Scholars agree that teachers’ subject matter knowledge is an important component of the expertise they need to teach mathematics (An et al., 2004; Ball et al., 2008; Copur-Gencturk & Tolar, 2022; Li & Kaiser, 2011). Although there are many conceptions of what constitutes subject matter knowledge (e.g., Ball et al., 2008; Copur-Gencturk & Tolar, 2022; Krauss et al., 2008; Tattoo et al., 2008; Tchoshanov et al., 2017), for this paper, we ground our understanding of subject matter knowledge in one of the most comprehensive frameworks for mathematical understanding, the National Research Council’s (NRC) strands of mathematical proficiency (p. 115, 2001). Although not specifically designed for teachers, the NRC’s conceptualization is one of the most thorough conceptualizations of subject matter knowledge as it encompasses conceptual focus along with procedural fluency while also including ancillary foci such as logic of making sense of and solving problems.

Teachers’ subject matter knowledge has found to be related to the learning environment they create for their students (e.g., Borko & Livingston, 1989; Copur-Gencturk, 2015; Lui & Bonner, 2016). A substantial amount of work has been conducted on teachers’ subject matter knowledge of mathematics, the majority of which has focused on teachers’ understanding of a particular topic or a content area (e.g., Copur-Gencturk & Ölmez, 2022; Harel & Behr, 1995; Izsák & Jacobson, 2017; Newton, 2008; Orrill & Brown, 2012; Post et al., 1991; Siegler & Pyke, 2013; Ucar & Bozkus, 2018). While this work has enhanced our understanding of the nuances of teachers’ knowledge of particular topics, little is known about how teachers’ understanding of one topic is related to their understanding of other mathematically related topics they are expected to teach.

This question warrants investigation for several reasons. First, mathematics concepts are built on one another. However, if teachers’ knowledge of these concepts is fragmented, they are unlikely to create coherent learning experiences for their students to develop the connectedness among these concepts. Given that the learning environment teachers create is closely tied to how teachers understand these concepts (e.g., Borko & Livingston, 1989; Lui & Bonner, 2016), exploring the relationships among teachers’ understanding of content areas allows us to shed light on how students are given opportunities to see these connections. For instance, teachers who are unaware of the relationship between special right triangles and the unit circle miss an opportunity to demonstrate how having a hypotenuse of one unit makes it easy to determine the values of different coordinate points along the unit circle. Instead, teachers may just have students memorize the unit circle without understanding it on a more conceptual level. Second, exploring the nature of teachers’ understanding of different yet related topics could provide new insights into teachers’ misunderstandings. For instance, if two adjacent domains have common misunderstandings, analyzing the domains together could reveal that the misunderstandings stem from the same underlying issue. As an example, it is well documented that fractions are a particularly difficult concept for teachers and students. It is possible that teachers show similar struggles with a theoretically connected topic, ratios, and proportions and analyzing their understanding of both simultaneously could reveal that the misunderstandings are related. Third, understanding the structure of teachers’ knowledge of school mathematics could help identify the unique elements of the knowledge they need to teach mathematics. This, in turn, would allow researchers to investigate the role of teachers’ subject matter knowledge more accurately on a large scale, given that studies investigating the role of teacher knowledge in teaching and student learning could have limited items from all the topics from school mathematics. Thus, this knowledge would allow researchers to reliably measure teachers’ knowledge by focusing on distinct content areas while still covering the breadth of teachers’ subject matter knowledge. Finally, teachers from different educational backgrounds may have had different learning opportunities, leading them to have differing levels of understanding of mathematical concepts. For instance, teacher certification programs for elementary teachers might provide prospective teachers more opportunities to learn about the content areas taught in elementary school, whereas teachers in mathematics specific certification programs might have more opportunities to learn the mathematics concepts in more nuanced ways. Thus, teachers from different programs with different certification types might develop different understandings of mathematical concepts.

Despite the importance of teachers’ subject matter knowledge, research investigating the structure of teachers’ subject matter knowledge in different mathematical content areas is limited. This study intends to address this gap in the literature by exploring the nature of the relationship between two content areas: fractions and ratios and proportional relationships. We selected these two content areas because they are theoretically connected concepts. If there is an area where teachers would be expected to have a coherent understanding, and thus form a single knowledge domain, this is a probable area. On the other hand, these concepts have different levels of emphasis in varying contexts (e.g., elementary and middle school), which could lead teachers to form different levels of understanding, depending on their educational background. We aimed to test our hypotheses that:

  1. 1)

    Teachers’ subject matter knowledge of fractions, ratios, and proportional relationships can be modeled as unidimensional.

