$\begin{array}{l}{v}_X=\frac{AB}{t}=\frac{11.0830}{0.5}=22.166\mathrm{m}/\mathrm{s}=79.8\mathrm{km}/\mathrm{h}\\ v_Y=\frac{AB}{t}=\frac{11.0830}{0.4}=27.707\mathrm{m}/\mathrm{s}=99.7\mathrm{km}/\mathrm{h}.\end{array}$$

4.2.2. By Right Triangles

The oblique triangle ABC in the second problem can be divided into two right triangles by drawing a line from point B perpendicular to line AC intersected at point D (Figure 5). For right triangle BCD,

$$BD=BC\sin C\mathrm{and}CD=BC\cos C$$

For right triangle ABD,

$$\begin{array}{l}AD=AC-CD=AC-BC\cos C\\ AB=\sqrt{B{D}^2+A{D}^2}=\sqrt{B{C}^2{\sin }^{2}C+{(AC-BC\cos C)}^2}\\ =\sqrt{B{C}^2{\sin }^{2}C+A{C}^2-2\cdot AC\cdot BC\cos C+B{C}^2{\cos }^{2}C}\\ =\sqrt{A{C}^2+B{C}^2-2AC\cdot BC\cos C}=\sqrt{{60}^2+{50}^2-2\cdot 60\cdot 50\cos 5°}=11.0830\mathrm{m}.\end{array}$$

The average speed for vehicles X and Y can be calculated similarly as above.

4.3. The Pre-Service Mathematics Teachers’ Performances

4.3.1. The Overall Performance

Only six out of the nineteen students solved this problem correctly, which was about 31% of the total. Three more students presented correct solutions to solving the oblique triangles but were wrong or miscalculated the speed of the vehicles, hence only partly correct. Another three students were completely wrong in solving this problem, which were due to their inability to use the law of cosines. Surprisingly, seven out of the nineteen students did not attempt to solve this problem. The combination of the incorrect solutions and no attempts was about 53% of the entire students. The overall performances of the students are summarized in

Table 5. Overall performances of students in solving the second word problem.

Outcome	Number	Percentage (%)
Correct	6	31
Partly correct	3	16
Incorrect	3	16
No attempt	7	37
Total	19	100

.

Gender seemed affected the performance significantly for this problem (

Table 6. Overall performances of students with different genders in the second word problem.

Outcome	Male	Female
Correct	4 (57.1%)	2 (16.7%)
Partly correct	1 (14.3%)	2 (16.7%)
Incorrect	1 (14.3%)	2 (16.7%)
No attempt	1 (14.3%)	6 (50%)
Total	7	12

). Among the six students who correctly solved this problem, four were male students (67%) and two were female students (33%). At the other end of the seven students who did not attempt the problem, six were female students (86%). A ratio of 2 females to 1 male was also found in both the partly correct and incorrect categories. For this case, in terms of the whole class, female students would be more likely to present incorrect solutions or to give up attempt to the problem (42%) than the male students (11%). Since the number of students correctly solved the problem was small, it should not be stereotyped that male students were outperformed the female students for this case.

There was a significant difference in the performance between the two age groups (

Table 7. Overall performances of students with different ages in the second word problem.

Age	Correct	Partly Correct	Incorrect	No Attempt	Subtotal
18–22	5 (50%)	1 (10%)	3 (30%)	1 (10%)	10
≥23	1 (11.1%)	2 (22.2%)	0 (0%)	6 (66.7%)	9

The percentage inside the parentheses is the relative value against the subtotal of an age group.

). Among the eleven students in the younger group (17–22 years old), five students (also out of all six students who solved the problem correctly) were correct whereas only two did not attempt the problem. This was a stark contrast to the elder group in which only one student presented a correct solution whereas five out of eight students (63%) did not attempt this problem, this being 71% of the seven students who did not attempt this problem in the whole class.

4.3.2. The Effect of Different Strategies on the Performance

Overweeningly, by excluding the seven students who did not attempt this question, eleven out of the twelve students (92%) used the law of cosines to directly solve the oblique triangle ABC. Only one student from the younger group applied the Pythagorean rule directly to the oblique triangle ABC to solve this problem by $A{B}^2=A{C}^2+B{C}^2$, which is obviously wrong (

Table 8. Effect of different strategies on the performances in solving the second word problem.

