Unpacking Mathematics Pedagogical Content Knowledge for Elementary Number Theory: The Case of Arithmetic Word Problems

: “Number” is an important learning dimension in primary mathematics education. It covers a large proportion of mathematical topics in the primary mathematics curriculum, and teachers use most of their class time to teach fundamental number concepts and basic arithmetic operations. This paper focuses on the nature of mathematics pedagogical content knowledge (MPCK) concerning arithmetic word problems. The aim of this qualitative research was to investigate how well the future primary school teachers in Hong Kong had been prepared to teach mathematical application problems for third and sixth graders. Nineteen pre-service teachers who majored in both mathematics and primary education were interviewed using two sets of scenario-based questions. The results revealed that innovative approaches were suggested for teaching third graders while the strategies suggested for teaching sixth graders were mostly based on a profound understanding of mathematical content knowledge. Many participants demonstrated sound knowledge about the sixth grader’s mathematical misconception, but most of them were unable to precisely indicate the third grader’s error in presenting a complete solution for a typical mathematics word problem. A deep understanding of elementary number theory seems to be a precondition for developing pre-service teachers’ MPCK in teaching arithmetic word problems.


Introduction
The core of primary mathematics is the understanding of number concepts and operations. Problem solving and daily life applications as part of mathematics teaching and learning in primary schools are highlighted in reform mathematics curricula across countries [1]. Apart from in-service professional development, pre-service teacher education is also a key concern to ensure the success of an innovative mathematics curriculum [2].
In reviewing the research literature on initial teacher preparation, there has been a longstanding interest in pre-service primary teachers' knowledge of elementary number theory [3,4]. Nevertheless, far too little attention has been paid to their pedagogical knowledge for teaching arithmetic word problems although primary students always demonstrated difficulty in solving multi-step word problems involving whole numbers in international assessments, for example, the Trends in International Mathematics and Science Study (TIMSS) [5]. Primary students in Hong Kong can struggle with mathematics word problems even if they maintain an outstanding performance in TIMSS [5,6]. The purpose of this research was therefore to ascertain the nature of Hong Kong pre-service primary teachers' mathematics pedagogical content knowledge (MPCK) on teaching lower-and upper-grade students to solve arithmetic word problems. The contributions of this paper would be of interest to mathematics educators, researchers, and curriculum developers.

The Concept of MPCK
Mathematics pedagogical content knowledge (MPCK) refers to pedagogical content knowledge (PCK) in mathematics education. The fundamental notion of PCK is attributed to Shulman's [18] theory of teacher knowledge. This particular type of knowledge is defined as the "special amalgam of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding" [19] (p. 8). It is widely accepted that PCK is a domain of subject-specific knowledge for teaching.
Based on the work of Shulman [19], various models and theories regarding knowledge of mathematics teachers have been produced. For example, Ernest [20] proposed a model of cognitive structures connecting mathematics teachers' knowledge, beliefs, and attitudes. He claimed that both pedagogical and curricular knowledge were regarded as practical knowledge of mathematics teaching. Ball and her colleagues developed a "practice-based theory of mathematical knowledge for teaching" [21] (p. 389). In their theory, PCK includes knowledge of content and student, knowledge of content and teaching, and knowledge of content and curriculum.
In 2008, the term MPCK initially appeared in the Teacher Education and Development Study in Mathematics (TEDS-M) project [22]. TEDS-M was the first large cross-national study into teacher education systems but Hong Kong did not participate. It used questionnaires to collect large-scale data from 17 participating countries to measure pre-service primary and lower-secondary teachers' mathematical competencies. A total of 32 question items were designed to investigate MPCK in the TEDS-M study. Pre-service teachers' abilities of generating fruitful questions, explaining or representing mathematical concepts or procedures, analyzing and diagnosing students' responses, providing appropriate feedback, and responding to unexpected mathematical issues were assessed [22]. Since there are different perceptions of MPCK, it is necessary to clarify the conception of MPCK by specifying its components for the present study.
An extensive search of literature indicated that MPCK is usually made up of two principle components. The first component is knowledge of instructional strategies for mathematics, such as the use of "analogies, illustrations, examples, explanations, and demonstrations" [19] (p. 9) in mathematics teaching. The second component is knowledge of students' conceptions about mathematics [21,23]. Researchers are especially interested in how teachers deal with students' mathematical errors and misconceptions [24]. For instance, the TEDS-M researchers found that pre-service teachers had a difficulty in assisting students to overcome or reduce their misconceptions in primary mathematics [25,26]. In regard to enactment of mathematics teaching, originally addressed by the TEDS-M study [22], it is believed that researching both components can foster a comprehensive understanding of MPCK among pre-service teachers.