  2. 2)

    Teachers’ professional backgrounds (e.g., certification types and routes) are related to teachers’ understanding of fractions, ratios, and proportional relationships.

In this paper, we first outline the theoretical framework linking fractions, ratios, and proportional relationships. Then we review the prior literature investigating the structure of teachers’ knowledge of these topics, followed by our study design, methodology, and finally our results and explanation of the findings.

Theoretical Framework

One of the most referenced conceptualizations of the relationship between fractions, ratios, and proportional relationships is the multiplicative conceptual field (MCF) (Vergnaud, 1983, 1988). The MCF encompasses “all situations that can be analysed as simple and multiple proportion problems and for which one usually needs to multiply or divide” (Vergnaud, 1988, p. 144) and has provided a theoretical understanding for many researchers studying fractions, ratios, and proportional relationships (e.g., Izsák & Beckmann, 2019; Long et al., 2010). The MCF asserts that mathematical concepts are interrelated to one another and that in solving mathematical problems individuals use many concepts together rather than one isolated concept, hence the need for teaching them in an integrated way (Vergnaud, 1983, 1988). Vergnaud goes so far as to say:

It would be misleading to separate studies on multiplication, division, fraction, ratio, rational number, linear and n-linear function, dimensional analysis, and vector space; they are not mathematically independent of one another, and they are all present simultaneously in the very first problems that students meet. (Vergnaud, 1983, p. 127)

The connections of these topics through their multiplicative structure make them intrinsically related such that teachers may be expected to have consistent knowledge across the MCF. As mentioned, we focus on two of the key components of the MCF, (1) fractions and (2) ratios and proportional relationships. We specifically focused on these topics because they are critical components of elementary and middle school mathematics and have demonstrated difficulties for students and teachers (e.g., Ball, 1990; Copur-Gencturk, 2022; Izsák & Jacobson, 2017; Ni & Zhou, 2005).

In this study, we define fractions similar to standards documents, as a/b, where a refers to the number of pieces and b refers to the size of the pieces making up the referent whole (National Governors Association Center for Best Practices & Council of Chief State School Officers [NGACBP & CCSSO], 2010). We define ratios as the multiplicative relationship between two quantities (NGACBP & CCSSO, 2010; Thompson & Thompson, 1994) and proportional relationships as a relation between all sets of ratios that make a proportion true. To understand proportional relationships, one must understand the covariance and invariance of quantities (Copur-Gencturk et al., 2023; Lamon, 2007). By covariance, we mean that as one quantity varies by a set amount, the second quantity will change in response (Lamon, 2007). Invariance refers to the intensive quantity of proportional relationships (Thompson & Thompson, 1994), where regardless of the covariance, there is a constant quotient (Copur-Gencturk et al., 2023).

We understand the relationship between the two as highly related, both involving multiplicative structures, as supposed by the MCF (Vergnaud, 1983, 1988). We recognize that both topics deal with the coordination of two variables. For fractions, the first variable shows the number of parts (i.e., numerator), and the second variable shows the number of equally partitioned units making up the whole referent unit (i.e., denominator). For ratios, the first variable shows the size of the first quantity, whereas the second variable shows the corresponding size of the second quantity. For proportional relationships, the quantities can change, while the quantity of one item per another is invariant, extending the ratio relationship to encompass equivalent ratios. Thus, for both topics, a teacher must consider how the two variables change simultaneously while also understanding that the quotient of the two stays the same.

