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Abstract

Problem posing has long been recognized as a critically important teaching method and goal in the area of mathematics education. However, few studies have used problem posing to assess in-service teachers’ mathematical understanding. The present study investigated in-service teachers’ mathematical understanding of fraction division, which is often considered challenging content in elementary school, from three angles: computation, drawing, and problem posing. Two studies involving 66 and 193 primary and middle school teachers were conducted to reveal the in-service teachers’ mathematical understanding and whether drawing and problem posing affected each other. Although the in-service teachers rarely had the opportunity to pose mathematical problems in their daily teaching, they were able to pose mathematical problems in this study. In addition, problem-posing tasks were more useful in diagnosing the in-service teachers’ conceptual understanding than were computation or drawing. Thus, problem posing seems to have contributed to their conceptual understanding of fraction division on the drawing task.

Keywords

problem posing conceptual understanding drawing in-service mathematics teachers

Teachers’ mathematical understanding directly affects students’ mathematics learning (Campbell et al., 2014). Although problem solving is an important means to evaluate teachers’ mathematical understanding and a method used in many research studies, an individual's mathematical understanding need not solely be assessed from problem solving. Problem posing has been increasingly adopted in the mathematics research field as a different way to assess individuals’ mathematical understanding. Indeed, researchers have found that problem posing can be an effective way to assess individuals’ mathematical understanding and it can even be used to foster individuals’ mathematical understanding (Cai et al., 2020; Silver, 1994; Singer et al., 2017).

Although researchers have begun to consider the function of problem-posing assessment, research in this area is still lacking. Therefore, the purpose of the present study was to examine in-service elementary and middle school mathematics teachers’ understanding of fraction division, which is commonly considered a difficult subject for primary school students, by using computation, drawing, and problem-posing tasks. We also explored the relationship between in-service teachers’ understanding of mathematical problem solving and their responses to drawing and problem-posing tasks.

1. Literature review

1.1 Conceptual and procedural understanding in mathematics

Mathematical understanding has been extensively studied since at least Brownell and Chazal (1935; see also Cai & Ding, 2017; Fariana, 2017). Although an abundance of theories and research findings exists on the meaning of mathematical understanding, most researchers agree that it generally includes two components: procedural understanding and conceptual understanding (e.g., Hiebert & Carpenter, 1992; Rittle-Johnson et al., 2015; Skemp, 1978).

According to the literature review of Osana and Pelczer (2015), the definitions of and distinctions between procedural and conceptual understanding of mathematics remain unclear and have been defined and measured differently by different researchers (Newton, 2008). Some have defined procedural understanding and conceptual understanding as related but with a significant difference: Conceptual understanding refers to understanding the relationships among concepts which are highly integrated (Baroody et al., 2007; Hiebert & Carpenter, 1992) whereas procedural understanding has been characterized as the “knowing how” to do something or “goal-directed action-sequences” (Byrnes, 1992). Specifically, learners who have conceptual understanding of arithmetic exhibit understanding of the means of and reasons for algorithms. Learners with procedural understanding can execute algorithms or rules linearly but only consider the procedures independent of meaning, meaning that a learner using a procedure does not need to reflect on what the elements implemented in the procedure mean (Hallett et al., 2012).

Researchers in mathematics cognition have long attempted to understand how individuals use their conceptual and procedural understanding in answering mathematics questions. Although studies have looked at the order in which these two understandings arise and the relationship between them, it has been difficult to make a definitive decision (Hallett et al., 2012; Vamvakoussi et al., 2019). For example, some studies have found that the distinction between conceptual and procedural understanding seems particularly sharp when applied to learners’ performance on fraction division (Lenz et al., 2019; Yao et al., 2021). Researchers (e.g., Hallett et al., 2012; Lenz et al., 2019; Rittle-Johnson & Schneider, 2015) have claimed that these findings can be explained by considering individual differences in the way that learners combine conceptual and procedural understanding. They have also proposed that constructing appropriate measures or scales would help to answer the questions with which previous studies have struggled.

In the present study, we aim to reveal the relationship between the conceptual and procedural understanding of in-service mathematics teachers using problem solving and problem posing, which are vital pedagogical goals for students.

1.2 Investigating learners’ conceptual understanding using problem solving

Mathematical problem solving is not just a goal of mathematics education but a way to learn and understand mathematics (Cai, 2003). Students are encouraged to explore and learn through problem solving, which involves the processes of using representations, communication, reasoning, and connections (National Council of Teachers of Mathematics [NCTM], 2000). In fact, school students are required to understand mathematics, be proficient at using various mathematical representations, and have problem-solving skills. A sound understanding of fraction division also requires working with representations of the meanings (Toluk-Uçar, 2009). Graphical representation, as an external representation and a form of problem solving, not only helps to represent but also to promote learners’ mathematical understanding (Son & Lee, 2016; Stylianou, 2011).

However, Toluk-Uçar (2009) reported that learning and thinking about fractions and their arithmetic meanings is constrained by the identification of the unit and the process of unitizing as well as by the symbolic or graphical representations of the meanings. It can be speculated that the representation of the meaning of fraction division of different denominators may present an even greater challenge to teachers. In this study, visual representations, what we call “drawing tasks,” were used not only to assess in-service teachers’ conceptual understanding of fraction division but also to examine the interactions between problem-solving tasks and problem-posing tasks in assessing teachers’ understanding.

1.3 Investigating learners’ conceptual understanding using problem posing

Problem posing has long been recognized as a critically important intellectual activity in scientific investigation (Einstein & Infeld, 1938), in research on reading (Rosenshine et al., 1996), and in research on mathematics education (Cai et al., 2015; Silver, 1994; Singer et al., 2013). Educators and researchers have become increasingly interested in problem posing (Brown & Walter, 1983; Cai et al., 2015; Kilpatrick, 1987; Silver, 1994; Singer et al., 2013) and previous studies have shown that mathematical problem posing can be used to make sense of how students understand mathematics (Cai et al., 2015; Harel et al., 2006; Kotsopoulos & Cordy, 2009; Silver, 1994).

