Enhancing mathematics teacher professional learning through a contextualized professional development program

ABSTRACT

This study investigated the impact of a year-long professional development (PD) program on mathematics teachers’ knowledge, beliefs, and practice. The PD program of the study was designed to connect mathematics teachers’ learning and teaching practice through integrating situated aspects of the teaching profession and teachers’ reflections. Data was collected from three cohorts of teachers across three program years. Results from repeated-measures ANCOVA show changes in self-efficacy, epistemological beliefs, and math pedagogical knowledge. Teachers’ content knowledge decreased in the first-year group but increased for teachers in Year 2 and 3 cohorts. Teaching practice improved about one standard deviation along the indices measured. Multiple regression results show that teachers’ self-efficacy and epistemological beliefs influenced their practice. The discussion includes assessing links between PD design and its effectiveness in influencing knowledge, belief, and practice change, as well as making the case for future PD programs to consider targeting multiple interconnected teacher outcomes.

KEYWORDS:

• Mathematics teacher education
• teacher competencies
• contextualized professional development program

Introduction

With professional development, teachers are expected to improve their teaching practice, but only incremental or short-term changes are usually reported. Research shows that mathematics teachers’ enactment of effective pedagogies is influenced not only by a teacher’s knowledge but also by their beliefs about teaching and learning (Ball, Lubienski, and Mewborn 2001; Berlin, Young, and Cohen 2021; Hill, Rowan, and Ball 2005; Remillard 2005; Richardson 1996; Shulman 1986; Thompson 1992). Even if teachers support theories of better teaching approaches, enactment does not always occur, possibly owing to other competing priorities or lack of capacity to do so (Golding 2017; Leatham 2006). Studies on teacher characteristics that influence teachers’ enactment of pedagogies found that change in practice demands changes in teachers’ knowledge, commitment, positive affective traits (e.g., emotional response – enthusiasm and energy), and external support by leadership and colleagues (Clarke and Hollingsworth 2002; Döhrmann, Kaiser, and Blömeke 2012; Golding 2017; Remillard 2005). This suggests the importance of teacher knowledge, beliefs, and disposition in changing their practice, and our study investigates the impact of a professional development on these teacher characteristics.

Another barrier to the enactment of teaching practice is the ‘double discontinuity’ in mathematics teacher education: ‘the missing connection of school mathematics and university mathematics and the lack of impact of university education on teaching practice in school’ (Kaiser et al. 2017, 162). Recognizing the missing link between a teacher’s learning and teaching practice, recent studies on mathematics teacher education pay increased attention to 1) situated aspects of the teaching profession, 2) teachers’ reflections, and 3) the general conditions of teacher education (Kaiser et al. 2017; Krainer and Llinares 2010). In order to foster mathematics teachers’ comprehensive professional competencies, these three trends need to be considered and should be integrated in a productive way in designing professional development programs for mathematics teachers (Kaiser et al. 2017; Krainer and Llinares 2010). The present study incorporated the three aforementioned considerations within the PD design.

The aim of this paper is to analyze the impact of a year-long professional development program (PD) on participating teachers’ perceived knowledge, beliefs about teaching and learning of mathematics, and self-reported instructional practice. We present our argument with data from three groups of teachers who participated in the year-long professional development program, during which we examined teacher perceived knowledge, beliefs, and self-reported instructional practice outcomes of the PD program.

Theoretical background

Many studies described classroom learning outcomes as being closely related to aspects of teaching practice (Berlin, Young, and Cohen 2021; Brophy and Good 1986; Wang, Haertel, and Walberg 1993) that are dependent on what teachers bring to the classroom, including their professional competence, beliefs, and attitudes (Campbell, McNamara, and Gilroy 2004; Shulman 1987). Researchers agree that cognitive, affective, and behavioral domains are important in teacher education and mutually influence each other (e.g., Jones et al. 2012; Jussim, Robustelli, and Cain 2009; Pajares 1992; Stipek et al. 2001).

Teacher knowledge

Professional competence is a critical factor in classroom practices (OECD 2009). Studies focusing on a cognitive perspective of teacher competence stress the importance of teacher knowledge. Shulman (1986, 1987) created a landscape of professional knowledge categories – content knowledge (CK), pedagogical content knowledge (PCK), and general pedagogical knowledge (PK) – and the framework is a guide to help develop teachers’ competence and effective teaching in practice. Scholars have suggested that these knowledge categories should be developed in domain-specific ways (e.g., mathematics versus science). For example, mathematics teachers should have opportunities to develop knowledge about mathematics content, teaching of mathematics, and students’ mathematical thinking (Darling-Hammond et al. 2009). Teacher education programs valuing a cognitive perspective of teacher competence have attempted to evaluate the effectiveness of their program through measuring knowledge needed for mathematics teachers to teach mathematics (e.g., the Mathematical Knowledge for Teaching, MKT; Ball, Thames, and Phelps 2008; Fauskanger 2015), which emphasizes the importance of mathematics teachers’ pedagogical knowledge specific for teaching mathematics.

Other scholars have emphasized that teachers’ knowledge development is context specific and involves multiple facets. For example, the framework Teachers’ Knowledge Developing in Context used by Fennema and Franke (1992) describes teachers’ knowledge as situated. That is, teacher knowledge is inexplicably tied to the classroom contexts – such as content domains, student characteristics, and school environments – in which it can be applied. Other influential studies on teachers’ knowledge characterizes teaching as a knowledge-intensive domain with different knowledge and motivational facets (Schoenfeld 2011; Schoenfeld and Kilpatrick 2008).

