Designing and Implementing Sustainable Professional Development Programs: Embodied Curriculum and Instruction for Kindergarten Teachers

Abstract

Embodied design is a theory-to-practice and pedagogical framework and emphasizes the role of embodied and situated activity. Since embodiment has a powerful impact on young children’s learning, implementing embodied deign in kindergartens is essential and beneficial to their future mathematical learning and performance. Therefore, this qualitative study aimed to develop a professional learning community composed of eight kindergarten teachers and university teacher educators (researchers) and to co-construct a professional development model for curriculum design and instructional implementation of embodied design in mathematics. Accordingly, the main research objectives were to obtain a sustainable professional development model and summarize principles of embodied mathematics curriculum design and strategies for embodied instructional activities, which can be used both in practical settings (i.e., kindergarten and preschool) and early childhood teacher education programs in higher education. Data were gathered through participative observations, including PD meetings and classroom observations, in-depth and follow-up interviews, teachers’ lesson design and reflection notes, and children’s learning profiles, and then were analyzed qualitatively using a descriptive and explanatory approach with replication logic. Based on the data collection and analysis, two major sections of the findings and discussion were included in this report, which were as follows: first, a sustainable “task design professional development model” was generated for collaboratively future improvements of both in-service and pre-service teacher training programs in higher education; second, two principles of embodied mathematics curriculum design and two effective embodied teaching strategies were proposed to promote kindergarten teachers’ professional knowledge and capabilities and their young children’s mathematics learning capabilities.

1. Introduction

1.1. Children’s Mathematics Learning Has a Profound Impact on Future Learning

There is abundant research evidence on young children’s mathematics learning showing that mathematics learning in kindergarten/preschool is essential and achievable [1,2]. In the literature on young children’s learning trajectories addressed by Sarama and Clements [3], it is confirmed that from birth, children have concepts including numbers, spatial sense, and graphics. Based on curiosity about life, young children will naturally practice mathematics, discover patterns from various phenomena, and know how to make inferences based on observation. Everyone is a born mathematician [4,5]. Clements and Sarama [6] studied a preschool teacher who had been devoted to mathematics learning for a long time. They found that the children he taught naturally used mathematics to think about problems and integrate them into situations in daily life. For example, a group of children was counting people on the school bus one day (e.g., now, 11 are here and 7 are absent; now, 13 are here and 5 are absent; now…). It could be seen that allowing children to spontaneously connect mathematics with life can be a powerful tool for laying the foundation of mathematics. Preschool children’s mathematical learning ability can be used as a predictor of many future academic achievements, including reading achievement and performance [7], literacy ability, and future mathematics learning [8]. Given the importance of young children’s mathematics learning, bringing appropriate mathematics learning into kindergarten is a necessary educational behavior [9].

Furthermore, since young children learn with their whole bodies [10] and their learning is the most important stage of embodied development [3,6], providing adequate learning opportunities with embodied teaching in kindergartens is fundamental and favorable to young children’s understanding of mathematical concepts and their future mathematical learning. Grounded in Dewey’s “learning by doing” philosophy, the embodiment advances the traditional “hands-on and real-life” learning activities to an integration of mind–body in a situated action [11]. For example, children typically use their fingers when counting and performing initial calculations, which may enhance their numerical understanding [12]. This embodied learning manner shows a link between mathematical concepts or ideas and bodily experiences, which indicates that the same brain regions are working while moving fingers as when just thinking of numbers. Since the use of fingers is a valuable tool for young children learning to count, teachers need to understand how these bodily experiences contribute to learning mathematics and designing and implementing correspondent embodied teaching in the classrooms. Consequently, employing embodied teaching and learning approaches, which makes mathematics more accessible, are authentically beneficial for young children’s comprehension and future learning.

1.2. Insufficient Professional Development for Kindergarten Teachers Deserves Improvements

Steele, Brew, Rees, and Ibrahim-Khan [13] pointed out that teachers needed more professional knowledge and skills to teach science and mathematics. The same situation occurs in early childhood teacher education in Taiwan. Within the pre-service teacher training program, subject-matter content related to mathematics, social studies, and science is often provided in “elective” courses, while mathematics is the least chosen subject by pre-service teachers [14]. In fact, there is only one required course for pre-service kindergarten teachers’ 4-year training program [in one author’s university—a public system] that is related to mathematics and science education, which is “Mathematics and Science for Young Children” (two credit hours per week). Moreover, compared to elementary teacher education, there is no “mathematics teaching materials and methods” course for furnishing pre-service teachers with applicable knowledge and capability of mathematics teaching and learning in real settings. Even though there are some courses—e.g., courses about young children’s development, curriculum design, and learning assessment that will cover certain topics related to mathematics teaching and learning—the amount of content and/or context specific to young children’s mathematics learning is still inadequate for cultivating pre-service teachers to be familiar with the design and implementation of embodied mathematics activities. Indeed, a considerable number of pre-service teachers refuse to learn mathematics [15,16] and are not efficacious in learning mathematics. As Schunk and Grootenboer [17] claimed, pre-service kindergarten teachers often have an avoidant attitude towards mathematics. When selecting courses, they avoid mathematics-related courses. Most of them have negative experiences while learning mathematics in the past and are frightened of mathematics [18,19].