However, Isik and Kar (2012) found that teachers’ problem posing can be hindered by inadequate conceptual understanding. Thus, some researchers have attempted to use problem posing to gain insight into students’ and teachers’ mathematical understanding. For example, Tichá and Hošpesová (2013) assessed preservice teachers’ understanding of fractions using problem posing and found that they exhibited some misunderstandings. Moreover, Yao et al. (2021) and Toluk-Uçar (2009) attempted to assess preservice primary school mathematics teachers’ understanding of the operational principle and fractions through problem posing, showing that using problem posing in the classroom could accurately reveal students’ thinking. Kwek (2015) asserted that mathematical complexity is an important attribute of the posed problem and reflects students’ mathematical understandings and cognitive processes. This indicates that problem posing can serve as a window into students’ thinking when learning mathematics (Silver, 1994).

1.4 Fractions and fraction division

The field of mathematics education has long emphasized teaching for students’ understanding (Brownell & Chazal, 1935). As NCTM (2014) stated, “effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency” (p. 42). There is no doubt that content knowledge is a crucial component of what mathematics teachers need to know to achieve this educational goal (Siegler & Lortie-Forgues, 2015). As an important content area and one that students often struggle with in the process of learning, fractions and fraction arithmetic have received increasing attention from mathematics education researchers.

Fractions take on different meanings across different contexts (Behr et al., 1992; Kieren, 1993). Correspondingly, there are also different types of understanding and interpretations regarding fraction arithmetic. Fraction division with different denominators involves different units for the divisor and dividend, which requires both conceptual and procedural understanding. Conceptual understanding in this context includes the equal sharing (or partitive) and equal grouping (or quotative) concepts. Equal sharing refers to dividing a quantity (the dividend) into a given number (the divisor) of equal parts. Equal grouping refers to how many groups of a given size (the divisor) there are in the dividend. In addition, procedural understanding in this context includes transposing the numerator and the denominator of the divisor and then multiplying the dividend by the transposed divisor (typically referred to as “invert and multiply”).

Fraction arithmetic is crucial for later mathematics achievement and the ability to succeed in many professions (Booth et al., 2014). Unfortunately, understanding fraction arithmetic represents a major hurdle for many children as well as adults (Lortie-Forgues et al., 2015). Ever since Carpenter et al. (1980) found that only 24% of 20,000 U.S. eighth-grade students chose the correct answer to the sum of 12/13 + 7/8 in 1978, many efforts have been made to improve mathematics education. For example, national organizations of education (e.g., Ministry of Education of China, 2022; NCTM, 2007), widely adopted textbooks (e.g., Everyday Mathematics; Jia & Yao, 2021), and innumerable research articles (e.g., Cai & Hwang, 2020; Lortie-Forgues et al., 2015) have advocated greater emphasis on conceptual understanding, especially conceptual understanding of fractions.

Although studies focusing on elementary school mathematics teachers’ content knowledge have found that these teachers have insufficient knowledge of fractions (Copur-Gencturk, 2021; Lortie-Forgues et al., 2015; Toluk-Uçar, 2009; Tröbst et al., 2018), some studies reported that Chinese teachers, especially expert teachers, typically have a better mathematical understanding of fractions compared to teachers in other countries (An et al., 2004; Cai & Ding, 2017). However, in general, the results from previous studies involving elementary teachers in China do not seem to support the hypothesis that elementary teachers in China have a strong preparation in mathematical understanding of fractions and fraction division (e.g., Li & Huang, 2008; Kang & Liu, 2018).

Given that most previous studies adopted problem-solving tasks to examine teachers’ mathematical understanding of fractions and fraction division, this study aims to investigate in-service teachers’ mathematical understanding of fraction operations exhibited on drawing and problem-posing tasks. The in-service teachers’ mathematical understanding of fraction division will be presented in detail in the method section of Study 1. In Study 1, three types of tasks were designed to investigate in-service teachers’ computational ability at and understanding of fraction division. To reveal the possible influence of task order on in-service teachers’ understanding, we conducted Study 2, which also included three types of tasks but with a different order and prompt compared to the tasks in Study 1.

2. Study 1

2.1 Method

2.1.1 Participants

This study involved 66 participants consisting of 52 elementary school and 14 middle school mathematics teachers who were members of a workshop, Using Problem Posing to Learn and Teach Mathematics, in a province in eastern China. The data were collected before the problem-posing workshop. We combined the data of the two sets of participants for our analysis because there was no difference in the performance on all the tasks between the elementary and middle school teachers.

2.1.2 Instrument

The instrument consisted of three tasks: a fraction division computation task, a graphical (drawing) task, and a problem-posing task (see the Appendix).

In this study, two different fraction division expressions were used ( $1 \frac{3}{4} \div \frac{5}{8} =$ and $1 \frac{3}{4} \div \frac{1}{2} =$ ) to prevent in-service teachers from working on the drawing task according to how they calculated the computation task.

2.1.3 Data coding and inter-rater reliability

For the computation task, we coded the answer as correct or incorrect. For the drawing and problem-posing tasks, we used three codes: no understanding, procedural understanding, and conceptual understanding.

The codes indicate the type of understanding of fraction division that was indicated by the drawing or posed problem (see Table 1). Drawings that illustrated the operation of doubling 1¾ (either through addition or multiplication) and posed problems that characterized the situation as the same operation, meaning that the participants considered it from the perspective of a fraction arithmetic procedure, were coded as procedural; drawings that illustrated the meaning of the expression of fraction division and posed problems that included an interpretation of the division in terms of equal grouping (how many ½ are in 1¾), meaning that the participants made a connection to the meaning of the fraction operation, were coded as conceptual.

