We describe the design of the institute and our analysis of teacher performance on a mathematics and CT exam, changes in TPACK, and participant reflections about how the institute impacted their knowledge of mathematics and ways to integrate CT with an emphasis on coding or programming into instruction.<\/p>\n\n\n\n
Literature Review<\/h2>\n\n\n\nNational Standards Emphasize Change<\/h3>\n\n\n\n
Both the Next Generation Science Standards<\/em> (NGSS; NGSS Lead States, 2013) and the CCSS-Mathematics emphasized computational thinking practices (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010; National Research Council, 2012). The NGSS included using mathematics and computational thinking as one of the eight recommended science and engineering practices. Furthermore, the CCSS-Mathematics emphasized eight Standards for Mathematical Practice, including making sense of problems and perseverance, reasoning abstractly and quantitatively, constructing viable arguments and critiquing the reasoning of others, modeling with mathematics, using appropriate tools strategically, attending to precision, looking for and making use of structure, and expressing regularity in repeated reasoning.<\/p>\n\n\n\n The CCSS-Mathematics were developed to address the need to prepare students for college and career expectations based on observations that, nationally, academic progress has been stagnant and there is a high need for mathematics remediation at the college level (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Key shifts from previous standards included a greater focus on fewer topics, coherent progressions from grade to grade, rigor with a pursuit toward conceptual understanding, procedural skills and fluency, and application of mathematics to real-life scenarios.<\/p>\n\n\n\n The ISTE standards for educators and students emphasized computational thinking competencies (ISTE, 2016, 2017, 2018).<\/p>\n\n\n\n Similarly to how technology is used by educators to deepen content area learning while building digital learning skills, teachers can integrate CT practices in their instruction to introduce computational ideas. This will enhance student content knowledge and build confidence and competence in CT. (ISTE, 2018, p. 1)<\/p><\/blockquote>\n\n\n\n The CSTA (2016) developed the K-12 CS Framework based upon five core concepts with benchmarks provided for K-2, 3-5, 6-8, 9-10, and 11-12, together with five crosscutting concepts or themes integrated with the core concept statements and seven core practices that demonstrate engagement with the core concepts. As written, this framework serves as a conceptual guide for developing standards for independent CS courses or for integration with mathematics, science, and other subjects throughout the K-12 path.<\/p>\n\n\n\n Of the seven core practices, four are specific CT practices, including recognizing and defining computational problems, developing and using abstraction, creating computational artifacts, and testing and refining computational artifacts. Three of the practices are general practices of CS that support CT, including fostering an inclusive computing culture, collaborating around computing, and communicating about computing.<\/p>\n\n\n\n CT is defined by some as the \u201cthought processes involved in formulating problems and their solutions so that the solutions are represented in a form that can be effectively carried out by an information processing agent\u201d (Wing, 2011, p. 20). CT is a human ability characterized by problem solving, designing solutions, communicating thoughts in a creative, organized way, and debugging thoughts with or without a computer (CSTA, 2016, p. 69). Consensus has been achieved for the need to nurture and develop CT natives. In addition CT has gained enough importance for a suggestion to make \u201c’rithms,\u201d short for algorithms, the fourth \u201cr\u201d for 21st-century literacy (Barr & Stephenson, 2011; CSTA, 2016; Grover & Pea, 2013; ISTE, 2016).<\/p>\n\n\n\n Current research endeavors have recognized the need to contextualize CT within specific disciplines (Gadanidis et al., 2017; Grover & Pea, 2013; Qin, 2009; Weintrop, et al., 2016; Ya\u015far, 2013). Benefits of embedding CT into mathematics and science classrooms include addressing the changing nature of these disciplines, such as bioinformatics and computational statistics, in the professional world; developing the reciprocal relationship between computational contexts and science and mathematics learning; and addressing the issues of underrepresentation of women and minorities in computer science fields (Weintrop et al., 2016).