The Dialog: A Journal for Inclusive Early Childhood Professionals 6 Examining the Influence of Practice- Based Teacher Education Approaches on Primary Grades Teacher Candidates’ Development of Inclusive and Equity- Based Mathematics Teaching The Dialog: A Journal for Inclusive Early Childhood Professionals 2025, Volume 28, Issue 2 https://doi.org/10.55370/thedialog.v28i2.1807 Contact: Drew Polly drew.polly@charlotte.edu Copyright © 2025 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (https://creativecommons. org/licenses/by/4.0/). CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS ABSTRACT This article proposes the construct of Inclusive and Equity-Based Mathematics Teaching (IEBMT), a synthesis of theory and research for effective math- ematics teaching for all students. There is a critical need for educator preparation programs to inten- tionally design course activities and clinical prac- tice experiences to provide teacher candidates (TCs) with ample, worthwhile experiences to develop the knowledge and skills required to be an effective teacher for all students. The study examines how practice-based teacher education (PBTE) approaches in two elementary education mathematics pedagogy courses influenced TCs enactment of IEBMT. The inductive analysis of teacher candidates’ projects and reflections indicated that they were adequately prepared to support learners who had knowledge of the counting sequence and were able to accurately count a set of 5 objects. However, teacher candidates reported and demonstrated in their project a desire and a need for further opportunities to develop class- room-based skills at observing or listening to primary grade learners and quickly making effective instruc- tional decisions about future activities and questions based on what they notice. The article concludes with implications for both course activities and clinical practice experiences for teacher candidates to support their capacity to teach mathematics effectively. KEYWORDS Early childhood education, elementary education, mathematics education, problem solving Introduction Framing Inclusive, Equity-Based Mathematics Teaching (IEBMT) There is compelling evidence that early childhood learners (Pre-Kindergarten through Grade 2) vary in their mathematics achievement based on multiple factors, such as their previous opportunities to learn, access to effective teachers teachers and research- Drew Polly, Ph.D. University of North Carolina at Charlotte The Dialog: A Journal for Inclusive Early Childhood Professionals 7 -based learning experiences, as well as ethnic and racial backgrounds (Domingo-Martos et al., 2022; Musu-Gillette et al., 2017). This article includes inclusive and equity-based mathematics teaching (IEBMT), a synthesis of empirically- and theoret- ically-based constructs designed to support learn- ing for all students (Table 1). IEBMT includes two primary pillars: 1) Access to grade-level aligned, re- search-based experiences and 2) Opportunities for the exploration of problems embedded in meaning- ful contexts. Within each of those pillars, there are two aspects, for a total of four aspects. Access to Grade Level-Aligned, Research-Based Practices Aligned to grade-level content. All learners de- serve access to grade-level aligned, research-based practices (Musu-Gillette, et al., 2017; Unbound Ed, 2021a). For those learners whose data suggests that they need learning experiences on concepts from previous grade levels, learning activities should serve as access points to grade level content, and be clearly evident to grade level concepts (Unbound Ed, 2021b). Historically, studies have found that students whose data suggests that they lack founda- tional concepts often do not receive opportunities to engage in grade-level activities (Liljedahl, 2020; Unbound Ed, 2021a). Therefore, early childhood teachers and TCs must be equipped with the knowledge and skills related to knowing characteristics of grade-level aligned and developmentally appropriate activities. Further, teachers and TCs need to know how to modify activities so that they align to grade-level concepts and provide access to all learners (Bostic et al., 2021; CAST, 2024; National Association for the Education of Young Children [NAEYC], 2020). Access to research-based practices. Addition- ally, early childhood TCs must be given opportu- nities to learn what supports are research-based and how to use them in a way that supports learn- ers’ development (NAEYC, 2020; Zhang & Cutler, CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS Table 1 Aspects of Inclusive, Equity-Based Mathematics Teaching Pillar Aspect Description Access to grade-level aligned, re- search- based experiences Alignment to grade level content Learners should engage in activities aligned to current grade-level Standards (Gutiérrez, 2009; NCTM, 2014; Un- bound Ed, 2021a). Concepts prior to grade-level Standards should be connected to grade-level concepts (Tomlinson, 2017; Unbound Ed, 2021b) Access to research- based expe- riences Learners will engage in activities that are aligned to re- search-based teaching practices proven to increase student learning and achievement (Cioè-Peña, 2017; Gutiérrez, 2012). Opportuni- ties for the exploration of problems embedded in meaningful contexts Exploring problems When appropriate, learners will explore problems and select the strategies that they will use to solve problems (Buchheister et al., 2019; NCTM, 2014). Gutiérrez (2009) describes this as Power. Meaningful mathematics contexts Learners engage in activities that are contextualized in real-life situations that are meaningful to learners and build upon their cultural and academic assets (Buchheister et al., 2019; Chao et al., 2015; Domingo-Martos et al., 2022). The Dialog: A Journal for Inclusive Early Childhood Professionals 8 2024). Part of the process of preparing TCs to en- act IEBMT with young learners in mathematics is ensuring that they are prepared to incorporate hands-on, concrete learning experiences with ma- nipulatives to support children’s understanding of mathematics concepts (NAEYC, 2020). Piaget (1952) posited that young learners must first expe- rience mathematics using concrete objects before learning mathematics with pictures and abstract symbols. Early childhood teacher education pro- grams need to ensure that TCs know how to use manipulatives, when to use them, and when to help students transition to