Furthermore, this predicament of deficient “pre-service” training also occurs in the “in-service” teacher training system, since in-service teachers usually do not have enough training in mathematics teaching at the university level nor receive sufficient professional development (PD) programs related to mathematics education [20]. This has furthered in-service kindergarten teachers’ resistance to mathematics. Coupled with the lack of motivation in learning mathematics, these two factors have contributed to the incapability of most kindergarten teachers to teach mathematics to young children. In Taiwan, universities responsible for “pre-service” teacher training (i.e., 4-year program) are also accountable for the professional development of “in-service” kindergarten teachers. As a result, both “pre-service” and “in-service” kindergarten teachers face a parallel predicament of deficiency training and have analogous needs in teaching young children mathematics. As Wu and Lin mentioned, “For a long time, children in Taiwan have not been interested in or even have been afraid of learning mathematics, which has led us into putting more effort in resolving the obstacles of indifference and fear” [20] (p. 843). As a consequence, how to resume and upgrade both “pre-service” and “in-service” kindergarten teacher training programs in Taiwan for early childhood teacher educators at the higher education level in Taiwan with a sustainable manner stands at the center of our reform efforts. Grounded in this argument and the benefit of employing embodied teaching and learning for enhancing young children’s mathematics learning, sustainable PD programs must be established for promoting both “pre-service (at higher education level, for undergraduate and/or graduate students)” and “in-service (in practical settings, such as kindergarten and/or preschool)” kindergarten teachers’ professional knowledge and capabilities of embodied mathematics teaching and learning.

1.3. Embodied Design Is the Focus of Modern Mathematics Education Research

Mathematics education is professional knowledge that includes interdisciplinary integration in fields such as humanities, social sciences, and natural sciences. The diversity of its connotations is an inevitable trend, and it can also be defined as the study of psychology and cognitive science. In fact, it has the closest connection with cognitive science [21]. This is because modern cognitive science focuses on understanding how information and knowledge are reorganized and integrated in the brain and how individuals understand language, add experience construction, and express and transform the content of knowledge. Related to information processing viewpoints of cognitive learning theory, evidence that is based on neuroscience studies reveals both children’s thinking activities connected to human brain functioning while they are learning and their learning process when facing specific learning content. An increasing number of studies [22,23] have provided substantial evidence that demonstrates the modern mathematics learning theory that focuses on cognitive learning. Cognitive theory has become a prominent scientific research topic in recent years. Coupled with active discussions on the thinking and application of meta-cognition, more diverse research methods have been put into practice. Exploring the embodiment of individuals’ mental learning is essential in the field of cognitive science today, especially in mathematics education, which has gradually developed into a new research direction. A series of research evidence and integrated handbooks, published by Hutto, Kirchhoff, and Abrahamson [24], Abrahamson and Sánchez-García [25], and Abrahamson [26], put forward innovative and reformative thinking on embodied cognitive learning (i.e., embodied design). Furthermore, exploring the embodiment of individuals’ mental learning opens a new paradigm for scientific research on mathematics learning, which allows us to rethink how humans learn.

In addition to focusing on understanding how children learn, the model of modern organizational research on early childhood mathematics education should focus on researchers and teachers forming professional learning organizations or a community to develop an efficacious and sustainable PD program where the designated curriculum and instruction can be developed to meet both the academically diverse and neurodiverse needs of young children and benefit their mathematics learning. Grounded in the literature review on the internationally emerged argument for cognitive learning of embodied mathematics in recent years, there is a lack of research on embodied mathematics curriculum and instruction in Taiwan. Most of the relevant foreign studies focused on students in elementary and secondary schools [26,27], but few studies focused on early childhood education (i.e., in preschools or kindergartens). Further, those studies related to mathematics for young children mostly used “pre- and post-tests’” experimental research design [28] or applied brain imaging technology to observe young children’s performance and/or their reflections in the brain during cognitive and physical “embodied mathematics” activities [27,29]. Embodied design is a long-term educational research program committed to advancing the understanding and improvement of teaching and learning processes [30] and is a design-based research program investigating the phenomenology of designing, teaching, enacting, and learning a culture’s cognitive practices [31,32]. Consequently, a research program developing a professional learning community emphasizing embodied design stands at the center of furnishing kindergarten teachers’ professional growth for sustainable improvement in embodied teaching and learning, especially for mathematics. This action, initiated by teacher educators (researchers) in the field of early childhood teacher education at a higher education level (i.e., university), is significantly beneficial for all participants including teacher educators and in-service kindergarten teachers. Based on the sustainable PD model co-constructed by “early childhood teacher educators” and “in-service kindergarten teachers”, it is also favorable for pre-service kindergarten teachers who are undertaking a 4-year undergraduate teacher training program and those who are in graduate programs (some of them are in-service programs), since the design and implementation of the targeted PD model and the operating mechanism of the professional learning community can be merged into the original programs in higher education institutions and provide sustainable impacts on all students’ mathematics learning and their future careener development.

1.4. Research Purposes and Questions

Since young children’s learning is the most important stage of embodied development [21,33,34,35,36,37], implementing embodied teaching in kindergartens is essential and beneficial to young children’s future mathematical learning and performance. Further, as embodied teaching and learning is based on new theory and practice in mathematics education, teachers’ PD that focuses on this topic is the basis for providing high-quality teaching both in the practical settings (i.e., kindergarten and preschool) and early childhood teacher education programs in higher education. Therefore, this qualitative study aimed to develop a professional learning community composed of eight kindergarten teachers and university teacher educators (researchers) with professional practice experience and enthusiasm and to co-construct a PD model for curriculum design and instructional implementation of embodied design in mathematics. It was also expected that the proposed PD model and its operational method could be sustainably used by both pre-service and in-service kindergarten teacher training programs.

Grounded in this argument, the targeted kindergarten teachers’ professional development process and, later, their design and implementation of embodied mathematics were recorded through intensive observations, interviews, and other documentation (such as lesson design, reflection notes, and learning profiles). Accordingly, the main research question was as follows: what is the sustainable professional development model that promotes kindergarten teachers’ embodied mathematics teaching and their children’s learning?

2. Embodied Design in Mathematics and PD

2.1. Embodied Design in Mathematics

Since young children learn with their whole bodies, both their minds and brains, embodied cognition becomes critical in their mathematics learning [10]. Mathematics deals with the logical and abstract properties of numbers, quantities, shapes, and patterns [38], and learning mathematics is actually about portraying the properties, relationships, and patterns of the physical world [10]. In addition, because of the human brain’s fundamental functions of predicting and modeling [39], mathematics develops the backbone of emerging cognition in all children. Therefore, keeping young children’s mathematics experience and education fully embodied becomes a core task in efficiently supporting their cognitive development. Based on the views of Abrahamson et al. [32], embodied design is “a theory-to-practice approach to mathematics education” and “a pedagogical framework” that “draws on principles of genetic epistemology, Enactivism, ecological dynamics, and cultural-historical psychology to engage students’ naturalistic sensorimotor capacity and stage opportunities for guided negotiation between grounded ways of knowing and mathematical forms and practices” (p. 2). This embodied design approach is the key solution for teachers to assist young children’s mathematics learning.