Literature Review

Teachers’ Content Knowledge of Fractions, Ratios, and Proportional Relationships

Despite a substantial amount of work investigating teachers’ knowledge of these topics separately, limited attention has been given to teachers’ understanding of the relationship between these two topics (e.g., Copur-Gencturk, 2021; Izsák & Jacobson, 2017; Lehrer & Franke, 1992; Newton, 2008; Siegler & Pyke, 2013; Ucar & Bozkus, 2018). Prior work has documented that teachers show similar struggles to students with understanding these topics (e.g., Copur-Gencturk, 2022; Harel & Behr, 1995; Orrill & Brown, 2012; Post et al., 1991). Specifically, teachers demonstrate difficulty with understanding the relationship between quantities in proportional relationships (e.g., Copur-Gencturk et al., 2023; Lamon, 2007; Lobato et al., 2010; Orrill & Brown, 2012; Simon & Blume, 1994) and struggle to distinguish between proportional and nonproportional situations (e.g., Cramer et al., 1993; Fisher, 1988; Izsák & Jacobson, 2017; Masters, 2012). Similarly, teachers seem to have many challenges with fraction arithmetic (e.g., for a review, see Olanoff et al., 2014), and they often solve fraction problems without understanding why the algorithms work (e.g., Ball, 1991; Copur-Gencturk, 2021; Copur-Gencturk & Doleck, 2021; Ma, 2010; Newton, 2008; Tirosh, 2000).

We found only two studies that investigated how teachers conceptualize the relationship between fractions, ratios, and proportional relationships. Of those two, one was a phenomenological study that was focused on understanding how ratios and fractions are defined broadly (Clark et al., 2003). The study had a much broader span, gathering data from textbooks and pre- and in-service teachers to generate theoretical models depicting the varied understandings of the relationship between fractions, ratios, and proportional relationships. The results of their study generated multiple models depicting this relationship, and the authors mentioned the varied conceptions teachers hold, but it was unclear how many teachers held each conception making it difficult to understand the overall trend of teachers’ understanding versus a description of every different conception they may have held (Clark et al., 2003).

The most related study was conducted by Pitta-Pantazi and Christou (2011) with 238 pre-service kindergarten teachers from three universities in Cyprus. The study aimed to assess the structure of preservice kindergarten teachers’ proportional reasoning to better understand what critical components are necessary to train and develop teachers in so that they can support students in developing proportional reasoning at a young age. The researchers used a similar method to this study of confirmatory factor analysis to test different indicators of teachers’ proportional reasoning. They compared two second-order models, where the second-order factor was proportional reasoning. For the first model, the 10 primary factors came from Lamon’s (2005) proposed structure of proportional reasoning. The 10 first-order factors were “sharing/comparing, operator, quotient, rate, measure, unitizing, quantities and covariance, relative thinking, measurement, and reasoning up and down” (Pitta-Pantazi & Christou, 2011, p. 155).

For the second model, they compared Lamon’s (2005) proposed structure of proportional reasoning with the aforementioned ten subcomponents to their conception that included missing value as an additional subcomponent of proportional reasoning (Pitta-Pantazi & Christou, 2011). Through their analysis, they found that their own model, with 11 subcomponents, fit the data better. All but one of the subcomponents (i.e., measure) were found to be significant indicators of teachers’ proportional reasoning. Yet, based on the fit indices reported, neither model was an excellent fit to the data. Thus, there is the possibility that teachers’ proportional reasoning could be better modeled by more than one construct, which requires more research.

A major contributor to their results could be that Pitta-Pantazi and Christou (2011) used a sample of pre-service teachers who were training to be kindergarten teachers. Their sample likely did not receive as much explicit instruction on fractions, ratios, and proportional relationships as teachers who will teach the content and they used a sample of pre-service teachers who had not yet experienced teaching the content. Prior work has shown that pre-service and in-service teachers’ understanding of mathematics, including their subject matter knowledge, are different, meaning that the patterns that emerge from studies with pre-service samples may not generalize to in-service teachers (e.g., Charalambous, 2016; Krauss et al., 2008). This difference in knowledge could be even more apparent since research has shown that teachers of different grade levels have different levels of knowledge (e.g., Depaepe et al., 2015; Senk et al., 2012), so a sample of kindergarten pre-service teachers is likely to have very different knowledge than a sample of in-service teachers who teach these concepts.

Overall, the prior literature is vague regarding teachers’ understanding of these topics and whether their understanding may or may not conform to the theoretical expectation. However, there are no studies with in-service teachers who teach the content and empirical assessments of the structure of their knowledge. Thus, the present study furthers the research in this area by exploring the relationship between teachers’ subject matter knowledge of these two theoretically related topics with a sample of Grade 3–7 in-service teachers (ISTs). We assess the structure of teachers’ knowledge of fractions, ratios, and proportional relationships and attempt to make our results more generalizable by using an education research company to recruit teachers so that the sample came from different educational backgrounds and taught at different grade levels and schools.

