The Dialog: A Journal for Inclusive Early Childhood Professionals 88 Practice Makes Progress in Mathematics: A Research to Practice Summary The Dialog: A Journal for Inclusive Early Childhood Professionals 2025, Volume 28, Issue 2 https://doi.org/10.55370/thedialog.v28i2.1919 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/). RTP PRACTICE MAKES PROGRESS ABSTRACT This research-to-practice article supplements the research article Practice Makes Progress: Lever- aging Practice-Based Teacher Education in Math- ematics Pedagogy Courses for Primary Grade Learners published in the journal The Dialog. This article presents suggestions based on a research study that provided a research-based framework, Inclusive and Equity-based Mathematics Teaching (IEBMT), and examined how future teachers (re- ferred to as teacher candidates) effectively planned for and taught mathematics activities in kinder- garten classrooms. This work is relevant to early childhood education professionals since research supports benefits in using inclusive and equi- ty-based practices when teaching mathematics. KEYWORDS Early childhood education, elementary education, mathematics education, practice-based teacher educa- tion, problem solving Classrooms today include a more diverse group of students than ever before, which includes children from varied linguistic, socioeconomic and cultural backgrounds (Domingo-Martos et al., 2002). It is crit- ical that programs that prepare and support future and current early childhood educators adequately help individuals teach classrooms of young children that are increasingly diverse. Additionally, the number of individuals in early childhood education programs who demonstrate learning differences continues to increase (Schaeffer, 2023), requiring early childhood education professionals to have background knowl- edge and experiences supporting learners who have learning differences. Further, in an era where student learning and achievement data continues to be a focal point, student achievement data continues to show discrep- ancies between students based on their cultural and linguistic backgrounds across all grade levels (Domin- go-Martos et al., 2022; Musu-Gillette et al., 2017). Drew Polly University of North Carolina at Charlotte The Dialog: A Journal for Inclusive Early Childhood Professionals 89 Early childhood educators need support and ad- equate resources to effectively teach all learners regardless of students’ linguistic and cultural back- grounds. Inclusive and Equity-Based Mathematics Teach- ing: What is It? The ideas included in inclusive and equi- ty-based mathematics teaching (IEBMT) come from research on teaching mathematics to learn- ers from diverse backgrounds as well as children who demonstrate learning differences. IEBMT has two main pillars or parts: 1) Access to grade-level aligned, research-based experiences and 2) Oppor- tunities for the exploration of problems embedded in meaningful contexts. Table 1 describes aspects of IEBMT. What We Know about Educator Preparation Pro- grams Experts who conduct research and influence policy have made recommendations for course work for teacher candidates to align with the work that they do in schools for clinical experiences (Put- man & Polly, 2021; Zeichner, 2021). Early child- hood TCs who hope to work with primary grade learners need ample experiences working with and learning about children in clinical practice settings in classrooms (Matengu et al., 2020; Polly, 2021). Practice-Based Teacher Education Practice-based teacher education (PBTE) is a process used to describe four phases to prepare TCs to learn about and enact research-based ped- agogies (Grossman et al., 2009). Table 2 describes the four phases of PBTE: Learn, Practice, Enact, and Reflect (McDonald et al., 2013). In a nutshell, TCs learn about a research-based pedagogy, plan to use that pedagogy and practice using it, then they go and use the pedagogy with children, and final- ly reflect on their experiences (Colonnese & Polly, 2022). RTP PRACTICE MAKES PROGRESS Table 1 Aspects of Inclusive, Equity-Based Mathematics Teaching Pillar Aspect Description Access to grade-level aligned, research- based experi- ences Alignment to grade level content Learners should engage in activities aligned to current grade-level Stan- dards (Gutiérrez, 2009; NCTM, 2014; Unbound Ed, 2021). Connect concepts prior to grade-level Standards to grade-level concepts (Tomlinson, 2017) Access to research- based experiences Learners will engage in activities that are aligned to research-based teach- ing practices proven to increase student learning and achievement (Ci- oè-Peña, 2017; Gutiérrez, 2012). Opportunities for the exploration of problems embed- ded in meaning- ful contexts Exploring problems When appropriate, learners will explore problems and select the strategies that they will use to solve problems (Buchheister et al., 2019; Sinha & Kapur, 2021). 