## 1 Introduction

Mathematical word problem solving is commonly used in schools to cultivate students’ abilities in applying mathematical knowledge and skills to everyday life. However, the complexity of cognitive processes demanded by this activity leads to a prevalence of difficulties in students . Diagram use is generally considered one of the most effective strategies for promoting success in this kind of problem solving . However, students tend not to use diagrams spontaneously, and even when they do use them, such use does not always lead to obtaining the correct solutions.

One reason for the failure to solve despite the use of diagrams is a lack of correspondence between the problem schema/type (i.e., its structure and requirements) and the kind of diagram that students construct . Word problems are included in the school curriculum to facilitate understanding of target learning contents, and diagrams are used as a way of “scaffolding” to make solving easier. However, inadequate attention is placed on teaching “appropriate diagram use” to match the problems given because the curriculum focuses on teaching “mathematical contents”. Thus, much of the knowledge that students develop about diagram use remains implicit.

To address this problem, ways to effectively facilitate understanding of the correspondence between problem schema/type and appropriate diagrams need to be investigated. While it would be difficult to capture and evaluate the wide range of methods that teachers use to promote such understanding, examining how the textbooks might facilitate that understanding is much more manageable. In Japan, all schools use textbooks certified by the Ministry of Education in line with the national guidelines (issuance law of Japan). Examining such textbooks could provide valuable insights into how those correspondences are portrayed, which had not been examined in previous research. Therefore, this study aimed to clarify: (i) the types of word problems used in math textbooks, (ii) the kinds of diagrams deemed useful for solving those problems, and (iii) the correspondence between problems and diagrams.

## 2 Method

The textbooks used were one of six mathematics textbook series eligible for use in Japan at the elementary school level (6 books corresponding to Grades 1 to 6, each containing 13–19 chapters) . Each chapter contains example problems with detailed explanations. The main purpose of those example problems is to teach the learning contents set by the national standard, assuming that children would be learning them for the first time. The present study focused on all the example problems that take the form of word problems, defined here as problems comprising two or more sentences with a background story, and which requires a mathematical solution. The word problems were examined and categorized according to their problem schema (i.e., type of word problem), and the kind of diagram deemed appropriate for solving them.

In line with previous studies , the word problems found were classified into one of six types: Change (e.g., Ken had 3 candies. Naomi gave him 5 more candies. How many candies does Ken have now?), Combine (e.g., Ken has 4 candies. Naomi has 3 candies. How many candies do they have altogether?), Compare (e.g., Ken has 4 candies. Naomi has 12 candies. How many times are Naomi’s candies greater than Ken’s?), Vary (e.g., Ken packs 5 candies in each box. How many boxes will he need to pack to fit 40 candies?), Organize (i.e., organizing data given to find an answer by using tables or graphs; e.g., The following data shows students’ reading times in a month. What number appears the most often?), and Visualize (visualizing the conditions given in the problem with figures, graphs, tables, etc., to facilitate search for rules difficult to find based only on superficial details; e.g., There are 3 children in front of Ken, and 4 behind him. How many children are there in total?).