pictures and abstract symbols (Zhang & Cutler, 2020). This includes ensuring that TCs are aware of various levels and progressions of learning mathematics concepts in- cluding the use of concrete manipulatives, pictorial representations, and the use of abstract symbols. Opportunities for Exploring Problems Embedded in Meaningful Contexts The second pillar of IEBMT focuses on provid- ing learners with opportunities to explore problems, when appropriate, that are embedded in meaning- ful contexts. Student exploration of problems. Syntheses of research indicate that students benefit by having meaningful experiences to explore mathematics concepts and develop a solid understanding of con- tent before learning specific procedures (NCTM, 2014; Sinha & Kapur, 2021). The process of provid- ing learners with opportunities to explore and the autonomy to choose strategies does not mean that learners have free reign while the teacher sits back. Rather, students use mathematics tools (e.g., ma- nipulatives, paper/pencil for drawings or symbolic work) and explore problems while the teacher scaf- folds and guides them with specific questions and feedback (Buchheister et al., 2019; NCTM, 2014). In her influential work on equity-based mathemat- ics practices Gutiérrez (2009, 2012) referred to this idea as giving students Power, and recommended that teachers give learners opportunities to explore and the freedom to select which strategies they want to use to solve problems that are relevant to them. Syntheses of research find benefit in allowing students to engage in productive struggle (failure) prior to discussions of specific strategies of solving problems (NCTM, 2014; Sinha & Kapur, 2021). Activities in meaningful contexts. Additional- ly, these activities must be relevant to learners and embedded in meaningful contexts that learners can relate to (Unbound Ed, 2021a, 2021b). Constructs, such as culturally responsive teaching, are ground- ed in evidence that students are more likely to per- sist and engage in challenging activities when they are grounded in relevant contexts related to their cultural assets and interests (Paris & Alim, 2017). Similar to the other aspects of IEBMT, teachers and TCs need opportunities to develop knowledge about their students’ cultural assets and interests of their students (Domingo-Martos et al., 2022). Context of Educator Preparation Programs There have been long-running recommenda- tions to intentionally align the course work within educator preparation programs, teacher candidates’ (TCs) clinical practice experiences, and the actual work done daily by teachers (American Association for Colleges of Teacher Education [AACTE], 2016; Putman & Polly, 2021; Zeichner, 2021). Historical- ly, research has documented discrepancies between what TCs experience during education courses, specifically those focused on pedagogies, and their experiences with clinical practice in school settings (Zeichner, 2021). Additionally, recommendations in the past decade have explicitly asked for educa- tor preparation programs to intentionally prioritize the design and implementation of clinical practice experiences that are intended to provide TCs with worthwhile, intensive but supported opportunities to plan instructional activities, teach them, and then reflect on their experiences (AACTE, 2016; National Association for Professional Develop- ment Schools, 2022). This study frames early childhood education broadly to encompass education for learners from Birth through Grade 3 with a specific focus on learners in Grades Kindergarten through Grade 3. In teacher education this involves clinical practice experiences occurring in multiple settings includ- ing early childhood education centers, community centers, as well as formal schools which often start in the United States in Pre-Kindergarten or Kin- dergarten. When considering the preparation of early CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 9 childhood educators, scholars speak to the essential nature of intentionally aligning course activities with clinical practice experiences is critical (Burns et al., 2016; Zeichner, 2021; Putman & Polly, 2021). His- torically, university-based course instructors include research- and theory-based approaches to teaching that do not align to what TCs are observing in PK- 12 school settings (AACTE, 2016). Additionally, the clinical practice activities that TCs engage in in class- rooms do not include the depth needed to adequately prepare them for full-time student teaching intern- ships as well as having their own classroom (AACTE, 2016). Early childhood TCs who hope to work with pri- mary grade learners need ample experiences working with and learning about children in clinical practice settings in classrooms (Matengu et. al., 2020). These clinical practice activities should include both infor- mal time in classrooms observing and interacting with children as well as more formal opportunities to co-lead or lead instructional activities and reflect on their experiences. Table 2 Phases of Practice-based Teacher Education Phase Description Learn Learn research-based pedagogies by participating as learners in an example les- son, watching videos, and/or other experiences. Prepare Prepare to enact the research-based pedagogies with young learners. This may include selecting activities, writing lesson plans, and rehearsing/practicing teach- ing with peers (aka other TCs) and receiving feedback on their rehearsal. Enact Enact research-based pedagogies with students in a school setting. This experi- ence may include the collection of artifacts from the enactment such as student work samples, audio recordings, video recordings, or observation notes from an observer. Reflect Reflect on the enactment based on TCs’ experiences, student data, or recordings of the enactment. Practice-Based Teacher Education Practice-based teacher education (PBTE) is a construct used to describe four phases to prepare TCs to learn about and enact research-based pedagogies (Grossman et al., 2009). Table 2 describes the four phases of PBTE: Learn, Practice, Enact, and Reflect (McDonald et al., 2013). McDonald and colleagues wrote in their introductory article about the learning cycle (p. 382), This cycle intends to offer guided assistance to candidates to learn particular practices by in- troducing them to the practices as they come to life in meaningful units of instruction, preparing them