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; Domingo-Martos et al., 2022). The Dialog: A Journal for Inclusive Early Childhood Professionals 90 As a result of the need to prepare future educa- tors to use aspects of IEBMT practices with children, this study examined how two different courses used PBTE to influence TCs use of IEBMT practices. Description of the Research Study The study included TCs from an in-person course offered during a spring semester (Course A) and an online course offered in a summer semester (Course B). All TCs were placed in a kindergarten setting (Course A) or had spent time in kindergar- ten in the past year (Course B). Description of Mathematics Activities All TCs had to plan and teach a number sense activity (aka a number talk) and a 3-lesson unit about addition and subtraction word problems. Number Sense Activity TCs in both the in-person and the online course experienced number sense activities as learners during class. Here is an example. The au- thor, who was the course instructor, displayed Fig- ure 1 and then asked TCs, “What do you notice in this picture?” TCs usually state ideas such as: • “I see dots in each row.” • “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.” In the course meeting after the activity, TCs spent time talking about their experiences as learn- ers and the benefit of these activities. Part of the discussion focused on the specific questions that the instructor (the author) asked during the activi- ty with a focus on how those questions elicited stu- dents’ thinking about the mathematical concepts RTP PRACTICE MAKES PROGRESS Table 2 Phases of Practice-based Teacher Education Phase Description Learn Learn research-based pedagogies by participating as learners in an example lesson, 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 teaching with peers (aka other TCs) and receiving feedback on their rehearsal. Enact Enact research-based pedagogies with students in a school setting. This experience 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 enact- ment. FIGURE 1 Screen shot of image from number sense activity The Dialog: A Journal for Inclusive Early Childhood Professionals 91 embedded within the activity. In a future class meeting in both courses TCs had planned their own number talk, practiced teaching it to class- mates, and received feedback from their classmates and the course instructor. TCs then went and did the activity with a small group of children and re- flected on their experience. Problem Solving Lessons During both courses the instructor provided examples of ways to support primary grades stu- dents while solving word problems. 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 problem-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 situation. 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 number path. • Instructor: What is the action in our prob- lem? • 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. Findings The findings focused on two areas: 1) TCs pro- vided access to grade-level aligned research-based experiences and 2) TCs providing opportunities for young children to explore problems embedded in RTP PRACTICE MAKES PROGRESS FIGURE 2 Picture of Number Path and Counters The Dialog: A Journal for Inclusive Early Childhood Professionals 92 meaningful contexts. In the first area, TCs effectively planned for and ask questions during the number talk that gradual- ly increased in difficulty- from asking students to share their strategy towards more high-level ques- tions where TCs asked young children to explain why they chose specific strategies. Examples of these questions are: “How is this strategy of find- ing the total number similar to the strategy we saw earlier?” and “How do you know that your thinking is correct?” TCs also effectively planned for and taught activities where learners used manipulatives and visuals. All TCs were aligned with the desired pedagogies by using manipulatives such as count- ers or cubes along with Number Paths (Figure 2). While 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, reported a lot of un- certainty across both courses on how to help stu- dents transition from manipulatives to pictures. In the second area, TCs effectively wrote and taught word problems that were embedded in con- texts that were meaningful and relevant to young learners. These problems were about topics that young learners could relate to, such as Latin Amer- ican food and cultural events for students whose family was from Latin America. However, there were mixed findings on TCs during their teaching. TCs were supposed to pose word problems and guide young children’s mathematics work by ask- ing questions; however, a few of the TCs directly taught children with step-by-step instructions that children just mimicked and copied. Tips and Suggestions Based on This Study This study found that the practice-based teach- er education (PBTE) activities such as Learning, Preparing, Enacting, and Reflecting helped support teacher candidates (TCs) enactment of Inclusive and Equity-Based Mathematics Teaching (IEBMT) with kindergarten learners. Specifically, TCs