Problem-based task in teacher training program: Mathematics teachers’ beliefs and practices

Research on mathematics teacher professional development has emphasized primarily teaching quality enhancement. Mathematics task as one of the vital components in mathematics teacher education has been sparsely investigated. The purpose of this research is to explore mathematics teachers’ beliefs and practices in applying a problem-based task after attending one-semester online teacher professional training funded by the Ministry of Education. This is a mixed-method research design with data collected by distributing questionnaires to 105 mathematics teachers from 24 provinces in Indonesia after accomplishing an online professional training program to determine their self-reported beliefs about mathematical problem-based tasks. Data were also collected by conducting focus group interviews with 15 mathematics teachers with different teaching experiences. Teachers' relevant documents, such as course contents, task forms, and notes, were analyzed and used to enrich the qualitative interpretation. The quantitative result indicated that mathematics teachers' beliefs on the problem-based task were low. Meanwhile, the qualitative findings indicated this category of teachers constructed their problem-based tasks with no theoretical framework. Three misconceptions in designing problem-based tasks were also reported. Therefore, future research is recommended for a more acceptable and applicable problem-based tasks framework and design a specific teacher professional development training to promote teachers’ competencies.

The ability to solve problem-based tasks has become one of the primary concerns of mathematics education (NCTM, 2000). Learning activities are expected to develop students' 21st-century skills and competencies (Bell, 2010). However, some previous research reported that students have low abilities in terms of solving problem-based tasks (Napitupulu et al., 2016;Haruehansawasin & Kiattikomol, 2018;Hendriana et al., 2018). Napitupulu et al. (2016) further stated that they lacked the ability to translate problem representations into mathematical equations, while Haruehansawasin and Kiattikomol (2018) reported that it is difficult for them to solve questions that are not properly structured. Hendriana et al. (2018) stated that initially, students had low abilities before adopting innovative teaching. More specifically, several research proved that the majority were unable to solve mathematical problems that required complex strategies (Nelson & Powell, 2018;Bardy et al., 2021) rather they find routine problems easier to solve (Yeo, 2017;Prendergast et al., 2018;Lavie et al., 2019;Verschaffel et al., 2020). PISA test results showed similar outcomes regarding students' low mathematical problem-solving abilities. For example, the 2018 PISA evaluation, which was launched on December 3, 2019, reported that approximately 76% of the test samples obtained from 65 different countries were categorized in level two. Ironically, the analysis results in 24 countries proved that more than 50% of students scored below the standard average (Schleicher, 2019). The 2018 PISA test results showed that most Indonesian students have low abilities in solving problem-based tasks. This depicts that 80% of them failed to reach level two. The outcome of this research led to the need to ascertain whether these students have been familiarized with problem-based task-solving activities. One of the possible causes of low achievement is the type of assignments given by teachers, which is far from the international mathematics standards, such as in the PISA tests (Andrews et al., 2014).
At all levels of education in Indonesia, the use of problem-based assignments, has been recommended as in other countries (Kemendikbud, 2016). It promotes and motivates students to develop 21st-century competencies and skills (Capraro & Slough, 2013;Lee & Blanchard, 2019). Previous research stated that adopting problem-based tasks positively affected students' academic achievement and conceptual development (Karaçalli & Korur, 2014;Trinter et al., 2015). Karaçalli and Korur (2014) specifically reported that they tend to build their knowledge during this process. Trinter et al. (2015) designed a model that provided students with the opportunity to be able to express their mathematical ideas. This caused the government to make significant efforts to motivate teachers to design problembased learning through professional development programs (Lee & Blanchard, 2019). However, the type of assignments given during the initiative is yet to be studied. Most professional development programs usually focus on learning innovation, strategies, and materials, as well as Technology Pedagogy Content Knowledge (TPACK) (Egert et al., 2018;Sims & Fletcher-Wood, 2021). Bosica et al. (2021) stated that the quality of these tasks perceived as an essential part of learning is often neglected in implementing teacher professional development programs.
Consequently, the implementation of the task design during the professional development programs needs to be investigated to be adequately understood. This can be explored in depth if an exploratory investigation is carried out on teachers' perceptions of these practices. CrossFrancis et al. (2015) stated that their beliefs shape the importance of achieving student learning goals. This research aims to ascertain teachers' perceptions of designing problem-based tasks after participating in the professional development program. The next goal is to identify factors that explain the suitability or discrepancy between their beliefs and approaches applied in the classroom.

