# Mathematical thinking: how to develop it in the classroom

Developing mathematical thinking is one of major aims of mathematics education. Mathematical thinking: how to develop it in the classroom, by Masami Isoda and Shigeo Katagiri, is the first volume in the series Monographs on Lesson Study for Teaching Mathematics and Sciences. This book has the potential to make a significant positive contribution to elementary education and the practice in the classroom.

Katagiri was the president of the Society of Mathematics Education for Elementary Schools in Japan. His life work has been devoted to analyze and classify mathematical thinking from 1960’s. Mathematical thinking consists on a proposal of mathematical ideas, methods, and attitudes which support the process of thinking mathematically. His proposal tries to enhance teachers’ questions for eliciting mathematical thinking from children.

The book consists of two parts. Part I, entitled Mathematical thinking: theory of teaching mathematics, explains Katagiri’s theory about how to develop children who learn mathematics for themselves according to the authors. Part II of the book, entitled Developing mathematical thinking with number tables: how to teach mathematical thinking presents 12 examples of lessons from the viewpoint of assessment.

The two parts are very different in their structure and content. The first part includes seven chapters. There is a preface of the book, and an introductory chapter by Isoda which is very helpful. Chapter 1 is referred to as Katagiri’s theory. In Chapter 2 the authors give arguments in favour of the development of mathematical thinking. In Chapter 3, it is explained the structure and a categorization of the components of mathematical thinking, which they name: Mathematical Methods, Mathematical Ideas, and Mathematical Attitudes. These components will be described in Chapters 4, 5 and 6. Finally in Chapter 7 the authors offer to the teachers a way for questioning to enhance the students. Questioning called “Hatsumon” in Japanese has three major usage in a mathematics in Japan. The first type of Hatsumon is relative to develop, recognize or reorganize mathematical knowledge, method and value. The second type is aimed to change phases of teaching in the whole classroom. And, finally the third type of Hatsumon is aimed to do the internalization of those two types of Hatsumon into children’s mind.

The second part of the book consists of 12 examples of lesson studies over the past 50 years (that develop and assess students’ mathematical thinking). These 12 chapters have a similar structure with sections: types of mathematical thinking to be cultivated, grade taught, preparation, overview of lesson process, worksheet, lesson process, summarization on the blackboard, evaluation and further development. I would like to note that from the 12 chapter-lessons, six of them are concerned with the arrangement of numbers on the tables, another are devoted to sums of numbers, two consider squares of numbers on the tables and the relationships among the numbers within the chosen squares, and the last two of them deal with the arrangement of multiples and common multiples.

As reviewer I would like to highlight two topics: Thinking mathematically and mathematical thinking in the curriculum and Mathematical Attitudes.

Thinking mathematically and mathematical thinking in the Japanese curriculum

The ideas relative to the main topic of this book are familiar to the majority of readers from the Western culture, as we can find in Polya, Schoenfeld, and Mason and colleagues writtings. For instance, the introductory Chapter 1 explains the pedagogical approach, which is referred to as ‘‘Problem Solving Approach’’. The approach is presented in five phases: Posing the Problem, Planning the Solution, Executing Solutions, Discussion (Validation and Comparison), and Summarization and Further Development. The reader can find resemblances to Polya’s (1945) description of the four steps of problem solving which, despite numerous extensions and critique (e.g., Schoenfeld 1985; Schoenfeld 1992) is still a major reference in instructional materials for teachers (e.g. NCTM 2000). However, I would like to underline the effort made by the authors to introduce the culture of Japanese teachers and Japanese teaching into English. Likewise, the reader would like to know more about the influence of cultural factors on the integration of these ideas. Finally, I consider that it is quite worth to adapt the solving problem ideas to a class of elementary school and also to offer the description of the 11 types of Mathematical Methods that are exemplified and discussed: inductive thinking, analogical thinking, deductive thinking, integrative thinking, developmental thinking, abstract thinking, simplifying, generalization, specialization, symbolization, and quantification and schematization. Although, it is missing in the book a closer match between the theoretical and the practical parts.