Factors Associated with Teachers’ Understanding of Fractions, Ratios, and Proportional Relationships

The role of teachers’ background and its relation to teachers’ subject matter knowledge of fractions and ratios has been investigated separately in prior work (e.g., Copur-Gencturk, 2021; Copur-Gencturk et al., 2023; Izsák et al., 2019; Weiland et al., 2019). The results across these studies depict inconsistent trends for the relationships between background characteristics and subject matter knowledge, suggesting that more work is needed to fully understand the relationship or lack thereof. In general, teachers’ education including their undergraduate degree type and the number of relevant courses taken, the route teachers take to enter the profession, and years of experience have commonly been used in some combination as predictors of teachers’ subject matter knowledge.

From the previous studies, the most consistent result appears to be the positive relationship between teachers’ subject matter knowledge and years of teaching mathematics (Copur-Gencturk, 2021; Izsák et al., 2019; Weiland et al., 2019). The other common trend is that mathematics education, both in terms of the number of courses and degrees, does not seem to relate to teachers’ content knowledge of these domains (Izsák et al., 2019; Weiland et al., 2019). Being credentialed in mathematics as opposed to special education or science has had inconsistent results with some studies finding a positive relationship with content knowledge (Izsák et al., 2019) and some finding no relationship (Copur-Gencturk, 2021). For context, in the USA, teachers can receive their bachelor’s degree in any number of fields, with many mathematics teachers receiving their degree in education or mathematics, where the latter would have more classes focused on higher level mathematics. Prospective teachers then complete a credential either with their degree or afterward. The credential is typically subject specific for grades 7–12 (e.g., mathematics certification), but K–6 teachers are certified as generalists, and special education has a unique certification.

In this study, we wanted to build on the prior literature to understand more clearly how teacher preparation and background may relate to subject matter knowledge. We pulled many of the same background characteristics from the previous literature, namely, around certification and education, to further investigate these relationships. It should be noted that we do not include years of experience because of our sample. This sample was entirely novice teachers meaning that there was not a reasonable spread of experience to understand the relationship.

One background characteristic that we were particularly interested in that is not always captured was certification pathway. In the USA, there are two major ways to become a certified teacher, alternative and traditional. For alternative certification programs, teachers are not required to have trained or studied as a teacher before teaching; they can get certified while teaching or with minimal preparation. Traditional certification requires teachers to have gotten an undergraduate degree or master’s degree in education or a related field, and in doing so, they become certified to teach. Alternative certification is contentious with proponents arguing for its necessity in attracting more diverse teachers and in allowing more entryways into the profession (Zeichner & Schulte, 2001), whereas critics reference a lack of preparation, high attrition, and low performance scores as major flaws with the approach (Walsh & Jacobs, 2007). Alternative pathways have grown in other countries as well, making this an important background characteristic to consider, as it affects a larger number of countries and their teachers. Understanding the relationship between certification pathway and content knowledge could have implications for the expansion of alternative certification programs, either providing more evidence for critics to stop their growth or evidence to promote it. The relationship between teacher knowledge and certification pathway is similarly unclear, so we investigate that for fractions, ratios, and proportional relationships.

Taken together, research is limited regarding teachers’ understanding of the relationship between fractions, ratios, and proportional relationships. In this study, we aimed to explore the structure of this relationship by using data collected from in-service teachers who were teaching these concepts. Additionally, we explored the extent to which the background characteristics of teachers were related to their subject matter knowledge of these topics.

Methods

Participants

Participants in this study were 246 teachers teaching Grades 3–7 in the USA who answered questions on fractions, ratios, and proportional relationships and provided their educational background data. The analytical sample included the 226 teachers who completed the survey in its entirety. Each teacher was sent an invitation to participate in the online survey and was provided a gift card for their time. These teachers were recruited across states through an educational research company, which provided teachers’ contact information. The survey was untimed, and teachers were allowed to take breaks while completing the survey if needed.

As shown in Table 1, the sample was predominantly White (69.5%) and female (83.6%), which reflects the demographics of US teachers (De Brey et al., 2021). Approximately 24% of the teachers were teaching either Grade 6 or 7 when the data was collected, and 71.2% of the teachers were certified through a traditional teacher preparation program, with around one-fifth of the teachers holding a mathematics teaching credential. As mentioned above, all of these teachers were novice teachers at the time of data collection, meaning that they were in their first four years of teaching.