Method	Correct	Partly Correct	Incorrect	Subtotal
Oblique triangle	6 (55%)	3 (27%)	2 (18%)	11
Right triangles	0 (0%)	0 (0%)	1 (100%)	1

The percentage inside the parentheses is the relative value against the corresponding subtotal.

). Among the eleven students who solved the problem using the law of cosines, considering the three partly correct cases where the students missed or miscalculated the speed of vehicles, technically, nine were correct in solving the oblique triangle and only two were incorrect. One correct example is recaptured in Figure 6.

Except the inappropriate use of the Pythagorean rule mentioned earlier, two other incorrect cases were due to miscalculation in using the law of cosines.

5. Discussion

5.1. Performance by Age Groups

Since only a small number of student teachers enrolled in each term, this may affect the analysis of students’ performances in and actions on the word problems in each term alone. Hence, data from the combined cohort of 45 students over these two terms would support a more objective analysis on student’s performances in and actions on solving the word problems involving triangles.

For the younger and elder groups of students, the frequencies of their attempts resulted in correct, partly correct, and incorrect solutions are summarized in

Table 9. Frequencies of student’s results and no attempts by age groups.

Age	Correct	Partly Correct	Incorrect	No Attempt	Subtotal
Younger (Y)	12	3	4	2	21
Elder (E)	10	2	2	10	24
Subtotal	22	5	6	12	45

, along with the number of students who did not attempt the question to reflect the actions of some students among the combined cohort. Intuitively, the two age groups seemed to show a similar pattern of performance distribution from those students who attempted the questions (Figure 7). However, by adding the number of inactions on the questions, the combined inclusive performance-action pattern for the two age groups would be different from each other.

To verify the significance of this difference in the combined inclusive performance-action pattern for the two age groups, a composite chi-test is applied to the inclusive frequencies of the two age groups, which includes firstly the distribution fitness of the Elder Group (E) with respect to the distribution model of the Younger Group (Y), and secondly that of the Younger Group (Y) with respect to the Elder Group (E). The chi-value of 29.083 for the Younger against the Elder is significant at α = 0.01 whereas the chi-value of 10.2 for the Elder against the Younger is significant at α = 0.025, respectively. The composite average of 19.642 is significant at α = 0.01 (

Table 10. Results of chi-test on student’s results and actions from the two age groups.

Test Type	Younger-Elder	Elder-Younger	Average	Critical Chi-Value	d.f.
Exclusive	0.476	0.821	0.649	9.21 (α = 0.01)	2
Inclusive	29.083	10.200	19.642	9.348 (α = 0.025) 11.345 (α = 0.01)	3

). This indicates that elder students would be more likely to give up an attempt to the problems (hence, more likely to drop their study) compared with the younger students, despite that both the younger and elder students would be able to achieve the similar learning outcomes should they attempt the questions shown as the Exclusive case in Table 10.

This analysis seems reasonable considering that the elder students, 10 out of 12 students who did not attempt the questions, are likely to have more concurrent commitments to work and family in addition to learning, compared with the younger students, even if not considering the advantage of the fresher mathematics background possessed by the younger students who recently graduated from high schools.

5.2. Performance by Gender Groups

The gender groups exhibit the similar pattern as the age groups in terms of the combined inclusive performance-action distribution shown in

Table 11. Frequencies of student’s results and actions by gender groups.

Gender	Correct	Partly Correct	Incorrect	No Attempt	Subtotal
Male (M)	13	1	3	2	19
Female (F)	9	4	3	10	26
Subtotal	22	5	6	12	45

and Figure 8, by mapping the Male Group to the Younger Group and the Female Group to the Elder Group. The significance of this difference in the combined inclusive performance-action pattern for the two gender groups is verified by the composite chi-test. The composite chi-value, the average from the two mutual fittings between the Male and Female groups with 28.976 and 11.691, respectively, is 20.33, against the critical value of 11.345 at α = 0.01 (

Table 12. Results of chi-test on student’s results and actions from the two gender groups.

Test Type	Male-Female	Female-Male	Average	Critical Chi-Value	d.f.
Exclusive	10.808	3.732	7.270	9.21 (α = 0.01)	2
Inclusive	28.976	11.691	20.333	11.345 (α = 0.01)	3

). This indicates that female students would be more likely to give up an attempt on the problems (hence, more likely to drop their study) compared with the male students, despite that both the male and female students would be able to achieve the similar learning outcomes should they attempt the problems shown as the Exclusive case in Table 12.

