Exploring Teachers’ Pedagogical Content Knowledge in the Instruction of Fractions and Fraction Operations

Prepared by the researche  : Dr. Samer Saif El-Din – Saint joseph university of Beirut

DAC Democratic Arabic Center GmbH

Journal index of exploratory studies : Nineteenth Issue – September 2025

A Periodical International Journal published by the “Democratic Arab Center” Germany – Berlin

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Abstract

This study examines the pedagogical content knowledge (PCK) of math teachers in teaching fractions and associated operations to learners in the second cycle of basic education in Lebanon.  The study examines how participants react to typical student misunderstandings and instructional scenarios, based on a comprehensive survey given to coaches, coordinators, and teachers. Based on the results, most instructors exhibit high levels of PCK, especially when it comes to using conceptual explanations and visual models instead of just procedural ones. For teaching fraction multiplication, division, and comparison, visual aids were used over rote algorithmic training, with a focus on deep comprehension. Additionally, a strong correlation was discovered between the individuals’ PCK levels and their professional jobs, with coordinators exhibiting the highest levels. These results demonstrate how crucial it is to increase conceptual understanding in teacher preparation programs in order to enhance mathematics education and successfully address enduring student misconceptions.

1. Introduction

There are four primary stages in the Lebanese educational system: kindergarten, basic education (grades 1–9), secondary education (grades 10–12), and higher education.  The Ministry of Education and Higher Education (MEHE) is in charge of overseeing it administratively, while the Center for Educational Research and Development (CERD) created the national curriculum (CERD, 2021).  With each cycle, mathematics instruction advances in complexity from an early start.  Standardized national tests, such as the Baccalaureate (Grade 12) and Brevet (Grade 9), are essential for student progress and school accountability.  However, the system still strongly emphasizes information and exams, with little integration of exploratory and conceptual learning strategies, including those required for fraction comprehension.  Curriculum reform calls have brought attention to the need for more competency-based and learner-centered teaching methods ( Chahine & King , 2013).

A fundamental but difficult topic in elementary mathematics is fractions. According to research, students who struggle with fractions may face long-lasting challenges that hinder their progress in later subjects like fraction computation, decimals, percentages, and even algebra (Singh et al, 2021). Fractions present significant difficulties for students far into the middle grades, according to national tests. Students sometimes carry over misconceptions from whole-number thinking because fractions entail notions that are different from whole numbers (Jarrah et al., 2022). For instance, many students in elementary classes, even after years of learning, see the numerator and denominator as distinct whole numbers and initially think that there are no numbers between two successive whole numbers. They also do not perceive a fraction like ¾ as a single number. Because of this, common mistakes made by students include expecting the product of two fractions to be greater than each fraction or adding fractions by adding numerators and denominators (Harriyani et al., 2022).

Fractions are mainly difficult for students since fractions are multi-faceted. They may represent: a part-whole relationship, a ratio, a division operation, a measure on the number line, or an operator, among other interpretations. Understanding all of these representations is necessary to master fractions, although part-whole interpretation is typically the only one that is emphasized in traditional education and textbooks (Ahl & Helenius, 2022).

            These difficulties make it necessary for teachers to possess strong pedagogical content knowledge (PCK) related to fractions in order to teach fractions effectively. In-depth knowledge of fractions and insight into students’ conceptions and misconceptions of fractions are essential for teachers. Shulman (1986) coined the term “pedagogical content knowledge,” which describes the specific knowledge educators acquire to connect pedagogy and content (Zolfaghari et al., 2021).

2. Rationale

Examining teachers’ pedagogical content knowledge in teaching fractions is obviously necessary given the significance of fractions and the recognized difficulties that both students and teachers have with this subject. Numerous studies have examined pre-service (student teacher) understanding of fractions and have frequently discovered notable deficiencies in both topic knowledge and PCK among new instructors. Fewer studies, nevertheless, have looked more closely at the PCK of in-service elementary teachers for fractions and fraction operations. It is possible that many current educators lack in-depth training that addresses widespread misconceptions about fractions or conceptual teaching techniques beyond what they themselves encountered as students (Abubakri, 2023). Critical insights can be gained by examining elementary teachers’ present PCK in this area. For instance, do primary school teachers identify the type of common errors that young learners make when working with fractions? When teaching fraction concepts, do they use a range of representations? To what extent do they grasp fractions and operations conceptually?