Table 1.
Codes for mathematical understanding on the drawing and problem-posing tasks.

Mathematical understanding Drawing task Examples Problem-posing task Examples

No Understanding Participants didn’t draw anything or drew an incorrect graph. N/A Participants didn’t pose any problem or posed an incorrect problem.
Two places A and B are 1¾ km apart. Two construction teams M and N build at the same speed from A and B separately. How many kilometers will team M finally complete?

A group of children are giving out apples. Each time, they give out four parts out of seven from that group. How many apples did they give out after two times?

Procedural Understanding Participants primarily illustrated the operation of doubling 1¾ (either through addition or multiplication).
Participants posed a problem that primarily characterized the situation as doubling 1¾, thereby considering it from the perspective of a fraction arithmetic procedure.
Zhang has 1¾ kilos of apples, Lee said he has twice as many apples as Zhang has. So how many kilos of apples does Lee have? [Students clearly converted the division problem into a multiplication problem.]

Conceptual Understanding Participants illustrated the division as forming groups of ½ from 1¾ (equal grouping).
Participants posed a problem that included an interpretation of the division in terms of equal grouping, thereby making a connection to the meaning of the fraction operation.
How many ½'s are there in 1¾? [This is a typical equal-grouping problem and, thus, is a conceptual understanding problem.]

Given two numbers 1¾ and ½, how many times larger is the larger number as compared to the smaller one? [This relates to both the relationship between multiplication and factoring and the meaning behind equal grouping. The latter can be obtained by interpreting the question as a division problem and the former by interpreting the magnitude of the relationship between 1¾ and ½.]

Two mathematics education researchers who are not authors of this paper were trained to code the tests through three steps: (a) They were introduced to the definitions of conceptual and procedural understanding as defined with respect to the drawing and problem-posing tasks for the given fraction division; (b) they coded 20 tests separately and discussed differences in their coding to establish consensus and ensure coding consistency; and (c) they coded all the tests separately. We used this process in Study 2 as well. The values of Cohen's Kappa for the two raters on the drawing task and problem-posing task were .92 and .90, respectively.
2.2 Results

Mathematical understanding	Drawing task	Examples	Problem-posing task	Examples
No Understanding	Participants didn’t draw anything or drew an incorrect graph.	N/A	Participants didn’t pose any problem or posed an incorrect problem.	Two places A and B are 1¾ km apart. Two construction teams M and N build at the same speed from A and B separately. How many kilometers will team M finally complete? A group of children are giving out apples. Each time, they give out four parts out of seven from that group. How many apples did they give out after two times?
Procedural Understanding	Participants primarily illustrated the operation of doubling 1¾ (either through addition or multiplication).		Participants posed a problem that primarily characterized the situation as doubling 1¾, thereby considering it from the perspective of a fraction arithmetic procedure.	Zhang has 1¾ kilos of apples, Lee said he has twice as many apples as Zhang has. So how many kilos of apples does Lee have? [Students clearly converted the division problem into a multiplication problem.]
Conceptual Understanding	Participants illustrated the division as forming groups of ½ from 1¾ (equal grouping).		Participants posed a problem that included an interpretation of the division in terms of equal grouping, thereby making a connection to the meaning of the fraction operation.	How many ½'s are there in 1¾? [This is a typical equal-grouping problem and, thus, is a conceptual understanding problem.] Given two numbers 1¾ and ½, how many times larger is the larger number as compared to the smaller one? [This relates to both the relationship between multiplication and factoring and the meaning behind equal grouping. The latter can be obtained by interpreting the question as a division problem and the former by interpreting the magnitude of the relationship between 1¾ and ½.]

We begin by noting that 100% of the in-service teachers were able to correctly complete the computation task, indicating that they could handle fraction division operations very well. The 66 participants provided 66 drawings and posed 132 problems. Figure 1 shows the in-service teachers’ mathematical understanding through problem solving (drawing). Over one half (54%) of the in-service teachers exhibited a procedural understanding of fraction division on the drawing task, whereas only 29% exhibited a conceptual understanding on this task. Figure 1 also shows the teachers’ mathematical understanding on their first and second posed problems. For the two problems, respectively, 82% and 88% of the in-service teachers exhibited a conceptual understanding, 9% and 3% of them exhibited a procedural understanding, and 9% and 9% of them exhibited no understanding of fraction division.

Figure 1.

In-Service teachers’ mathematical understanding on the drawing task and the two posed problems in study 1.

The results of a Chi-square analysis showed that there was no difference between the two problems the in-service teachers posed on the problem-posing task ( $χ^{2}$ = 3.21, p > .025)¹. Thus, either of them could be used to represent the in-service teachers’ mathematical understanding via problem posing. In comparing the in-service teachers’ mathematical understanding on the drawing and the first problem that they posed, we found a significant difference between the in-service teachers’ mathematical understanding of fraction division on the drawing task and the problem-posing task ( $χ^{2}$ = 59.91, p < .001). More specifically, the percentage of in-service teachers who showed a conceptual understanding on the problem-posing task was significantly higher than the percentage who did so on the drawing task (82% vs. 29%); the percentage of in-service teachers who displayed a procedural understanding on the drawing task was significantly higher than that for those who displayed a procedural understanding on the problem-posing task (54% vs. 9%). In sum, there was a huge difference in the mathematical understanding of the participants as assessed by their problem posing and drawing.