to actually enact those practices, requiring them to enact the practices with real students in real classrooms, and then returning to their enactment through analysis. Depending on the goals and purposes of the teacher educator, it is possible to start this learning cycle in any of its four quadrants. CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 10 “The learning cycle is rooted in theory that posits that learning is situated within meaningful learning experiences and refined through empirical studies on teacher candidates’ learning.” The learning cycle is rooted in theory that posits that learning is situated within meaningful learning experiences (see Rogoff, 1997; Wenger, 1998) and refined through empirical studies on teacher candidates’ learning (Grossman, 2013; Ka- zemi et al., 2009; Windschitl et al., 2012). While the learning cycle has four distinct phases, based on TCs background, teacher educators may begin the cycle with any phase. In the Learn phase TCs are introduced to spe- cific research-based pedagogies through various means including, but not limited to, an instruction- al activity where the teacher education models the pedagogies as TCs take on the role of learners, con- ducting live observations of a teacher enacting this practice, watching a video of a teacher implement- ing the pedagogy with students, or analyzing writ- ten vignettes from teachers or researchers about classroom-based enactments of the pedagogies. The priority in this phase is that TCs gain a deeper understanding of what the practice looks like with learners in actual schools. Typically, the Prepare phase also occurs during a course meeting when TCs practice using the re- search-based pedagogies with peers in what is often called rehearsals. During this phase, the course in- structor and peers give feedback to help refine the use of the pedagogy before TCs enact it with chil- dren in classrooms. The Enact phase occurs in a classroom when TCs teach an instructional activity and use the re- search-based pedagogies with students. Lastly, the Reflect phase includes TCs analyzing student work, recordings, or thinking about their experiences during the enactment phase. Research on PBTE approaches have indicat- ed that this process, especially the Practice phase with Rehearsal activities have positively influenced TCs’ perceptions of feeling prepared to enact re- search-based pedagogies (Colonnese et al., 2022; Shaughnessy & Boerst, 2018; Shaughnessy et al., 2019). Additionally, the PBTE phases contributed to TCs posing more questions about elementary school students’ mathematical thinking and ideas instead of just asking more basic questions about the answer that they received (Polly, 2021). In light of the context of educator preparation programs and the need to support early childhood educator preparation programs in preparing TCs to be more inclusive of primary grades learners, PBTE has a lot of potential that needs to continue to be explored as a possible approach to support TCs enactment of research-based pedagogies. To that end, this paper explores the following broad question: How do course activities in early child- hood mathematics pedagogy courses influence TCs’ performance during Practice and Enactment activities? Methods Context This study examined data sources from two ele- mentary education mathematics pedagogy courses focused on teaching primary grades learners (Kin- dergarten through Grade 2) taught by the author at a large university in the southeastern United States. Course A is a traditional face-to-face course taken by undergraduate students who are earning their undergraduate degree in Elementary Education and their initial teaching license in Grades Kinder- garten through Grade 6. Course A required TCs to attend face-to-face courses for 3 hours each week for a 16-week semester. The course also included clinical practice activities in which TCs enacted a number sense activity (aka a number talk) and three lessons focused on word problems. Meanwhile, Course B was an online course with both synchronous and asynchronous course- work taken by post-baccalaureate students during a 10-week summer session. Each of the TCs in Course B had a degree in a non-education related field and were earning their initial teaching license in Grades Kindergarten through Grade 6. During this section TCs completed 8 asynchronous mod- ules and participated in 3 synchronous sessions, two of which involved practice and rehearsal activ- ities. Enactment and reflection were not required in this course since the course occurred in the summer. CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 11 Due to the differences in the course formats, the purpose of this study is to examine each course separately in light of the research question: How do practice-based teacher education (PBTE) course activities in early childhood mathematics pedago- gy courses influence TCs’ enactment of Inclusive and Equity-based Mathematics Teaching (IEBMT)? Both courses were included since PBTE approaches have not been studied thoroughly in both face-to- face and online, bichronous settings (Polly, 2021; Shaughnessy et al., 2021). Description of Course Activities Number Sense Activity Course A. In Course A TCs participated in number sense activities (aka number talks) as learn- ers three times during the first two weeks of the se- mester. Each of these activities focused on having TCs look at and talk about pictures of dots that were arranged differently based on the research-based ac- tivity of subitizing (Clements, 1999; Figure 1). The goal of the activity was to think about different ways to count the total number of dots. While there are various ways to arrange manipulatives inside of a ten frame the purpose of these arrangements was to promote conceptual subitizing (Clements, 1999) so that TCs could clearly see two distinct groups with a constant total amount. In the activity all the pictures had a total of 10 dots with varying dots in each of the ten frames. In Figure 1 when asked about what they notice, TCs may state ideas such as: • “I see 5 on the top and 5 on the bottom and I know that 5 plus 5 is 10.” • “I see that the bottom dots can be moved up to fill the empty boxes so that all 10 boxes on top are full to make a total of 10.” • “I went from left to right and counted by 2s. I landed on 10 which is the total.” During the discussion after the activity, TCs spent time talking about their