Problem Posing

The investigation of problem-posing abilities among students and future teachers is becoming increasingly prominent in mathematics education research. However, tasks designed specifically to develop this competence are rarely found in both public education and teacher training. Research has shown that engaging in problem-posing activities enhances mathematical thinking and problem-solving skills, and it is also included among the basic competencies in the National Core Curriculum. For future mathematics teachers, developing problem-posing skills is essential because they need to create or find mathematically appropriate, relevant, and appropriately dressed tasks for their students. Hungarian mathematics education is renowned for its problem-centered approach, with numerous renowned mathematicians and high-quality math competitions (e.g., KöMaL and the Arany Dániel Mathematical Competition) serving as examples. Developing problem-solving skills can be achieved not only through solving sets of problems and competition tasks but also through the development of problem-posing abilities. As György Pólya writes, a critical step in expert-level problem-solving is reframing, varying, and posing new questions and problems based on the given task (Duncker, 1945; Pólya, 1957).

Modern research on problem-posing was laid out by Kilpatrick (1987) and Silver (1994), and over the past two decades, it has become one of the central topics in mathematics education research. In 2015, Singer, Ellerton, and Cai published a comprehensive study titled "Mathematical Problem Posing: From Research to Effective Practice," which discusses the state of research on problem-posing and its key questions. The first chapter of the book, authored by Cai, Hwang, Jian, and Silber, lists and examines the fundamental questions of problem-posing research in public education. According to them, the most important questions in problem-posing research currently include the following (italicized questions indicate those addressed in our own research):

1. Why is problem posing important in school mathematics? 
2. Areteachersandstudentscapableofposingimportantmathematicalproblems?
3. Canstudentsandteachersbeeffectivelytrainedtoposehigh-qualityproblems?
4. What do we know about the cognitive processes of problem posing? 
5. How are problem-posing skills related to problem-solving skills? 
6. Is it feasible to use problem posing as a measure of creativity and mathemat- ical learning outcomes? 
7. How are problem-posing activities included in mathematics curricula? 
8. What does a classroom look like when students engage in problem-posing activities? 
9. How can technology be used in problem-posing activities? 
10. What do we know about the impact of engaging students in problem-posing activities on student outcomes? 

Our initial research on problem-posing was motivated by the observation of how many dull and poorly designed mathematics tasks we encountered during our years learning and teaching mathematics. We believed that mathematics teachers must acquire the ability to select or create engaging, thought-provoking tasks that can be effectively used in their lessons. Therefore, our first research topic focused on examining the problem-solving and problem-posing abilities of prospective mathematics teachers. Later on, seeing the positive experiences related to problem-posing and task creation in mathematics lessons, we also planned and initiated experiments among students in public education. During these experiments, we developed our own evaluation criteria based on our experiences and the existing literature (Rékasi, Stirling 2021; Rosli et al. 2015).

The aim of our current research is to create and present methods that enable students and teacher trainees to engage in problem-posing and task creation. Additionally, we aim to establish a comprehensive researcher perspective for evaluating the tasks created, based on our experiences and international literature.

Bibliography

• Cai, Jinfa & Hwang, Stephen & Jiang, Chunlian & Silber, Steven. (2015). Problem- Posing Research in Mathematics Education: Some Answered and Unanswered Questions. 10.1007/978-1-4614-6258-3_1. 
• Czeglédi Csaba, Stirling Anna Krisztina (2022). Adott témakörben különböző módszerekkel történő problémafelvetés vizsgálata a szakképzésben. TDK dolgozat
• Duncker, K. (1945). On problem solving. Psychological Monographs (Vol. 58). New York, NY: American Psychological Association.
• Kilpatrick, J. (1987). Formulating the problem: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive Science and Mathematics Education (pp. 123- 147). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Muzsnay Anna, Szabó Csaba (2017). Dressed up problems - the danger of picking the inappropriate dress. TEACHING MATHEMATICS AND COMPUTER SCIENCE 15: (1- 2) pp. 77-94.
• Pólya György (1957). How to solve it. Princeton, NJ: Princeton University Press.
• Rékasi Anna, Stirling Anna Krisztina (2019). Matematika tanárszakos hallgatók problé- mafelvetési és problémamegoldási készségeinek összehasonlítása. TDK dolgozat
• Rékasi Anna, Stirling Anna Krisztina (2019). Matematika tanárszakos hallgatók problé- mafelvetési képességeinek vizsgálata fejlesztési céllal. TDK dolgozat
• Rékasi Anna, Stirling Anna Krisztina (2021). Közoktatásban tanuló diákok feladatkészítési képességének vizsgálata. TDK dolgozat
• Rékasi Anna, Stirling Anna Krisztina (2021). A problémafelvetés és feladatkészítés egy lehetséges megjelenítése matematikaórán. TDK dolgozat
• Rosli, Roslinda & Capraro, Mary & Goldsby, Dianne & Gonzalez, Elsa & Capraro, Robert & Onwuegbuzie, Anthony. (2015). Middle-Grade Preservice Teachers’ Mathematical Problem Solving and Problem Posing. DOI: 10.1007/978-1-4614-6258-3_16.
• Silver, E. A. (1994). On mathematical problem posing. – In: For the Learning of Mathematics 14 (No.1), p. 19-28.
• Singer, F.M., Ellerton, N.F., Cai, J. (2015, Eds.). Mathematical Problem Posing: From Research to Effective Practice, NY: Springer