Examining TPACK Enactment in Elementary Mathematics with Various Learning Technologies

Abstract

As schools continue to invest more resources into learning technologies and their infrastructure, there is a need to consider ways to support teachers’ professional learning related to the appropriate uses of learning technologies. The construct of TPACK continues to provide a framework for the field to help make sense of the knowledge and skills related to effectively integrating learning technologies into teaching. The article includes three vignettes that describe research-aligned enactments of TPACK using three different learning technologies: an internet-based activity focused on lower-level fluency, an internet-based simulation/game on budgeting, and an internet-based open-ended tool. The three vignettes suggest that TPACK can be enacted regardless of the learning technologies and that the culture and context of schools greatly influences whether and how teachers integrate technology into their mathematics teaching.

1. Background

1.1. Effective Teaching with Technology: The Role of the TPACK Framework

The purpose of this article is to describe aspects of Technological Pedagogical Content Knowledge (TPACK) and describe vignettes from classroom observations where TPACK is enacted in elementary school mathematics classrooms. Vignettes provide both scholars and practitioners with tangible examples of how constructs and theories are operationalized in realistic settings [1]. Practitioners have been shown to benefit from the use of vignettes focused on specific instructional practices [2,3]. Scholars have found that technology has the potential to positively impact student learning in mathematics [4,5,6,7]. Technology, when used in specific ways, has proven itself as a tool for improving student learning in mathematics [4,5,6]. Specifically, research shows that the pedagogy that accompanies the technology used is more important than the technology itself [6,7]. These vignettes provide examples for scholars and practitioners of how technology and research-based pedagogies can support the teaching and learning of mathematics.

In this article, I position Technological Pedagogical and Content Knowledge (TPACK) as a framework and construct that describes the knowledge that teachers have and can apply in their classroom. The construct enactment of TPACK or enacted TPACK reflects the idea that TPACK can be applied and put into practice by teachers when they or their students use learning technologies. Mishra and Koehler, two of the original TPACK scholars, advanced the idea that effectively teaching with technology requires teachers to have and be able to apply knowledge related to various discrete aspects (see Figure 1) [8]. These three distinct aspects include using technology (Technological Knowledge [TK]), knowledge of the content that they are teaching (Content Knowledge [CK]), and knowledge of effective research-based pedagogies (Pedagogical Knowledge [PK]). Additionally, the seminal TPACK model advances the idea that there are additional aspects represented in the integration of these aspects such as using technology with specific research-based pedagogies (Technological Pedagogical Knowledge [TPK]), using technology to teach specific content (Technological Content Knowledge [TCK]), and using specific research-based pedagogies to teach specific content (Pedagogical Content Knowledge [PCK]). The center of the Venn diagram reflects the aspiration for effective teaching with technology, which is using technology along with research-based strategies to teach specific content (Technological Pedagogical Content Knowledge [TPACK]).

As the center of the Venn diagram continues to be the goal, it is important to consider factors that may present a challenge for teachers as they attempt to integrate technology into their mathematics teaching. Below, I discuss teachers’ beliefs and their school’s context and culture.

What Does TPACK Look Like? An Example from Second-Grade Mathematics

The purpose of this section is to provide an example of how the different aspects of TPACK are enacted in a mathematics classroom. The context is a second-grade classroom, where the teacher has asked students to solve the problem “102 take away 26” (often written as “102 − 26 = ___”). A commonly used tool is physical place value blocks or virtual place value blocks. Figure 2 shows a process that students used to build 102 and use the virtual place value blocks to show how to solve “102 take away 26”.

Table 1. Example of aspects of TPACK in the context of elementary school mathematics.

Aspect	Description
Technological Knowledge (TK)	A teacher uses virtual place value blocks on a device to represent numbers and break blocks up into smaller pieces.
Pedagogical Knowledge (PK)	A teacher introduces an activity to students and pose questions to facilitate students’ completion. OR A teacher introduces an activity, provides directions, and directs teaching to facilitate students’ completion and then provides opportunities for students to practice those skills on follow-up problems.
Content Knowledge (CK)	A teacher solves “102 − 26” and explains how they found the answer. The teacher is familiar with various strategies that students may use to solve this problem.
Technological Pedagogical Knowledge (TPK)	A teacher introduces an activity where students use place value blocks, and the teacher poses questions to facilitate student completion. OR A teacher introduces an activity where students use place value blocks, provides directions, and directs teaching to facilitate students’ completion.
Technological Content Knowledge (TCK)	A teacher uses virtual place value blocks themselves to represent 123 and show how to use the blocks to show “102 − 26”. The teacher is familiar with various strategies that students may use to solve this problem both with and without technology.
Pedagogical Content Knowledge (PCK)	A teacher introduces an activity without technology and poses questions while students solve “102 − 26”. The teacher is familiar with various strategies that students may use to solve this problem. OR A teacher introduces an activity and provides direct instruction about how a student should solve “102 − 26” without technology and then provides opportunities for students to practice those skills on follow-up problems. The teacher is familiar with the strategy that they are providing to their students.
Technological Pedagogical Content Knowledge (TPACK)	A teacher has second-grade students use virtual place value blocks on a device to solve “102 − 26”. The teacher introduces activities and poses questions to facilitate student completion. The teacher is familiar with various strategies that students may use to solve this problem. OR A teacher has second-grade students use virtual place value blocks on a device to solve “102 − 26”. The teacher provides direct teaching on a device while students follow along on their device and then provides opportunities for students to practice those skills on follow-up problems. The teacher is familiar with the process that they are using to teach students.