experiences as learn- ers and the benefits of the number sense activities. Part of the discussion during the course focused on the specific questions that the instructor (the au- thor) asked during the activity with a focus on how those questions elicited students’ thinking about the mathematical concepts embedded within the activ- ity. During the fourth class meeting TCs came to class with a number sense activity prepared and questions planned. The TCs spent time practicing their number sense activity in small groups of 5 or 6 people. After each round of practice each TC who practiced received feedback from their peers. Course B. Since Course B was an online course with asynchronous modules and a few synchronous sessions, TCs experienced two examples of a num- ber sense activity during the first synchronous ses- sion and then watched a video example of a number sense activity during a subsequent online module. In the second synchronous session the TCs prac- ticed their number sense activity with colleagues. Similar to Course A, after each round of practice each TC received peer feedback. Problem Solving Lessons Course A. In the undergraduate face-to-face course TCs spent two class periods in the Learn phase where they analyzed and solved different types of word problems. During this time the in- structor modeled how to teach a word problem using an inquiry-based approach that involved the teacher posing the word problem, asking a series of questions about the problem, and then allowing stu- dents to solve the problem. CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS FIGURE 1 Screen shot of image from number sense activity The Dialog: A Journal for Inclusive Early Childhood Professionals 12 During the course meetings the instructor (the author) provided examples of ways to support pri- mary grades students. For example, consider the word problem: There are 4 dogs in the park. Then 3 more dogs show up. These scaffolds that were shown to TCs included asking questions about the problem to guide students through the prob- lem-solving process and providing them with a number path and hands-on counters to help stu- dents who need help keeping track of the numbers in the problems (Figure 2). The instructor walked through the process with TCs in the following way: • Instructor: I want us to think about this situa- tion. There are 4 dogs in the park. Then 3 more dogs show up. • Instructor: How many dogs are in the park? • TCs: There are 4. • Instructor: How can we use our counters to show that? • TCs cover the numbers 1, 2, 3 and 4 on the num- ber path. • Instructor: What is the action in our problem? • TC: 3 more dogs showed up. • Instructor: Use your counters to show the 3 dogs that showed up. • TCs put counters that are a different color on the numbers 5, 6, and 7. • Instructor: The question we are going to answer is “how many dogs are now in the park? What is the answer? • TC: The answer is 7. • Instructor: How do you know? • TC: I have 4 counters and 3 counters. That is a total of 7. TCs then spent time during a class meeting creating word problems to use in each of their three lessons. The template that was given to stu- dents required five word problems per lesson. TCs were required to create an opening word problem and then four follow up word problems. TCs were asked to create two of the four follow-up problems to be easier than the opening problem, meaning the numbers were smaller OR there would be more guidance from the TC while teaching young learn- ers. Additionally, two of the follow-up problems were expected to be more challenging than the opening problem with larger numbers and possibly focusing on the use of pictures instead of hands- on manipulatives and less teacher guidance. After students created word problems, they had a class session where they practiced teaching one of their word problems using the process that was detailed above. Similar to the number sense activities, peers provided feedback. Course B. Course B, the 10-week online bi- chronous course, was markedly different from Course A regarding the Learn phase of the prob- lem-solving process. TCs examined different types of word problems in the first two asynchronous modules and received feedback about their word problems from the author via an assignment that they submitted. In the second module TCs wrote their five word problems for each lesson. These word problems were similar to Course A with one opening problems and four follow-up problems, two that were easier and two that were harder com- pared to the opening problem. In the course’s sec- ond synchronous session, which occurred during Week 4 of the course, TCs practiced teaching one CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS FIGURE 2 Picture of Number Path and Counters The Dialog: A Journal for Inclusive Early Childhood Professionals 13 of their word problems using the same process de- scribed in Course A. TCs attended Zoom sessions in groups of four to six where each TC taught their peers as if they were young learners. After each re- hearsal peers gave glows and wonderings. Glows were positive things from the rehearsal and re- hearsals included things that peers had questions about. Most peers gave glows and there were few wonderings. The author, as the course instructor, also gave glows. All suggestions for their tasks or teaching were communicated privately by the au- thor after the practice rehearsal teaching. Participants, Data Sources, and Data Analysis This section describes the participants, data sources, and data analysis included in this study. Participants In order to examine the research ques- tion with a focus on the early childhood grades this study includes data from TCs who completed course activities related to Kindergarten. While the course included 22 TCs in Course A and 28 students in Course B, focusing only on TCs who had placements in Kindergarten reduced the num- ber of participants to seven TCs in Course A and five TCs in Course B. All TCs in Course A com- pleted clinical practice activities in urban schools where over 85% of the students within the school were experiencing poverty and over 60% of the stu- dents identified as Black or Latinx. While TCs in Course B did not complete clinical practice activ- ities during the summer course, each of them had completed clinical practice activities in previous semesters in either urban or rural schools where over 80% of the students within