As mentioned above, cognitive theory has become a popular scientific research topic in recent years. Since empirical evidence revealed the importance of cognitive learning in modern mathematics learning theory [22,23], understanding the embodiment of individuals’ mental learning in mathematics education has flourished into a contemporary research path in the field of cognitive science. Hutto, Kirchhoff, and Abrahamson [24]; Abrahamson and Sánchez-García [25]; and Abrahamson [26] published a series of studies and integrated handbooks that addressed innovative and reformative thinking on embodied cognitive learning, i.e., “embodied design” [31,40,41]. As theory-to-practice and pedagogical framework, the “embodied design” approach focuses on the role of embodied and situated activity, where young children’s full understanding of cognitive abilities (e.g., their early mathematical concept) is authentically scaffolded by certain practices. In such an action-based embodied design, educators are seeking to establish mathematical ideas in sensorimotor experiences in which they think of mathematical learning as moving in new manners [40]. In fact, they want “kids first to figure out how to move in some new way, before they come to realize that this way of moving will become the meaning of some mathematical idea” [30] (p. 9). Based on this argument, kindergarten teachers need to provide sufficient opportunities for young children to practice significant skills by engaging in diverse action-based “embodiments” that bridge real-world experiences to abstract mathematical concepts.

Grounded on Piagetian theory, this theory of mathematics learning was proposed by Dienes [42] and is composed of four principles (i.e., the constructivity principle, multiple embodiment principle, dynamic principle, and perceptual variability principle) and six developmental stages (i.e., free play, free experiments, comparison, representation, symbolization, and formalization). From his perspective, the constructivity principle recommends that teachers need to design hands-on activities with real-life objects and concrete manipulatives or teaching aids. In this way, young children “can be stimulated both physically and mentally to sense the structural relationships firsthand” [20] (p. 846). In addition, the multiple embodiment principle brings about furnishing children’s learning in multiple contexts that may help them to make predictions from one structural position to another. Moreover, since children like to use their body (such as hands or arms) to explore the physical world, it is suggested that teachers can design task-based learning activities where kids can use their figures to manipulate objects, count, and make comparisons and use gestures for comprehend algebraic concepts [10,32], as well as to physically move by engaging spatially with their environments for geometric understanding and abstract mathematical knowledge [10]. In short, as educators (including teacher educators and kindergarten teachers), we must provide our children embodied mathematical experiences and activities with real-life situations while learning to promote their cognitive development in mathematics for future life and learning. Particularly, the “embodied design” approach is going to serve as the core concept and action of enhancing kindergarten teachers’ PD on designing and implementing adequate embodied teaching and learning in the real settings.

2.2. PD for Embodied Mathematics Teaching and Learning

For teachers, PD not only provides various opportunities to acquire and update their professional knowledge and abilities but is an essential factor in reducing the possible gap between theory and practice [43,44]. In addition, teachers can positively improve their beliefs, professional knowledge, teaching strategies and actions through the PD process [45]. Except focusing on teaching content (e.g., mathematics for young children), Ratnayake, Thomas, and Kensington Miller [44] claimed that effective PD programs need to provide active learning opportunities and collective participation to strengthen the impacts on teachers’ future performance. For example, teachers can be given designated learning tasks to acquire, apply, and evaluate their new knowledge and abilities, allow extended group discussions, and use different cases to guide them for future applications. Similarly, Philip and Gupta [46] argued that the organization of a PD learning environment should involve how the shared activity is locally structured to address the participating teachers’ problems or issues with their kids’ learning collectively. In teaching mathematics, how children understand mathematical concepts and how teachers teach to solve children’s possible learning problems must become teachers’ daily interactive tasks [47].

An influential teacher PD community needs to rely on good relationships among peers and a combination of training and consulting services (e.g., collaborative with teacher educators), a focus on improvement and evaluation functions that collaborate with external experts for learning new knowledge and receiving challenges, and all activities and programs lasting a longer time [48]; these actions, especially for mathematics or science teachers, are beneficial to promote teachers’ professional growth of teaching and learning in different ways. In addition, the most favorable forms of PD emphasize a substantial set of curriculum design and instructional strategies, are directly connected to teachers’ practice or a problem within their classrooms (e.g., mathematical learning issues), and occur over an extended period [49]. It is also crucial to support a collaborative working environment that encourages peer-to-peer feedback, and to provide evidence-based protocols that ensure the collaborative learning remains focused on instructional development [50], which may lead to their children’s learning improvement such as mathematical understanding. Therefore, a professional learning community that is collaboratively constructed with external support and empirical research evidence will be sustainably beneficial for promoting teachers’ professional knowledge and abilities. Based on this argument and the importance of embodied deign for young children, how to design and implement embodied mathematics teaching and learning must be embedded in a co-constructed learning community for kindergarten teachers, where sustainable PD programs are furnished for their professional development and future improvements. Consequently, PD programs of the professional learning community in this study for embodied mathematics teaching and learning are planned for advancing the targeted kindergarten teachers’ professional knowledge and abilities, which, in turn, will lead to the betterment of their young children’s mathematics understanding and future learning.