Problem-Based Tasks in Mathematics
Problem-based assignments are an aspect of problem-based learning, but they have a different focus on providing students with certain academic experiences. It is important to note that not all tasks are problem-based (van Barneveld & Strobel, 2009;Bosica et al., 2021). It is one of the mathematical instructional forms that aims not only to solve certain arithmetic questions (Uden, 2003;Takahashi, 2008). This also entails the reconstruction or representation of similar problems related to students' lives. The designed model also emphasizes the importance of relevant issues encountered daily, and in the future, it tends to help them resolve real challenges. In the context of this research, problem-based assignments are designed based on the activities of mathematics teachers who have participated in professional development programs. These aim to ensure that students can solve complex and contextual or realworld tasks (Hmelo-Silver, 2004).
Several previous research on similar topics reported positive and negative results. The positive results indicate the optimistic influence of assignments on students independence, motivation, creativity, and learning outcomes (Nicol & Krykorka, 2016;Günter & Alpat, 2017;Wijnen et al., 2017). Meanwhile, despite its successful implementation (van Barneveld & Strobel, 2009;Bosica et al., 2021), the shortcomings of problem-based tasks are also criticized. It was highlighted that there were no firm theoretical frameworks in the available research on problem-based tasks that potentially influenced students' learning processes and outcomes. Based on these voids, it is evident that there are still several gaps in problem-based tasks that previous research and teachers failed to investigate. Previous research models that concerning problem-based tasks, were rarely investigated at the primary and secondary education levels (Westwood, 2011;Merritt et al., 2017). Although the analysis results on PBL and taskbased learning have been widely acknowledged, its successful implementation, failures, and challenges have not been widely investigated. This research aims to discern the mathematics teachers' perception and implementation of problem-based tasks in the context of professional development.
The proposed frameworks involve two crucial components, namely mathematical problems, and tasks. Delisle (1997) first initiated a mathematical problem, its design, and how teachers' role in implementing problem-based learning has been properly explained. This research adopted the framework designed by Delisle (1997), which mandated that: a) the designed problem-based task needs to consider students' previous experiences, b) the given problems relate to their daily activities, c) problems are developed by connecting students' real-life activities and the curriculum, d) they are illstructured, and e) instructions or questions should promote their engagement.
This section elaborates on mathematical tasks, a central aspect of the teaching and learning activities. In this context, it denotes what is expected of students. Meanwhile, this activity affects their learning outcomes, including students' interaction and involvement (Sullivan et al., 2012). The nature of the chosen or designed task is also another determinant factor that enhances students' learning engagement (Grevholm et al., 2005). Swan (2011) designed five types of mathematical tasks, namely classification of objects (asking students to make certain designs, and classify them), interpreting multiple representations (being able to determine the relationships between these representations), evaluating mathematical statements (the ability to determine their validity), creating problems for others to solve (Students create problems for their classmates), as well as analytical reasoning and solutions (Students can identify errors and compare solutions). Tasks: a. Narrate the steps for solving the given problems b. Investigate the validity of the steps c. What are the possible causes of the different results?
Furthermore, the problem-based task model is based on the following criteria: 1) ill-structured problems, 2) issues related to students' real-life contexts and curriculum, and 3) the instructions fulfill one or more of the five criteria of math assignments (designing classifications, interpreting multiple representations, evaluating the validity of mathematical statements, creating problems, or identifying errors and comparing solutions). These are compared in Figures 1 and 2.  Figure 1 is a problem-based task where students interpret several representations, decide the validity of the mathematical statements, and identify the cause of the error. Meanwhile, in Figure 2, they routinely practice procedural skills previously learned in class. This is completely different in terms of the learning experiences and thinking processes. According to Sullivan et al. (2012), these are portrayed when they work on a series of problem-based tasks, not on those completed and used as an example by teachers.