Frameworks for Analyzing MPCK
Many studies have utilized qualitative research methods to uncover the successes and challenges that pre-service and in-service teachers are likely to experience in their teaching practice [27,28]. In terms of qualitative measurement, various frameworks have been developed to unpack the nature of MPCK. All such frameworks can be construed as progressions of the concept of PCK devised by Shulman [19,29].
Rowland and his colleagues in the United Kingdom collected data from classroom observations and proposed a "knowledge quartet" framework for the analysis of mathematics teaching [30][31][32]. Their framework consisted of four categories: foundation, transformation, connection, and contingency. Noting that the second category of the quartet "transformation" echoes Shulman's [19] PCK. Undoubtedly, the observation of actual classroom practice is an ideal data collection method to identify teachers' MPCK. However, this may not be practicable for cross-sectional studies of pre-service teachers with a large sample size.
Chick and her colleagues at the University of Melbourne developed a framework for analyzing Australian primary teachers' MPCK in teaching decimals and subtraction algorithm, respectively [33,34]. They intended to apply their framework to not only data obtained from classroom observations but also information gathered from questionnaires and interviews. Their framework encompassed three categories and 20 sub-categories [33]. The evidence that identifies each sub-category of the framework is provided in Appendix A.
In Chick, Baker et al. [33] three-category MPCK framework (see Appendix A), the first category is "clearly PCK", in which pedagogy and content are thoroughly intertwined. The contents of its 11 sub-categories can be summarized as follows: "knowledge of teaching strategies for mathematics, knowledge of students' conceptions and misconceptions, knowledge of representations in mathematical concepts, and knowledge of resources and curriculum" [35] (pp. 68-69). The second category, namely "content knowledge in a pedagogical context", involves five sub-categories. It covers the profound understanding of fundamental mathematics, ability to deconstruct content knowledge to its key components, awareness of mathematical structure and connections, procedural knowledge, and methods of solutions [33]. The third category of the framework is "pedagogical knowledge in a content context". It consists of four sub-categories referring to "knowledge of learning goals, knowledge of strategies for getting and maintaining student focus, and knowledge of classroom techniques in a specific content area" [35] (p. 69).
The analytical framework proposed by Chick, Baker et al. [33] has been applied by other researchers to investigate pre-service primary teachers' MPCK in teaching various mathematical topics [36,37]. Its explicit descriptions of the components allow researchers to examine qualitative data in detail. Therefore, the current study utilized Chick, Baker et al. [33] framework (see Appendix A) to identify strengths and limitations in the MPCK held by the sampled pre-service primary teachers in Hong Kong.
There were two research questions guiding this study: 1.
What is the nature of Hong Kong pre-service teachers' MPCK in teaching arithmetic word problems at the third-and sixth-grade levels?

2.
To what extent do Hong Kong pre-service teachers recognize and properly comment on primary students' specific mathematical errors?

Materials and Methods
This cross-sectional study was designed to explore pre-service primary teachers' MPCK across the four academic years of a teacher education program in Hong Kong. It aimed to determine how well the participating pre-service teachers had been prepared to teach arithmetic word problems for third and sixth graders. A qualitative research method was employed in the study. A series of two-phase scenario-based interviews was conducted in order to explore the nature of participants' MPCK and knowledge of student errors. The following four sub-sections elaborate in more detail on the methods of data collection.

Participants
Nineteen individual interviews were carried out (with four each from Year 1 to Year 3, and seven from Year 4). Of the 19 interviewees, 14 were female and five were male. These participants had selected mathematics as their major in the undergraduate teacher education program with a concentration in primary education. As the sampled participants were mathematics major students, they had to do 20 mathematics modules throughout the four academic years. Apart from subject matter knowledge, the pedagogy of primary mathematics was also included in their studies. Under the program structure, the pre-service teachers were required to complete particular field experiences and placements: classroom observations during the first year, a 10-day attachment in a local primary school during the second year, as well as six-and eight-week practicums during the third and final years, respectively. It was anticipated that recruiting participants from different year groups would provide rich insights into the research questions.