Table 1 Descriptive Statistics for Teacher-Level Variables
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Teacher Knowledge Measures

Teachers’ fraction, ratio, and proportional relationship knowledge were measured by 14 items (see electronic supplementary materials for a full list of the items used). Of the 14 items, six focused on fraction knowledge and eight on ratio and proportional relationship knowledge. All the items were adapted from prior literature and teacher knowledge assessments to capture teachers’ subject matter knowledge as conceptualized in our study (Izsák et al., 2019; Siegler & Lortie-Forgues, 2015; Tatto et al., 2012; Van de Walle et al., 2010; Van de Walle et al., 2019).

The items on fractions and fraction operations covered relevant topics such as flexibility with referent units, comparing fractions, creating equivalent fractions, and estimating fraction operations (Izsák et al., 2019; Kazemi & Rafiepour, 2018; Siegler & Lortie-Forgues, 2015; Tatto et al., 2012; Van de Walle et al., 2019). For instance, to measure teachers’ attention to referent units, teachers were asked whether it is possible for 1/3 to be greater than 1/2 and to explain their reasoning. This item targeted referent units because teachers who are not as aware of referent units would likely say no, thinking only of the magnitude of the values. Conversely, teachers who fully understand referent units can recognize that the answer fully depends on what the fractional parts are referring to.

The eight items that measured ratio and proportional relationship knowledge encompassed theoretically important concepts, such as appropriately differentiating proportional and nonproportional situations, understanding the covariance and invariance of quantities in a proportional relationship, and key topics represented in the Common Core State Standards, such as finding missing values given a ratio table (Arican, 2019; Brown et al., 2020; Lamon, 2007; Van Dooren et al., 2005; Weiland et al., 2019). As an example, the items measuring teachers’ ability to distinguish proportional relationships had different scenarios and asked teachers to identify if the scenario described was directly proportional or not and to explain their reasoning. One of the scenarios mentioned two people running laps at the same rate, where one started before the other, which is a situation that is commonly assumed to be directly proportional but is linear (Van Dooren et al., 2008).

Coding Scheme

Each constructed-response item had an associated rubric to assess teacher responses for correctness and reasonableness of the provided explanation. Constructed response items were scored on a 3 point scale from 0 to 2. A score of 0 indicated the teacher’s response was either incorrect or the teacher did not explain the answer. A score of 1 indicated that the key idea of the concept was not explicitly stated or only procedural explanations were given. A score of 2 indicated the teacher provided the correct answer along with a conceptual explanation of the key idea about the concept being measured by the item (see Fig. S1 and Table S1 in the supplementary materials for an example item and explanation of the coding). The rubrics for each constructed-response item were created by the third author who then trained the other rater. The two raters scored a set of responses simultaneously and discussed the rating for each response. Once there was a consensus on how to score each item, the remaining responses were scored independently by both raters. After scoring independently, the raters compared their codes and when there was disagreement, they discussed the discrepancy and agreed on a final score. The interrater reliability was above 90% for every item, and Cronbach’s alpha, which is an estimate of the reliability of the entire fraction, ratio, and proportional relationship scale, was 0.75 (0.60 for the fraction scale and 0.63 for the ratio scale). Table 2 provides descriptive data about teachers’ performance on each item.

Table 2 Descriptive Data of Teacher Performance by Item
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Analytical Approach

To investigate the latent structure of teachers’ knowledge of fractions, ratios, and proportional relationships and the relationship between teachers’ subject matter knowledge and educational backgrounds, we used structural equation modeling (SEM). SEM is appropriate for answering our research questions as it combines path models (i.e., the relationship between teachers’ content knowledge and educational backgrounds) involving latent variables (i.e., teachers’ content knowledge of fractions, ratios, and proportional relationships). The measurement model (i.e., confirmatory factor analysis), one part of SEM, is a theory-driven approach to test our hypothesis that teachers’ content knowledge of fractions, ratios, and proportional relationships is unidimensional based on the theory of the multiplicative conceptual field (Vergnaud, 1983, 1988). The path model, the other part of SEM, tests our hypotheses that teachers’ professional backgrounds are associated with teachers’ content knowledge of fractions, ratios, and proportional relationships. Teachers’ professional background variables include whether teachers have a master’s degree, the path entering the teaching profession (traditional or alternative certification; the latter is the reference group), types of credentials (mathematics, special education, and K–6; K–6 as the reference group), and fields of their bachelor’s degree (education or not; education as the reference group). To evaluate the extent to which our hypothesized model fits our sample data, we referred to multiple model fit indices including the comparative fit index (CFI), root mean square error of approximation (RMSEA), Akaike information criterion (AIC), and Bayesian information criterion (BIC) (Akaike, 1987; Hu & Bentler, 1999; Kline, 2005; Schwartz, 1978). As recommended by Kline (2005) and Hu and Bentler (1999), we use CFI ≥ 0.9 and RMSEA ≤ 0.05 as our evaluation cutoff for a reasonable model fit. The AIC and BIC do not have as set guidelines, but can be used to compare model fit, with lower values being preferred (Akaike, 1987; Schwartz, 1978). All analyses were conducted with Mplus 8 (Muthén & Muthén, 1998–2017).