Mathematics Teacher Professional Development
In the past 20 years, discussions regarding the issue of mathematics teachers' professional development in various traditional, hybrid, and online approaches heavily relied on learning innovation and media, teaching strategies, and materials, including syllabus design. The majority usually ignore task design which is a core aspect, and rather focus on pedagogy, content, and knowledge (PCK). Hiemstra and Brockett (1994) initially investigated traditional teachers' professional development programs and focused on how they get credit for certification requirements. Subsequently, Gabriel (2004) reported similar results and further asserted that this program is mainly based on government initiatives. It focuses more on learning administration than content knowledge.
In the next phase, the programs were applied using mixed offline and online platforms, known as blended teachers' professional development. This initiative focuses on technological, pedagogic, and content, knowledge (TPACK) and tends to ignore task content. Desimone (2009), Glava and Glava (2010), Salmon (2012), and Chikasanda et al. (2013) reported that the focus of blended professional development is to design learning instructional activities for both perspectives. Previous research (Borko, 2004;Sims & Fletcher-Wood, 2021) highlighted five effective criteria, namely content, active learning, coherence, duration of program implementation, and learning content. Mathematics content was taught in a general manner within a limited time. Meanwhile, task design was neglected during hybrid professional development.
In this digital era, professional development program has completely evolved into an online platform which focus is centered on equipping teachers to adapt to this new learning environment, how to design it (Tschida et al., 2016), and highlight its impact on learning (Elliott, 2017;Parsons et al., 2019;Powell & Bodur, 2019). Elliott (2017) specifically reported how teachers were able to overcome the challenges encountered during the transformation of traditional learning to online and how it was designed to manage distance and time constraints. Parsons et al. (2019) further stated that professional development is focused on equipping teachers with technological skills to adapt to the online learning environment. Powell and Bodur (2019) reported similar results that emphasized teaching strategies, media, motivation, and lesson plan design, although task quality was excluded because it was not Solve the quadratic equation 2 + 2 − 3 = 0 evaluated. This is not in accordance with Ratnayake et al. (2020), that investigated how to improve the use of digital technology for learning mathematical concepts in the classroom. Ratnayake et al. (2020) focused on designing these tasks by drawing graphs of quadratic functions using Geogebra. This research provided teachers with an idea of how to design quality assignments, which optimize students' thinking processes by leveraging digital technology. The professional development program lasted for a relatively short period, only four days. Therefore, the significant impact on teachers' ability to design quality assignments could not be revealed.

Participant
The subjects of this research involved 105 secondary school mathematics teachers who participated in the online professional development program for three months in East Java Province, Indonesia. It was implemented in the Mathematics Education Department of one of the private universities, which has been accredited with an "excellent" rating by the National Accreditation Board. This online professional development program was designed for five sessions over three months. Teachers independently learned mathematical content for two weeks in the first session using five modules. Meanwhile, in the second session, which lasted for 30 days, the participants learned the module tasks and analyzed possible misconceptions. They were also taught the theory of innovative learning models, including problem-based tasks.
In the third session, teachers were instructed to design lesson plans supervised by the senior lecturers and tutors for 14 days. The designed learning activities started with problem identification. These participants were able to prepare their lesson plans, teaching media, content, and evaluation instruments or tasks, based on the formulated problems. This learning set was designed using a problem-based approach.
Afterward, the participants conducted two-day peer-teaching practices using their prepared lesson plans. During this session, one of them acted as a teacher, while two others were the observers, and ten participants acted as learners. In the fifth session, the participants underwent a comprehensive test to determine their eligibility for participating in the online classroom teaching practice for 31 days. The next session involves a final review, reflection, and discussion of their previous teaching practices performed using video conferencing before they took a competency test involving mastery of math content and their performances. Teaching practice videos and portfolios were used to determine whether they deserved to hold a professional certificate.