This analysis seems reasonable considering that 10 out of the 12 students who did not attempt the questions were females. This further illustrates that mature female students have to prioritize family commitment over learning under tough situations.

5.3. Performance by Using Different Strategies

Since both word problems could be solved by using either right triangles (R) or oblique triangles (O), the students’ performances in solving the problems using these two different strategies can be analyzed by comparing the mutual fitness of the distribution patterns to each other. For this purpose, the frequency of no attempt is excluded in the chi-test. The two strategies exhibit similar patterns of performance shown in

Table 13. Frequencies of student’s results by using different strategies.

Triangle	Correct	Partly Correct	Incorrect	Subtotal
Right	13	2	4	19
Oblique	9	3	2	14
Subtotal	22	5	6	33

and Figure 9. The significance of this similarity in the performance pattern for the two strategies is confirmed by the composite chi-test at significance α = 0.01. The chi-value against the critical value of 9.21 is 1.92 and 1.713, respectively, with an average of 1.817 (

Table 14. Results of chi-test on student’s results by using different strategies.

Test Type	Right-Oblique	Oblique-Right	Average	Critical Chi-Value
Exclusive	1.920	1.713	1.817	9.21 (α = 0.01)

). This indicates that there would be no significant difference in solving the word problems using either right triangles or oblique triangles among the students who attempted the questions. Pedagogically, enabling students to choose the method they are mostly comfortable with from multiple options, even the chosen method may not be the preferred one to solve the problem, should be encouraged by the teachers, just like what was reported in [22]. Solving real-world problems correctly is the ultimate goal of learning mathematics as long as the correct solution is obtained in terms of applications.

This overall statistical outcome is different from the intuitive view of the second word problem alone where only 31% of the students solved the oblique triangle correctly (Table 5), compared with the 61% who correctly solved the right triangles in the first word problem (Table 1). This might be partly influenced by the fact that the first word problem was obviously biased to the right triangles whereas the second word problem could be easily connected to solving oblique triangles. In other words, it was not a tough decision for students to choose the preferred method from the two obvious options.

5.4. Key Factors Affecting Study Progress of Per-Service Mathematics Teachers in the RRR Areas

This study indicates that solving problems involving triangles is indeed more challenging to pre-service mathematics teachers in the RRR areas, shared by the student mathematics teachers in cities [13,16]. Including all enrolled students, an overall rate of correction on solving the word problems from the 45 RRR student teachers was just below 50%, which is roughly similar to the average level of satisfaction reported by Nabir et al. [13] and Walsh et al. [14]. By excluding no attempts, the rate of correction on solving the word problems from the 33 RRR student teachers would be increased to 67%, which is significantly higher than that of the metropolitan students who could satisfactorily solve word problems involving triangles reported by Walsh et al. [14]. This means that the quality of education provided to the RRR student mathematics teachers at the regional university would result in superb learning outcomes once the students determined to stay the course to engage with learning mathematics. However, the distinguishable factors that more likely to affect the performance of the RRR student mathematics teachers compared with the metropolitan students seem from the two major issues: the higher proportion of no attempt and the weaker background in basic mathematics.

For the 45 enrolled students, twelve or 27% of them, did not attempt the questions, and among the twelve students more than 80% were elder females. These students are likely to take most of the family responsibilities and some work commitment in addition to their degree study. Due to unpredictability of a family emergency, these female students are vulnerable to the rigid study schedule at any stage during the course, particularly if engaging with studying two or more courses at the same time. Once an emergency occurs, the mathematics course would be the first one to be dropped by the student because a halt in the mathematics learning sequence would mean that the student needs to spend much more amount of time to catch up and then keep going with the required pace in learning mathematics. This is a stark contract to other subjects in social sciences on which common practices and experiences would be the foundation of reasoning and conclusion. Hence, a gap in the learning sequence of social sciences would be relatively easier to be filled with the pre-knowledge or experiences in a relevant area.

Therefore, it might be useful for such elder students who want to become a mathematics teacher in the future to carefully craft their study plan from the very beginning by taking the required commitment to their mathematics learning into account. In particular, one should not underestimate the importance of making a consistent effort to learning mathematics weekly in a logical way of knowledge building from previous knowns to future unknowns over the entire course. If uncertainties are likely to negatively affect the progress of learning in a term, one should consider choosing one mathematics course as the only course in that term so that there is only one course to focus when an emergency happens. Meanwhile, in an emergency, the student should communicate with the teaching team and the learning support team in the institution to seek extra assistance for easing the stress on learning in a tough situation. Although not every request could be met by the institution, every possible effort would be made by the institution, particularly the regional universities, to help students out of extraordinary circumstances.