3. Purpose

            Examining and characterizing elementary teachers’ pedagogical subject knowledge in the teaching of fractions and fraction operations is the aim of this study. The study specifically focuses on three important components that collectively make up instructors’ PCK for fractions: teaching strategies for fractions, knowledge of students’ misconceptions and teachers’ conceptual and procedural knowledge of fractions.

4. Research Questions

The overarching research questions are:

• What instructional strategies do teachers employ when teaching fractions?
• What common fraction misconceptions do teachers recognize, and how would they respond instructionally?

How strong is teachers’ own content knowledge (both conceptual and procedural) of fractions and fraction operations?

5. Significance

            Mastery of fractions in the middle and upper elementary grades is a reliable indicator of success in high school algebra and general mathematical accomplishment (Spitzer & Moeller, 2022). This study directly addresses a factor that may have an impact on students’ long-term academic progress by examining instructors’ proficiency in teaching fractions. Teachers need to have the skills and resources they need to develop students’ conceptions of fractions on a large scale. The results of this study can help determine whether teachers presently have such expertise and what areas may require development. Second, by offering actual data in a particular content area, the study adds to the body of knowledge on pedagogical content knowledge. Although PCK is a well-established idea, topic-specific PCK continues to be of interest.

6. Methodology

A quantitative survey research design is used in this study. In particular, it is a cross-sectional descriptive study designed to provide a snapshot of teachers’ self-reported fractional knowledge and practices. As the main tool for gathering data, a questionnaire was created and distributed online using Google Forms. Since the goal is to collect data from a sizable sample of teachers in a consistent manner without changing any parameters, survey research is suitable in this case. Without attempting to test an intervention empirically, the descriptive and exploratory design aims to record the present state of instructors’ PCK in fractions. As such, it continues the legacy of previous teacher knowledge surveys. For instance, Lee & Lee (2023) investigated primary teacher applicants’ PCK using a descriptive survey and questionnaire.

7. Participants

In-service primary school math instructors, particularly those with prior expertise instructing fractions and fundamental fraction operations in the classroom, were the study’s target audience. 261 instructors, mostly at the upper primary level (grades 3–6, where fractions are intensively emphasized), make up the sample, which was gathered via convenience and voluntary sampling techniques. These instructors are a mix of new aneducators, with varying years of teaching experience and backgrounds from different universities. Because it represents a range of experience levels and maybe educational backgrounds or training, such a broad sample improves the findings’ generalizability. Apart from the requirement that participants have taught fraction topics, no stringent exclusion criteria were used to make sure the questions would be pertinent to their practice. The study has sufficient power to identify common trends and even investigate sub-group comparisons (e.g., comparing less experienced vs. more experienced instructors, or analysing any variations across grade levels) because to the comparatively high sample size (n=261). Every participant consented to the use of their answers for research after being made aware of the study’s objectives. In order to promote honesty and lessen social desirability bias, the poll was anonymous.

There were several sections in the questionnaire:

• Section A: Teaching Strategies. In order to assess instructors’ methods for teaching fractions, this section mostly employed Likert-scale items with 5-point measures ranging from “strongly disagree” to “strongly agree”). Items such as “I use visual area models (like pie charts or fraction circles) to introduce fractions” and “Using a number line is essential for teaching students that fractions are numbers” asked instructors to score how much they agreed with certain assertions. Other sections asked instructors to assess how often they utilize or how effective they think certain teaching scenarios or tactics were.
• Section B: Knowledge of students’ misconceptions. A combination of multiple-choice and open-ended questions were used in this part to test teachers’ knowledge of typical student mistakes. One inquiry can, for instance, ask the instructor to point out a typical mistake made by students when solving a fraction issue (such as adding fractions by adding numerators and denominators) or to explain why the student might have given that answer. For example, “What mistake are students likely to make when comparing?” may ask teachers to anticipate the most frequent error that students will make on a certain fraction exercise. In order to ensure content validity, certain items were modified from familiar test items(Basturk, 2016). The survey measures not only whether teachers are aware of a particular misunderstanding but also how effectively they can explain or address it by using both multiple-choice (to assess misconception awareness) and open-ended (to allow teachers to express their thoughts) questions.
• Section C: Conceptual and Procedural knowledge of fractions. This part had a number of content-focused questions designed to evaluate instructors’ own comprehension of fractions. These included both simple computations and more complex conceptual issues, much like the kinds of questions that could be found on a math test. Teachers may be asked to calculate the outcomes of fraction operations (such as adding, subtracting, multiplying, or dividing given fractions) or find equivalent fractions, among other tasks, in order to assess students’ procedural understanding. Inquiries about reasoning explanations (e.g., “Explain in words or with an example why dividing by 1/2 gives the same result as multiplying by 2”) or problems requiring comprehension beyond rote procedures (e.g., sorting a set of fractions by size, or deciphering a fraction word problem) were examples of items for conceptual knowledge.