describes the aspects of TPACK as it relates to using virtual place value blocks to solve “102 take away 26”. While it is definitive that students are expected to obtain the correct answer, 76, and use the place value blocks to find the answer, the process of teaching students differs based on which body of research teachers subscribe to. In the mathematics education communities, advocates would promote the idea of supporting inquiry-based experiences and productive struggle, which embodies a guided discovery approach in which a teacher provides access to the virtual base-ten blocks and the task and poses questions as students explore ways to find the answer [9,10]. Meanwhile, in the special education communities, scholars would advocate for a more direct teaching approach where students would watch the teacher model how to solve the problem and then complete multiple practice problems with support from the teacher using the teachers’ specific strategy [11]. With the aspects of TPACK related to pedagogy, there is content or domain-specific considerations that must be taken into account. In the case of mathematics pedagogy, both scholars and practitioners must take into account the inconclusive research on the use of guided exploration [9,10] or direct instruction [11]. Specifically, in the case of TPACK, it is possible that teachers may enact evidence of TPACK in their classrooms in ways that promote students’ exploration of how to solve problems or more direct teaching approaches.

As stated in the examples above, TPACK has multiple aspects that each can be seen when TPACK is enacted in a classroom. In the case above, examples provided in Table 1 provide alternatives for aspects related to pedagogy, since research is conflicting between inquiry-based and teacher-directed approaches to mathematics teaching and learning.

1.2. The Case for TPACK and Revised Bloom’s Taxonomy

Just as the research on how to teach students mathematics has mixed findings, there is inconclusive evidence about the types of technologies that are most effective at influencing student learning. Specifically, studies have found positive impacts when technology has been used as a tool to promote problem solving and students’ exploration of open-ended, non-routine problems [6,12] as well as more basic skills such as computation [12]. One way to frame ways in which technology is used in mathematics classrooms is through the content-agnostic framework of Revised Bloom’s Taxonomy [13].

Table 2. Levels of Revised Bloom’s Taxonomy related to mathematics (synthesized from [11,12]).

Level	Related Actions	Examples
Remember	Identifying or recalling a fact or concept	What is the product of 7 × 6? Type the product in the box on the iPad.
Understand	Sorting, classifying, and summarizing	Use the virtual tool to make different shaped rectangles with 12 tiles. Based on your rectangles, is 12 prime or composite? Explain how you know.
Apply	Representing and/or solving a problem	Use the virtual place value blocks to solve 1205 − 747. Write a few sentences to explain your process.
Analyze	Examining and comparing strategies, quantities, and patterns	The picture shows two ways to solve 1205 − 747 = __. Record a video where you explain how the two strategies are similar and how the two strategies are different. Now use those strategies to solve 1403 − 576 = __.
Evaluate	Evaluating patterns and relationships	Use the virtual geometry tool to make a rectangle. What do you notice about the size of each angle? Now drag the top left vertex to the left. What happens to each of the angle measurements? Now drag the top right vertex to the left so that you have 2 sets of parallel sides. What happens to each of the angle measurements?
Create	Synthesizing and constructing something new from various concepts	Use technology to determine the price it would take to prepare a meal for 4 people. Then determine the price it would take to feed 4 people at a restaurant. How much money do you save by cooking at home compared to a restaurant?

provides the Revised Bloom’s Taxonomy framework with examples of technology-rich activities that are relevant to elementary school grades.

Revised Bloom’s Taxonomy provides a way to critically examine teachers’ use of higher-level thinking skills during all learning activities. Since mathematics education research has found benefits in the use of technology in ways that align with higher-level and lower-level thinking skills [13], examinations of teachers’ enactment of TPACK must also include discussion of the extent to which students are using higher-level thinking skills.