3. Materials and Methods

3.1. Methodology and Participants

A qualitative case study approach was employed in this study. Eight kindergarten teachers were the main participants of this study, in which their main tasks were to design the designated embodied mathematics curriculum and instruction (for 5-year-old children) based on the PD program developed and executed by the researchers (teacher educators) with the employment of the “embodied design” approach. These teachers worked in a public kindergarten and had 10~20 years teaching experience. Data were gathered through participative observations, including PD meetings [e.g., PD-101720] and classroom observations [e.g., OB-102020]; in-depth and follow-up interviews; and various kinds of documents (such as teachers’ reflection notes [e.g., RF-D-011521] and children’s learning profiles [e.g., pictures/images]) and then analyzed qualitatively by a descriptive and explanatory approach with replication logic. The targeted kindergarten implemented “thematic teaching” promoted by the new “Curriculum Guidelines of Kindergarten Activities” [51], and mathematics learning activities were designed based on themes and implemented in the classrooms. Thematic teaching involves organizing the curriculum and instruction around a theme and having interdisciplinary and learner-centered feature (not subject matter), where learning activities are designed with rich and engaging topics along with the theme and young children’s real-life experience. In this study, a major theme was determined by teachers before the semester started, and corresponding learning activities were designed aligned to the theme and the guidelines. Grounded in its emergent characteristic, certain topics or concepts (e.g., mathematical concept) realistically and naturally emerged in the original lesson plans; for example, a measurement activity was added based on several children talking during recess time (see Section 4.1.2 for details) involving an emergent design and implementation of embodied mathematics teaching and learning.

3.2. PD and Experimental Embodied Teaching and Learning

In this study, Zaslavsky and Leikin’s [52] three-layer model of teachers’ professional growth (including teacher educators and kindergarten teachers) and Jaworski’s [53] teaching triad (i.e., management of learning, sensitivity to students, and mathematical challenge) were used as the framework of the PD programs for embodied mathematics teaching and learning. Under this framework, the major content of all PD programs, which employed the embodied design approach, is summarized in

Table 1. PD program for embodied mathematics teaching and learning.

Teaching Triad	PD Module	PD Content (Brief Description)
Management of learning (ML)	Theory and Practice of Embodied Learning	Starting from the concept of embodied learning and further discussing the theory and practice of embodied cognitive science, which can promote teachers’ professional development of embodied design and implementation
	Instructional Activity Design of Embodied Mathematics with Thematic Teaching	Through drawing of thematic concept and activity maps, integrating mathematics learning content into embodied mathematics design for 5-year-old children
Sensitivity to students (SS)	Mathematical Learning Development of 5-year-old Children	Integrate and analyze young children’s mathematics learning indicators in the United States, the United Kingdom, and Taiwan and comprehend findings of the relevant literature to provide feedback on all embodied activities designed
	“Student-centered” Activity Design of Embodied Mathematics with “Real-life Experience”	Assess children’s prior knowledge (prerequisite of targeted learning concept), interests, and needs in mathematics and design “real-life” interdisciplinary learning activity and authentic assessment tools
Mathematical challenge (MC)	Fundamental Principles of Mathematics Development in Early Childhood Education	Discuss and analyze constructivism (e.g., Piaget and Vygotsky), conduct literature reviews, and summarize possible implication for children’s mathematics learning and development
	Major Content of Young Children’s Mathematics Learning	Analyze real teaching cases based on the new curriculum standards of the United States, the United Kingdom, and Taiwan and early childhood mathematics-related content in Taiwan
	Embodied Teaching Strategy of Young Children’s Mathematics Learning	Conduct literature reviews and embodied teaching experiments and then discuss practical embodied teaching strategies

. As mentioned above, thematic teaching was employed for designing and implementing the learning activities. Although they are used to include concrete objects and hand-on activities while teaching, embodied design in mathematics was still new to them. Before the semester started, we (teacher educators) provided PD activities based on the PD modules and assisted the targeted kindergarten teachers to explore how to design embodied mathematics teaching and learning activities. After the semester started, they began to test their embodied mathematics teaching and learning activities in real settings. Through the whole process of the PD program, including PD meetings and classroom observation and after-class discussions, they gradually adjusted their embodied design and implementation and came up with their own understanding for future improvements. Examples of their embodied mathematics design and implementation are provided in the following sections along with the task design PD model generated from the data collection and analysis. Basically, the PD meetings were arranged every Wednesday afternoon (no class for children), while extra meetings were inserted based on teachers’ needs. In order to fully record the implementation process, intensive classroom observations (usually in the morning, 5 days a week) were organized along with their original embodied mathematics activities. Additional observations were added for emergent activities.

4. Findings and Discussions

Grounded in the data collection and analysis, the following two major parts of findings and discussions were included in this report for answering the research question: the first section is “task design professional development model”, which provides empirical evidence during the whole PD process to co-construct a sustainable PD model and its operational mode that will benefit the field of the early childhood teacher education (both pre-service and in-service training). The second section further provides empirical evidence of “designing principles of embodied mathematics curriculum and instruction”, which is grounded in the “embodied design” PD approach for embodied mathematics teaching and learning. Based on the design, implementation, and revision of the PD programs, sustainable principles of embodied mathematics curriculum design and effective embodied teaching strategies were generated to furnish both pre-service and in-service kindergarten teachers’ knowledge and capabilities for promoting their children’s mathematics learning in the future.

4.1. Task Design Professional Development Model

4.1.1. Scaffolding Kindergarten Teachers’ PD with “Horizontal Content Knowledge (HCK)” and “Knowledge of Content and Students (KCS)” of Embodied Design

Before participating in this study, targeted kindergarten teachers stated that learning activities related to mathematics were randomly arranged in their daily routines and primarily focused on singing and counting. Teachers rarely consider the diversity of mathematical content for young children and the learning theories that may be connected to their children’s development. The PD program in this study emphasized Jaworski’s teaching triad theory [52], which provided targeted kindergarten teachers adequate prerequisite knowledge in designing appropriate mathematics learning tasks. The teaching triad theory and its practice are similar to the model and concept of “Horizon Content Knowledge (HCK)” and “Knowledge of Content and Students (KCS)” listed in the “domains of mathematical knowledge for teaching” [54] (p. 403). Through professional dialogue within the PD activities, a great deal of empirical evidence was provided by the university research team (i.e., teacher educators) to guide these teachers in understanding the horizontal mathematics content knowledge of 5-year-old children’s mathematical capabilities. As Teacher A said,

In the past, I only knew that concrete objects are essential for young children’s mathematics learning without understanding why they are essential. In these PD activities, we authentically discussed the potential problems our children might have while learning mathematics. These kinds of experiences were different from those in-service training activities nor the pre-service courses (PD-032521).