Two experts in mathematics education examined the questionnaire to make sure it was clear and had solid information. To make sure the questions were understood as intended and to gauge completion time, a brief pilot test was carried out with a few instructors who were not included in the main sample. Minor changes were made to the phrasing and example clarity in response to criticism. Teachers took an average of 20 minutes to complete the final survey, which included roughly 30–40 items altogether. Data collection was effective and geographically broad since the survey was delivered using Google Forms, allowing instructors from various schools or areas to reply online. To guarantee the veracity of their knowledge and opinions, participants were told to do the survey on their own, without seeking advice from others.

8. Data Collection

Elementary school teachers were invited to take part in the study through professional networks and emails. The link to the survey was available for almost three weeks. Periodically, reminders were distributed in an effort to increase response rates. 261 replies have been recorded by the end of the data collection period. After downloading the data from Google Forms into a spreadsheet, its correctness was verified. Each respondent was given an anonymous ID code for analysis, and all answers were kept private.

9. Data Analysis

Both descriptive and inferential approaches are used in the analysis because this study focuses on three constructs. Descriptive statistics were calculated for each item and scale in the first phase. When it comes to category items, this involves figuring out frequencies and percentages (averaging replies on each teaching approach item to assess general agreement levels, etc.). Overall success rates (e.g., the percentage of teachers who correctly answered a specific fraction problem or who gave a valid explanation for a concept) were tabulated for the conceptual/procedural knowledge questions. In essence, these descriptive results outline the PCK profile in the sample by giving a general overview of instructors’ knowledge and practices.

10. Results

Table 1 presents analysis of strategies teachers suggested for instruction. It includes the question and suggested answers with frequencies and means.

Table 1. Teaching Strategies Answers

In Sara’s case, the majority of teachers (78%) recognized that Sara’s misunderstanding stems from comparing visuals based on unequal wholes and correctly recommended using unit fractions and visual models to reconstruct equivalent representations. This choice reflects a solid grasp of conceptual teaching through visual models, rather than simply relying on symbolic or numerical comparisons. While a small number of teachers preferred converting to decimals or teaching common denominators procedurally. As for Mazen’s thoughts that multiplying numerators needs a common denominator, a clear majority of teachers (59.4%) opted for using visual models (e.g., area models, shading grids) to show how fraction multiplication works, emphasizing conceptual understanding. Only a small proportion (11.4%) suggested reinforcing Mazen’s misconception by teaching the common denominator method, which is unnecessary and misleading for multiplication. Nearly one-quarter (26.6%) gave a mathematically correct but less instructional response (stating common denominators aren’t needed) without recommending how to build understanding. Overall, the responses indicate that most teachers understand the importance of visual, conceptual representation in multiplication—a key component of PCK. As for addition and subtraction of fractions with different denominators, a small group (15.4%) favoured procedural correction (common denominators), while very few relied on decimal conversions. The majority highlighted the need to anchor students’ reasoning in visual, whole-based representations when correcting misconceptions.

Table 2 explores how teachers interpret and respond to common fraction-related misconceptions.