Another group of studies on teachers’ knowledge support the relationship between teachers’ knowledge and their teaching experience. Elbaz (1983) viewed that practical knowledge is based on first-hand experiences and is represented in practice. Schön (1983, 1987) also suggested the importance of experience in developing professional knowledge using the concept of knowledge-in-action, meaning professional knowledge is developed within (instructional) action. Even though Schön’s theory may be interpreted variously, most agree with his contribution in recognizing that teachers learn from experience.

Teacher beliefs

Across many years, research has revealed a well-established link between teachers’ mathematical beliefs and their instructional practices (Bird et al. 1992; Buehl and Fives 2009; Handal and Herrington 2003; Philipp 2007; Philippou and Christou 1998; Raymond 1997). Despite such growing interest in teacher beliefs, there is still a great deal to be learned about the nature and role of these beliefs in teachers’ professional learning and knowledge development. Two belief constructs relevant to this study include self-efficacy beliefs and epistemological beliefs of teaching mathematics.

The first belief construct, self-efficacy, is grounded in Bandura’s (1986; Bandura 1997) social learning theory. Teaching efficacy is considered to be a teacher’s belief in their ability to teach effectively (Tschannen-Moran and Woolfolk Hoy 2001), which influences their instructional practices and student achievement (Nurlu 2015; Yilmaz‐Tuzun and Topcu 2008). Efficacy beliefs are categorized as teachers’ beliefs about controlling student achievement and motivation (Rotter 1966) or beliefs about their capacity to affect student performance (Bergman et al. 1977). Teacher education programs should be designed with the consideration that ‘teachers who believe student learning can be influenced by effective teaching (outcomes expectancy beliefs) and who also have confidence in their own teaching abilities (self-efficacy beliefs) should provide a greater academic focus in the classroom’ (Gibson and Dembo 1984). Some researchers have specifically found teaching efficacy to be a relevant factor of good instructional practices and student outcomes (Borko et al. 1992; Farncis, Rapacki, and Elker 2014; Pajares 1992; Peterson et al. 1989). While existing studies have confirmed the importance of developing math teaching efficacy, this belief is content dependent and may require specific strategies for teachers from different content domains (Bandura 1997; Tschannen-Moran and Woolfolk Hoy 2001). Our PD program focused on this type of domain-specific development of teaching efficacy for math teachers.

The second belief construct related to this study is epistemological beliefs, which is related to pedagogical beliefs including the beliefs that teachers have regarding how they should teach and how students should learn. They are rooted within teachers’ conception of the nature of knowledge, how we come to know something, and how knowledge is justified (Hofer 2004). Epistemological beliefs can develop from naïve to more sophisticated conceptions. Specifically, epistemological belief change can be described as the development of one's meaning making to adapt to expanding notions of the complexity of knowledge (Bendixen 2002). That is, moving from perceiving knowledge as dualistic (i.e., true versus false) and originating from experts to an understanding that knowledge may be tentative and mutually validated.

Studies on epistemological beliefs can be divided in two areas, domain-general beliefs or domain-specific beliefs (i.e., general beliefs across disciplines vs. discipline-specific beliefs). Domain-general epistemological beliefs (Cooney and Shealy 1997; Schommer 1990; Spillane and Zeuli 1999) are ‘individuals’ beliefs about the nature of knowledge and the process of knowing’ (Hofer and Pintrich 1997, 117). Domain-specific (subject-specific) epistemological beliefs emphasize that each academic subject includes epistemological issues regarding what knowing means in the subject and how knowledge in the subject should be developed (Calderhead 1996; Schoenfeld 1992). The importance of developing both domain-general and domain-specific epistemological beliefs is addressed in many studies (Buehl and Alexander 2001; De Corte, OptEynde, and Verschaffel 2002; Schraw 2001). For example, the notion of teaching mathematics for conceptual understanding through problem solving can be promoted by developing both general and specific epistemological beliefs (Gill, Ashton, and Algina 2004; Hiebert et al. 1996; NCTM 2000, 2014; Yurekli et al. 2020). The assumption is that domain-general epistemological beliefs play an important role in promoting conceptual change (Pintrich 1999) and have an influence on specific epistemological belief change in mathematics (Cooney and Shealy 1997). Thus, our PD program aimed to foster math teachers’ development of positive domain-general and domain-specific epistemological beliefs.

Development of the PD program of the study

The PD of the present study – Teaching Algebra in Context, Community, and Connections (TACCC) – aimed to improve mathematics teachers’ competencies by removing the gap between what teachers learn and how teachers teach. As illustrated in Figure 1, TACCC was designed to fill the gap and to improve teacher knowledge and aid development of teacher beliefs by providing teachers with Opportunities to Learn and Opportunities to Reflect, which puts in practice the three conditions linking teachers’ learning and practice: (1) situated aspects of the teaching profession, (2) teachers’ reflections, and (3) the general conditions of teacher education (Kaiser et al. 2017; Krainer and Llinares 2010). Opportunities to Learn and Opportunities to Reflect are delivered through PD content and activities as well as structured application of PD knowledge.

Figure 1. Conceptual framework of the study.

Situated aspects of the teaching profession

Situated learning theory suggests that teaching should take place in the social context where new knowledge is to be applied (Korthagen 2010; Langer 2009), and teachers’ professional learning ‘should be embedded in their everyday context while they are engaged in daily instruction’ (Amendum and Liebfreund 2019, 343). TACCC implemented the situated aspects by responding to the data collected from participating teachers and the mathematics education field, as well as connecting ‘learning to teach’ and ‘learning by teaching,’ that is, connecting in-university and on-site PD activities (Lee et al. 2018).