Instrument
Three instruments were used to obtain valid data on teachers' beliefs and practices of problem-based math tasks. First, a questionnaire consisting of 10 statement items was developed to measure their beliefs. Based on this instrument, item numbers 1 to 4 explain the teachers' perception of problem-based math assignments, procedures, and their opinions regarding the adequacy of the problem-based tasks in the handbooks used. In addition, items 5 to 10 were designed to investigate the practices regarding problem-based math tasks. The question items were designed based on frequency ranges, such as how often teachers designed and modified problem-based tasks from the textbooks when they felt that some were unsuitable.
Second, a semi-structured interview protocol was used to support the quantitative data acquired from the questionnaire. Its objectives were achieved by carefully designing it based on the research. Four sets of questions were developed based on the task structures, information type, relevance to the students, and the availability of problem-based tasks in the teacher's Handbook. Furthermore, three experts evaluated the checklist of the semi-structured interview guide. Their assessments, notes, and suggestions were used to determine the content validity and appropriateness of the questions. This semistructured interview was given to the teacher at the end of the professional development program. Third, related documents were analyzed to support its qualitative findings, namely the assignments designed by teachers during the program.

Data Collection
First, it was ensured that all the subjects had completed a five-month professional development program. They were given a questionnaire consisting of 10 statements related to teachers' beliefs and practice of problem-based tasks. Second, semi-structured interviews were held with nine teachers selected based on the results of the questionnaire and by considering the distribution of demographics and their teaching experiences. Third, as long as they participated in the professional development programs, their products, including problem-based task designs and teaching materials, were collected and analyzed.

Data Analysis
This research involved three types of information, namely, first, quantitative data derived from teachers' frequency in applying problem-based tasks and their beliefs. These were descriptively analyzed using SPSS 22.0 to determine how often it was promoted in front of students. The highest score indicated frequent implementation. Furthermore, the data concerning teachers' beliefs were then compared with their frequency reports of problem-based tasks to determine whether the information matched. Second, qualitative data acquired from semi-structured interviews were used to support the information obtained from the questionnaires. Nine teachers were purposely selected and interviewed based on the demographic distribution of different provinces and their experiences which ranged from ≤5 years, 5<PM≤10 years, PM>10 years. Third, data acquired from problem-based tasks and teaching materials used during the professional development programs were collected to support the qualitative obtained from the semi-structured interviews. Those who attended the program produced three cycles and 105 problem-based tasks. The data from teachers' practices in designing these tasks were evaluated using the analytical framework developed by the authors, as shown in Table 1. The tasks should lead the students to decompose the problem or create a similar representation, identify errors and compare solutions.

Straightforward tasks
The tasks could be directly calculated or offered after an explanation.

Teachers' Beliefs about Problem-Based Tasks, the Nature of Problems, Clarity, and Completeness of Information to Find Solutions and Task Instructions
Regarding teachers' beliefs, the majority reported that their practices did not support students' abilities to solve tasks, their tasks indicating did not reflect the concept of problem-based tasks. Figure 3, related to the provision of completion steps (fifth statement), shows that 86% or 90 out of 105 teachers, consisting 24 agree and 66 strongly agreed, agreed to provide transparent information regarding the adopted mathematical procedures. Based on the information type acquired (first statement), approximately 64% (67 of 105) agreed to support students in solving problem-based tasks only when the required data were included in the assignment. In accordance with task instruction, 92% of teachers (97 of 105) significantly agreed that the task instruction was provided to help students solve the problem. This is possible because teachers lack a proper understanding of the task, which is in line with the research conducted by de Araujo and Singletary (2011), despite participating in the professional development. The tasks designed just to get the job done tend to provide opportunities and meaningful learning experiences for students (Schoenfeld, 2022).

Teachers' Reported Teaching Practice
Figure 4 describes teachers' reported teaching practices. According to them, they rarely give problembased tasks to students. Only 19% stated that they gave such assignments weekly. A few designed their problem-based task, while the rest adopted it from Handbook (19 of 105 or 18%). Furthermore, some teachers also tried to modify those obtained from the textbooks. It was reported that 21 teachers engaged in this activity weekly, while 24 of them assigned it to the learners every semester. This is possible because teachers' adopted practices are highly influenced by their beliefs (CrossFrancis et al., 2015).