Interview Protocol
The interviews comprised two sets of mathematics classroom scenarios. Each scenario consisted of two core questions concerning pre-service teachers' MPCK in teaching arithmetic word problems (see Table 1). The interview questions aimed to investigate the types of teaching strategies the pre-service teachers preferred and their knowledge of students' mathematical errors, especially regarding word problems for third and sixth graders. Hong Kong Examinations and Assessment Authority (HKEAA), although the answers are correct in both cases. Such shortcomings in responses to mathematics word problems are culturally referred to as "student errors" in the Hong Kong setting.  [38][39][40]. Two word problems and the answers given by the primary students displayed in Table 1 were shown to the interviewees. Each interviewee was required to describe methods to teach the third and sixth graders how to solve these mathematical application problems, and to suggest approaches that could be used to correct students' mistakes.

Classroom Scenario Interview Question
To determine the usefulness of the interview protocol, the draft interview questions were trialed with one pre-service teacher from another program. After the pilot interview, a follow-up question was added after Question 1b-"If a third grader gives this following answer, what will you do?", and Question 2b-"If a sixth grader gives this following answer, what will you do?" The additional question was considered necessary because the interviewee in the pilot study was able to indicate the issues with the students' answers but he could not clearly express whether the answers were correct or not. Thus, an additional question-"Do you think the answer is correct?" was added in order to clarify interviewees' thinking. The interview was semi-structured in design, allowing the interviewer to add follow-up questions if the need arose.

Procedures and Ethical Considerations
Three documents including information sheets, consent forms, and interview materials were given to the pre-service teachers who agreed to participate in the study and attend an individual interview. All 19 participants were identified through the program director in order to protect their rights to privacy. Each of the semi-structured interviews took about 20 minutes. The interviews were conducted in Cantonese (a mainstream dialect of Hong Kong people) and audio recorded. The interviewees remained anonymous in the recordings. Their personal information only including gender and year of study at the time of the data collection were reported in the research.

Data Analysis
The interview data were thematically analyzed with the assistance of NVivo software package [41]. A content analysis approach was adopted to generate categories in an inductive manner [42]. Hong Kong Examinations and Assessment Authority (HKEAA), although the answers are correct in both cases. Such shortcomings in responses to mathematics word problems are culturally referred to as "student errors" in the Hong Kong setting.  [38][39][40].

Classroom Scenario
Interview Question Two word problems and the answers given by the primary students displayed in Table 1 were shown to the interviewees. Each interviewee was required to describe methods to teach the third and sixth graders how to solve these mathematical application problems, and to suggest approaches that could be used to correct students' mistakes.
To determine the usefulness of the interview protocol, the draft interview questions were trialed with one pre-service teacher from another program. After the pilot interview, a follow-up question was added after Question 1b-"If a third grader gives this following answer, what will you do?", and Question 2b-"If a sixth grader gives this following answer, what will you do?" The additional question was considered necessary because the interviewee in the pilot study was able to indicate the issues with the students' answers but he could not clearly express whether the answers were correct or not. Thus, an additional question-"Do you think the answer is correct?" was added in order to clarify interviewees' thinking. The interview was semi-structured in design, allowing the interviewer to add follow-up questions if the need arose.

Procedures and Ethical Considerations
Three documents including information sheets, consent forms, and interview materials were given to the pre-service teachers who agreed to participate in the study and attend an individual interview. All 19 participants were identified through the program director in order to protect their rights to privacy. Each of the semi-structured interviews took about 20 minutes. The interviews were conducted in Cantonese (a mainstream dialect of Hong Kong people) and audio recorded. The interviewees remained anonymous in the recordings. Their personal information only including gender and year of study at the time of the data collection were reported in the research.