Responding to data

The ultimate goal of the PD program was to improve the quality of mathematical learning and teaching in our schools. The quality of student learning is not only influenced by what a teacher does in the classroom, but also the whole package ‘that a teacher brings into the classroom. Enthusiasm, energy, self-awareness, and open-mindedness, have a tremendous influence on students and their learning’ (Curtis and Szesztay 2005, 4). As discussed in the previous section, teachers’ knowledge, beliefs, and dispositions are largely formed during an individual’s past experiences with previous teacher education program activities; successful performances strengthen these beliefs while failures weaken them. Therefore, PD contents and activities should be selected to challenge teachers’ existing knowledge, beliefs, and dispositions about mathematics and eventually improve their practice.

Also, in order to motivate and engage teachers to learn, PD activities should be learner centered (Polly, Neale, and Pulalee 2014) and respond to the needs of participating teachers and their students (Lee 2004). The focal mathematical concepts and pedagogical skills of TACCC were selected based on the TACCC pre-survey data and recommendations of the mathematics education field that respond to test outcomes. Both local teachers and the larger community of mathematics education recognized the need for improving algebraic skills of middle school students. The results from American Diploma Project (ADP) and National Assessment of Educational Progress showed that students who take higher-level mathematics courses (algebra) in earlier grades perform better than students who take the same course(s) later in their high school career (Achieve 2010; Rampey, Dion, and Donahue 2009). As part of the educators’ endeavor to prepare our students for the new world, the ADP Network in the USA has developed new requirements: students need to take four units (20 credits) including one unit of Algebra II, or its equivalent, to get a high school diploma. In order to respond to the requirements, changes need to be made not only in high schools, but also in middle schools. Some middle schools have to teach an Algebra course in their building. For example, teachers have to move some of the eighth-grade state standards (indicators) to seventh grade, seventh-grade indicators to sixth, and so on. The requirements and all these consequent adjustments require both middle school and high school teachers to acquire more mathematics content knowledge. TACCC’s focal concepts were geared toward mathematics concepts needed to improve middle school students’ algebraic thinking skills.

Connecting learning to teach and teaching practice

Principles to Actions (National Council of Teachers of Mathematics (NCTM) 2014) provide a vision and recommendations of the instructional practices in mathematics classroom, and Standards for the Preparation of Middle Level Mathematics Teachers (NCTM 2020) offer standards to determine teacher candidates’ readiness. Both standards recommend that the role of teachers is to hear student ideas, let students construct their own meaning, and assess student progress in alternative ways, which require teachers’ mathematical knowledge as well as ability to notice students’ learning. As mathematical problems can be approached in many different ways, students at many different levels of mathematical maturity can work on similar (or identical) problems, each in their own way. In order to make sense of formal mathematics, students need opportunities to make connections to social or personal contexts, as well as to other subjects or within mathematical areas (Lee 2007). Thus, teachers (facilitators) themselves should also be able to see connections between mathematical concepts within mathematics, with other subject areas, and to our daily lives. TACCC teachers involved themselves in revisiting standards, exploring vertical progression of mathematics concepts, investigating horizontal connections of mathematics concepts with other subjects, and examining how algebraic concepts are used in other subjects and in our daily life.

The most effective ways to change teaching approaches occur if teachers themselves do mathematics differently and can transform their learning to teach into their teaching practice. TACCC had teachers involved in the PD as learners (learning to teach) and teachers (learning by teaching). As learners, teachers would solve a mathematics problem in various ways and then would analyze their own work before using the same or similar problem in their own teaching. Then, the same or similar problem was adopted in their own classroom, and students’ performance on the problem was analyzed. This activity could improve teachers’ understanding of how to develop rich tasks and ability to notice students’ learning, improve teachers’ understanding of their students’ developmental level, and lead them to change teaching practice (Lee 2016).

Ongoing reflections

Changing teachers’ beliefs is hard, and that contributes to the difficulties in changing practice (Ball 1990; Bird et al. 1992; Handal and Herrington 2003; Philipp 2007). Farrell (2001) argued that teachers would ‘take more responsibilities for their actions in the classroom’ (23) if they became reflective practitioners and learned about their own beliefs of teaching and learning. However, reflection does not occur naturally, and ‘reflective capacity … has to be learned and encouraged’ (Gelter 2003, 337). Reflection is highly personal and represents a teacher’s epistemological and self-efficacy beliefs. Thus, teachers should have Opportunities to Reflect on their own beliefs including their ability to do mathematics, beliefs about mathematics teaching and learning, and beliefs about mathematics (Drake and Sherin 2006).

TACCC promoted reflection, in both verbal and written form, as teachers engaged in ‘learning to teach’ and ‘learning by teaching.’ More specifically, TACCC teachers reflected on their own learning of mathematics, teaching approaches, and beliefs about teaching and learning. As shared earlier, teachers solved mathematical problems, reflected on their own mathematical solutions, and discussed alternative solutions with other teachers. As teachers engaged in these activities, they reflected on their mathematical learning and developed a better understanding of how student learning would progress with similar mathematical concepts/problems. Reflection on teaching involved observing one’s own recorded teaching and research-based lessons (video-taped, released by an National Science Foundation-funded project team). After observing each lesson, teachers analyzed the instruction, reflected on the recorded teaching, and shared their observations of instructional approaches with other teachers. Teachers also reflected on their own beliefs about mathematics, learning mathematics, and teaching mathematics through verbal communications and written responses to reflection prompts.