<\/p>\n\n\n\n Weintrop, et al. (2016) developed a computational thinking in mathematics and science practices taxonomy that includes four major categories: data practices, modeling and simulation practices, computational problem-solving practices, and systems thinking practices. Each of these categories is composed of a subset of five to seven practices that are interrelated and dependent upon one another. Data practices include collecting, creating, manipulating, analyzing, and visualizing data. Modeling and simulation practices include using models to understand a concept, using models to find and test solutions, assessing models, designing models, and constructing models. Computational problem-solving practices include preparing problems for computational solutions, programming, choosing effective computational tools, assessing different approaches\/solutions to a problem, developing modular computational solutions, creating computational abstractions, and troubleshooting and debugging. Systems thinking practices include investigating a complex system as a whole, understanding the relationships within a system, thinking in levels, communicating information about a system, and defining systems and managing complexity.<\/p>\n\n\n\n The TPACK Framework serves as the guiding theoretical framework for this project. Participants were challenged to use this framework as a guide as they intentionally incorporated computational thinking practices into their curriculum, instruction, and assessment. As described by Mishra and Koehler (2006), the TPACK framework explores how technology is integrated with teaching through the following seven categories: technology knowledge (TK), content knowledge (CK), pedagogy knowledge (PK), pedagogical content knowledge (PCK), technological pedagogical knowledge (TPK), technological content knowledge (TCK), and TPACK.<\/p>\n\n\n\n The TPACK framework builds on the work of Shulman (1986) and is based upon the need for teachers to build subject-specific PCK. Koehler et al. (2013) emphasized the context-specific nature of incorporating digital technology by stating, \u201cIntegration efforts should be creatively designed and structured for particular subject matter ideas in specific classroom contexts\u201d (p. 14).<\/p>\n\n\n\n Technology-based professional development for teachers should provide explicit opportunities for teachers to connect technology to curriculum, assessment, and instruction rather than teach technology skills in isolation (Hughes, 2005; Ndongfack, 2015). To prepare their students for the technological demands of the 21st-century, teachers need to develop the same skills as the students, which calls for sustained professional development opportunities (Goode et al. , 2014; Matherson et al., 2014). To develop TPACK, teachers need opportunities to collaborate, plan, and reflect on their experiences on both learning and teaching with technology (Goode et al., 2014; Olofson et al., 2016).<\/p>\n\n\n\n Niess et al. (2009) described a five-stage Mathematics Teacher TPACK Developmental Model through which teachers can progress when they are learning to integrate technology in teaching and learning mathematics. This process begins when a teacher has sufficient PCK. As the teacher progresses through the stages using a particular technological tool TPACK is developed. These levels are as follows:<\/p>\n\n\n\n Descriptors and examples are provided for the Mathematics Teacher TPACK Development model for four themes, including curriculum and assessment, learning, teaching, and access.<\/p>\n\n\n\n Richardson (2009) described a professional development project for 20 eighth-grade Algebra I teachers focused on developing their TPACK knowledge by integrating technology such as TI-Nspire, virtual manipulatives, and GeoGebra. The teachers\u2019 exit interviews indicated that they advanced in their development of TPACK; however, they needed more training in making a transition from using technology-based manipulatives for illustrating content to using these tools to help students develop conceptual understanding of concepts. Common challenges included helping teachers focus on technology, content, and pedagogy collectively rather than focusing solely on the technology in and of itself.<\/p>\n\n\n\n Olofson et al. (2016) defined TPACKing, which is \u201can active process carried out by the teacher in which s\/he constructs knowledge in the technology-rich setting\u201d (p. 189). This process is continuously modified because teachers draw upon context, experience, and knowledge of students as they build TPACK. Then, as they enact TPACK in their unique setting with students, they change their TPACK construct. This change influences their understanding and beliefs about technology, pedagogy, and content.