were able to plan mathematics activities that were rele- vant to learners and appropriately aligned to their grade level and TCs. Here are some takeaways: • Practice does make progress. TCs did number talk activities and problem-solving activities as learners, planned their own activities, and then did practice teaching before using them with children. This practice helped TCs be- come more comfortable with the activities and how to teach them. • Focusing on questioning. TCs demonstrated, in a lot of instances, their skills in asking ques- tions about children’s math ideas as well as follow-up questions that included more how or why questions about children’s use of strate- gies (see Colonnese et al., 2022). • Guiding versus directing children. While the approach of practice-based teacher education helped TCs, some TCs reverted back to direct- ing students’ process of solving word problems when they sensed that children were struggling instead of guiding them. A lot of practice and discussion may be needed with TCs related to supporting children’s problem solving when they are striving to figure out what to do with a word problem. RTP PRACTICE MAKES PROGRESS “Practice does make progress. TCs did number talk activities and problem- solving activities as learners, planned their own activities, and then did practice teaching before using them with children.” The Dialog: A Journal for Inclusive Early Childhood Professionals 93 References Buchheister, K., Jackson, C., & Taylor, C. (2019). “Slid- ing” into an equitable lesson. Teaching Children Mathe- matics, 25(4), 224-231. https://doi.org/10.5951/teacchil- math.25.4.0224 Cioè-Peña, M. (2017). The intersectional gap: How bilingual students in the United States are excluded from inclusion. International Journal of Inclusive Education, 21(9), 906-919. https://doi.org/10.1080/13603116.2017.1296032 Colonnese, M. & Polly, D. (2022). Using practice-based teaching experiences to leverage teacher candidate effective- ness. PDS Partners: Bridging Research to Practice, 17(2), 65-83. Colonnese, M., Reinke, L. T., & Polly, D. (2022). An analysis of the questions elementary education teacher candidates pose to elicit mathematical thinking. Action in Teacher Education.44(3), 196-211. https://doi.org/10.1080/01626620 .2021.2020696 Domingo-Martos, L., Domingo-Segovia, J., & Pérez-García, P. (2022). Broadening the view of inclusion from a social justice perspective. A scoping review of the literature. Inter- national Journal of Inclusive Education, 1–23. https://doi.org /10.1080/13603116.2022.2095043 Grossman, P., Hammerness, K., & McDonald, M. (2009). Re- defining teaching, reimagining
teacher education. Teachers and Teaching: Theory and Practice, 15(2), 273 - 289. Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.” Teaching for Excel- lence and Equity in Mathematics, 1(1), 4-8. Gutiérrez, R. (2012). Context matters: How should we conceptualize equity in mathematics education? In Choppin, J., Herbel-Eisenmann, B., & Wagner, D., (eds.), Equity in discourse for mathematics education: Theories, practices, and policies, pp. 17-33. Springer. Matengu, M., Ylitapio-Mäntylä, O., & Puroila, A. M. (2020). Early Childhood Teacher Education Practicums: A Literature Review. Scandinavian Journal of Educational Research, 65(6), 1156–1170. https://doi.org/10.1080/00313831.2020.1 833245 McDonald, M., Kazemi, E., Kelley-Petersen, M., Mikolasy, K., Thompson, J., Valencia, S. W., & Windschitl, M. (2014). Practice makes practice: Learning to teach in teacher educa- tion. Peabody Journal of Education, 89(4), 500-515. Musu-Gillette, L., de Brey, C., McFarland, J., Hussar, W., Sonnenberg, W., & Wilkinson-Flicker, S. (2017). Racial and Ethnic Groups Status and Trends in the Education of 2017 (NCES 2017-051). U.S. Department of Education, National Center for Education Statistics. http://nces.ed.gov/pubsearch National Council of Teachers of Mathematics (2014). Princi- ples to action: Ensuring mathematical success for all. Author. Polly, D. (2021). Advancing equity-based mathematics teach- ing in the primary grades: The case of two clinical practice experiences. International Journal of Teacher Education and Professional Development, 4(1), 68-88. Putman, S. M., & Polly, D. (2021). Examining the develop- ment and implementation of an embedded, multi-semester internship: Preliminary perceptions of teacher education can- didates, clinical educators, and university faculty. Peabody Journal of Education, 96(1), 99-111. https://doi.org/10.1080/ 0161956X.2020.1864250 RTP PRACTICE MAKES PROGRESS https://doi.org/10.5951/teacchilmath.25.4.0224 https://doi.org/10.5951/teacchilmath.25.4.0224 https://doi.org/10.1080/13603116.2017.1296032 https://doi.org/10.1080/01626620.2021.2020696 https://doi.org/10.1080/01626620.2021.2020696 https://doi.org/10.1080/01626620.2021.2020696 https://doi.org/10.1080/13603116.2022.2095043 https://doi.org/10.1080/13603116.2022.2095043 http://nces.ed.gov/pubsearch https://doi.org/10.1080/0161956X.2020.1864250 https://doi.org/10.1080/0161956X.2020.1864250