The general conditions of teacher education

To impact teachers’ beliefs and knowledge and eventually change their practice, PD developers and providers should consider programmatic structure, delivery methods, participants’ role, duration of the PD, systematic aids, collective participation, and communities of practice (Ball 1996; Brown 2012; Darling-Hammond 2009; Desimone et al. 2002; Desimone 2009; Garet et al. 2001; Lee 2004, 2007; Loucks-Horsley et al. 2010).

Ongoing longer duration

TACCC provided an ongoing year-long hybrid PD, including face-to-face monthly workshop meetings and online PD between monthly meetings in spring, a week-long intensive summer institute, monthly face-to-face workshop meetings and online PD between monthly meetings in the fall, and follow-up data collection during the winter. Thus designed, TACCC adopted a job-embedded PD to connect teacher learning with classroom practices – an intervention designed to be embedded within the daily practice of the profession.

Collective participation

Collective participation is considered as an effective strategy for teacher learning, which extends what they learn from formal professional development experiences, helps to build trust and supportive relationships, and motivates working through problems of practice together (Desimone et al. 2002; Garet et al. 2001; Little 1993). TACCC recruited two to four teachers from the same school building or district, which promoted teachers’ professional communities of practice and collaborative learning.

Supporting sustained learning through coaching/mentoring

Sustaining teachers’ learning beyond the PD is as important as helping teachers gain knowledge and change their practice during the PD. While teachers’ growth rate and amounts of new learning retained following PD vary (Goldschmidt and Phelps 2010), assisting teachers through job-embedded and ongoing support such as coaching shows promise (Artman-Meeker et al. 2015). TACCC provided mentoring/coaching to help teachers implement what they learned in the PD into their classroom practice. PD providers were paired with a current classroom teacher, a retired classroom teacher, and a higher education mathematics education faculty. PD providers played the role of mentor and coach during three site visits – one each semester – during the PD. Having current and former classroom teachers was a source of meaningful guidance in improving teaching (Lieberman and McLaughlin 1992).

Methods

Research questions

Employing a mixed-method research design, this study examined how middle school mathematics teachers, who participated in a year-long professional development program, Teaching Algebra in Context, Community, and Connections (TACCC), perceive their knowledge, beliefs, and teaching practice. The study was grounded in the following research questions:

1. Are there pre/post PD differences for teachers’ perceived knowledge, beliefs, and self-reported instructional practice?

2. Do the pre/post PD differences for teachers’ perceived knowledge, beliefs, and practice vary over three iterations of the program?

3. How do teachers’ post-program perceived knowledge and beliefs influence their enactment in practice?

Participants

Eighty middle school math teachers from 17 school districts participated in the TACCC program. Each teacher participated in only one year of the program, thus there were three cohorts of teachers across three program years (Year 1: n = 27, Year 2: n = 23, Year 3: n = 30). Each year, data was collected on the first day and at the end of the program. Weblinks for the electronic survey were emailed to teachers, and they completed the surveys anonymously. Most participants were female, and all but one teacher identified themselves as White-Caucasian. There was a diversity in participating teachers’ teaching grade levels, roles (position), and classroom types. The middle school license in Ohio covers Grades 4–9, so, while the PD program targeted the traditional middle school Grades 5–8, 13 of the teachers enrolled taught 4th grade and 12 taught 9th grade.

Data collection: measures

To measure teachers’ knowledge, efficacy beliefs, epistemological beliefs, and practice, the study used three sets of surveys and an open-ended reflection: a modified Mathematics Teaching Efficacy Beliefs Instrument (MTEBI; developed by Enochs, Smith, and Huinker 2000), a state-developed survey (https://www.ohiohighered.org/itqp), and an open-ended self-report reflection.

Knowledge

Teachers’ perceived knowledge was measured using items from the degree of confidence subsection in the Improving Teacher Quality Program survey from the Ohio Board of Regents. Two items measured teachers’ math pedagogical knowledge, and one item measured math content knowledge. Items were assessed using a Likert scale from 1 = strongly disagree to 5 = strongly agree. Higher scores represented higher perceived knowledge. A sample item for math content knowledge was: ‘I have a good understanding of fundamental core content in my discipline.’ In addition, teachers were asked to reflect on growth/changes in their knowledge in the final reflection.

Epistemological beliefs

Teachers’ epistemological beliefs were measured using items from the Beliefs subsection in the Improving Teacher Quality Program survey from the Ohio Board of Regents. Three items measured epistemological beliefs toward teaching and assessed the extent to which teachers supported student-centered instruction. Three additional items measured epistemological beliefs toward learning and assessed a teacher’s views on the active role of students in their own learning. Items asked teachers to mark from 1 to 5 the extent to which the viewpoint at each end of the continuum matches their teaching approach. A sample item for epistemological beliefs toward teaching was at one end, 1 = classroom interaction consists of teacher-led lecture with limited response from students, to the other end, 5 = classroom interaction involves a dialogue among teachers and students.

Self-efficacy

Teachers’ self-efficacy toward teaching math was measured using 12 items from the Personal Math Teaching Efficacy (PMTE) subscale of the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI). The Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) was developed by Enochs, Smith, and Huinker (2000) to investigate subject-specific teacher efficacy. The PMTE subscale asked from the first-person perspective about the degree to which the teacher was confident in teaching mathematical concepts. Items were measured using a Likert scale from 1 = strongly disagree to 5 = strongly agree, with higher scores representing higher perceived efficacy. A sample item is, ‘I know how to teach mathematics concepts effectively.’