To further understand the different functions of the two methods in assessing in-service teachers’ mathematical understanding of fraction division, we examined the mathematical understanding on the drawing task of in-service teachers who exhibited a conceptual understanding of fraction division on the problem-posing task (Table 2). There were 54 and 39, respectively, in-service teachers who exhibited a conceptual understanding on the first and second problems they posed; however, only 20 and 13 of them, respectively, demonstrated conceptual understanding on the drawing task. It is notable that, even in the subset of in-service teachers whose drawing did not provide evidence of understanding, many of them were still able to pose problems reflecting conceptual understanding of fraction division.

Table 2.

Mathematical understanding on the drawing task of in-service teachers who exhibited conceptual understanding on the problem-posing task.

Drawing	No understanding	Procedural understanding	Conceptual understanding
Posed conceptual understanding problem
First posed problem (n = 54)	5 (9%)	29 (54%)	20 (37%)
Second posed problem (n = 39)	5 (13%)	21 (54%)	13 (33%)

However, whereas the teachers were asked to draw one graph to show fraction division, they were asked to pose two mathematical problems based on fraction division. Is it possible that more teachers would have exhibited conceptual understanding on a second drawing if they had been asked to produce two drawings? Furthermore, the task order advantage warrants consideration, whereby drawing first could have affected the teachers’ problem posing. Thus, to further clarify whether in-service teachers did indeed exhibit better conceptual understanding on the problem-posing task than on the drawing task, we designed a second study using two versions of the test, Version 1 with two drawings and Version 2 with different task orders (problem-posing task first and drawing second) to examine the in-service teachers’ mathematical understanding on the drawing and problem-posing tasks.

3. Study 2

3.1 Method

3.1.1 Instruments

Two versions of the test were adopted in Study 2 to reveal whether making two drawings would lead to more conceptual understanding of fraction division than in Study 1, which we expected to be consistent with their performance on the problem-posing task, and whether problem posing would influence the participants’ mathematical understanding on the drawing task. Version 1 of the test in Study 2 also had three items, two of which are identical to those on the test in Study 1: The first item involved division computation (1¾ ÷ 1/2) and the third item involved posing two different mathematical problems for a given expression. The only difference between Version 1 of the test in Study 2 and the test of Study 1 was the second item: The test used in Study 1 asked in-service teachers to draw a graph to represent the meaning of the given expression, and Version 1 of the test used in Study 2 asked the in-service teachers to draw two graphs to represent the meaning of the given expression. In addition, Version 2 of the test in this study also had three items identical to those in Version 1. The only difference between the two versions was that we switched the order of the last two items in Version 2 (see the Appendix).

The purpose of designing two versions of the test was to determine whether one more opportunity to draw would increase the participants’ conceptual understanding on the fraction division task and whether the problem-posing task would affect the participants’ mathematical understanding on the drawing task.

3.1.2 Participants

Study 2 involved a total of 193 elementary school in-service teachers from another province in eastern China different than the one in which we collected data in Study 1; specifically, the number of in-service teachers who took Version 1 of the test was 115 and the number of in-service teachers who took Version 2 of the test was 78. The two provinces we chose to collect data from in Studies 1 and 2 have similar levels of economic development and education.

3.1.3 Data coding and inter-rater reliability

The data were collected in Study 2 using the same method used in Study 1. The two mathematics education researchers who coded all the tests of Study 2 were the same as those in Study 1, and the values of Cohen's Kappa for the two raters on the drawing task and problem-posing task were .90 and .91, respectively.

3.2 Results

About 99% of the in-service teachers in Study 2 were able to correctly complete the computation task of fraction division, which is almost the same as the results of Study 1.

Figure 2 shows the in-service teachers’ mathematical understanding on the drawing and problem-posing tasks. Compared to the number of in-service teachers who exhibited conceptual understanding on the drawing task in Study 1 (29%), which involved only one drawing opportunity, there was a slightly lower percentage of in-service teachers who exhibited a conceptual understanding on the drawing task used in Version 1 of the test in Study 2, which requested two drawings (first drawing: 12%; second drawing: 17%; see Figure 2). Therefore, we cannot say definitively that the more opportunities to draw they have, the greater the conceptual understanding the in-service teachers might have. There was also no significant difference in the in-service teachers’ mathematical understanding on the drawing task used in Version 1 of the test in Study 2, which required them to make two drawings (χ²= 2.60, p = .27 > .05).² To confirm this conclusion more rigorously, we compared the participants’ mathematical understanding on the drawing task in Study 1 with the mathematical understanding exhibited on the second drawing on Version 1 of the test in Study 2. The results of a Chi-square analysis showed that there was also no significant difference for this comparison (χ²= 6.51, p = .04 > .0083).

Figure 2.

In-Service teachers’ mathematical understanding on the two drawings and the two posed problems in study 2.

Similarly, the results of a Chi-square analysis showed that the in-service teachers’ mathematical understandings on the problem-posing task on Versions 1 and 2 of the test were not significantly different (χ² = 2.97, p = .23 > .0083; χ² = 0.31, p = .86 > .0083). These results imply that in-service teachers with more opportunities to draw or pose problems do not seem to demonstrate better conceptual understanding of fraction division on drawing and problem-posing tasks. Thus, the best mathematical understanding that each in-service teacher had on the two tasks was then considered, given that they had two opportunities to draw and pose problems on the two versions of the test (see Figure 3).

Figure 3.

In-Service teachers’ best mathematical understanding on the drawing and problem-posing tasks in study 2.