Data Analysis
The interview data were thematically analyzed with the assistance of NVivo software package [41]. A content analysis approach was adopted to generate categories in an inductive manner [42]. These arithmetic word problems were selected from the mathematics papers of a large-scale assessment compulsory for all primary and secondary schools in Hong Kong, namely Territory-wide System Assessment (TSA) [38,39]. This was because the errors made by the primary students for these two multi-step word problems were typical. The errors such as "no descriptive statement, explanations, or conclusions" and "inversion of dividend and divisor" have been particularly discussed in the report of students' mathematics performances on TSA [40], which was published by Hong Kong Examinations and Assessment Authority (HKEAA), although the answers are correct in both cases. Such shortcomings in responses to mathematics word problems are culturally referred to as "student errors" in the Hong Kong setting.
Two word problems and the answers given by the primary students displayed in Table 1 were shown to the interviewees. Each interviewee was required to describe methods to teach the third and sixth graders how to solve these mathematical application problems, and to suggest approaches that could be used to correct students' mistakes.
To determine the usefulness of the interview protocol, the draft interview questions were trialed with one pre-service teacher from another program. After the pilot interview, a follow-up question was added after Question 1b-"If a third grader gives this following answer, what will you do?", and Question 2b-"If a sixth grader gives this following answer, what will you do?" The additional question was considered necessary because the interviewee in the pilot study was able to indicate the issues with the students' answers but he could not clearly express whether the answers were correct or not. Thus, an additional question-"Do you think the answer is correct?" was added in order to clarify interviewees' thinking. The interview was semi-structured in design, allowing the interviewer to add follow-up questions if the need arose.

Procedures and Ethical Considerations
Three documents including information sheets, consent forms, and interview materials were given to the pre-service teachers who agreed to participate in the study and attend an individual interview. All 19 participants were identified through the program director in order to protect their rights to privacy. Each of the semi-structured interviews took about 20 min. The interviews were conducted in Cantonese (a mainstream dialect of Hong Kong people) and audio recorded. The interviewees remained anonymous in the recordings. Their personal information only including gender and year of study at the time of the data collection were reported in the research.

Data Analysis
The interview data were thematically analyzed with the assistance of NVivo software package [41]. A content analysis approach was adopted to generate categories in an inductive manner [42]. Before analyzing the qualitative data, the 19 audio-recorded interviews were transcribed and translated from Chinese to English and entered into NVivo.
In the data analysis, the interview transcripts were coded as a set of nodes under each core interview question. These nodes could exist in different forms such as a word, a phrase, or a sentence representing interviewees' ideas. Hence, a list of sub-categories was generated by grouping nodes which were similar in meaning. Each sub-category was associated with an overarching category for further analysis. Moreover, the analysis of interviewees' MPCK concerning word problems followed the analytical framework developed by Chick, Baker et al. [33]. Therefore, specific categories and sub-categories were used for classifying the nature of their MPCK (see Appendix A).

An Arithmetic Word Problem for Third Graders
Question 1a shown in Table 1 aimed to determine the extent of pre-service teachers' MPCK in teaching a class of third graders. All responses to Question 1a were classified into 10 types of teaching methods. Among the 19 interviewees, 17 provided more than one teaching method. On the basis of the MPCK categories and sub-categories proposed by Chick, Baker et al. [33], the teaching methods were classified into eight sub-categories under three categories. One example for each teaching method is presented in Table 2. In this particular analysis task, Chick, Baker et al. [33] framework for investigating MPCK of Australian primary mathematics teachers fits the teaching strategies suggested by the pre-service primary teachers in Hong Kong. A total of 45 methods for teaching a mathematics word problem in a third-grade classroom were coded. Seventeen teaching methods suggested by 13 pre-service teachers were placed in three sub-categories under the first category "clearly PCK". These methods were referred to as reform-oriented approaches such as role-playing, addressing student thinking, and using appropriate representations of the concept of addition. The content being included in the third-grade curriculum was also mentioned. Twenty-one teaching methods provided by 15 pre-service teachers were classified into three sub-categories under the category of "content knowledge in pedagogical context". The interviewees demonstrated sound procedural knowledge. Other methods covered by the second category included deconstructing content to key components, procedural knowledge, and methods of solutions. In the category of "pedagogical knowledge in content context", seven teaching methods given by seven pre-service teachers were grouped into this sub-category. These methods included questioning and group discussion to maintain students' focus.
The second question of the first classroom scenario required interviewees to make comments on the work of a third grader in solving the word problem (see Table 1, Question 1b). Six sub-categories emerged from an analysis of the interview data. Responses from each interviewee were grouped into more than one sub-category. These sub-categories were further classified into two categories. The categories were labeled as "comments on the student's presentation" and "comments on the student's working steps". Table 3 presents the categories and shows a sample response for each sub-category. Table 3. Examples for interviewees' comments on a third grader's work.