Practice

Teachers’ classroom practice was measured using a self-reported final reflection on changes in their teaching and changes noticed in their students, as well as a professional activities subsection in the Improving Teacher Quality Program survey from the Ohio Board of Regents. Practices listed in the survey were subdivided into five categories: using recommended practices (3 items), encouraging student engagement (4 items), connecting content to students’ experiences (4 items), using alternative assessment methods (1 item), and engaging in continued professional learning (4 items). Items were measured using a frequency Likert scale from 1 = never to 3 = frequently. Higher scores represented more frequent engagement in adaptive practices.

Data analysis

In order to explore the first two research questions, pre- and post- scores on knowledge, belief, and practice indices were analyzed using repeated-measures Analysis of Covariance (ANCOVA; Wilcox 2017) with STATA ver. 12. Mean differences in the target constructs over time (pre vs. post) were examined, controlling for the implementation year. Further, interactions between time and implementation year were specified to test whether improvement in scores differed for each iteration of the program. The Greenhouse–Geisser correction was used to account for non-independent observations from repeated measures data. Teachers’ teaching grade level (secondary level vs. lower) was added as a control variable.

To answer the third research question, multiple regression analysis (Keith 2019) was carried out with STATA ver. 12. Five regression models were specified, each with one type of practice entered as the dependent variable. Practice scores after teachers participated in the program were used. Focal predictors included perceived knowledge, epistemological beliefs, and self-efficacy assessed at post-program. Standardized composite of math content and math pedagogical knowledge makes up the perceived knowledge index used in the regression. A standardized composite of epistemological beliefs for teaching and learning was used as the epistemological beliefs index. Study year and teaching grade level were first included as control variables. However, teaching grade level was not a significant predictor of any practice outcome and was removed from the final regression models for parsimony.

To supplement quantitative data, teachers’ responses in their final reflection data were also analyzed. Qualitative responses were not analyzed to test any pre-existing hypotheses or theories. Instead, we sought patterns in teachers’ noticing of own learning, teaching, and their students’ learning in order to find anecdotal statements showing the relationship between knowledge, beliefs, and practice.

Results

Table 1 presents descriptive statistics on our variables of interest. Results showed that teachers perceived to have more math content knowledge compared to math pedagogical knowledge, although both types of knowledge perception improved after program participation. In terms of beliefs, teachers scored high on average on self-efficacy for mathematics teaching, scoring 4.22 out of the possible 5 before program participation and 4.65 after program participation. Scores for epistemological beliefs for teaching and learning matched one another and also changed over the course of the program. In terms of practice, teachers scored relatively high on using recommended practice, connecting content to students’ experiences, and engaging students. They scored lower relative to other types of practice on using alternative assessment and engaging in professional learning.

Table 1. Descriptive statistics.

	Pre-Program			Post-Program
	Mean	SD	Min–Max	Mean	SD	Min–Max
Knowledge
Math Content	4.21	0.52	3.00–5.00	4.42	0.62	2.00–5.00
Math Pedagogical Knowledge	3.83	0.64	2.00–5.00	4.21	0.53	3.00–5.00
Beliefs
Epistemological Belief–Teaching	3.70	0.60	2.00–5.00	4.07	0.59	2.50–5.00
Epistemological Belief–Learning	3.60	0.69	2.00–5.00	3.99	0.68	1.70–5.00
Self-Efficacy	4.22	0.66	2.60–5.30	4.65	0.51	3.40–6.00
Practice
Using Recommended Practice	2.37	0.40	1.00–3.00	2.66	0.32	1.70–3.00
Connecting Content	2.48	0.32	1.50–3.00	2.79	0.29	1.80–3.00
Using Alternative Assessment	1.82	0.58	1.00–3.00	2.23	0.65	1.00–3.00
Engaging Students	2.19	0.38	1.00–3.00	2.67	0.33	1.80–3.00
Professional Learning	1.47	0.48	1.00–3.00	1.92	0.54	1.00–3.00

All scores are averaged across items measuring the same construct.

Perceived knowledge change

In answer to our first research questions, ANCOVA results showed that teachers’ math pedagogical knowledge improved pre- to post-program participation (Table 2, Figure 2). The mean score increase was calculated in terms of standard deviation units for ease of interpretation and indication of effect size. The change was significant and of relatively large effect size, ranging from 0.32 to 1.01 standard deviation units increase across years. ANCOVA results also showed that changes in math pedagogical knowledge differed across implementation year. Teachers in year three appeared to improve more on this index compared to those who participated in the first two years of the program. There was also a significant change in teachers’ math content knowledge pre- vs. post- program, and the interaction between time and implementation year was significant. However, whereas teachers’ perceived content knowledge decreased in Year 1, teachers who participated in the program in Years 2 and 3 improved in their perceived math content knowledge score. Although all teachers in Year 1 started the program saying that they had a good understanding of fundamental core content in their discipline, this dropped slightly by year-end (11%), which may suggest that the course made them aware of some gaps in their knowledge. We were unable to confirm why only Year 1 but not Years 2 and 3 teachers’ perceived knowledge decreased but suspected that it could be due to the grade level they were assigned to teach. According to the baseline data, more teachers in Year 1 were teaching upper-level mathematical concepts than teachers in Years 2 and 3.

Table 2. Repeated-measures ANCOVA of knowledge, belief, and practice indices.