Teacher D also expressed her appreciation for participating in the PD activities. She said, “Thank you, professor. We learned a great deal about children’s mathematics learning; for example, we used gestures to help our children while learning the ratio concept” (PD-061721). Furthermore, Teacher F mentioned in her reflections,

Young children’s learning should not be divided into separate subject areas; instead, they need to learn in an integrated manner. Once the learning tasks focus on solving real-world problems, the designated mathematical concepts will naturally merge into the learning process. Then, they can clarify those mathematical concepts through the problem-solving process (RF-D-011521).

In short, the targeted teachers recognized the benefits of the PD program.

The PD programs aimed to promote these teachers’ understanding and implementation of embodied mathematics teaching for young children. They must strengthen children’s instincts and interest in mathematics to authentically understand the relationship between the content they learn and its real-life application. These actions include “connecting children’s mathematics knowledge and daily life experience and adding fundamental information about their family and cultural background at this informal mathematics learning stage” (PD-010821). In particular, young children’s cognitive, language, physical, and social-emotional practices should be integrated with their mathematics learning. Opportunities for problem-solving must be equipped in the embodied mathematics learning tasks so that their “capabilities of reasoning and communication can be simultaneously cultivated while interacting with the mathematics concepts and peers” (RF-B-011521). In the meantime, sufficient time and suitable materials are beneficial for young children’s mathematics learning while involving embodied teaching. Teachers must closely observe their children’s learning progress during this learning process to provide further sustainable strategies or revise the learning tasks to support their learning.

Thematic teaching was employed in these teachers’ classrooms, associated with the embodied design, where mathematics related to children’s daily lives and the content of various subject areas were integrated with the problem-solving process. For example, in the “garbage is not just garbage” activity designed by Teacher D at the end of the semester, a picture book, Maltose Recycling Truck, was used to motivate their students’ learning interest, which led to a recycling truck being designed and built by children. Through building this recycling truck and the process of recycling, all children could count the amount of all the bottles involved, which resulted in the establishment of the concept of 10 (i.e., one-by-one counting of “10 bottles in a row” and recognizing that the total amount was “10”) (OB-052521). Of course, children were confused at the beginning of this recycling and counting activity, but they were gradually familiarized with the concept, since more clarifications were provided through continuing to practice this recycling and counting task. Later on, grounded in this type of “learning through playing” and “real-life experience” activity, they progressively learned the concept of 1–100 and counting to 10, as well as strengthening their capability for sorting various types of garbage (Figure 1).

Therefore, children’s understanding and coping strategies in daily lives are the basis of their mathematical concept development. It was found that children’s curiosity could promote them to make inferences based on their observation. As a result, children can acquire a better understanding of mathematical concepts and acquire corresponding mathematics abilities. In short, the content of the PD program in this study efficaciously develops these teachers’ growth mindset in the embodied curriculum design and instructional implementation. Teacher F said, “I think we don’t need the technical ‘step-by-step’ mathematics teaching anymore; this [i.e., embodied design] is definitely a better way of teaching young children mathematics” (PD-061721). This was echoed in the similar feelings expressed by Teacher B, who shared the following in her reflection note:

In the past, I would design the learning tasks from a “teacher-centered” perspective, which I always wanted to design in a more rigorous manner because I thought the counting concept is more cognitive and technical. Thus, the better way of learning it was to keep counting, and then they would memorize the process. However, I had totally different thoughts about my own teaching and more positive attitude toward their learning. Creating an embodied “learning by doing” environment was authentically beneficial to our kids where they could learn through observing, operating, problem-solving and eventually reach robust understating of the mathematics concepts (RF-B-062321).

As shown above, scaffolding by the PD program, the targeted teachers’ professional development on their HCK, and KCS of the embodied mathematics design and implementation was effectively advanced, which led to more desirable teaching practices. These instructional activities, in turn, promoted their children’s mathematics understanding and further inspired the potential of these children’s future mathematics learning.

4.1.2. Task Design PD Context with Emerging Issues Promotes Conceptual, Theoretical, and Practical Thoughts and Actions

This PD program emphasized teachers’ task design and reflection during the teaching and learning process. After implementing the teaching and learning tasks, it also focused on the targeted teachers’ professional dialogue for revisions or designing follow-up activities. In one thematic activity, children tried to measure the height of “the Mama Tree” (i.e., a very tall tree, at the front of the classroom, which children played around daily). This measurement idea came from several children’s talking during the recess time, which was observed by one teacher and merged this topic to the original lesson plans. The teachers brought Montessori teaching aids (i.e., counting sticks) to promote children’s understanding. Through the concrete operation of these teaching aids, the children were able to gradually comprehend this abstract mathematics concept (e.g., measuring with informal tools). The children became active learners and experienced self-corrections of their errors. Since the Montessori teaching aids follow a standard structuralism and “self-discovery” model, they allow young children to operate step-by-step and build up their understanding and skills for later learning. Further, children were better able to detect the base unit of measurement themselves, and the aids assisted the children in understanding continuous quantities (PD-101720). In fact, this PD process was designed based on an understanding of structural theory, specifically, the process of “Knowing What”. Later, children discussed various ways of measuring the Mama Tree and decided on the following two major plans: using balloons and taking pictures. These measuring experiments were conducted in a problem-solving manner with the embodied design to assimilate and adapt their learning schema in “doing mathematics”. This problem-solving and embodied learning process echoed “Dennis’s game theory” and allowed children to discover mathematics rules in free play (PD-101720), which aimed to merge “Knowing What” into the learning task context. In addition, teachers must understand their children’s prior knowledge and abilities while designing learning tasks. In this study, when the children did not have the concept of a “base unit” for measurement at the beginning of the task, teachers were instructed not to directly tell the child the answer; instead, they should assist their students in finding out and solving the problem through observations and discussions. Accordingly, students reached a consensus and chose a child with a height of about 110 cm as a “base unit” for measurement (see Figure 2). The teachers promptly provided cognitive conflict, making learning a concrete process of experiments, revisions, thinking, and discussions (OB-102020). This learning process was “Knowing Why”, which made good use of understanding children’s development and thinking to design the learning tasks. Moreover, these teachers discussed the embodied curriculum’s design and implementation in the PD process, which triggered reflection on teaching and developing new strategies. Whether in the Mama Tree measuring activity or the recycling and counting activity, teachers endeavored to motivate their kids to understand the concept of 10 (which is ten ones) through counting and corresponding tasks and how to measure the big tree by using informal tools (PD-103120). In this way, the teachers were able to brainstorm under the task design context and then improve their teaching capabilities, which was the “Knowing How” process.