Table 2. Knowledge of Students’ Misconceptions

Each item presents a realistic student error and invites teachers to identify both the nature of the misconception and the most appropriate instructional response. For the first misconception that the fraction increases as its denominator increases, the vast majority of teachers (79.8%) accurately identified the whole-number thinking misconception and proposed a conceptual and visual remedy. This indicates strong awareness of one of the most well-documented misconceptions in fraction comparison. A smaller proportion (12.2%) focused on procedural correction without addressing the misconception, suggesting a minority still rely on rule-teaching as a response. Overall, the results reflect strong PCK for fraction magnitude and comparative reasoning. Another misconception is adding fractions by adding numerators alone and denominators alone.  Once again, nearly 80% of teachers correctly identified the misconception as improper addition of numerators and denominators, stemming from whole-number reasoning. Their recommended instructional strategy—using area models or visual aids—is fully aligned with research-based best practices for teaching fraction addition. The 10.6% recommending rote practice highlight a smaller subset of teachers with a procedural orientation. These results suggest a high level of diagnostic awareness and appropriate pedagogical responses for addition misconceptions. Multiplication of fractions always yields a greater faction seems to probe teachers’ recognition of a common overgeneralization—that multiplication increases values. Over three-quarters (77.6%) of teachers correctly identified the issue and proposed conceptual instruction with visual models (e.g., showing half of a quarter). The relatively small portion advocating pure rule-teaching (14.4%) suggests that most respondents favour concept-building strategies. This is consistent with high PCK in addressing multiplicative reasoning errors. Moreover, the fraction division misconception targets conceptual misunderstanding in fraction division. A clear majority (73.4%) diagnosed the misconception as whole-number transfer and recommended real-world or hands-on approaches (e.g., serving problems or measurement scenarios). Notably, 14.4% defaulted to procedural instruction without exploring student reasoning. This confirms that while most teachers aim to build conceptual understanding, a small subset still relies on algorithms, potentially missing opportunities to deepen comprehension. However, for the misconception of accepting unequal parts in fractions, about 64.5% recognized that the student’s drawing with unequal parts indicates a conceptual error about what fractions represent. Their preferred solution—demonstrating equal partitions using visual models—is pedagogically sound. Meanwhile, 26.5% chose to simply state the rule, which, while mathematically accurate, may not fully address conceptual misunderstanding. These results suggest that while many teachers understand the importance of visual and experiential learning, others still lean toward verbal rule reinforcement.

Table 3 presents data from five conceptual and procedural items assessing teachers’ understanding of key fraction operations and representations.

Table 3:  Teachers’ Conceptual and Procedural Knowledge

Each item offers multiple response options, and the frequency and percentage distribution of choices indicate patterns in teachers’ mathematical knowledge.

For the first question about explaining the conceptual understanding of 2/3 × 3/4, the most frequently selected response was the procedural answer (“Multiply numerators and denominators”), chosen by 43.7% of teachers. Only 38% selected the conceptual explanation (“Two-thirds of three-fourths is one-half”), indicating that while some teachers do understand how to conceptually interpret multiplication of fractions, a larger proportion still rely primarily on procedural knowledge. The relatively low selection of the reasoning-based statement (“The product is smaller because both < 1”)—which suggests magnitude understanding—further reinforces that conceptual depth is less prevalent among respondents.

For the second question on explaining 3/4 ÷ 1/8 conceptually, the majority of teachers (57.4%) selected the rule-based procedural strategy (keep-change-flip), while only 36.9% opted for the conceptual interpretation (“how many 1/8 portions fit in 3/4?”). This indicates a strong preference for algorithmic teaching rather than sense-making strategies. Conceptual understanding of fraction division—widely acknowledged in research as challenging yet essential—is still not dominant among respondents, despite being pivotal to developing student reasoning.

For the third question about explaining 7/4 as a fraction, it shows a relatively balanced distribution of conceptual and procedural knowledge. While 51.3% of teachers chose the accurate step-by-step approach (division to mixed number), a substantial proportion (40.3%) opted for the measurement interpretation (“7 fourths = 1 whole and 3/4”), which is conceptually stronger. Only a small number relied on a decimal approach. This suggests that for fraction representation (unlike operations), many teachers demonstrate a deeper understanding, possibly due to the more intuitive nature of mixed numbers.

When teachers were asked to choose a truth statement about fractions, most teachers (64.6%) correctly identified a mathematically valid generalization about multiplication of fractions. However, a significant 28.5% still held the misconception that a larger denominator always means a smaller fraction—this is not always true unless numerators are equal. This indicates that while many teachers understand multiplicative properties of fractions, some still carry overgeneralizations from part-whole models, which may impact their instructional clarity.

As for the simplification question of (2/3) × (3/4) ÷ (1/2), it assesses procedural fluency, and the results are overwhelmingly positive. Nearly all teachers (94.7%) arrived at the correct result. This indicates strong procedural competence in handling multi-step fraction operations. However, since this item does not evaluate reasoning or interpretation, it reflects algorithmic knowledge rather than conceptual depth.

Teachers in this study demonstrate strong procedural fluency in fraction operations, especially in straightforward computations. However, conceptual understanding is less prevalent, as seen in questions 1 and 2. A majority prefer rule-based strategies such as “keep-change-flip” over explanatory or visual models that enhance student understanding. In questions involving interpretation (e.g., Q3), a relatively higher percentage showed conceptual grasp, suggesting that teachers might better understand representations of improper fractions than operations. Misconceptions persist, particularly in generalizations about denominators and value comparisons (Q4).