1.3. Developing TPACK in Teachers

School districts in the United States continue to spend enormous amounts of funding on learning technologies and the infrastructure to support the use of learning technologies in classrooms. This continuous investment is justified by educational leaders in an attempt to help prepare their students to be college- and/or career-ready and be proficient in technological skills and knowledge. Still, though, there is a lack of quality professional learning opportunities that teachers can receive to be adequately prepared to have the knowledge and skills needed to effectively integrate learning technologies in their classrooms [14,15]. As the goal for TPACK enactment is the center of the Venn diagram in Figure 1, it is critical to consider factors that may present a challenge for teachers as they attempt to integrate technology into their mathematics teaching [16,17,18]. Below, I discuss teachers’ beliefs and their schools’ context and culture, critical factors related to teachers’ capacity and willingness to teach with technology [16,17,18].

1.3.1. Teacher Beliefs About Teaching with Technology

Teachers’ beliefs about technology have always been a critical part of the conversation about teachers’ use of technology [18,19]. In some efforts, teachers engaged in professional learning focused on mathematics pedagogy where technology is situated as a tool to support teaching and learning [20]. Additionally, teachers have been situated in degree programs or professional learning efforts where they are explicitly taught how to use the learning technologies first and then given ideas and examples about how those learning technologies can be used in classrooms [21,22]. Regardless of whether pedagogy or technology comes first, there is a need to continue to examine ways for addressing teacher beliefs about teaching with technology [23]. In the case of teachers’ enactment of TPACK in mathematics classrooms, teachers’ beliefs about how to teach mathematics as well as their beliefs about using technology both influence the extent to which teachers will use technology as well as how students will use it in their classrooms. Alignment with beliefs is the construct of self-efficacy and teachers’ perceptions on the usefulness of technology. Teachers will not use technology if they do not feel comfortable using it themselves, so there is a critical need for professional learning that develops teachers’ technological knowledge and skills [18,19]. In prior work, researchers found that teachers were less likely to use technologies that would not enhance their teaching or improve student learning [23,24].

1.3.2. Context of the Teacher

Recent ideas around TPACK have proposed an updated model to the seminal TPACK model in order to better reflect how contexts and cultures of school districts, schools, and classrooms influence teachers’ applications of TPACK [17,25]. Specifically, without a school context that encourages and helps teachers to teach with learning technologies, teachers’ use of technology is likely to be less frequent or in ways that do not reflect research-based uses of technology [18,19,24,25]. Some of these contextual factors include teachers’ autonomy or mandates on what resources and tools to use to teach standards; emphasis or lack of emphasis on specific data sources, such as high-stakes test scores; and the amount of support and collaboration with teachers and their colleagues [17,20,25].

Regardless of the TPACK that a teacher has, their context will influence how a teacher uses technology [16,17,24]. Specifically, while two teachers may hold similar types and amounts of TPACK, the contexts in which they teach may lead to varied application of how each teacher integrates technology into their teaching. To this end, the next section of this article includes three vignettes of how elementary school teachers have enacted TPACK in elementary mathematics classrooms.

2. Vignettes of Teachers Using Technology in Elementary Mathematics Classrooms

2.1. Overview of Vignettes

These three vignettes provide examples of elementary school teachers’ efforts to integrate technology into their mathematics teaching. The vignettes were purposefully selected [26] to provide descriptive pictures of how teachers’ TPACK plays out in their classrooms in schools in which the organizational culture encourages and supports teachers’ use of technology to support the teaching and learning of mathematics. Based on the current literature, there is a need to examine how teachers enact and apply TPACK in various content areas and settings. Based on Revised Bloom’s Taxonomy, these vignettes reflect variance in the types of higher-level thinking. The first vignette describes a multiplication and division fluency internet-based activity, the second vignette describes a budget simulation, and the third vignette describes an open-ended fraction tool. These vignettes were purposefully included to provide insight into TPACK in two different ways: (1) an activity that has primarily lower-level skills (fluency activity), (2) a structured activity that includes higher-level thinking (budget simulation), and (3) an open-ended activity that includes higher-level thinking (fraction).

Each vignette includes an overview of the activity, the role of the learning technology, teachers’ enactment of TPACK, and connections to student learning. Each vignette is from three different elementary schools near a major city in the southeastern United States that the author observed during his service and research work in local partner schools. The author’s notes for classroom observations and post-observation conversations with the classroom teachers informed the vignettes; an inductive approach was used to summarize information. Each vignette is organized into the following sections: (1) overview of context and activity, (2) role of learning technology, (3) enactment of TPACK, and (4) connections to student learning.