In summary, the researchers (teacher educators) did not deliberately explain the relevant mathematics concepts, theories, and the design and implementation of embodied curriculum and instruction in a traditional manner. Instead, the task design context with emerging issues was employed to integrate the conceptual (philosophical), theoretical, and practical thoughts and actions into the whole professional development process for furnishing these kindergarten teachers’ capabilities of designing and implementing the embodied design in their classrooms under an ongoing “feedback and reflection” dialogue system. This is naturally different from the PD program of many previous studies in which the targeted teachers received the professional knowledge and skills in a highly pre-arranged and well-structured manner where those PD activities were designed in advance and tended to put the theories and practices into incoherent sections [55,56]. The professional development model used in this study is shown in Figure 3. It integrated the “Knowing Why”, “Knowing What”, and “Knowing How” processes into the task design context with emerging issues. This task-based PD program promoted the targeted kindergarten teachers to produce conceptual (philosophical), theoretical, and practical thoughts and actions that aligned with the embodied design. Therefore, the embodied design’s philosophy, theory, and practice were originated collectively by all teachers in response to the task design context with emerging issues but independent from the context. In addition, it was impossible to generate the embodied design’s philosophy, theory, and practice separately. However, this study found that it took a long time for the targeted teachers to recognize that the embodied design and implementation philosophy, theory, and practice were inseparable within this task-based PD model. For these teachers, a complete and continuous PD program could sustainably assist them in being skillful in embodied design and implementation in their own classrooms.

4.2. Embodied Mathematics Curriculum and Instructional Design

4.2.1. Basic Principles of Embodied Mathematics Curriculum Design

1.

Designing learning tasks with real-life situations

Learning tasks in the classroom should be designed and implemented as consistently as possible in children’s daily lives, and these tasks must provide adequate opportunities for children to learn and perform in a real-life manner. In addition, kindergarten teachers need to be aware that children’s life experiences originally contain many signs of mathematical concepts. Therefore, teachers can naturally integrate thematic learning activities in various fields without deliberately creating them, bringing children more life-related learning possibilities and experiences. More importantly, teachers have to step back from being a busy instructor to an observer and researcher. Echoing the “Common Core State Standards for Mathematics (CCSS-M)” [57], when teaching number concepts, the learning context should focus on representing, connecting, and manipulating integers and provide concrete objects (hands-on operations) and practical experience (real-world situations). For example, Teacher B designed and implemented a “purchasing” task, where the children first learned about the decomposition and synthesis within “10” through a poker game; then, they discovered the “price tag” of different items and expressed how they felt about the use of the concept of 10 in the poker game (OB-031621). As Teacher C said,

In fact, mathematical concepts such as classification, integers, addition and subtraction of numbers, and comparisons of different numbers were all included in this series of embodied tasks. The main task was integrating these concepts to make sense to our kids. We would discuss this together after every class period (RF-C-041221).

In short, it is evident that the PD program was beneficial for assisting their professional growth in teaching practice and regarding children’s learning effectiveness.

2.

Using pictures, images, manipulatives, and embodied operations to promote thinking

In addition to positioning the focus on learning content meaningfully, embodied teaching emphasizes guiding children to a robust understanding of the content from multiple perspectives and solving real-life mathematical issues with embodied design rather than just instilling them with fixed knowledge in a traditional manner [58]. In the “Garbage is not just garbage” activity in the second semester, children tried to make orange detergent essential oil with the following steps (see Figure 4) by peeling the orange skin into small pieces, putting these into a glass bottle, and soaking them in 95-degree alcohol for seven days. This causes the oil in the orange skin to be dissolved by the alcohol, producing orange essential oil (OB-052921). This activity was designed by Teacher F, and the children could learn authentically by carrying out the activity and discussing the whole process of this hands-on experience.

Another example of embodied teaching focused on the concept of capacity retention. In this activity, children needed to understand the concept of capacity and make sense of the meaning of “capacity retention” through a measuring task. First, Teacher A showed two measuring cups of different sizes (see Figure 5): the large one was 1000 cc., and the small one was 100 cc. At the same time, the children were asked to express their opinions about the cups (focusing on the size difference, #1 of Figure 5). Secondly, an experiment was conducted for comparison of the two cups. Water (100 cc.) dyed with orange color (for observation) was poured into the small cup, and then the water in the small cup was poured into the large one (#2 of Figure 5). Through observation, kids discussed whether the amount of water (100 cc.) in the large cup was the same as in the small cup (i.e., 100 cc. originally). In fact, after this discussion, all children agreed that the amount of water was the same even though the size of the two measuring cups differed (OB-121420). In addition, four measuring cups of different sizes were provided for further experiments (#3 of Figure 5), where the teachers confirmed that they possessed the concept of capacity retention based on this embodied learning experience. Later, the children worked in small groups to freely experiment with how to measure the capacity.