Teachers' Beliefs of Reported Teaching Practices in Designing Problem-Based Tasks
The extraction of teachers' beliefs through questionnaires indicated that most teaching practices did not support students' abilities to solve problem-based tasks and did not align with the mandated framework. Approximately 86% agreed to provide relevant information regarding the required mathematical procedures. The results of interviews with nine selected teachers regarding problem-based task designs during their enrolment in the professional development program are shown in Table 2.

A Group of Teachers with Less than Five Years of Teaching Experience
This section describes three task designs produced by SN, LZS, and AH. These participants had less than five years of teaching experience. The results of the developed tasks are presented in Figures 5, 6, and 7, respectively.

Figure 5. Mathematical relation task by SN
SN designed the task in Figure 5, however, it does not contextually reflect a problem-based assignment. It is similar to the result of SN's questionnaire, where SN scored 3 in the first item questions regarding unclear information in designing problem-based tasks. This simply implied that SN did not understand the context.
Consequently, SN was unable to apply a relevant and acceptable approach to the developed task. Based on the nature of the problem, its contents regarding Mr. Idris, Mr. Sugandar, and Mr. Adhim lacked a clear directive. Second, the instruction "…find the possible mathematical relations…" was not welldefined in discerning whether the connections were between parents or among children in a family. This practice was considered inappropriate for task openness, as proposed by Yeo (2017). It was further stated that its limitations had to be well-defined to determine a possible solution. The task instructions need not make students focus on the context of the problem.
Meanwhile, during the interview held with SN concerning 'Mathematical relation task', two misconceptions were discovered in the developed task, namely that of 'problem' and instruction or questions. First, SN did not understand the concept of an ill-structured task and failed to follow the stipulated rules. Not all information in terms of guiding students on how to solve these problems was provided, and the available data are not interrelated. Previous research on the ill-structured characteristics of problem-based tasks was properly described by Delisle (1997). It was the first to characterize problem-based learning. Delisle further stated that most teachers made mistakes while defining ill-structured tasks at the conceptual level but not in practical design. The findings of this research provide empirical evidence that the designed assignments are still far from the ill-structured concept. The following is an excerpt from the interview held with the subject SN.

Researcher : From the "Mathematical relation" task you designed, what would you like students to do?
and what information do they need to be able to execute the assignment? SN : I want students to find as many relations as possible, besides the information provide is the number of members in each set, for example I = {Faisal, Alu, Riski}; S = {Sunaida, Firman}; A = {wafi} Researcher : Was adequate information provided? SN : Hmmm…yes, because for this assignment the information is usually incomplete, students need to be creative The following results were obtained from the second subject, named LZS, who had problems related to 'context' as part of a problem-based task. The context was literally interpreted. Based on her understanding, it was perceived as an object easily encountered by students in their daily activities. Therefore, it was interpreted as a type of fruit, although she failed to explain the acceptable focus of the task. When they have completed the assigned task, the context seems to disappear because its focus relies only on the calculated process, causing the problem to disappear. Based on this finding, Clarke and Roche (2018); and Wijaya et al., (2015) asserted that 'context' should help students solve the problem, and not only see it as an object related to their lives.

Figure 6. SPLDV task by LZS
In contrast to the two misconceptions from subjects 1 and 2, the third also experienced it at the routine tasks levels. She was also unable to understand the concept of a problem-based task because her task was easily solved using direct calculations. This is a straightforward task because it had no impact and did not motivate students to engage in such problems.

A Group of Teachers' Teaching Experience between Five to Ten Years
The following findings are derived from the subjects with teaching experience between five to ten years. The results indicated they had problems related to context design and the completeness of the information provided to find solutions. This is because teachers understood the task with the aim that students can complete it, not on how they develop their thinking processes (de Araujo & Singletary, 2011). The SPLDV task designed by GH, as shown in Figure 8, had two misconceptions.

Figure 8. SPLDV task by GH
The first was about the context, and it was similar to LZS. Surprisingly, both subjects were unable to identify the context accurately. The second was concerned with task data or information required to find a solution to GH's assignment that every variable with "The price of a book is Rp. 3.000, and one pencil is Rp. 4.000".