the school were experiencing poverty. The students in the schools where TCs in Course B previously had completed clinical activities included diverse populations of students including those who identified as White, Black, and Latinx. Data Sources and Data Analysis In order to examine the research question mul- tiple data sources were examined (see Table 3). The data from each course was examined separately since the courses included different populations (undergraduates and post-baccalaureate teacher candidates) and activities. These data sources came from the two primary course activities that includ- ed practice (both courses) and enactment (only Course A). CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS Table 3 Data Sources Used the Study Course A Undergraduate Face-to- Face Course Course B Online Post Baccalaureate Course Number Sense Activity Practice: TC’s written reflection Practice: TC’s written reflection Enactment: TC’s written reflection No Enactment Data Problem Solving Lessons Practice: Planning Map of word prob- lems and questions TC’s written reflection Practice: Planning Map of word prob- lems and questions TC’s written reflection Enactment: TC’s written reflection No Enactment Data The Dialog: A Journal for Inclusive Early Childhood Professionals 14 In order to examine the research question, data was analyzed using an inductive process us- ing open coding (Bingham, 2023; Bogdan & Biklen, 2006). Bingham (2023) proposed a five-stage pro- cess which was used. First, data was organized with all the data sources inserted into google doc- uments. Second, data was sorted into categories based on the course. Third, the author coded data. The codes were informed by the two pillars on in- clusive, equity-based mathematics teaching (Table 1). Codes were assigned to excerpts of the data by the author. Examples of codes include TC’s actions such as posing word problems, posing questions, supporting students and direct teaching. Examples of codes related to student actions included solving word problems, using manipulatives, and choosing strategies. Fourth, codes were organized by similar topics to generate themes. The themes were then checked by revisiting the original data excerpts in order to help establish trustworthiness. Lastly, findings were explained. Due to the limited context of the study, Bingham’s suggestion of advancing theory was not included. As stated earlier, based on the limited context, this was an exploratory study to better understand the influence of PBTE activities in early childhood teacher education programs, and the study was not designed to advance theory or make generalizations. Findings This study was framed around the research question: How do practice-based teacher educa- tion (PBTE) course activities in early childhood mathematics pedagogy courses influence TCs’ en- actment of Inclusive and Equity-based Mathemat- ics Teaching (IEBMT)? The findings are organized by the two pillars of Inclusive and Equity-Based Mathematics Teaching ([IEBMT]; Table 1): 1) ac- cess to grade-level aligned research-based experi- ences and 2) opportunities for exploring problems embedded in meaningful contexts. Access to Grade-level Aligned Research-based Experiences Teacher candidates (TCs) in both courses planned learning experiences that were aligned with the Kindergarten grade-level standards. All candidates wrote word problems and number sense activities that provided scaffolds to support primary learners’ access to the word problems. Spe- cific themes related to this pillar of IEBMT includ- ed progressions of questions in the number sense activity and the use of visuals and manipulatives. Progression of questions in number sense activity. In terms of the number sense activity, all seven TCs in Course A had planned for a series of questions that progressed from low-level to more higher-level. When the seven teacher candidates did not ask the questions that they had come pre- pared with; instead, they made up other questions on the spot that were basic and low-level questions. In Course B all five TCs planned for and asked questions that progressed from low-level to high- er-level questions. Examples of the higher-level questions that were asked were similar to those asked during the examples provided by the instructor (the author) during the Learn phase, such as: “How is this strategy of finding the total number similar to the strategy we saw earlier?” And “How do you know that your thinking is correct?” Similar to what hap- pened during the Learn phase, TCs in both Course A (face-to-face) and Course B (online) mimicked what they saw the instructor (the author) do and implemented the progression from low-level to higher-level questions with a high degree of fidelity. One TC from Course A who decided to make up her own questions wrote in her reflection, “I did not feel comfortable asking the harder ques- tions since I was still unsure what a correct answer would look like and how I would respond to my classmates.” Despite the activities that this TC went through she still chose not to ask higher-level ques- tions due to her uncertainty about the correct an- swer and how to respond to answers. A TC from Course B wrote, “Even though the synchronous time before we practiced was brief, it helped to see an example on video and also be part of one that [the instructor] led. These activities definitely helped me feel prepared to practice my own activity.” Only TCs in Course A enacted this activity with primary grade learners. In their reflection, six of the seven TCs reported a positive experience from CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 15 teaching the activity to a small group of primary grade learners. One TC who had a positive experi- ence wrote, “The questions that I had planned for worked very well and my students were able to do the activity and share their thinking.” The TC who did not report a positive expe- rience was one of the TCs who abandoned her planned questions during the Practice phase. She wrote in her reflection, “When I was teaching my students the activity was too easy, so I had to make up more problems and questions on the spot.” This reflection speaks to the idea that her planned ac- tivity did not align well to students’ strengths and needs. In summary, TCs in both courses were able to practice the number sense activity in ways that aligned with the instructor had intended. TCs in Course A