Although the in-service teachers had two opportunities to draw pictures to demonstrate their mathematical understanding of fraction division on both Versions 1 and 2 of the test, we also found that there were more teachers who exhibited conceptual understanding on the problem-posing task than on the drawing task (69% vs. 21% on Version 1; 67% vs. 29% on Version 2). Meanwhile, they exhibited greater procedural understanding on the drawing task than on the problem-posing task (61% vs. 15% on Version 1; 59% vs. 26% on Version 2). A similar proportion exhibited no understanding of fraction division on the drawing and problem-posing tasks (17% vs. 18% on Version 1; 12% vs. 8% on Version 2). The results of a Chi-square analysis showed that the in-service teachers’ mathematical understanding on the problem-posing task was significantly greater than that on the drawing task, which is consistent with the results obtained in Study 1 (Version 1: χ² = 60.9, p < .001; Version 2: χ² = 22.06, p < .001). In addition, around 15% of the in-service teachers could only pose one problem on both Versions 1 and 2 of the test.

Meanwhile, to determine whether there was a task order advantage, we compared the in-service teachers’ mathematical understanding on the drawing task on Versions 1 and 2 of the test. The results of a Chi-square analysis didn’t show a significant difference in mathematical understanding of fraction division on the drawing task between the in-service teachers who took Version 1 and Version 2 of the test; however, we still found that the in-service teachers who took Version 2 of the test, which required them to draw after problem posing, had a greater conceptual understanding than those who took Version 1 of the test, which required them to draw before problem posing (29% vs. 21%). Moreover, considering Figure 2, the percentages of in-service teachers who exhibited no understanding on both drawings in Version 1 were higher than for those who took Version 2 of the test (25% & 31% vs. 14% & 24%). Therefore, future studies will need to verify the effect of problem posing on drawing or other kinds of problem-solving approaches.

4. Conclusion and discussion

In this study, we investigated in-service teachers’ mathematical understanding of fraction division from three angles: computation, drawing, and problem posing. It is clear that the teachers in this study could compute fraction division quite accurately. However, there was still a large number of them who failed to display conceptual understanding of fraction division, which supports the results of many previous studies showing the difficulty of developing learners’ conceptual understanding of fraction division (Castro-Rodríguez et al., 2016; Lortie-Forgues et al., 2015; Yao et al., 2021). In fact, the authors of previous studies have revealed that one of the most important reasons that students struggle with understanding fractions and their operations is because their teacher might have a weak understanding of fractions as well, whether it is in the United States or China (Lortie-Forgues et al., 2015; Ma, 1999; Siegler & Lortie-Forgues, 2015; Simon et al., 2018; Tichá & Hošpesová, 2013). We not only confirmed the conclusions of previous studies but also found that the in-service teachers in this study lacked a conceptual understanding of fraction division but had good computational skills and procedural understanding.

Based on the results of the two studies, we found that problem posing can be an additional avenue to reveal in-service teachers’ mathematical understanding in addition to problem solving and graphical representation. This finding is consistent with findings of several previous studies that have viewed problem posing as a window into students’ thinking while they learn mathematics and proposed it as a method that could help teachers understand and assess students’ understanding of mathematics (Buchholtz et al., 2013; Cai et al., 2013; Silver, 1994). On the computation task, although all of the in-service teachers were able to compute the fraction division expression correctly, about one half of them did not exhibit a conceptual understanding of fraction division, and around 8–31% of them, respectively for Studies 1 and 2, even exhibited no understanding of fraction division on the drawing task. On the drawing task, which is a form of problem solving, the in-service teachers were only able to solve the given problems regardless of the situations they referred to or their difficulty levels. Thus, the more problem-posing opportunities that an educator provides to learners, the more possibilities the learners will have to understand them.

Ma (1999) claimed that a teacher's ability to pose fraction division problems was strongly associated with their understanding of the meaning of fraction division; thus, we must determine whether teachers and students are capable of posing important and worthwhile mathematical problems. Although most of the in-service teachers in this study could pose at least one problem, it was difficult for them to pose two different types of problems. Several empirical studies have also shown that even though students and teachers are capable of posing interesting and important mathematical problems, some students and teachers pose nonmathematical problems, unsolvable problems, and irrelevant problems (e.g., Cai & Hwang, 2020; Silver & Cai, 1996). Therefore, the quality and quantity of the problems posed by teachers, which not only represent their mathematical understanding but also their creative thinking (Silver & Cai, 2005), need improvement.

In addition, some previous studies have shown that graphical representations, as external representations, not only help to represent learners’ mathematical understanding but also to promote their mathematical understanding (e.g., Goldin, 2003; Lesh et al., 1987; Stylianou et al., 1999). Son and Lee (2016) found that most of the preservice teachers in their study could explain their thinking about fractions using graphical representations. In the current study, however, we not only found that the in-service teachers rarely demonstrated conceptual understanding via drawing, which is consistent with the results of the study by Isik et al. (2011), but we also found that providing one additional drawing opportunity did not lead to more in-service teachers having a conceptual understanding of fraction division on the drawing task in Study 2. As Toluk-Uçar (2009) explained, learning and thinking about fractions and their arithmetic meanings are constrained by the identification of the unit and the process of unitizing as well as by the symbolic or graphical representations of those meanings. Therefore, although some studies have shown that external graphical representations influence and reflect the internal representations of the mathematical knowledge possessed by learners, drawing might not be suitable for all learners or situations.

As a form of problem solving, drawing seems not to have contributed to the in-service teachers’ mathematical understanding of fraction division on the problem-posing task. Although previous research has shown that students’ mathematical problem-solving abilities are related to their problem-posing abilities (Cai & Hwang, 2002; Silver, 2013), drawing, as a graphical presentation and a special form of problem solving, creates challenges to represent in-service teachers’ mathematical understanding of fraction division, a finding which validates the results of the study by Toluk-Uçar (2009). Graphical representations, as a common teaching method used by teachers in classrooms, are helpful for learners to understand mathematical concepts, but an issue that deserves further investigation is determining why it might be hard for some in-service teachers to represent their mathematical understanding through drawing. Thus, further studies are needed to reveal the role that graphical representation plays in learners’ mathematical understanding and how educators can use it to teach mathematics.