I am wondering if the good results achieved by this country in international assessments are a result of this methodological proposal. The PISA mathematics literacy test asks students to apply their mathematical knowledge to solve problems set in real-world contexts. To solve the problems students must activate a number of mathematical competencies as well as a broad range of mathematical content knowledge. When PISA results in 2012 are examined, Japanese students’ mean scores are typically above the OECD average on the top. The place of Japan in mathematics is the 7th with scores 536, In sciences Japanese position is 4th, 547 and reading position is 4th, 538 (34 OECD member countries). Also, in 2003 and 2009 Japan got a good the mathematics position (in 2003 position 4, 534 (30 OECD member countries) in 2009 position 9, 529 (34 OECD member countries)).

Part of the reason pupils do so well in Japan, according to the OECD's delegate director of education, Andreas Schleicher, is that they handle their confidence to achieve their potential. In Japan -- which ranked 7th overall -- more than 80% of students disagreed or strongly disagreed that they proposed problems were difficult, and 68% disagreed or strongly disagreed that they gave up easily when confronted to a problem.

In spite of Japanese students are interested in inquiry-based learning, whereas science teaching at the upper secondary level does not catch their interest. This fact could contribute to an understanding of why Japanese students in PISA show relatively low levels of positive attitudes toward science (Yasushi, 2009, 175). For example, in this country higher mean performance has lower average levels of mathematics interest. Interest in math remains low. In the latest survey, 38 percent of Japanese students said they were interested in the things they learn in math class. The figure is five points higher than the survey carried out in 2003, but it is 15 points lower than the average for students in the OECD. The increment of Japanese students who are interested in mathematics was a good trend, even though the percentage of attitudes, those feel pleasure, interest and motivation in solving mathematical problems remains low and it raises the need of the development of mathematical attitudes, as was outlined in this book.

Mathematical Attitudes

The Mathematical Attitudes considered in Chapter 6 are named: Objectifying, Reasonableness, Clarity and Sophistication. Here, it is important to note that the authors consider attitude as a ‘‘mindset’’, a mathematical disposition, where one examines the obtained answers and seeks for ‘‘better’’ ways to describe a situation.

While the attitudes toward mathematics have long been studied (Kulm, 1980, McLeod 1992, Di Martino & Zan, 2011; Hannula, 2002; Ruffell et al., 1998), the study of mathematical attitudes has been less thoroughly developed.

As early as 1969, Aiken and Aiken suggested two classical categories, attitudes toward science (when the object of the attitude is science itself) and scientific attitudes (when the object is scientific processes and activities, i.e., scientific epistemology), which were later adapted by a number of authors (Hart, 1989; NCTM, 1989; Gómez-Chacón, 2000) to mathematics and denominated attitudes toward mathematics and mathematical attitudes.

Although developments in attitudes have focused more on attitudes toward mathematics, I would like to underline the importance of developing mathematical attitudes as this book tries to promote. I missed in the book a major discussion about both categories and why they focus on one of the categories. In the following, consequently, this discussion will deal with the distinction that should be drawn, in teaching and learning, between attitudes toward mathematics and mathematical attitudes such as made in previous studies (e.g., Gómez-Chacón, 2011, p. 149).

Attitudes toward mathematics refer to the valuation of and regard for this discipline, the interest in the subject and the desire to learn it. They stress the affective component - expressed as interest, satisfaction, curiosity, valuation and so on – more so than the cognitive component. Mathematical attitudes, by contrast, are primarily cognitive and refer to the deployment of general mathematical disposition and habits of mind. Disposition refers not simply to attitudes but to a tendency to think and to act in positive ways. Students’ mathematical attitudes are manifested in the way they approach tasks such as flexible thinking, mental openness, critical spirit, objectivity and so on, which are important in mathematics (see NCTM Standard 10 (1989), for instance). Due to the predominantly cognitive nature of mathematical attitudes, to be able to be regarded as attitudinal, they must also comprise some affective dimension: i.e., a distinction between what a subject can do (mathematical disposition and habits of mind) and what a subject prefers to do (positive attitude toward mathematics). In 1992 Schoenfeld coined the term enculturation to mean that becoming a good mathematical problem solver may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. Enculturation is entering and picking up the values of community or culture (Schoenfeld, 1992, p. 340). According to Schoenfeld, students need a socialization process, to be imbued with certain habits of mind and mathematical attitudes such as those mentioned above. Consequently, it is incumbent upon teachers to create environments favouring the inquisitive spirit, pursuit of ideas, research and questioning associated with the practice of mathematics.