		Model Statistics				Pre vs. Post (Time)				Time * Year Interaction
		SS	MS	R^2	F(df)	Std diff	SS	MS	F(df)	SS	MS	F(df)
KNOWLEDGE
	Content Knowledge	35.16	0.47	0.77	2.67(75)***	−0.17–0.71	0.86	0.86	4.88(1)*	2.51	1.25	7.14(2)**
	Pedagogical Knowledge	41.53	0.55	0.79	3.08(75)***	0.32–1.01	3.40	3.40	18.90(1)***	1.65	0.83	4.59(2)*
BELIEF
	EP Belief – Teaching	38.30	0.51	0.72	2.10(75)**	0.42–0.70	5.10	5.10	20.99(1)***	0.47	0.23	0.96(2)
	EP Belief – Learning of Math	60.02	0.80	0.84	4.36(75)***	0.31–0.73	4.95	4.95	26.98(1)***	0.68	0.34	1.86(2)
	Self-Efficacy	38.07	0.51	0.78	2.86(74)***	0.63–0.76	3.58	3.58	19.88(1)***	0.18	0.09	0.50(2)
PRACTICE
	Using Research-based Recommended Practice	13.98	0.19	0.71	1.95(74)**	0.40–1.04	2.68	2.68	27.65(1)***	0.24	0.12	1.24(2)
	Connect the Lesson/Content with Students	10.70	0.14	0.74	2.25(74)***	0.51–1.14	2.70	2.70	41.99(1)***	0.17	0.09	1.33(2)
	Alt. Assessment	38.34	0.52	0.70	1.84(74)**	0.40–0.94	3.99	3.99	14.21(1)***	1.03	0.52	1.83(2)
	Engage Students	20.71	0.28	0.83	4.03(74)***	0.95–1.39	5.89	5.89	84.83(1)***	0.12	0.06	0.87(2)
	Sustained Active Professional Learning	32.37	0.44	0.80	3.27(74)***	0.66–1.17	5.16	5.16	38.52(1)***	0.03	0.02	0.12(2)

*p < 0.05; ** p < 0.01; ***p < 0.001. Repeated measures performed with the Greenhouse–Geisser correction, controlling for teaching grade level. Std diff represents the range in standardized scores difference from pre vs. post surveys (post score – pre score) across three years of intervention. SS refers to the model sum of squares whereas MS refers to the mean squares. F refers to the F-statistic and df to the degrees of freedom in each model. Critical F statistics for hypothesis testing depend on the model degree of freedom and alpha level, and can be accessed in statistical texts such as Wilcox (2017).

Figure 2. Teacher-perceived knowledge change.

Facilitative beliefs change

In addition, all indices of teachers’ beliefs significantly changed after participation in the program (Table 2, Figure 3). Change in self-efficacy pre- to post- program was the largest in terms of standardized scores, ranging from a 0.63 to 0.76 standard deviation-unit increase across three years. Epistemological belief for teaching and learning also changed over time. Increase in epistemological beliefs for teaching ranges from 0.42 to 0.70 standard deviation units, whereas increase in epistemological beliefs for learning ranges from 0.31 to 0.73. Teachers’ beliefs changed toward increased agreement with the positive relationship between students’ achievement with teacher’s extra effort, attention, and instructional approaches. Also, these findings show that teachers became more confident in their own instructional skills, but their views about how students learn math effectively changed to a smaller degree. The opportunities that the TACCC PD provided to learn and reflect on how students learn math appeared to change teachers’ belief in the effectiveness of student-centered teaching and the importance of students taking active roles in their own learning.

Figure 3. Teacher belief and practice change.

Teaching practice improvement

In terms of practice, ANCOVA results showed that all indices of practice we measured significantly improved after program participation, including utilizing research-based recommended resources (learned from PD), encouraging student engagement, connecting the lesson/content with learners, using alternative assessment, and sustained active professional learning. Overall, improvements in practice scores showed relatively large effect sizes, ranging from an increase of 0.40 to nearly 1.5 standard deviation units. Comparatively, teachers showed the largest gains in engaging students relative to other types of practice change, showing 0.95 to 1.39 standard deviation-unit increase across years. This reflects the focus of the TACCC PD to help teachers encourage students to solve mathematical problems in their own way and to see connections between math and their daily lives. Teachers made the least, although still substantial, gains in using alternative assessments, improving 0.40 to 0.94 standard deviation units across three years.

Perceived knowledge and beliefs predict teaching practice

To answer the third research question regarding the predictive quality of perceived knowledge and beliefs on post-program self-reported practice, we built five multiple regression models with each of the five practice indices specified as the outcome variable, in turn. We examined how perceived knowledge, epistemological beliefs, and self-efficacy assessed after program participation predicted these teaching practices. Results showed that perceived knowledge consistently predicted all five types of practice (Table 3), including using recommended practice (standardized β = 0.23, all p < 0.05), connecting content to students’ experiences (standardized β = 0.17), using alternative assessments (standardized β = 0.16), engaging students (standardized β = 0.22), and participating in professional learning (standardized β = 0.18). Also, even after perceived knowledge was controlled, teachers’ epistemological beliefs also predicted practices in some models. Epistemological beliefs positively and significantly predicted the extent to which teachers connect math content to students’ experiences (standardized β = 0.19), engage students in math learning (standardized β = 0.18), and participate in continued professional learning (standardized β = 0.15). Self-efficacy was not a significant predictor of practice when the influence of perceived knowledge and epistemological beliefs were also accounted for.

Table 3. Multiple regression of teacher practice.