Among the students who did not attempt the questions, a few of them might be really struggling with the Pythagorean theorem or laws of sines and cosines, hence not able to understand the scenarios described in words. Since many students in the Elder Group had been out of schools for five years or even more than 30 years, their knowledge in algebraic operations, basic geometric features, and trigonometric concepts obtained in schools had long faded away, well below their perceived level. They had to drop the course once realized such discrepancy between the reality and the perceived expectation in their mathematics background.

Thus, institutions should maintain the minimum entrance standard for potential applicants who want to be trained as future mathematics teachers. For those applicants who are academically not eligible to enter a formal tertiary program in mathematics but still keen to become a future mathematics teacher, the institution may need to adjust the curriculum to include an extra elective course in preparatory mathematics to lay the foundation for the students to smoothly progress to the formal mathematics curriculum, similar to the effort for the transitional engineering students reported recently in [21]. This arrangement of including a preparatory mathematics course does not require students to complete other non-credit preparational mathematics courses over 1–2 years before eligible to enter a formal program. This would be critical in terms of retaining most of these students in the RRR areas because many would not be able to keep doing the non-credit preparation courses over 1–2 years without earning credits for their future study, concurrently with other family and/or work commitments.

6. Conclusions

By using the comparative method and statistical analysis on the performances in solving word problems involving triangles by two groups of first-year pre-service mathematics teachers who enrolled in an undergraduate education program with a regional university in Australia, this exploratory case study confirmed that solving word problems involving triangles and trigonometry is also challenging as one of the most difficult topics for the RRR pre-service mathematics teachers, similar to what has been reported in solving problems involving triangles and trigonometry by the pre-service mathematics teachers engaging study and training in metropolitan institutions. However, there was about 27% of the RRR students who did not attempt any of the word questions, which is the distinctive difference between the metropolitan and RRR student mathematics teachers. Once determined to stay the course to solve the problems, the age and gender of the RRR student mathematics teachers do not statistically affect their performances in solving the problems, and such RRR students seemed performed better than the metropolitan students in solving word problems involving triangles.

This exploratory case study also indicated that most of the students who did not attempt any of the questions were female and elder students. This may be because the elder female students in the RRR areas are likely to take more family responsibilities concurrently with their study. On one hand, the regional institutions, the governments, and the broad communities should make every possible effort to support a student’s endeavors to become a future school mathematics teacher. On the other hand, the student must be fully prepared to make the required effort to the mathematics learning journey that is most time challenging for most student mathematics teachers given the nature of mathematics. Tertiary education is not only an equal right for every citizen, but also a privilege that allows one to be trained as a competent professional serving the society with required quality standard. This is particularly important for pre-service mathematics teachers whose competency in mathematics not only affects their individual performance in mathematics teaching, but also has a profound impact on mathematics learning of many generations of school students. Hence, the student must be fully prepared and committed to the challenging course of mathematics study. Otherwise, it would be a waste of time, money, emotion, and resources for the student and family, the institution and educators, the communities and governments.

Although this exploratory case study has achieved its goals, it should be noticed that this study lacks support from qualitative data analysis in terms of students’ cognitive development and reflection on solving word problems involving triangles. The qualitative data collected from questionnaires or one-to-one interviews would enrich the understanding of students’ learning processes and hence enable educators to create new pedagogical plans or modify existing practices to target the common issues hindering students’ mathematics learning and applications. A class for a mathematics course in Australian regional universities usually has about 15–30 students, among whom only one (sometimes zero) or a few may be able to attend either the physical class or online class regularly during a teaching term, due to concurrent commitments to study, work, and family responsibilities for the RRR students. By university policy for research involving people, any research involving collecting data through questionnaires or interviews must be pre-approved by the ethics committee of the university and no compulsory action should be allowed to force the students to participate in such research project even it is approved. The seriousness of such a formal process, though necessary, would likely to drive many RRR students away from participating in any interview or completing a questionnaire. Even though, the qualitative component would be incorporated into new case studies as much as possible in the future.