In this study, we showed how problem posing is not only helpful in accurately assessing in-service teachers’ mathematical understanding but also in promoting their problem-solving performance. Graphical representations, oppositely, did not have the same influences on the in-service teachers’ conceptual understanding. Therefore, it can be speculated that problem posing can affect in-service teachers’ mathematical understanding of fraction division on drawing but not vice versa. This study reported an initial result on the relationship between drawing and problem posing and an insight into the relationship between problem posing and problem solving. Future studies should examine the interaction between problem posing and other forms of problem solving.

4.1 Limitations and future studies

The present study has several limitations, each of which suggests directions for future research. One limitation is that our study only involved one type of situation for the problem, with the computation first, drawing task second, and the problem-posing task third, to examine teachers’ mathematical understanding. Using only one type of situation might make it difficult to represent a teacher's mathematical understanding. In addition, although we found that the problem-posing task increased the in-service teachers’ conceptual understanding of fraction division on the drawing task, we do not know how problem posing interacts with drawing or other forms of problem solving. To understand the interaction between problem posing and problem solving, more types of problem situations and different orders of tasks on different tests should be used to investigate learners’ mathematical understanding and examine the relationship between problem posing and problem solving.

In addition, although we know that problem posing can play an important role in supplying different opportunities to reveal in-service teachers’ mathematics understanding, there is much work to be done to better understand what problem posing looks like in various contextualized situations, how we can improve problem-posing competencies in these contexts, and how problem posing can be used to teach mathematics in these contexts in the future, as proposed by Liljedahl and Cai (2021). As they suggested, research on problem posing should be considered from three perspectives: problem posing as a cognitive activity, problem posing as a learning goal unto itself, and problem posing as an instructional approach. These perspectives parallel the problem-solving categories proposed by Stanic and Kilpatrick (1989). Given this, researchers should not only aim to exhibit teachers’ mathematical understanding through problem posing but also consider it as a learning goal and an instructional approach (Cai, 2022).

As with many other studies, this study was limited to analyzing only some teachers from eastern China. Although the samples were representative to a certain extent, due to large differences in education levels among provinces in China, the conclusions of this study need to be refined with further studies in multiple regions or countries. Meanwhile, whether and how in-service teachers’ problem-posing ability affects students’ problem-posing and problem-solving performance also warrants future study.

Footnotes

Acknowledgments

We gratefully acknowledge support from “The cultivated project of Advanced discipline of Hangzhou Normal University”, Project number is 18JYXK044; from “Hangzhou philosophy and social science planning project”, Project number is Z23JC052.

Contributorship

Yiling Yao built the framework of the study and finished the manuscript. Suijun Jia supplied some suggestions for revising the paper and did some revision work. Jinfa Cai designed the study and collected the data. He also contributed to the conceptualization of data coding and the drafting of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the The cultivated project of Advanced discipline of Hangzhou Normal University, Hangzhou philosophy and social science planning project, (grant number 18JYXK044, Z23JC052).

Informed consent

The participants provided informed consent to partake in the study. The participants were made aware that the data would be used for publication.

ORCID iD

Jinfa Cai

Correction (December 2024):

Article updated to add Informed consent section.

Notes

Author biographies

Yiling Yao is a lecture at Hangzhou Normal University and a researcher at the Chinese Education Modernization Research Institute. Her research interests are teacher education, problem posing, and mathematical affect.

Suijun Jia is the professor at Zhejiang International Studies. His research interests are teacher education, mathematics textbooks and history of mathematics.

Jinfa Cai is the Kathleen and David Hollowell Professor at University of Delaware, and a Fellow of American Educational Research Association. He served as the Editor-in-Chief for the Journal for Research in Mathematics Education (2017-2020) and as Program Director at the USA National Science Foundation. He was the Editor of the 2017 Compendium for Research in Mathematics Education, by NCTM and an Editor-in-Chief for the Research in Mathematics Education book series by Springer (He is always looking for high quality book proposals). Currently, he is working on a 4-year longitudinal research project on problem posing. He loves reading, travel, and scholarly collaboration.

Appendix

Fraction Division Test

References

Kulm

(2004). The pedagogical content knowledge of middle school mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145–172. https://doi.org/10.1023/B:JMTE.0000021943.35739.1c

Baroody

A. J.

Feil

Johnson

A. R.

(2007). Research commentary: An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115–131. https://doi.org/10.2307/30034952

Behr

Harel

Post

Lesh

(1992). Rational number, ratio and proportion. In Grouws

(Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 296–333). New York, NY: Macmillan.

Booth

J. L.

Newton

K. J.

Twiss-Garrity

L. K.

(2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. https://doi.org/10.1016/j.jecp.2013.09.001

Brown

S. I.

Walter

M. I.

(1983). The art of problem posing. New Jersey, NJ: Lawrence Erlbaum Associates.

Brownell

W. A.

Chazal

C. B.

(1935). The effects of premature drill in third-grade arithmetic. The Journal of Educational Research, 29(1), 17–28. https://doi.org/10.1080/00220671.1935.10880546

Buchholtz

Leung

F. K.

Ding

Kaiser

Park

Schwarz

(2013). Future mathematics teachers’ professional knowledge of elementary mathematics from an advanced standpoint. ZDM – The International Journal on Mathematics Education, 45(1), 107–120. https://doi.org/10.1007/s11858-012-0462-6

Byrnes

J. P.

(1992). The conceptual basis of procedural learning. Cognitive Development, 7(2), 235–257. https://doi.org/10.1016/0885-2014(92)90013-H

Cai

(2003). What research tells us about teaching mathematics through problem solving. In Lester

F. K.

Jr. (Ed.), Research and Issues in Teaching Mathematics Through Problem Solving (pp. 241–254). Reston, VA: National Council of Teachers of Mathematics.