	Rec. Practice	Connect Content	Alt. Assessment	Engage Students	Prof. Learning
Knowledge	0.23** (0.07)	0.17* (0.07)	0.16* (0.07)	0.22** (0.07)	0.18* (0.07)
Epistemological Belief	0.02 (0.07)	0.19** (0.07)	0.13 (0.07)	0.18* (0.07)	0.15* (0.07)
Self-Efficacy	−0.03 (0.13)	−0.16 (0.13)	−0.12 (0.13)	−0.08 (0.13)	0.01 (0.13)
Study Year 2	0.54 (0.28)	0.34 (0.28)	0.02 (0.30)	0.19 (0.28)	0.26 (0.28)
Study Year 3	0.64* (0.25)	0.60 (0.25)	0.27 (0.26)	−0.04 (0.25)	−0.14 (0.26)
Constant	−0.40* (0.8)	−0.34 (0.18)	−0.09 (0.19)	−0.04 (0.18)	−0.02 (0.19)

Standardized Regression Coefficient (standard error): *p < 0.05; ** p < 0.01; ***p < 0.001

Study Year 1 used as reference group. Rec. Practice refers to using research-based recommended practice. Connect Content refers to connecting the lesson content with students’ experiences. Alt. Assessment refers to using alternative assessment methods. Engage students refers to encouraging student engagement. Prof. Learning refers to engaging in continued professional learning.

Qualitative trends in teacher reflection

At the end of the program, teachers were asked to reflect on growth in their knowledge. Common themes found were improvements in own understanding of mathematics, self-confidence in mathematics, and confidence in teaching. Some teacher comments noted that changes in teaching approaches were due to their own improved understanding of mathematics: ‘I now have a better understanding of algebra and other mathematical concepts. Math makes more sense to me,’ ‘I am able to see how depth is better than width in relation to teaching concepts. If you go deep enough, you’ll “hit” the outlying concepts somehow,’ and ‘It helped explain the why instead of just knowing formulas.’ These teacher responses reflected quantitative trends which showed that many teachers (particularly in Years 2 and 3) reported improvement in content knowledge after participating in the program. Some teachers also reported a ‘shift’ in their understanding of math: ‘I was taught to look at mathematical content differently than I had done before,’ and ‘I realized I had forgotten some important content knowledge, which would help students understand the whys of concepts.’ By understanding math content in a different way or noticing math concepts they had forgotten, teachers may have been more aware of the gaps in their content knowledge after program participation. This potentially explained why some teachers (particularly in Year 1) reported a slight decrease in content knowledge after the program.

Some teachers’ comments emphasized the growth in their content knowledge and changes in teaching as results of improved confidence: ‘I am feeling much more confident about my content knowledge after this class. … I still have much to learn and am eager to learn more’ and ‘I am more confident with varying teaching strategies and trying new things. I have a greater understanding of what I need to do to help my students be successful.’ These teacher responses corroborate the relatively large increase in self-efficacy teachers reported between pre- and post- surveys.

Participants were also asked to reflect on changes observed from their students after they participated in the project. Encouragingly, by year-end, several revealed that they had succeeded in getting their students to buy in to the changes. Some teachers noted changes in students contributing to a richer learning environment and engaging in deeper learning: ‘[S]tudents were resistant at first because I was requiring higher-level thinking responses and made even those who did not know the answer respond. My class has now become a much richer learning environment with very positive discussion,’ and ‘[M]any of my students still struggle with mathematical concepts, but they are becoming much better at showing their work and explaining their thought processes. This makes it much easier for me to pinpoint the problem and help them to better understand.’

Additionally, some teachers appreciated changes in students promoting more active student engagement and more mathematical discourse: ‘[M]y students are really enjoying math. They are becoming much more comfortable with communicating with each other and using good math vocabulary,’ and ‘[M]y students have a clearer understanding of concepts and have conveyed that to me by extended responses, performance, and discussion. Their confidence levels have also grown,’ ‘[T]hey are talking more. I am talking less,’ and ‘[S]tudents like to be involved. They enjoy showing their work and explanations, including the wrong answers!’

In summary, teachers’ reflection focused more on their students’ attitude toward mathematics and learning of mathematics than improved test scores. They noticed that students participate in mathematical dialogue and share actively, focus more on process than immediate answer, demonstrate growing confidence in mathematics, and have less fear with new situations or concepts. The improvement TACCC teachers reported in students’ attitude and engagement was also reflected in quantitative results, with teachers reporting the largest increase in engaging students among all practice indicators.

Discussion

The ultimate goal of a professional development is to enhance a teacher’s instructional practice in the classroom environment. Math education scholars agree that teaching practice is guided by teachers’ self-efficacy beliefs and epistemological beliefs. Most studies pay attention to either self-efficacy or epistemological beliefs to measure the effectiveness of a program. This study examined the effectiveness of the program in inducing changes in teacher knowledge, beliefs (efficacy & content-specific epistemological beliefs), and teaching practice. We also integrated situated aspects of the teaching profession and contextualized effective PD strategies for mathematics as a specific content domain and for participating teachers’ personal characteristics. TACCC contextualized PD activities to connect participating teachers’ learning to teach and teaching practice, as well as to serve participating teachers’ teaching context and needs.