Based on the way the MCF describes the interconnectedness of topics (Vergnaud, 1983, 1988), specifically fractions, ratios, and proportional relationships, we hypothesized that teachers’ knowledge of these topics would be unidimensional. Meaning that the items would likely form a single construct of knowledge and thus teachers’ background characteristics would relate to their fraction, ratio, and proportional relationship knowledge similarly for the two major topics. Therefore, in our hypothesized model (see Fig. 1), teachers’ content knowledge of fractions, ratios, and proportional relationships is a unidimensional latent construct measured by teachers’ responses on 14 fraction and ratio items (i.e., measurement model). The relationships between teachers’ content knowledge and their professional backgrounds are examined by the path model on the left in Fig. 1. We contrasted this hypothesized model with a two-factor model that had the fraction items loading onto a latent factor and the ratio and proportional relationship items loading onto a separate factor to understand which more accurately represented the teachers’ content knowledge for this sample.

Fig. 1
figure 1

Hypothesized Model

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Results

First, we report the factor loadings of the items to understand which items effectively measured teachers’ knowledge of fractions, ratios, and proportional relationships. Then, we compare how the hypothesized model fits the data versus a two-factor model. Finally, we relate how the teachers’ background characteristics were associated with their content knowledge.

Measurement Model: the Structure of Teachers’ Fraction and Ratio Knowledge

Table 3 displays factor loadings of each item and model fit indices for our hypothesized latent structure of teachers’ content knowledge of fractions, ratios, and proportional relationships. PR4 and PR7 were found to have unsatisfactory loadings, so we tested whether their removal affected any of the results. We ran all the analyses with PR4 and PR7 included and excluded and found that there were no significant changes with or without their inclusion. Thus, we excluded PR4 and PR7 from the final analysis.

Table 3 Results of Measurement Model in the SEM Framework
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In general, the results of the analyses show that our hypothesized unidimensional model of teachers’ content knowledge of fractions and ratios was a reasonable fit to our data (CFI = 0.97 > 0.90; RMSEA = 0.02 < 0.05). For a robustness check, we also ran another SEM model in which we had the fraction items loading onto a fraction latent variable and the ratio and proportional relationship items loading onto a proportional relationship latent variable to test whether they are better represented by distinct constructs. Comparing these two models, we found model fit indices were similar for both models. Both the AIC and BIC were slightly lower for our hypothesized model (i.e., teachers’ knowledge of fraction and ratio as a unidimensional construct), indicating a better fit to the data. The result of the likelihood ratio test also showed that the two models did not differ statistically (p > 0.05). Taken together, the results from our measurement model in the SEM framework and the robustness check all favored the more parsimonious model, suggesting that teachers’ knowledge of fractions, ratios, and proportional relationships can be modeled as unidimensional in its latent structure. This result supports the MCF’s theorized structure of teachers’ knowledge of fractions, ratios, and proportional relationships as a single construct hereafter referred to as “multiplicative reasoning.”

Path Model: the Relationship Between Teachers’ Content Knowledge and Professional Backgrounds

Figure 2 presents the relationship between teachers’ knowledge of multiplicative reasoning and their professional backgrounds. The results suggested that teachers’ knowledge of multiplicative reasoning is associated with their path entering the teaching profession, type of teaching credential, and field of their bachelor’s degree. Specifically, mathematics teachers who entered the profession through a traditional preparation program outperformed teachers who entered the profession through an alternative path by 0.26 standardized deviations (p = 0.003 < 0.01) on average. Additionally, teachers who have their teaching credential in mathematics scored 0.31 standard deviation (p < 0.001) above teachers with a credential in K–6 on average. However, there was no significant difference in teachers’ knowledge between those with a credential in K–6 and special education. We found teachers with a bachelor’s degree in education scored 0.29 standard deviation below (p = 0.001 < 0.05) teachers who majored in other fields (e.g., math, science/engineering, psychology).