Sub-Category 1 Selected Response of Interviewees
Comments on the student's presentation The student should use "88 + (10 × 4)" for solving the problem so that he or she can get the answer quickly. (Year 4, male) 1 Numbers in brackets indicate interviewees whose responses were included in this sub-category.
As shown in Table 3, the first category "comments on student's presentation" involved three sub-categories. According to the HKEAA [40], the solution written by third graders was assessed to be "no descriptive statements, explanations, or conclusions" (p. 249). Of the 19 pre-service teachers, nine made a similar comment such as "no descriptions" and "missing a descriptive statement". Their comments were identified as belonging to the first sub-category-no descriptive statements. Other comments indicating the existence of problems with an aspect of presentation but not specifying a lack of descriptive statements, such as "missing a dollar sign" (Year 1, female) and "an unclear solution" (Year 4, female) were classified as belonging to the second and the third sub-categories-missing a unit of account and unclear presentation.
The second category "comments on the student's working steps" covered three sub-categories: incorrect use of brackets, correct solutions/answer, and slow methods for getting the answer. There were 20 responses in this category while some of them were problematic in terms of mathematics teaching. Examples of problematic comments are as follows: The solution seems to be too long. (Year 2, female) The bracket should be omitted. (Year 4, female) Even though these comments were unclear as to the intention of the approach, others were reasonable as long as a logical explanation was provided. For instance, the following comment was identified as belonging to the sub-category-incorrect use of brackets, but the interviewee provided a rationale: When the student used a bracket in the solution, the calculation inside the bracket should be done first but the student did not. The working steps on the right-and left-hand side did not match each other, so I can see this student could not correctly use the bracket. (Year 4, female) In short, fewer than half of the interviewees could correct the third grader's error. Most of the pre-service teachers paid attention to how the student worked out the solutions but were not concerned if the student could use appropriate mathematical language to illustrate the solution. The significance of this observation is that the pre-service teachers might embrace more conventional beliefs about mathematics learning with the focus on getting right answers. They were unaware of missing a description statement in the solution because they accepted the student performing procedures and finding answers without understanding what they meant.