<\/p>\n\n\n\n Olofson et al. (2016) used a multiple case study method and a radical constructivist lens to isolate four unique ways in which teachers develop TPACK, including interpersonal, environmental, and internal interactions and equilibration. The authors suggested that the TPACKing framework could be used during professional development to help analyze in-service teachers\u2019 progress, to identify practices that lead to TPACKing and how making the process explicit to teachers impacts their constructions.<\/p>\n\n\n\n Coding for the Core provided professional development for 22 middle school teachers (17 mathematics, two science, and three STEM), ranging from 1 to 36 years of experience, focused on computer programming and robotics, with specific connections to CCSS-Mathematics. The project was funded through the state of Tennessee Improving Teacher Quality (ITQ) grant program and included a 2-week summer institute held at a rural university in Tennessee along with two follow-up Saturdays during the fall semester for a total of 72 contact hours.<\/p>\n\n\n\n This institute aimed to improve teachers\u2019 mathematics content and pedagogical knowledge, with a focus on best practices for instruction as required by the Tennessee Educator Acceleration Model, or TEAM, which is the evaluation system required by the state (Tennessee Department of Education, 2019). Teacher participants received a $75 daily stipend, a Lego Mindstorms EV3 Core Set, hands-on mathematics manipulatives, and publications geared for middle level teachers on the topics of robotics and hands-on manipulatives.<\/p>\n\n\n\n Coding for the Core used a programming package developed by Code.org and Bootstrap (Schanzer et al., 2013) that requires students to write code that illustrates algebra and geometry concepts. Additionally, programming robots created using Lego Mindstorms allowed for writing code illustrating ratios and proportions as well as data analysis, statistics, and probability. The goal was to promote a change in teacher understanding of mathematics and TPACK with the use of CT practices emphasizing computer programming activities and additional instruction using hands-on and virtual manipulatives directly aligned with middle school content.<\/p>\n\n\n\n The summer institute was planned using characteristics of effective professional development, which included the following components: focused on clear goals, based on content and practice, provided active learning experiences, led by facilitators with appropriate expertise, aligned with state and district goals and standards, and enabled collaborative and collective participation of teams of teachers (Demonte, 2013; Desimone, 2009; Koba et al., 2013; Loucks-Horsley et al., 1996). Although we provided 72 contact hours over 4 months, due to the nature of the ITQ grant, we were unable to extend the grant for a longer period of time, which is a limitation of this study.<\/p>\n\n\n\n Coding for the Core was conducted jointly by education methods and mathematics and computer science faculty to effectively model pedagogy and focus on building mathematics content knowledge and practices within the context of embedding CT practices. This approach aligned with a principle of effective PD for mathematics and science education that suggests the importance of providing \u201cteachers with opportunities to develop knowledge and skills and broaden their teaching approaches, so they can create better learning opportunities for students\u201d (Loucks-Horsley et al., 1996, p. 1).<\/p>\n\n\n\n CT practices were directly linked to CCSS-Math standards for mathematics content and mathematics practices. Participants were engaged in TPACKing (Olofson et al., 2016) as they reflected upon their individual contexts, experiences, and knowledge of students and developed their beliefs about technology, pedagogy, and content. Participants were also engaged in active learning throughout each day using computers, robotics, and mathematics manipulatives.<\/p>\n\n\n\n As recommended by Loucks-Horsley et al. (1996), teachers worked in learning communities and were prepared to serve in leadership roles. During the workshop teachers worked in groups of three to four in grade-level teams as they experienced activities that embedded CT practices that mirrored methods they could use with students. Project staff explicitly addressed teacher\u2019s knowledge of TPACK within mathematics as well as the context of CT practices.<\/p>\n\n\n\n Each participating middle school identified at least two mathematics teachers to form a professional learning community (PLC). In some cases when only one mathematics teacher volunteered, we recruited a science or STEM teacher. These PLCs were also challenged to seek opportunities within their school and district to provide at least one professional development session for their peers.