Fig. 2
figure 2

Standardized Results of Path Model in SEM Model. Note: Only significant predictors were displayed; p < 0.001***, p < 0.01**, p < 0.05*

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Discussion

The goal of this study was to investigate the relationships between teachers’ knowledge of two theoretically linked topics that are assumed to belong to one general construct, multiplicative reasoning. Our findings confirmed that the structure of teachers’ content knowledge of fractions and ratios can be modeled with one dimension. This result reinforces some of the findings of the prior studies investigating how teachers conceptualize the relationship between fractions, ratios, and proportional relationships. For instance, Clark et al. (2003) generated five theoretical models describing the different conceptions teachers and instructional materials forwarded as the relationship between fractions and ratios. Of the five, four of the models expressed the relationship as overlapping in some way, with only one model finding them completely distinct. Their study demonstrated that fractions and ratios are inextricably linked in many teachers’ conceptions and within common instructional materials (Clark et al., 2003). This tight linkage is similar to our hypothesized model where they are so related as to be best modeled by a single construct.

Similarly, Pitta-Pantazi and Christou (2011) demonstrated that these topics were closely aligned for a very different sample of teachers, as they found that 10 of the 11 factors contributed to teachers’ proportional reasoning. Many of their factors such as quotient, missing value proportional problems, reasoning up and down, and sharing/comparing covered fraction and ratio topics, indicating that they also found that teachers’ knowledge of these concepts is highly related. Including our study, there is a consistent theme of the overlapping relationship between these topics, which supports how the MCF relates and describes these topics (Vergnaud, 1983, 1988).

It is important to note that we looked at only two major domains within the MCF even though the MCF includes many other topics relating to multiplicative relationships. Therefore, future work is needed to investigate the structure of the entire MCF and to ensure that the unidimensional model holds when other concepts are included. In addition, although we recruited teachers from locations across the USA, our sample was not a nationally representative sample of teachers. Thus, more work is needed to investigate the extent to which these patterns would hold true with a nationally representative sample.

One implication of our findings is regarding the development and assessment of teachers’ knowledge, particularly when teachers’ knowledge is measured across content areas. Teacher knowledge assessments have been widely used and played a major role in policy decisions and their implications. Thus, having an accurate measure of teacher knowledge that depicts its role in teaching and student learning is instrumental. However, the development of an assessment that covers all the content areas teachers are expected to teach is difficult. In this study, we found that when the concepts are theoretically linked, teachers’ knowledge of these areas is similar compared to their varying levels of understanding when the content areas are different such as in numbers and operations and algebra (Hill et al., 2004). This finding can allow for a more strategic creation of assessments such that related content areas, which should have consistent knowledge levels, can be measured with fewer items without sacrificing accuracy.

Another implication of the one-dimensionality of teachers’ understanding of fractions, ratios, and proportional relationships is related to how we prepare teachers to teach these topics. Given that these concepts are both theoretically and (as we found) empirically connected, we advocate for an increased focus in teacher preparation programs to make explicit the mathematical connections between these topics. Instead of teaching fractions and ratios separately, we see value in discussing fractions, ratios, and proportional relationships together, identifying their similarities and subtle differences. This connected approach has been advocated by mathematics teaching reform efforts and scholars (NGACBP & CCSSO, 2010; National Council of Teachers of Mathematics, 2000, 2014; NRC, 2001) as it facilitates students’ procedural learning along with conceptual learning.

We also explored the association between teachers’ professional backgrounds and their knowledge of multiplicative reasoning. We found that being traditionally certified and credentialed in mathematics was significantly related to teachers’ understanding of multiplicative reasoning. Having a master’s degree and having a bachelor’s degree in education were either not related to or negatively related to teachers’ scores. These results contribute to the literature but demonstrate that there is still a lack of consensus about the relationship between teachers’ background characteristics and content knowledge. For instance, with similar content, Izsák et al., (2019) also found that being certified in mathematics was related to higher subject matter knowledge, but Copur-Gencturk (2021) and Weiland et al., (2012) did not. The inconsistency of results may indicate that the proxy measures of teachers’ background indicators may not capture qualitative differences in the programs teachers attended or the experience they accumulated that is associated with different content knowledge levels.