Children actually learn with their whole bodies more than with their brains and minds. Such embodied cognition is critical in mathematics learning [10]. Embodied cognition involves a “perception-action cycle” [59] where one’s behavior comprises a series of adaptive motor reactions. These actions, such as hands-on movements, produce changes in external (e.g., manipulatives, teaching aids) and internal (e.g., interest, motivation) environments that consequently affect later actions as a circular process. This cycle, including the interaction among perception, environment, and action, is fundamental to a learner’s embodied learning activity [60]. Therefore, using real-life mathematics issues and hands-on experiments, the embodied teaching designed and implemented by the targeted teachers fulfilled the “perception-action cycle” and provided abundant “doing mathematics” opportunities for our children to learn meaningfully. Within this embodied learning process, the children could discover the essence of the designated mathematics issues through observations and proposed conjectures and revisions through collaborative peer interactions, which also promoted children’s mathematical thinking and capability [4,61].

In summary, once teachers possess the HCK and KSC, their curriculum design will be learner-centered, and they will be able to record their students’ complete hypothetical learning trajectories (HLTs) [62] that will contribute to later adjustment of their instruction. In the first embodiment world of young children, their mathematical cognitive development is based on the perception of the natural world, combining their senses and the experience of operational behaviors. Therefore, teachers must provide children with concrete, visible, and embodied objects and use understandable language and pictures or images to help them connect visual and other sensory actions to facilitate the development of physical and mental embodied learning.

Tall [37] emphasized that children’s cognitive development is a process of “set-before” and “met-before”, which is critical to the first embodiment world of their mathematics cognitive development and has a significant impact on the subsequent development of symbolization and formalization of the whole mathematics cognitive world. Numerous scholars [21,33,34,35,36,37] revealed that teachers need to provide children with a mental structure based on past learning experiences, which is recognized as the “supportive aspect” of using prior knowledge as the basis for learning new knowledge. In addition, the “problematic aspect” has to be furnished, where cognitive conflicts could be fundamental for them to further in-depth inquiry and problem solving. Both perspectives focus on providing children with opportunities to experience new things and develop their meta-experiences from the embodied curriculum and instruction provided by teachers, thereby supporting young children in generating new perspectives in real-world situations, leading to consequent problem-solving processes. Moreover, these two perspectives allow them to organize and transform their mental structures continuously. Sometimes, the more supportive challenging and/or cognitive conflicting a problem one has, the better it will help the child to develop different insights into concepts while learning mathematics. Lakoff and Núñez [63] used “subtraction” as an example; in mathematics, subtraction can be regarded as “taking away”. In real life, you cannot remove more objects than the original number; however, in algebra, you can quickly generate the formula “5 − 7 = −2”, which is why “taking away” sometimes brings young children more confusion and difficulties. Consequently, when there is a cognitive conflict between children’s innate experience and acquired mathematical learning experience, the embodied design becomes the essential nutrient for assisting our children to meaningful learning.

4.2.2. Effective Embodied Mathematics Teaching Strategies

1.

Establishing children’s active learning habits

Mathematics learning for young children is a critical enlightenment for their future learning. While teaching, mechanical learning methods such as piecemeal accumulation and cramming are in vain; on the contrary, young children must learn through active and collaborative participation and with inquiry and problem-solving tasks and acquire all concepts in unity. The interaction of children’s previous and new experiences becomes fundamental in constructing a new concept where the learning process is more important than the acquisition of learning outcomes. In this study, Teacher F discussed the garbage with her kids at the end of the first semester, and they buried some trash in the corner outside of their classroom on campus (taking pictures as well). Later, they dug out the buried garbage at the beginning of the second semester (taking pictures again) (OB-053121). By comparing these photos of buried garbage (before and after), observations, comparisons, discussions, and classifications were collaboratively conducted, and presentations of comparative results were also made where all the students reached the conclusion that there were two types of garbage: one type, buried garbage underground, would gradually disappear, while the other kind of buried garbage would not disappear. Further, many parents provided feedback to the teachers. They expressed that, with this real-life learning experience, their children could actively explore how to classify garbage at home and discuss how to classify garbage to protect the earth. Consequently, this type of active learning task, combined with real-life situations and practical observations, discussion, and classification, is the most significant essence of embodied mathematics learning and the quality required by future citizens.

2.

Providing children sufficient time and materials to explore the essence of mathematics

Children should be encouraged to use mathematical tools and strategies while learning. For instance, Teacher A asked her kids to connect the number 16 with the corresponding quantity by stringing 16 beads on a necklace and preparing eight plates in order for the children to place two biscuits on each plate. This specific operation helped the children to understand the meaning of the number “16” easily. Similarly, Teacher F provided many materials in a thematic “Zhuluo Chiayi” activity. Children were permitted to use a string of plastic rings to measure the buildings they constructed in the building block area, use toy banknotes to buy tickets at the drama center, and use cloth or string to measure a plant’s growth. There were other examples of how many mathematics learning activities can easily be integrated into different activities. Teacher F placed a scale in the outdoor loose-material area, and her children developed a game named “Find the Heaviest Stone”. Teacher B asked children to sort things into corresponding trash cans during cleaning time. Teacher D provided opportunities for her students to explore graphics and shapes while creating art products. Teacher A helped her students tell the time by using an itinerary that allowed them to see the time of the next activity. Teacher C had her students record and calculate the time it took them to pass the jungle gym on the playground. In these embodied mathematics learning tasks, children understood the relationships and connections among quantities to complete real-life tasks using embodied mathematical tools and real-life experiences. This kind of learning allows teachers and their children to act spontaneously, which, in turn, helps them to become active learners who can collaboratively interact with the environment and others.