As discussed in the Results section, there were statistically significant differences between pre- and post-surveys in all three areas. Results support that providing teachers with connecting PD activities in the university and on site, ongoing mentoring, collaborative participation, promoting ongoing reflection were effective ways to change teacher knowledge, beliefs, and practice. These PD approaches were combined with the long-term, job-embedded PD design allowing for integration of program recommendations with teachers’ own practice, followed by guided reflections of how their changing teaching practices are linked to student outcomes. This PD design appeared to be effective at changing both teacher knowledge and beliefs, each of which have been shown to be critical toward enabling a teacher to structure a student-centered learning environment. Trends in qualitative data especially pointed to teachers’ reflections being key for improved self-evaluation of knowledge gains and understanding of the importance of student-led instruction. Moreover, results also showed positive changes in all of the practice indices measured, especially in how teachers were able to engage students. Again, PD features allowing teachers to integrate PD content with their teaching may have helped facilitated practice change.

The findings of this study confirm the significant relationships among teacher knowledge, beliefs, and practice. In particular, teacher knowledge and epistemological beliefs were significant predictors of practice in our study. These results were consistent with previous research that found a close relationship between teacher beliefs and practice (e.g., Buehl and Fives 2009; Handal and Herrington 2003; Philipp 2007; Vongkulluksn, Xie, and Bowman 2018). Additionally, our results point to the effectiveness of targeting both teacher knowledge and beliefs in facilitating a change in teacher practice. When teachers have both the capacity to affect change thanks to increased knowledge, as well as a willingness to do so because adaptive practices are aligned with their beliefs, they are more likely to enact and integrate PD-recommended strategies in their classroom. Interestingly, our study found that teachers’ efficacy was not a significant predictor of teacher practice when knowledge and epistemological beliefs were controlled. Several factors may help explain this trend. First, teachers who participated in our study began with already high levels of self-efficacy, and additional increases may not have translated to improved practice. Second, as previous studies have pointed out, teacher efficacy and perceived knowledge are highly related (e.g., Abbitt 2011; Oppermann, Anders, and Hachfeld 2016; Swackhamer et al. 2009). It is possible that teacher knowledge was the more proximal predictor of teacher practice. Self-efficacy, when included in the same regression model as knowledge, may not have contributed to additional explained variance. To be clear, this is not to say that PD programs should not focus on improving self-efficacy. Rather, teacher knowledge and self-efficacy gains may go hand-in-hand, and the influence of one on teacher practice may be difficult to be distinguished from another. Our study added to the growing literature that aspects of teachers’ internal factors including knowledge and beliefs do have a significant influence on their approaches to teaching.

Results in the study also showed a decrease in teachers’ perceived knowledge in Year 1. This is not necessarily a negative outcome of the project because teachers’ knowledge improved in qualitative data-driven analysis, and teachers had different opportunities to learn to teach in their own classrooms. Additionally, qualitative data also showed that some teachers noticed a shift in their understanding of math content and may have been more aware of some gaps in their math knowledge after participating in the program. However, this finding suggested a need for a follow-up study with more specified interview questions in order to investigate why teachers’ perceived knowledge decreased even though the qualitative data showed otherwise.

Recently, more researchers have adopted evidence-based assessments to measure the quality of teacher competencies, such as evidence of teacher performance, evidence of teacher knowledge, skills, and practices, or evidence of student learning (Boyd et al. 2008; Chingos and Peterson 2011; Clotfelter, Ladd, and Vigdor 2006; Hill, Rowan, and Ball 2005; Kane and Staiger 2008). We agree with this suggestion, but also acknowledge that if teachers perceive a change in their own beliefs and practice, then that change is real and should be recognized (Curtis and Szesztay 2005). Future research investigating similar PD programs could add more to our understanding by conducting evidence-based assessments to measure changes in teacher competencies and teaching practice.

Implications and conclusion

Our work has clear implications for future research and practice. The TACCC program is unique because it offers two key ingredients, Opportunities to Learn and Opportunities to Reflect, embedded within an ongoing PD program highly integrated with teachers’ daily teaching practice. In our study, this PD design was shown to significantly change teachers’ knowledge, beliefs, and practice across three waves of implementation. More research can be conducted to investigate how this program can be applied in other contexts, serving teachers in other subject domains and grade levels. Furthermore, additional research can be done on how to faithfully replicate this program across sites, and to expand capacities by building in digitized components or using communication tools to facilitate PD delivery.

Our work also has practical implications for PD designers and teacher educators. Our study showed that providing teachers with continuous opportunities to reflect about what they learn from a professional development program and how it shapes student outcomes may be the key to changing both teacher knowledge and beliefs and, more distally, teacher practice. Teacher educators should consider adopting these reflective opportunities as a way to supplement other programs geared for in-service teachers. Moreover, our study suggests that targeting both knowledge and beliefs may be a good way to change teacher behaviors and improve student learning experiences. Others are encouraged to further investigate how professional development programs can be designed to target both knowledge gains and belief change in different contexts, leading to more comprehensive programs that have a great chance at positively influencing how students learn in their classrooms.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Ohio Board of Regents [09-31,10-35,11-33].

Notes on contributors

Hea-Jin Lee

Hea-Jin Lee is a Professor of Mathematics Education at the Ohio State University at Lima. Lee’s research areas are related to mathematics teacher education, including designing and evaluating professional development programs; assessing teacher growth; teaching mathematics with digital resources; teaching mathematics equitably; culturally responsive mathematics teaching.

Vanessa W. Vongkulluksn

Vanessa W. Vongkulluksn is an Assistant Professor in the Department of Educational Psychology, Leadership, and Higher Education at University of Nevada, Las Vegas. She earned her PhD in Education with a concentration in Educational Psychology and Quantitative Methods from Rossier School of Education, University of Southern California. Dr Vongkulluksn’s research utilizes a variety of statistical methods to examine factors associated with student motivation and engagement, especially in technology-integrated contexts.