<\/p>\n\n\n\n Our schedule typically allowed for four blocks of instruction each day, including 90 minutes each for Bootstrap Algebra Units and Lego Mindstorms challenges led by the mathematician\/computer scientist on our team, 2 hours for mathematics-specific content aligned with Bootstrap and Lego units led by our mathematician, and 1 hour for pedagogical strategies emphasized by TEAM (Tennessee Department of Education, 2019) led by our mathematics education specialist. Day 1 allowed for teachers to complete their preassessment content test, the TPACK survey, and a baseline TEAM survey. The teachers also completed an Hour of Code at code.org and an icebreaker in which they formed teams to construct artistic robots made using a plastic cup, a battery, wires, a motor, and markers. They were introduced to Lego Mindstorms, received their Lego kits, and had an opportunity to sort their bricks. Following Day 1 the teachers followed four blocks of instruction.<\/p>\n\n\n\n Even though the taxonomy developed by Weintrop et al. (2016) was published after the summer institute, its framework of practices (data, modeling and simulation, computational problem-solving, and systems thinking) was useful in describing our institute design. In the data practices category, Lego Mindstorms robots were used to generate experimental data to illustrate concepts including ratio and proportion, rates of change, and statistical concepts. These data were then manipulated and displayed graphically and the results were analyzed.<\/p>\n\n\n\n The Bootstrap programming also involved data practices. To test the code the participants were required to generate test data. In the modeling and simulation practices category, functional models were created using the data collected from the Lego Mindstorms robots, and these models were tested. The Bootstrap programming also involved developing functional models of the motion of game components.<\/p>\n\n\n\n In the computational problem-solving category, both the Lego Mindstorms and the Bootstrap activities involved computer programming. This programming often involved breaking the problem down into functional modules and then creating computational abstractions to create these modules.<\/p>\n\n\n\n Finally, the programs needed to be tested and debugged. The final category in Weintrop’s taxonomy is systems thinking practices, which were addressed in both the Lego Mindstorms and the Bootstrap programming. A robot, together with its programming, is a complex system, as is a computer game developed in Bootstrap. The relationships between the various components must be understood and managed for the system to operate correctly.<\/p>\n\n\n\n Bootstrap Algebra is a 20- to 25-hour module divided into nine units specifically focused on using computer science in algebra to construct a videogame around three elements: a player (the user\u2019s avatar), a target (something the player wants), and a danger (something the player must avoid). Each unit in the Bootstrap curriculum is designed to integrate and introduce three interrelated components: a new game feature, a programming concept, and a mathematics concept (Schanzer et al., 2015).<\/p>\n\n\n\n For example, in Unit 1 the game feature is locating elements on a screen, the programming concept is creating expressions with the use of \u201cCircles of Evaluation,\u201d and the mathematics concept is cartesian coordinates. The video game is built in a sequence of frames, and a function is written to locate each character within a frame to describe the character\u2019s change in position as the character moves. The Bootstrap curriculum allows for students to model three representations of functions: symbolic form, domain and range, and input\/output tables (Schanzer et al., 2018).<\/p>\n\n\n\n Bootstrap Algebra begins with learning how to diagram expressions and practice writing functions using a notation called Circles of Evaluation. This notation provides an organized means to express the order of operations. We asked the teachers to transform the circles into a \u201csyntactically valid textual code\u201d called Scheme using the Racket programming language (Schanzer et al., 2015, p. 2). The circles of evaluation provide a means to clearly connect algebraic functions and expressions to formal notation or written computer code.<\/p>\n\n\n\nDisciplinary Computational Thinking<\/h3>\n\n\n\n
Theoretical Framework<\/h3>\n\n\n\n
Studies Incorporating TPACK Mathematics<\/h3>\n\n\n\n
Professional Development Design<\/h2>\n\n\n\n
Professional Development Framework<\/h3>\n\n\n\n
Institute Schedule and Design<\/h3>\n\n\n\n
Block 1: Bootstrap Algebra<\/em><\/h3>\n\n\n\n
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