5. Conclusions and Implications

5.1. Task Design PD Model and Principles of Embodied Design in Mathematics Serves as Sustainable “Pre-Service” and “In-Service” Teacher Training Programs for Kindergarten Teachers

Grounded in the findings, the following two major parts of the findings and discussions were included in this report: first, a sustainable “task design professional development model” (see Figure 3) was generated for future improvements of both in-service and pre-service teacher training programs. In this model, PD modules that were designed based on the three-layer model of teachers’ professional growth [52] and the teaching triad [53] for teacher educators and kindergarten teachers served as the PD framework for embodied mathematics teaching (see Table 1). This model and its operational mechanism can be merged into both the practical settings (i.e., kindergarten and preschool) and early childhood teacher education programs in higher education, which is also useful for teacher educators (researchers) to authentically reflect and revise their actions while teaching and researching. Associated with this PD model, two principles of embodied mathematics curriculum design and two effective embodied teaching strategies were proposed for upskilling all teachers’ (i.e., both pre-service and in-service kindergarten teachers) professional knowledge and capabilities and, in turn, promoting young children’s mathematics learning effectiveness and performance.

As aforementioned predicaments of both a deficient “pre-service” and “in-service” kindergarten teacher training system, insufficient training in mathematics and lack of motivation in learning mathematics result in the incapability of most kindergarten teachers to teach mathematics to young children. Moreover, their indifference and fear in mathematics have a negative impact on young children’s mathematics learning for a long time [19]. Steele, Brew, Rees, and Ibrahim-Khan [13] pointed out that teachers need more professional knowledge and skills to teach science and mathematics. Because of this systematic issue, resuming and upgrading both “pre-service” and “in-service” kindergarten teacher training programs with a sustainable manner in Taiwan became essential tasks for early childhood teacher educators at the higher education level in Taiwan. In this study, we produced a sustainable “embodied design PD model” for young children’s mathematics teaching and learning from working with a group of in-service kindergarten teachers, as well as specific principles and strategies for embodied mathematics curriculum design and instructional implementation. This embodied mathematics PD model and its content and operational mode are also suitable for merging into the pre-service (at 4-year undergraduate level) and in-service (at graduate level) kindergarten teacher training program, which will be sustainably beneficial to effectively prepare both pre-service and in-service teachers’ knowledge and capability regarding embodied mathematics design and implementation in their future teaching. This, in turn, will promote their young students’ mathematics learning.

5.2. “Reflective Learning” of Teacher Educators and In-Service Teachers Co-Constructs a Sustainable Professional Development Program for Future Improvements

Research is a thinking framework with perspective. In addition to deliberating the limitations of the existing framework, we should analyze its nature more profoundly. In the past, teacher educators were primarily responsible for conducting research in practical settings and providing essential professional development for in-service teachers. Their roles were usually observers, PD program planners, analysts, and researchers. In previous studies in Taiwan, teacher educators mostly provided PD programs in a separate manner and then employed traditional quantitative (e.g., questionnaires) or qualitative (e.g., observations, interviews, and other documentation) ways to collect and analyze the data in order to reach the research objectives. In fact, when the teacher educator’s role is “only a researcher or planner”, the researcher is considered as an “outsider” of the study. It is inevitable to worry that the researcher’s role and subjectivity often have a decisive impact on the interpretation and decision making of the research. However, when the researcher “stands out” from the research and “shapes” the research from an outsider’s perspective, it is often difficult to clearly observe the phenomena, understand the participants (i.e., in-service teachers), and reach sustainable conclusions, especially for the qualitative research that requires “professional subjectivity” [64].

Currently, in more and more studies, researchers are no longer the “traditional others”. They tend to ask “reflective questions” based on the observed data and adjust their strategies and their way of thinking during the whole research process. In other words, teacher educators must employ classroom observations to adjust research strategies to accurately meet specific needs of in-service and pre-service teachers. Steele, Brew, Rees, and Ibrahim-Khan [13] conducted action research through collaborative teaching to enhance pre-service teachers’ confidence in science and mathematics teaching, emphasizing classroom understanding and experience outside the university classroom. During the research process, various researchers cooperated through reflection and raised essential questions after observing the targeted pre-service teachers, such as the following: “Why were the pre-service teachers not interested in mathematics? Were they even worried about any issue? Did their own experience and background influence their attitudes towards learning? What kind of activities were the pre-service teachers particularly interested in?” In addition, Kindle and Schmidt [65] studied two pre-service teachers by using the self-collaborative research model and used multiple scaffolding strategies, such as changing the pre-service teacher’s perspective, switching from being the authority to being the students’ peer, and raising questions to understand pre-service teachers’ behavior and examine their comprehension skills. During the whole research process, the two teacher educators (as researchers) constantly used “reflection, revision, and practice” as the basis for improving their own teaching in the university’s classrooms. The results showed that when teacher educators considered themselves as learners, they were no longer “strangers”; instead, their roles were both “teachers” and “reflective leaners”, so that they could successfully teach and learn with their students (i.e., pre-service teachers) together. In this way, they were collaborative “important others” of their pre-service teachers, where they co-constructed a professional development environment that combined both horizontal (peers) and vertical (teachers and students/pre-service teachers) learning manners, which could also promote their problem-solving and critical-thinking capabilities.

In this study, the teacher educators (researchers) were also “collaborative important others” of the targeted kindergarten teachers. Within the task design PD program, their roles included PD planners, researchers (observers and analysts), teacher educators, and reflective and collaborative learners (learning with those kindergarten teachers as peers). Through the whole reflective process of the PD program, the PD model (including the framework and the content) were revised and verified, during which the targeted kindergarten teachers’ professional development with “horizontal content knowledge (HCK)” and “knowledge of content and students (KCS)” of embodied design were advanced. Under this task design PD context, emerging issues promoted these kindergarten teachers’ conceptual, theoretical, and practical thoughts and actions. This co-constructed PD learning process was authentically beneficial for both teacher educators (researchers) and kindergarten teachers (in-service teachers). This “embodied design” PD model can serve as an efficacious manner for both “in-service” and “pre-service” training to sustainably improve teachers’ professional development and students’ learning performance, as well as an authentic reflection tool for teacher educators’ professional development and future improvements.