enacted the activity with students and had a positive experience, except for one individ- ual who reported that her students were bored and she had to plan her activity on the spot since the original activity was too easy. Use of manipulatives and visuals. All the TCs across both courses planned for and used manip- ulatives such as counters or cubes during each of the lessons. This was consistent and evident during each TCs Practice activity. During the reflection of the practice TCs wrote questions though about when they should have their students use manip- ulatives compared to drawing a picture about the math problem. One TC wrote, “I always make them use counters or can I give them the option of counters or a picture?” Another TC wrote, “I know my students will always want counters but some may not need them. How do I take them away and have them draw pictures?” All TCs were aligned with the desired peda- gogies by using manipulatives such as counters or cubes along with Number Paths (Figure 2). TCs re- ported confusion and uncertainty knowing when they should encourage students to transition from manipulatives to pictures in ways that are positive and non-threatening. Following the Practice this was discussed in a class session for Course A, and through an email to TCs in Course B. The instruc- tor gave ideas about how they may encourage stu- dents to create a representation with manipulatives and then draw a picture of the concrete objects, or when ready begin their work by trying to first draw a picture to solve the problem. When the TCs in Course A enacted their les- son, they reported that their students found the use of counters to be very helpful and some primary grade learners demonstrated misconceptions when they had to draw pictures instead of manipulatives. One TC wrote, “If my students have counters they get them all correct. If they have to draw a picture then 3 of my 5 students will need help just getting started.” This TC had firsthand evidence that the transition from manipulatives to pictures was hard and needed more support in the future. In summary, TCs demonstrated some degree of fidelity to the desired pedagogies by planning for and using manipulatives such as counters and cubes in their Practice activity. TCs, though, re- ported a lot of uncertainty across both courses on how to help students transition from manipulatives to pictures. Opportunities to Explore Problems Embedded in Meaningful Contexts Opportunities to explore problems. The anal- ysis of data related to the problem-solving lessons indicated that all 12 TCs across both courses had planned to allow students to explore word problems with TCs scaffolding by providing manipulatives and questions to support students. This matched both the face-to-face course session for Course A and the asynchronous modules in Course B. The planning and preparation of their lessons included scaffolds for learners since it went systematically step by step using questions to guide students. However, four of the seven TCs during the Practice phase of Course A; two TCs taught di- rectly how to solve a problem with direct teaching and did not ask any questions, and two TCs taught posed the word problem and provided no questions or guidance to support students. In all these four instances, the practice either provided too much scaffolding with direct teaching or not enough scaf- folding by simply giving the word problem. In their reflection of the practice, one TC who practiced in a direct instruction method that did not match the desired pedagogies wrote, “As soon as I got into my practice I know I needed to teach every step of the problem since my students will need to be given all CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 16 the information like that when I teach it.” This TC had a predetermined notion of her students that they needed direct instruction instead of inqui- ry-based instruction and she decided to practice using that approach. During the enactment with primary grades’ learners, TCs from Course A all used a step-by- step inquiry-based approach. The debrief of the Practice activity provided TCs with more examples and clarification about what the lessons should look like, which led to a higher degree of fidelity during the enactment with students. In her reflec- tion, one of the TCs who taught with direct instruc- tion during practice but inquiry-based during en- actment said, “I still was skeptical during my lesson with kids until I saw how my Kindergarteners had their own ideas and strategies on how to solve the problem. They had enough of a foundation to be successful and share their ideas with me and their classmates.” In terms of Course B, during a synchronous ses- sion, two of the five TCs practiced their word prob- lem with classmates in a step-by-step approach that provided scaffolds and an inquiry-based approach where the teacher taught by asking a lot of questions. There were three TCs who directly taught each step and did not use an inquiry-based approach. These TCs who directly taught how to solve word prob- lems included comments in their reflection about how their schools have completed clinical practice experiences in teaching mathematics this way with direct instruction about specific strategies instead of inquiry-based approaches. One TC who taught her practice lesson in a direct instruction manner wrote, “This is what my school does. They teach step by step while students listen and watch and then they will eventually get to practice.” Data analysis indicated that the number sense routines included plans for and evidence during the practice activities of students having Power and being able to determine which strategy they want to use to count the total number of dots. Additionally, the number sense activities all included alignment to grade level content since subitizing and the ques- tions posed aligned to the state mathematics stan- dards. Identity was not explicit since number sense activities are non-contextualized problems, but the scaffolds provided by the TCs were designed to em- power primary grade learners that they are capable of being successful in mathematics. In regards to the idea of Power, one TC from Course A wrote in her reflection after the enact- ment, “It was transformative to hear my students’ ideas during the number talk. They had such cre- ative, correct ways to find the answer.” One TC from Course B wrote in her reflection after