Problem-Solving as an Ideal Approach for Studying Mathematics
Research shows that giving students the opportunity to solve new problems on their own can help them achieve the demanding mathematical practice standards outlined by the CCSS. The National Council of Teachers of Mathematics emphasizes the importance of “building new mathematical knowledge through problem-solving” and defines problem-solving as “engaging in a task for which the solution method is not known in advance” (2000, p. 52). They argue that students should think mathematically to learn by solving novel problems in order to acquire knowledge of mathematical procedures. Independent problem-solving needs to be an integral part of all mathematics learning and not just an end of chapter activity.
Problem-solving has been a major focus of school mathematics education research for decades. Researchers corroborate that it’s better for students to learn new mathematical concepts by trying to solve problems on their own rather than by just imitating the work of others. For example, in 1945, Polya suggested in his famous book How to Solve It that teachers should help students discreetly and unobtrusively as they work independently to solve new problems (Polya, 1945). In 1970, Gattegno, the inventor of geoboards and largely responsible for the popularity of Cuisenaire rods, argued that teachers cannot simply impart their knowledge to students through the lecture method (Gattegno, 1970). According to Lesh and Zawojewski (2007) the Journal for
Research in Mathematics and Educational Study in Mathematics published one hundred and fifty-six research articles on problem-solving during the 1980s and 1990s. These articles addressed topics such as studies on how students think mathematically when grappling with new problems and how to nurture them to develop their problem-solving skills (e.g., Schoenfeld, 1985). In 1980, the NCTM proposed in Agenda for Action that problem-solving should be the focus of school mathematics for everyone, researchers as well as teachers and educators (National Council of Teachers of Mathematics, 1980). Researchers and experts agree that students need to explore new mathematical concepts through problem-solving in order to develop the ability to think mathematically.
However, despite recognizing the need to teach students how to think mathematically, there have been challenges in meeting this goal. International studies, such as the Trends in International Mathematics and Science Study (TIMSS), evaluate student achievement through the lens of Travers and West- bury’s (1989) three aspects of curriculum: “intended curriculum,” “implemented curriculum,” and “attained curriculum” (Mullis, Martin, &. Loveless, 2016; Travers, 2011; Travers & Westbury, 1989). The “intended curriculum” are the formal documents that describe what the students are expected to learn, such as CCSS and NCTM standards. The “implemented curriculum” are the lessons taught by a teacher. The “attained curriculum” is what the students actually learned (International Bureau of Education, 1995). Textbooks and other resources serve as potential curricula (e.g., Schmidt, McKnight, Valverde, Houang, &. Wiley, 1997). They are designed to address the intended curriculum, but effective implementation relies on the skills of the teacher (Figure 1.1.02). This is why the lecture method, or simply “teaching the textbook,” cannot successfully impart the intended curriculum by contemporary standards.
Figure 1.1.02 How effective curriculum implementation comes from the teacher. The three aspects of curriculum come from Travers and Westbury (1989).
The best way to ensure that the attained curriculum matches the intended curriculum has been the subject of much research. There is a significant amount of curricula available designed to teach students how to think mathematically. For example, the NCTM developed many guidelines and resources (e.g., National Council of Teachers of Mathematics, 1989, 2000). There are also several projects funded by the National Science Foundation, such as Everyday Mathematics developed by the University of Chicago School Mathematics Project (1992) and The Connected Mathematics (CMP) developed by Michigan State University (1996).
Still, the results have been uneven. Stigler and Hiebert (2009) argue there is no evidence that there was any improvement in teaching mathematics in the United States between 1995 and 1999. In 2018, Banilower et al. (2018) reported that more than 85% of teachers believed that students should learn mathematics by solving problems on their own and that students should also be able to explain their solutions. However, most American teachers do not give students such opportunities on a daily basis. Less than a quarter of classrooms make independent problem-solving and discussion a part of their everyday lessons (Banilower et al., 2018). It is still the exception and not the norm. As a result, student learning isn’t meeting expectations (e.g., Mullis, Martin, &. Loveless, 2016).
Teachers may be hesitant to switch their teaching methods. Leaving behind the lecture method requires a sophisticated pedagogical approach, which takes time to learn. Research shows that teachers may continue to rely on the lecture method due to a lack of professional development opportunities which would give them the chance to update their pedagogical skills (e.g., Stigler &. Hiebert, 2009; Wei, Darling-Hammond, Andree, Richardson, &. Orphanos, 2009). This struggle to shift to student-centered instruction doesn’t only exist in the United States. Many countries whose curriculum emphasizes the importance of teaching students how to think mathematically are also grappling with how to implement these values into their classrooms. A gap remains between the intended and implemented curriculum. The Japanese pedagogical approach of TTP can help bridge this gap.