Teaching Mathematics Through Problem-Solving

A Pedagogical Approach from Japan

Akihiko Takahashi

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000359862
• Edition: 1
• Topic: Éducation
• Subtopic: Enseignement des mathématiques

This engaging book offers an in-depth introduction to teaching mathematics through problem-solving, providing lessons and techniques that can be used in classrooms for both primary and lower secondary grades. Based on the innovative and successful Japanese approaches of Teaching Through Problem-solving (TTP) and Collaborative Lesson Research (CLR), renowned mathematics education scholar Akihiko Takahashi demonstrates how these teaching methods can be successfully adapted in schools outside of Japan.

TTP encourages students to try and solve a problem independently, rather than relying on the format of lectures and walkthroughs provided in classrooms across the world. Teaching Mathematics Through Problem-Solving gives educators the tools to restructure their lesson and curriculum design to make creative and adaptive problem-solving the main way students learn new procedures. Takahashi showcases TTP lessons for elementary and secondary classrooms, showing how teachers can create their own TTP lessons and units using techniques adapted from Japanese educators through CLR. Examples are discussed in relation to the Common Core State Standards, though the methods and lessons offered can be used in any country.

Teaching Mathematics Through Problem-Solving offers an innovative new approach to teaching mathematics written by a leading expert in Japanese mathematics education, suitable for pre-service and in-service primary and secondary math educators.

1 Development and Major Concepts of Japanese "Teaching Through Problem-Solving" (TTP)

The unique Japanese pedagogical approach revealed in the publication of The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom caught the attention of teachers and researchers around the world (Stigler & Hiebert, 1999). Its video analysis of Japanese mathematics classrooms showed how Japanese teachers teach new mathematical concepts by giving students compelling mathematical challenges to solve on their own and discuss. Researchers found that this progressive approach, called “Teaching Through Problem-solving” (Schroeder & Lester, 1989), or “Structured Problem-solving” (Stigler & Hiebert, 1999), not only differed from typical U.S. classrooms but from other Asian classrooms as well (Mullis, 2000; Stigler & Hiebert, 1999). How did Japanese educators come up with such a unique method? How did it spread throughout Japan as a major approach to teaching mathematics?

In this book, I will share my insights based on decades of being front and center in Japan’s education reform movement and the development of Teaching Through Problem-solving (TTP). My career has focused heavily on the promotion of TTP through research and personal practice. I have observed hundreds of lessons taught by fellow educators and spent years hosting public lessons and creating lesson plans. I have written this book based on my experience to show how teachers and schools outside Japan can use this approach to nurture their own students to become independent problem-solvers. This chapter will discuss the characteristics of TTP and outline how it became an established pedagogical practice in Japan.

1.1 The Need to Move Beyond the Lecture Method

1.1.1 The Purpose of the Study of Mathematics

Why do we have children study math? The most critical value of studying mathematics is to learn the process of mathematics, such as mathematical thinking and problem-solving. Viewing the world through a lens of mathematics gives young learners the chance to explore and make sense of the world around them. However, asking students to simply memorize facts and procedures, the results of others’ exploration and discovery, makes mathematics dreary. How can we teach students how to grapple and engage with problems using mathematical reasoning if we only give them the opportunity to memorize formulas and facts?

We must teach students how to think mathematically. In 1998, the National Research Council in the United States put together a committee to synthesize a wide variety of research on mathematics education. Their report, Adding it Up: Helping Children Learn Mathematics, begins with a strong statement, “All young Americans must learn to think mathematically, and they must think mathematically to learn” (National Research Council, 2001, p. 1). Facts and procedures are essential, but students must learn how to think mathematically to solve problems. Adding it Up breaks down what it means to think mathematically into several different strands of proficiency (2001). These abilities are described as five interwoven and interdependent strands (Figure 1.1.01). The report defines them as:

• conceptual understanding—comprehension of mathematical concepts, operations, and relations
• procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, represent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
• productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

(National Research Council, 2001, p. 116)

Figure 1.1.01 Intertwined strands of proficiency. Reprint of National Research Council (2001, p. 117)

However, the lecture method and routine repetition cannot nurture all five strands of proficiency.

Today’s teachers will be familiar with the current Common Core State Standards (CCSS), which emphasize the need for students to learn how to think mathematically. They outline eight Standards for Mathematical Practice (MP):

• MP1 Make sense of problems and persevere in solving them.
• MP2 Reason abstractly and quantitatively.
• MP3 Construct viable arguments and critique the reasoning of others.
• MP4 Model with mathematics.
• MP5 Use appropriate tools strategically.
• MP6 Attend to precision.
• MP7 Look for and make use of structure.
• MP8 Look for and express regularity in repeated reasoning.

(Common Core State Standards Initiative, 2010, pp. 6–8)

However, these are demanding expectations which many schools and teachers are still not sure how to accomplish in their classrooms. Helping students develop mathematical thinking skills requires highly developed pedagogical strategies.

1.1.2 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.

1.1.3 Challenges

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 Westbury’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 tea...