the practice, “It makes me nervous to ask questions about what my students are thinking because I do not know what they will say, but I know that it is good for them and I want to keep doing that even if it is a bit nerve wracking.” In terms of the problem-solving lessons, all of the TCs across both courses provided word prob- lems that were embedded in real-life situations which were intended to be relevant to learners, which helps to develop their identity. As stated above, not all TCs demonstrated during the Prac- tice pedagogies that give students power since some TCs taught using direct instruction the ex- act steps about how to solve a word problem. Ad- ditionally, when TCs (and teachers) teach using direct instruction where students are not actively doing mathematics for a time that has potential to negatively impact their identity since they may feel not capable of doing mathematics without the teacher first demonstrating each step. Problems embedded in a meaningful context. Additionally, each of the TCs were able to write word problems that included contexts that were meaningful to their students. In the lesson plans for the problem-solving lessons TCs provided a written rationale about how their word problems were embedded in a meaningful context. Examples from teachers’ rationales included references to students’ cultural backgrounds or businesses near their school. One TC in Course A wrote: “Many of my students speak Spanish at home so I decided to make word problems about Spanish food.” A TC wrote in Course B, “There is a park down the street from my school and so I wrote word problems about the park since students are always talking about the park.” In summary, there was evidence that TCs were able to write word problems in a meaningful con- text when they embedded the problems in students’ cultures or community assets such as parks or stores. In terms of allowing students to explore problems, CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 17 data only came from Course A since Course B TCs did not enact lessons in Kindergarten classrooms. For TCs in course A, some TCs had preconceived notions that their primary grade learners needed direct instruction of specific strategies instead of opportunities to explore, so they practiced that. Fortunately, for the TCs in Course A, the debrief after the practice activity provided clarity which led to all TCs teaching word problems in an inqui- ry-based approach during the Enact phase. Discussion and Implications The purpose of this inductive, exploratory qualitative study was to examine the broad research question, How do practice-based teacher education (PBTE) course activities in early childhood mathe- matics pedagogy courses influence TCs’ enactment of Inclusive and Equity-Based Mathematics Teach- ing (IEBMT)? Data from TCs reflections and in- structor’s notes were used as primary data sources. In this section, I connect the findings to current re- search and provide possible implications for future work. This section focuses specifically on the po- tential of practice-based teacher education (PBTE) and the consideration of inclusive, equity-based teaching in early childhood educator preparation programs. Potential of Practice-based Teacher Education (PBTE) In this paper, the data analysis led to findings that the process of Practice-based Teacher Edu- cation (PBTE) and its four phases led to direct alignment between desired research-based ped- agogies associated with IEBMT. Specifically, evi- dence of these practices were more clearly evident in TCs plans and their practice teaching during each course related to the number sense activities and number talks. TCs used activities that were aligned to state mathematics standards and a se- ries of questions that progressed from lower-level to higher-level which matched what TCs had ex- perienced when they were learning about these ac- tivities. In the case of number sense activities, the course activities and PBTE processes of Learn and Practice led to TCs using the desired pedagogies across both Course A (face-to-face) and Course B (online). This finding supported previous studies (Colonnese & Polly, 2022; Shaughnessy et al., 2021) in that there was documented evidence that the PBTE framework was linked with evidence of TCs learning as noted on both TC reflections on clin- ical documents (Colonnese & Polly, 2022) as well as performance-based tasks that embody the daily work of teaching mathematics (Shaughnessy et al., 2021). In this study, the PBTE process was associat- ed with TCs posing word problems that had em- bedded research-based scaffolds, such as number paths and manipulatives. Additionally, TCs in both courses wrote word problems that were embedded in meaningful contexts relevant to their students. However, there was evidence of a lack of alignment for some TCs in each course between the desired approach to teach these problem-solving lessons using an inquiry-based approach focused on ask- ing students questions, and a direct instruction approach that some TCs used during their prac- tice teaching. This direct instruction approach to teaching how to solve word problems was evident in both Course A and Course B. This complexity about the lack of alignment between course con- cepts and the actual enacted pedagogies that TCs observe in clinical classrooms affirms the critical need to better align pedagogies in clinical place- ments and the research-based practices taught in teacher education courses (AACTE, 2016; Zeich- ner, 2021). However, the alignment between course activ- ities and clinicals appeared to be influenced by the specific course. Since Course A included the Enact phase with primary grade learners and Course B did not, all TCs in Course A received feedback and developed more clarity about how to teach using an inquiry-based approach and used that approach with students in classrooms. In the case of TCs in Course B since it was a summer course without CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS “TCs in both courses wrote word problems that were embedded in meaningful contexts relevant to their students” The Dialog: A Journal for Inclusive Early Childhood Professionals 18 clinical practice experiences and an Enact phase there is no evidence about how the TCs would teach word problems to students. This finding con- tributes to the literature since it provides empirical data that while PBTE has potential as a process