10.

Cai

(2022). What research says about teaching mathematics through problem posing. Éducation & Didactique, 16, 31–50. https://doi.org/10.4000/educationdidactique.10642

11.

Cai

Chen

Zhang

Song

(2020). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ conceptions on problem posing and lesson design. International Journal of Educational Research, 102, 101404. https://doi.org/10.1016/j.ijer.2019.02.004

12.

Cai

Ding

(2017). On mathematical understanding: Perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, 20(1), 5–29. https://doi.org/10.1007/s10857-015-9325-8

13.

Cai

Hwang

(2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. The Journal of Mathematical Behavior, 21(4), 401–421. https://doi.org/10.1016/S0732-3123(02)00142-6

14.

Cai

Hwang

(2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101–391. https://doi.org/10.1016/j.ijer.2019.01.001

15.

Cai

Hwang

Jiang

Silber

(2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In Mathematical Problem Posing (pp. 3–34). New York, NY: Springer.

16.

Cai

Moyer

J. C.

Wang

Hwang

Nie

Garber

(2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83(1), 57–69. https://doi.org/10.1007/s10649-012-9429-3

17.

Campbell

P. F.

Nishio

Smith

T. M.

Clark

L. M.

Conant

D. L.

Rust

A. H.

DePiper

J. N.

Frank

T. J.

Griffin

M. J.

Choi

(2014). The relationship between teachers’ mathematical content and pedagogical knowledge, teachers’ perceptions, and student achievement. Journal for Research in Mathematics Education, 45(4), 419–459. https://doi.org/10.5951/jresematheduc.45.4.0419

18.

Carpenter

T. P.

Corbitt

M. K.

Kepner

H. S.

Jr. Lindquist

M. M.

Reys

(1980). Results of the second NAEP mathematics assessment: Secondary school. The Mathematics Teacher, 73(5), 329–338. https://doi.org/10.5951/MT.73.5.0329

19.

Castro-Rodríguez

Pitta-Pantazi

Rico

Gómez

(2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129–146. https://doi.org/10.1007/s10649-015-9673-4

20.

Copur-Gencturk

(2021). Teachers’ conceptual understanding of fraction operations: Results from a national sample of elementary school teachers. Educational Studies in Mathematics, 107, 525–545. https://doi.org/10.1007/s10649-021-10033-4

21.

Einstein

Infeld

(1938). The evolution of physics. New York, NY: Simon & Schuster.

22.

Fariana

(2017). Implementasi model problem based learning untuk Meningkatkan Pemahaman Konsep dan Aktivitas Siswa. Journal of Medives: Journal of Mathematics Education IKIP Veteran Semarang, 1(1), 25–33.

23.

Goldin

G. A.

(2003). Representation in school mathematics: A unifying research perspective. In Kilpatrick

Martin

W. G.

Schifter

(Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 275–285). Reston, VA: National Council of Teachers of Mathematics.

24.

Hallett

Nunes

Bryant

Thorpe

C. M.

(2012). Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience. Journal of Experimental Child Psychology, 113(4), 469–486. https://doi.org/10.1016/j.jecp.2012.07.009

25.

Harel

Koichu

Manaster

(2006). Algebra teachers’ ways of thinking characterizing the mental act of problem posing. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 241–248). Rehovot, Israel: The Weizmann Institute of Science.

26.

Hiebert

Carpenter

T. P.

(1992). Learning and teaching with understanding. In Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (pp. 65–97). New York, NY: Macmillan.

27.

Isik

Kar

(2011). Matematik ogretmeni adaylarinin sozel ve gorsel temsillere yonelik kurduklari problemlerin analizi. Pamukkale Universitesi Egitim Fakultesi Dergisi, 30, 39–49.

28.

Isik

Kar

(2012). An error analysis in division problems in fractions posed by pre-service elementary mathematics teachers. Educational Sciences: Theory and Practice, 12(3), 2303–2309.

29.

Jia

Yao

(2021). 70 Years of problem posing in Chinese primary mathematics textbooks. ZDM – Mathematics Education, 53(4), 951–960. https://doi.org/10.1007/s11858-021-01284-9

30.

Kang, R., & Liu, D. (2018). The importance of multiple representations of mathematical problems: Evidence from Chinese preservice elementary teachers' analysis of a learning goal. International Journal of Science and Mathematics Education, 16, 125–143.

31.

Kieren

T. E.

(1993). Rational and fractional numbers: From quotient fields to recursive understanding. In Carpenter

T. P.

Fennema

Romberg

T. A.

(Eds.), Rational Numbers: An Integration of Research (pp. 49–84). New Jersey, NJ: Lawrence Erlbaum Associates, Inc.

32.

Kilpatrick

(1987). Problem formulating: Where do good problems come from. In Schoenfeld

A. H.

(Ed.), Cognitive Science and Mathematics Education (pp. 123–147). London, UK: Routledge.

33.

Kotsopoulos

Cordy

(2009). Investigating imagination as a cognitive space for learning mathematics. Educational Studies in Mathematics, 70(3), 259–274. https://doi.org/10.1007/s10649-008-9154-0

34.

Kwek

M. L.

(2015). Using problem posing as a formative assessment tool. In Singer

F. M.

Ellerton

N. F.

Cai

(Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 273–292). New York, NY: Springer.

35.

Lenz

Dreher

Holzäpfel

Wittmann

(2019). Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions. British Journal of Educational Psychology, 90, 809–829. https://doi.org/10.1111/bjep.12333

36.

Lesh

Post

T. R.

Behr

(1987). Representations and translations among representations in mathematics learning and problem solving. In Problems of Representations in the Teaching and Learning of Mathematics (pp. 33–40). New Jersey, NJ: Lawrence Erlbaum Associates.