An Arithmetic Word Problem for Sixth Graders
The responses to Question 2a were identified as illustrating eight types of teaching strategies. Among the 19 interviewees, 15 suggested more than one teaching method. According to the Chick, Baker et al. [33] MPCK categories framework, these teaching strategies were classified into six sub-categories under three categories. An example for each teaching strategy is provided in Table 4. Table 4. Examples for interviewees' MPCK in teaching a word problem for a class of sixth graders. The teaching methods suggested by the pre-service teachers match Chick, Baker et al. [33] PCK framework. A total of 36 methods for teaching an arithmetic word problem in a sixth-grade classroom were coded. Five teaching methods provided by four pre-service teachers were coded into three sub-categories under the first category "clearly PCK". These methods included teaching with examples, preparing worksheets, addressing student thinking, and using appropriate representations of the concept of average. Nineteen teaching methods suggested by 16 pre-service teachers were classified as three sub-categories under the second category "content knowledge in a pedagogical context". These methods were referred to as pre-service teachers' profound understanding of fundamental mathematics and their knowledge of methods of solution. In the category of "pedagogical knowledge in a content context", interview data showed that 12 pre-service teachers suggested teaching by questioning and arranging group discussions in order to engage sixth graders in mathematics learning.
The second question of the second classroom scenario required interviewees to make comments on the work of a sixth grader for solving the word problem (see Table 1, Question 2b). Six sub-categories emerged from an analysis of the interview data. The responses from 10 interviewees were grouped into more than one sub-category. These sub-categories were classified into two categories. The categories were labeled "comments on the student's mathematical concepts" and "comments on the student's working steps". Table 5 presents the categories and shows a sample response for each sub-category.  The student should omit the bracket. It is not necessary.
(Year 4, female) 1 Numbers in brackets indicate interviewees whose responses were included in this sub-category.
In Table 5, the first category "comments on the student's mathematical concepts" involved three sub-categories. According to the HKEAA [40], the solution written by the sixth grader was evaluated as "inversion of dividend and divisor" (p. 269). Among the 19 pre-service teachers, three were able to indicate this specific mathematical error. Their comments were identified as the first sub-category-inversion of dividend and divisor. Two pre-service teachers' comments referring to the student's general thinking ability were identified as the second sub-category-problematic student thinking.
The comments relevant to general mathematical concepts were grouped into the third sub-category-misconceptions of division/average. Twelve pre-service teachers commented on the student's misunderstanding of concepts in division or average. However, half of them were not able to use appropriate mathematical language such as "dividend" and "divisor" to illustrate the student's error. For instance, Two sets of numbers "48 × 6" and "8" were inversed. (Year 4, female) Eight people cannot be divided by "48 × 6". (Year 4, female) Apart from the conceptual errors in numbers, many interviewees were concerned with the student's correct and incorrect steps for solving the word problem. Their comments were classified as the second category "comments on the student's working steps". It covered three sub-categories including incorrect methods of solution, correct answer, and incorrect use of brackets. It should be noted that nine pre-service teachers' responses were classified as both first and second categories. For example, the following response was classified as belonging to four sub-categories under two different categories: The student had a misunderstanding of concept of average. The numbers were inversed. The error in solution tells me his way of thinking was problematic. [ . . . ] He could find out a right answer but his method of solution was absolutely wrong. (Year 2, male) This interviewee made comments on the student's mathematical concepts and presentation of the solution. He could only point out that the "numbers" were inversed but did not clearly mention inversion of the dividend and divisor so that might confuse the student. Apparently, the interviewee was not able to address this student's error by using appropriate mathematical terms.
These findings showed that the pre-service teachers knew about the sixth grader's mathematical misconceptions. However, they seemed to have difficulties applying the correct mathematical terminology. Due to a lack of mathematical language for division, most of the pre-service teachers were not able to accurately specify the student's errors.

Discussion and Conclusions
These findings contributed to answering the research questions regarding the nature of pre-service teachers' MPCK and their knowledge of student errors. The results indicated that there was a significant difference of pre-service teachers' MPCK between third-and sixth-grade mathematics, although it could be due to differences in the items as much as the grades. For teaching lower-grade students in solving arithmetic problems, a variety of reform-oriented approaches was suggested. Different kinds of learning activities which aimed to motivate lower grade students to learn were involved in their teaching suggestions which are in accordance with the reform curriculum aimed at promoting student-centered approaches.
On the other hand, the strategies suggested for teaching students in upper grades were mostly categorized as "content knowledge in a pedagogical context". The majority of interviewees focused on demonstrating the fundamental mathematical concept, as well as the strategies for finding the correct answer. The findings also indicated that most of the pre-service teachers were not able to precisely address the lower grade student's error in presenting a complete solution for a typical mathematics word problem. However, many interviewees demonstrated concrete knowledge about the upper grade student's mathematical misconceptions even though only a few could specify the student's errors. Some interviewees were also able to discuss what caused the sixth grader's misunderstanding.
Furthermore, interview data also indicated that few pre-service teachers demonstrated sound knowledge of student errors. Of the 19 interviewees, 10 were not able to identify the third grader's error in setting out. They did not identify a missing descriptive statement in the student's work. Only three interviewees could specifically indicate a sixth grader's mathematical error. In fact, pre-service primary teachers have been criticized for a lack of conceptual knowledge in previous studies [43][44][45]. Prior research has documented that pre-service primary teachers use procedural knowledge rather than conceptual knowledge and are weak at dealing with student misconceptions in primary mathematics [25,46,47]. These findings are consistent with the interview results in the current study, showing that some interviewees were not able to use their conceptual knowledge to comment on a student's misconceptions of average.
In conclusion, a profound understanding of elementary number theory should be addressed in teacher education. It could be one of preconditions for the development of pre-service teachers' MPCK in teaching arithmetic word problems at the primary school level. In order to further develop their MPCK, more reflection activities should be provided for pre-service teachers before and after the teaching practicum [48] because reflection can improve their teaching skills and pedagogical understanding [49].