to support TCs adoption and use of research-based pedagogies, there is a need for the instructor to be attentive to TCs performance during Practice and look for ways to support pedagogical shifts be- tween the Practice and Enact phases. Recent work has started to look at the influence of mediated field experiences (MFEs) where the course instructor is in the clinical practice setting with TCs to provide ongoing support and coaching during enactment (Colonnese & Polly, 2022; Gesel et al., 2023). Future research is needed to exam- ine which models of supporting both Practice and Enactment are best, especially in the cases of on- line teacher education courses and programs. Ad- ditionally, the research base still needs studies that examine the influence of approaches like PBTE on the enactment of research-based practices as well as student learning outcomes. In order for these types of approaches to be more widespread in edu- cator preparation programs these topics need to be empirically investigated to associate these efforts to teacher candidate learning and, when possible, PK- 12 student learning. Preparing TCs to Enact (IEBMT) in Early Childhood Settings As stated earlier, inclusive, equity-based teach- ing practices were framed in two ways in this study- 1) providing access and support to grade level con- tent, and 2) integrating equity-based mathematics pedagogies that are relevant to primary grade learners. Providing access to grade-level aligned, re- search-based experiences The data analysis indicated that TCs in both courses were able to successfully create number sense activities and word problem lessons that aligned to grade level content, and TCs were also able to think about how to provide access to ma- nipulatives such as counters and cubes as a support to do the mathematics. As indicated in the findings, though, is that TCs asked questions and reported a lack of clarity on how to help students transition from manipulatives to pictures; more specifically, when students show that they may not need the supports of the manipulatives anymore they were uncertain on how to remove them in a non-threat- ening way. In some cases, TCs reported not know- ing when they should think about and consider removing the manipulatives from students, instead strongly encouraging the use of pictures. This find- ing supports a seminal mathematics education study that found that teachers tended to focus on isolated aspects of mathematics teaching such as manipulative use, but in a superficial way where hands-on manipulatives were used but teachers did not demonstrate efforts to help learners make con- nections between the manipulatives and the actual mathematics concepts (Cohen, 1990). Additionally, there is a need for educator preparation programs to ensure that ample experiences are included to support TCs enactment of IEBMT. Future research studies need to examine ways that educator preparation programs can provide both course and clinical practice experiences re- lated to providing support and scaffolds for learn- ers. This includes the process of preparing TCs to recognize in the moment of teaching or afterwards while examining student work information or data that scaffolds may no longer be needed or need to be modified. Additionally, research is needed about how to best support TCs use of transitions of scaffolds as they move from more intensive to less intensive supports. In the case of this study, TCs in Course A planned and enacted three problem solving lessons to small groups of students where manipulatives were used in nearly each case. In the teacher education program that Course A is part of, previous studies found that having TCs enact lessons and instructional activities to small groups of students makes these clinical practice activities easier to enact compared to requiring whole group teaching experiences (Polly, 2021; Colonnese & Polly, 2022; Putman & Polly, 2022). Part of the rea- son that small group activities appear to be a better context for TCs to enact inclusive, equity-based practices is that clinical educators who host TCs are more apt to give TCs control of a small group of students compared to a whole group of students. One equity-based,inclusion practice that was com- CANDIDATES’ DEVELOPMENT OF INCLUSIVE AND EQUITY-BASED MATHEMATICS The Dialog: A Journal for Inclusive Early Childhood Professionals 19 -plex in this study was TCs decision to use manip- ulatives or encourage young learners to represent their word problem with pictures on paper. In some cases, in Course A, TCs required manipula- tive use but reported that in some cases, in retro- spect, they likely should have encouraged pictures instead of manipulative use. As scholars continue to think about inclusion and meeting students’ ac- ademic needs, one-size-fits all approaches do not work, and teacher education programs have a re- sponsibility to develop TCs who are flexible and responsive to data when deciding how to best sup- port young learners. Opportunities for Exploration while Solving Problems in Meaningful Contexts The second aspect of inclusive, equity-based teaching focuses on providing learners with oppor- tunities for exploration while solving problems in meaningful contexts. The findings from this study provide evidence that TCs were able to successful- ly create (both courses) and enact (only Course A) word problems in contexts that were embedded in relevant and meaningful contexts to primary grade learners. However, in their reflections, some TCs shared that although they planned to engage learners in activities that provided student explo- ration and inquiry-based approaches, they were afraid it would be too challenging, and instead opted to use direct instruction pedagogies in their practice activities, which did not align to what the TCs had learned and used in the PBTE activities in the course. This dissonance between planning (intended practices) and how TCs taught (enact- ed practices) occurred only in Course A because TCs enacted their lessons. Meanwhile, in Course B, this potential disconnect was not as clearly evident since enactment was not required with early child- hood learners. This potential disconnect between PBTE and other course-based activities dealt with pro- viding early childhood learners with opportunities to explore problems and use their choice of strate- gies while solving problems. 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