37.

Huang

(2008). Chinese Elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: The case of fraction division. ZDM – The International Journal on Mathematics Education, 40, 845–859. https://doi.org/10.1007/s11858-008-0134-8

38.

Liljedahl

Cai

(2021). Empirical research on problem solving and problem posing: A look at the state of the art. ZDM – Mathematics Education, 53(4), 723–735. doi:10.1007/s11858-021-01291-w

39.

Lortie-Forgues

Tian

Siegler

R. S.

(2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221. doi:10.1016/j.dr.2015.07.008

40.

(1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New Jersey, NJ: Lawrence Erlbaum Associates.

41.

Ministry of Education of China (2022). China’s Compulsory Education Mathematics Curriculum Standards (2022 Edition). Beijing, China: Beijing Normal University Press.

42.

National Council of Teachers of Mathematics (2000). The principles and standards for school mathematics. Author. Reston, VA: NCTM.

43.

National Council of Teachers of Mathematics (2007). Mathematics teaching today: Improving practice, improving student learning (2nd ed.). Author. Reston, VA: NCTM.

44.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Author. Reston, VA: NCTM.

45.

Newton

K. J.

(2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110. https://doi.org/10.3102/0002831208320851

46.

Osana

H. P.

Pelczer

(2015). A review on problem posing in teacher education. In Mathematical Problem Posing (pp. 469–492). New York, NY: Springer.

47.

Rittle-Johnson

Schneider

(2015). Developing conceptual and procedural knowledge of mathematics. In Kadosh

R. C.

Dowker

(Eds.), Oxford Handbook of Numerical Cognition (pp. 1118–1134). Oxford, England: Oxford University Press.

48.

Rittle-Johnson

Schneider

Star

J. R.

(2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587–597. https://doi.org/10.1007/s10648-015-9302-x

49.

Rosenshine

Meister

Chapman

(1996). Teaching students to generate questions: A review of the intervention studies. Review of Educational Research, 66(2), 181–221. https://doi.org/10.3102/00346543066002181

50.

Siegler

R. S.

Lortie-Forgues

(2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909. https://doi.org/10.1037/edu0000025

51.

Silver

E. A.

(1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.

52.

Silver

E. A.

(2013). Problem-posing research in mathematics education: Looking back, looking around, and looking ahead. Educational Studies in Mathematics, 83(1), 157–162. https://doi.org/10.1007/s10649-013-9477-3

53.

Silver

E. A.

Cai

(1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539. https://doi.org/10.5951/jresematheduc.27.5.0521

54.

Silver

E. A.

Cai

(2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129–135.

55.

Simon

M. A.

Placa

Avitzur

Kara

(2018). Promoting a concept of fraction-as-measure: A study of the learning through activity research program. The Journal of Mathematical Behavior, 52, 122–133. https://doi.org/10.1016/j.jmathb.2018.03.004

56.

Singer

F. M.

Ellerton

Cai

(2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7. https://doi.org/10.1007/s10649-013-9478-2

57.

Singer

F. M.

Voica

Pelczer

(2017). Cognitive styles in posing geometry problems: Implications for assessment of mathematical creativity. ZDM – Mathematics Education, 49(1), 37–52. https://doi.org/10.1007/s11858-016-0820-x

58.

Skemp

R. R.

(1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9–15.

59.

Son

J. W.

Lee

J. E.

(2016). Pre-service teachers’ understanding of fraction multiplication, representational knowledge, and computational skills. Mathematics Teacher Education and Development, 18(2), 5–28.

60.

Stanic

Kilpatrick

(1989). Historical perspectives on problem solving in the mathematics curriculum. The Teaching and Assessing of Mathematical Problem Solving, 3, 1–22.

61.

Stylianou

D. A.

(2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265–280. https://doi.org/10.1007/s10649-010-9273-2

62.

Stylianou

D. A.

Leikin

Silver

E. A.

(1999). Exploring students’ solution strategies in solving a spatial visualization problem involving nets. In Zaslavsky

(Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 241–248). Psychology of Mathematics Education.

63.

Tichá

Hošpesová

(2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143. https://doi.org/10.1007/s10649-012-9455-1

64.

Toluk-Uçar

(2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175. https://doi.org/10.1016/j.tate.2008.08.003

65.

Tröbst

Kleickmann

Heinze

Bernholt

Rink

Kunter

(2018). Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic. Journal of Educational Psychology, 110(8), 1049. https://doi.org/10.1037/edu0000260

66.

Vamvakoussi

Bempeni

Poulopoulou

Tsiplaki

(2019). Reflecting on a series of studies on conceptual and procedural knowledge of fractions: Theoretical, methodological and educational considerations. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Utrecht, Netherlands.

67.

Yao

Hwang

Cai

(2021). Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM – Mathematics Education, 53, 937–949. https://doi.org/10.1007/s11858-021-01277-8

Beyond computation: Assessing in-service mathematics teachers’ conceptual understanding of fraction division through problem posing

Abstract

Keywords

1. Literature review

1.1 Conceptual and procedural understanding in mathematics

1.2 Investigating learners’ conceptual understanding using problem solving

1.3 Investigating learners’ conceptual understanding using problem posing

1.4 Fractions and fraction division

2. Study 1

2.1 Method

2.1.1 Participants

2.1.2 Instrument

2.1.3 Data coding and inter-rater reliability

3.1 Method

3.1.1 Instruments

3.1.2 Participants

3.1.3 Data coding and inter-rater reliability

3.2 Results

4.1 Limitations and future studies

Footnotes

Acknowledgments

Contributorship

Declaration of conflicting interests

Funding

Informed consent

ORCID iD

Correction (December 2024):

Notes

Author biographies

Appendix

References