Bearing in mind, then, that attitude is defined to be a psychological tendency expressed as the evaluation of an object from the two categories of attitude specified. Chapter 6 of Isoda and Kataragiri book (p. 111-120) is dedicated to mathematical attitudes and in the p. 50 is listed items to assess the student behaviour. The reader would like to know more about the cognitive-emotional processes involved in evaluation of students when doing mathematics. In addition, apart of the mentioned attitudes, it is missing other essential mathematical attitudes in the setting of classroom as flexible thinking, critical spirit, visual thinking, inductive attitude, curiosity, perseverance, creativity, independence, systematization, cooperation and teamwork and how those are monitoring.

Since the authors say that this approach is extended in Japan and Korea, the reader would like to know more about the effect of this implementation, independent of the results from PISA.

References

Di Martino P. & Zan R. (2001). Attitude toward mathematics: some theoretical issues. Proceedings of PME 25 (Utrecht, Netherlands), vol.3, 351-358.

Di Martino, P., & Zan, R. (2011). Attitude towards mathematics: a bridge between beliefs and emotions. ZDM, 43(4), 471–482.

Frenzel, A.C.; Goetz, T.; Lüdtke, O.; Pekrun, R. and Sutton, R. E. (2009). Emotional Transmission in the Classroom: Exploring the Relationship between Teacher and Student Enjoyment. Journal of Educational Psychology, Vol. 101, pp. 705-716.

Gómez-Chacón, I. Mª (2000). Affective influences in the knowledge of mathematics. Educational Studies in Mathematics, 43 (2), 149-168.

Gómez-Chacón, I. Mª. (2000). Matematica emocional. Los afectos en el aprendizaje matematico (Emotional mathematics. Affectivity in mathematics learning). Madrid: Narcea.

Gómez-Chacón, I. Mª (2011). Mathematics attitudes in computerized environments. A proposal using GeoGebra. In L. Bu and R. Schoen (eds.), Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra, (147-170). Sense Publishers.

Kulm, G. (1980). Research on Mathematics Attitude. In R.J. Shumway (Ed.), Research in mathematics education (pp.356-387). Reston, VA, NCTM.

Hart , L. (1989). Describing the Affective Domain: Saying What We Mean, en McLeod, D.B. y Adams, V.M. (eds.). Affect and Mathematical Problem Solving (pp. 37-45). Springer Verlag.

Hannula, M. (2002). Attitude toward mathematics: emotions, expectations and values. Educational Studies in Mathematics, 49, 25-46.

McLeod, D.B. (1992). Research on affect in mathematics education: A reconceptualization. In D.A. Grouws (Ed.) Handbook of research on mathematics teaching and learning (pp.575-596). New York: Macmillan.

N.C.T.M. (1989). Curriculum and Evaluation Standards. Reston: NCTM.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

OECD (2010). PISA 2009 Initial Report: Learning Trends (Volume V), PISA, OECD Publishing.

OECD (2013). PISA 2012 Assessment and Analytical Framework Mathematics, Reading, Science, Problem Solving and Financial Literacy. OECD Publishing. http://dx.doi.org/10.1787/9789264190511-en

Polya, G. (1945/1988). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.

Ruffell, M., Mason, J. & Allen, B. (1998). Studying attitude to mathematics. Educational Studies in Mathematics, 35, 1-18.

Schoenfeld, A. (1985). Mathematical problem solving. New York, NY: Academic Press.

Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.) Handbook of Research on Mathematics Teaching and Learning. (pp. 334-389). New York: McMillan.

Yasushi, O. (2009). Comparison of Attitudes toward Science between Grade 9 and 10 Japanese Students by Using the PISA Questions and Its Implications on Science Teaching in Japan, paper presented at the QEG Meeting, Offenbach, Germany, 19-21 October.

**Submitted by Anonymous |

**31 / Jan / 2014