Figure 9. SPLDV task by NH
Nani goes to the market to buy apples and rambutans. It was easy to predict the answer, and 100% could guess the answer correctly through a short process without involving any form of comprehension in this context. Similarly, Subjects NH and DA also developed their tasks, as shown in Figures 9 and 10.   Figure 10. SPLDV task by DA

A Group of Teachers with More than Ten Years of Teaching Experience
The following findings were obtained from senior mathematics teachers with longer teaching experiences who tended to apply straightforward tasks. This is due to the complexity of the problems and their experiences in terms of participating in professional development that was not equipped with characteristic tasks, specifically when the focus is on how to design good learning in general and not on the content (Clarke et al., 2009;Zaslavsky & Sullivan, 2011). In addition, senior teachers who are used to the application of direct tasks reluctantly find it difficult to change to more complex ones (Schoenfeld, 2022). The task in Figures 11, 12, and 13 could be solved directly or in a straightforward manner.  The subjects gave clear and specific information because they feared students were unable to solve the tasks. Jackson et al. (2013) reported that most teachers often instruct students to help them understand and find the solution to the problem.

Disharmony of Teachers' Beliefs and Practices
Regarding all the questionnaire items, teachers' beliefs about problem-based tasks had the highest In the cube ABCD.EFGH with the side length being 6 cm, the distance from point B to the diagonal of space AG is...
Determine the sum of (3 − 17 + 35 ) (4 + 23 − 9) agreement in statement 1. These questions have to provide clear information about the mathematical procedures needed to find possible solutions, of 105 teachers, 90 stated that they significantly agreed with the statement. This was compared to item 8. The results showed that only 4 of 105 teachers reported not designing a problem-based task using clear information regarding the mathematical procedures needed to find a solution. This was due to several reasons, first, the short duration of the training and teachers' workload during the program, and this tends to cause a less optimum understanding. Second, the concept of problem-based tasks was introduced a long time by Delisle (1997), but its frameworks were not explicitly written. The participants were unable to properly interpret them and practice the designed problems according to the right frameworks. Meanwhile, the highest unsuitability of tasks was found in questionnaire item 4 regarding problem based-tasks in the textbook. 73 out of 105 teachers stated that these assignments were not sufficiently available in their handbooks. This was followed by teachers' unwillingness to design or modify tasks already in their Handbooks (Ruthven & Hennessy, 2002). Pepin et al. (2013) stated that they tend to adopt tasks in the textbook. Even and Olsher (2014) also stated that teachers presume it is easier to give assignments using textbooks. It is in accordance with Casa et al. (2019), which stated that the majority designed their tasks based on the textbook without further adaptations. This finding emphasizes that teachers' beliefs about problem-based tasks are less available in the textbooks they use in class, but it has no clear procedures and contexts. They are unwilling to modify the tasks as it is time-consuming and challenging for them. Further, Glasnovic Gracin (2018) and van Zanten and van den Heuvel-Panhuizen (2018) stated the importance of mathematics books in fostering students learning through the tasks .

CONCLUSION
There are two significant findings, first, it was revealed that teachers' beliefs about problem-based math tasks were low, indicating their lack of understanding. These also gave insightful recommendations for professional development programs by the Indonesian government and similar initiatives to involve problem-based tasks as one of the crucial learning elements. At the initial stage, the government needs to map and analyze teachers' abilities to design a problem based-task. Second, those with relatively minor teaching experience (less than five years) have more complex problems. There is a misconception in the practice of developing problem-based tasks. This is similar to the problem structure, its context, and the design of instructions or questions.
The group of teachers with moderate teaching experience (between five to ten years) had the main problem, which is related to 'context'. Similarly, those in the senior group tend to apply straightforward tasks. Teachers need intensive and sustainable training to develop problem-based tasks, environment, and culture can be formed. Their common understanding of this conception may be due to a lack of appropriate designs. It is recommended that future research need to develop a more acceptable and applicable problem-based tasks framework and organize specific professional development training to promote teachers' competencies in designing problem-based tasks. Haruehansawasin, S., & Kiattikomol, P. (2018). Scaffolding in problem-based learning for low achieving learners.