Table 4 presents a detailed analysis of the Chi-Square Crosstabulation Table relating participants’ current role to their Pedagogical Content Knowledge (PCK) level, as inferred from the strategies they selected in a teaching fractions questionnaire.

Table 4. Current Role × PCK Level (Strategies Crosstabulation)

Higher professional roles (Coordinator, Coach) are associated with a greater likelihood of choosing high-quality, conceptually rich strategies. Teachers still formed the largest share of high PCK responses numerically (112), but proportionally, their high PCK percentage (57.7%) is lower than that of coordinators (72.3%) and coaches (71.4%). Low PCK responses were almost entirely from teachers (30 out of 37), suggesting that some classroom teachers may still rely on less effective instructional approaches.

11. Discussion

The results of this study provide important light on the kind and extent of teachers’ Pedagogical Content Knowledge (PCK) when it comes to teaching fractions, especially when it comes to conceptual comprehension, procedural knowledge, and identifying student misunderstandings. According to this research, there are still significant gaps, particularly across specific professional responsibilities, even if the majority of instructors appear to have a solid understanding of successful tactics.

Answering research question 1 related to the teachers’ suggested strategies, the majority of participants consistently preferred conceptual teaching strategies, as indicated in Table 1, especially the use of visual models (such as area diagrams, fraction bars, and unit fractions) to explain complex fraction concepts like converting between forms, multiplying and subtracting fractions, and comparing unlike denominators. For example, while teaching the addition of 1/2 and 1/3, 84.4% of instructors opted for visual modeling, and when teaching subtraction, 87.7% of teachers preferred to use common denominators and visual representations. These results are consistent with the literature (Basturk, 2016) that emphasizes the use of numerous representations in fraction learning.

Answering research question 2, teachers also shown a high level of understanding of typical misconceptions held by students. For instance, 77.6% of students correctly answered to a student who was astonished that 1/2 × 1/4 equals a lower number, and 79.8% correctly recognized the mistake causing a student to believe that 3/8 is higher than 3/5. These choices demonstrate a strong capacity to identify misunderstandings and offer corrective techniques that work, including using models and real-world situations. This is in line with Ahl & Helenius (2024), which highlighted the need of comprehending students’ thinking and mistakes in order to provide successful mathematics training. It appears that many participants have a strong PCK unique to fraction learning, as seen by the frequent detection of misunderstandings based on whole-number thinking and the choice of correction techniques that focus on the conceptual basis of mistakes.

Answering research question 3, fewer instructors selected explanations that showed profound conceptual reasoning, even if the majority of them properly answered computation-related issues (94.7% correctly simplified (2/3 × 3/4) ÷ 1/2, for example).  For example, 43.7% of respondents selected the procedural option of “multiplying the numerators and denominators,” however only 38% selected the conceptual interpretation of 2/3 × 3/4 as “two-thirds of three-fourths is one-half.”  Comparably, when asked to explain 3/4 ÷ 1/8, only 36.9% chose a conceptual interpretation (i.e., “how many 1/8s in 3/4?”), whereas 57.4% used the keep-change-flip approach.

The cross-tabulation in Table: Current Role × PCK Level, which demonstrates a substantial correlation between professional role and PCK level, is one of the most convincing results.  The majority of individuals with High PCK were classroom instructors (112 out of 158), however the PCK levels of coordinators and coaches were proportionately higher at 72.3% and 71.4%, respectively, than those of teachers, who had 57.7%.

 Given that coordinators and coaches are frequently more involved in planning, demonstrating, and analysing instructional strategies, this may be a result of experience, ongoing professional development, and involvement with curriculum leadership.  Professional positions including peer mentorship and instructional leadership may greatly improve pedagogical depth (Darling-Hammond & McLaughlin, 2011).

Overall, this study shows that instructors’ PCK in fractions is encouraging. The majority of teachers have a firm grasp of both common misunderstandings among students and successful teaching techniques based on conceptual knowledge. However, a fraction of respondents’ continued dependence on procedural techniques emphasizes the need for focused assistance, particularly for classroom teachers. Moreover, the clear link between professional role and PCK underscores the importance of ongoing learning, collaboration, and leadership roles in fostering advanced instructional knowledge.

Future studies should look into how teachers’ PCK growth is impacted by involvement in certain coaching programs or professional learning communities, and how these improvements affect student results and classroom practice.

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