For our sample, traditionally certified teachers on average had more mathematical content knowledge than alternatively certified teachers. This contrasts with the findings of Izsák et al. (2019), who found that there was no difference between teacher performance for traditionally and alternatively certified teachers. It also refutes one of the main purported benefits of alternative certification, which is that alternative certification recruits higher quality candidates who have higher content knowledge and majors in the subjects they teach (Kanstroom & Finn, 1999; McKibbin & Ray, 1994). This difference may be a product of traditionally certified teachers’ extended time in preparation. Traditional certification routes are typically longer than alternative routes, providing traditionally certified teachers more opportunities to learn the content knowledge at a deeper level. This supports the many scholars and organizations that argue for the increased professionalization of teaching (Darling-Hammond, 2000; Darling-Hammond & Sykes, 1999; National Commission on Teaching and America’s Future [NCTAF], 1996, 1997), who claim that teacher preparation is one of the most important components in education.

Similarly, teachers who were certified in mathematics were found to have higher knowledge levels. Teachers who chose a mathematics certification may have a stronger mathematical background or be more interested in the topics than those who chose not to get a specified mathematics credential. Being certified in mathematics is a specialized credential for those that teach at a higher grade (typically grade 7 and above) in part due to their more focused course work, which could explain the positive relationship between mathematics certification and knowledge (Hill, 2010).

The other significant predictor of teachers’ multiplicative reasoning was having a bachelor’s degree in education. Unlike being traditionally certified or certified in mathematics, having a bachelor’s degree in education was negatively associated with teachers’ multiplicative reasoning. We find two plausible explanations for this finding. Since teachers were grouped by having a degree in education and not, students who had degrees in a mathematics heavy field such as engineering were grouped into the non-group. Thus, that group may have been composed of teachers who had a strong background and multiple classes in mathematics. Since past studies demonstrated that the number of mathematics classes taken was related to performance (Izsák et al., 2019), that could explain the negative finding. Another related possibility explaining this relationship is that having a degree in education does not necessitate more mathematics classes or mathematics specialization. Teachers might get general education degrees or focus on pedagogical strategies rather than content in their degree programs. More knowledge of pedagogy does not lead to increased content knowledge (Kleickmann et al., 2013), so if teachers with a degree in education were not focusing on math, then it would make sense to not see a significant positive relationship. From this data set, we were not able to determine if the teachers with education degrees had a high number of mathematics or content courses, which could explain why there was no significant relationship with teacher knowledge.

Conclusion

Our results indicate that teachers’ knowledge of fractions, ratios, and proportional relationships can be modeled as a single construct. Meaning that for this sample, teachers’ knowledge of fractions and their knowledge of ratios and proportional relationships were very similar. This work provides information about the structure of teachers’ subject matter knowledge while substantiating the MCF’s purported structure of these topics, as proposed by Vergnaud (1983, 1988). The unidimensional structure of these topics suggests that, at least for this situation, the theoretically related content is in fact indistinguishable. This is important in understanding how researchers view and measure teachers’ content knowledge since the content chosen can impact the measurement. For instance, if multiple topics are used and it is assumed that teachers’ knowledge is consistent across topics, it may lead to an inaccurate measurement unless the content is theoretically linked. More work needs to explore the relationship among teachers’ knowledge of related content to better understand what domains can be considered unidimensional.

This connected structure of teachers’ knowledge of fractions, ratios, and proportional relationships also suggests that there should be coherence between these topics in teacher preparation programs and school curricula. Teachers must be able to connect these topics so that they can help their students develop a conceptual understanding of the MCF, which is necessary for later mathematics success. We find that certification type (alternative vs traditional), teacher credential type, and undergraduate major are associated with a teacher’s multiplicative reasoning knowledge. This combination of factors may suggest the importance of adequate mathematics preparation in building teachers’ knowledge, as teachers with a certification in mathematics had higher knowledge scores on average. It also underscores how certification type may play a role in building teachers’ content knowledge. In general, teachers need robust content knowledge to teach their students, and understanding the relationship between teachers’ knowledge of different content can help inform future research on the measurement and development of this knowledge.