2.2. Vignette 1: Developing Lower-Level Skills with Multiplication and Division Fluency in Grade 4

2.2.1. Overview of Context and Activity

Mrs. Paleo teaches fourth grade in a rural school that is 30 min from a major city. Approximately 60 percent of her students are primary Spanish speakers, and over 90 percent of the students in her school qualify for free and/or reduced lunch. Mrs. Paleo works in a context that allows teachers to decide what activities they use in their mathematics classroom as long as they align to their state’s standards. In an effort to develop her students’ basic fact fluency, Mrs. Paleo had her students use the internet-based resource MathFactLab.com. The resource includes a pre-assessment, and then, students are recommended to use the website multiple days a week to develop their fluency. The website provides various problems (e.g., 7 × 8 = __) along with various pictorial models that provide scaffolds to help students determine the answer.

Students work on this activity by themselves on their own devices, and teachers do not have to be involved in the activity at all. The teachers’ responsibility is just to create opportunities for students to use this internet-based activity during the day. The program is designed so that after students have spent 15 min on the program it logs students out automatically after they complete the next activity. Once the activity is finished, students work on different paper-based mathematics activities.

2.2.2. Role of Learning Technology Math Fact Lab

In this activity, students used the learning technology independently to practice their multiplication and division fluency. The website had embedded scaffolds through pictorial representations (Figure 3). The website provides an additional scaffold when the student pushes the “Hint” button OR when an incorrect answer is entered.

Since students have their own account, they engage in a personalized experience based on their past performance and the specific numbers that they are working on. Students, therefore, were completing activities that aligned to their mathematics needs.

2.2.3. Enactment of TPACK in Math Fact Lab

Based on her understanding of how important multiplication and division fluency are for her students to be successful (CK) and based on data that nearly all of her students needed additional practice, she had her students work on this activity multiple times a week. Mrs. Paleo had spent time with her grade-level colleagues learning about the benefit of developing fluency with pictorial representations and models (PCK), so she thought the website would be helpful since it included those within the learning technology (TPACK). In this case, all of the evidence of TPACK is in Mrs. Paleo’s selection of this tool before the activity since she did not interact with students at all while they were performing the activity.

2.2.4. Connections to Student Learning in Math Fact Lab

Fluency and students’ knowledge of multiplication and division facts is a foundational part to success in mathematics. Based on a conversation with Mrs. Paleo and data that she shared from the website, all of her students were demonstrating growth in their fluency, but they were growing at different rates. Students who needed more foundational work seemed to be growing at a faster rate and progressing faster than those students who were already doing well and just needed to grow in their multiplication and division fluency with only a few specific numbers.

2.3. Vignette 2: Exploring Economics in Grade 5

2.3.1. Overview of Economics Activity

Mrs. Herrara teaches fifth grade in an urban school in the center of a large city. Over 80 percent of her students have foundational misconceptions in mathematics from grade 3 and/or grade 4. As part of their unit on budgeting and financial literacy, Ms. Herrara taught a two-day lesson in which 5th-grade students explored the cost of living in different parts of their town and surrounding neighborhoods. Similar to the first vignette, Mrs. Herrera worked in a school in which teachers had autonomy and freedom about what activities to use to teach specific standards. Additionally, Mrs. Herrera worked with a grade level that shared a lot of activities, and her colleague told all of the teachers about this website.

On Day 1, pairs of students were randomly assigned a job with a weekly salary, a housing location with monthly housing costs, and transportation (truck with low gas mileage, car with high gas mileage, or no car). The students made a budget and discussed ways that they could possibly earn or save more money in pairs and during a class discussion. The teacher’s role was to introduce the activity and facilitate discussions among classmates. On Day 2, the students were given the opportunity to play The Budget Game (Figure 4), which is an online, internet-based budget simulation (https://natwest.mymoneysense.com/students/students-12-16/the-budget-game/, accessed on 1 June 2024). The students worked in pairs to play the game over a 45 min period. During this time, the students were required to keep track of all of their expenses and earnings on paper.

2.3.2. Role of Learning Technologies in Economics Activity

The internet-based budget activity provided a simulated environment for the fifth graders to explore concepts involved in budgeting and practice living on a budget. The technology provided an adaptive learning environment and gave students information that they used to make decisions. However, the students did all of the mathematics themselves and, on their own, kept track of their earnings, expenses, and the money that they had saved using paper and pencil.

2.3.3. Evidence of TPACK in Economics Activity

Mrs. Herrera made the decision to use the website as a way to let students learn about and practice budgeting (TCK). During the activity, Mrs. Herrera asked her students questions as she circulated the room, such as “How will your choices influence the amount of money that you will spend?” and “How can you make sure that you will have enough money in the future?” These questions reflect Mrs. Herrera’s understanding of PCK since she was facilitating students’ work by asking specific questions about budgeting and money. Therefore, during the lesson, there was evidence of TPACK as she made connections between the content of budgeting and the internet-based activity by using research-based pedagogies of asking questions and supporting students’ thinking.

2.3.4. Student Learning Outcomes in Economics Activity

A conversation with Mrs. Herrera indicated that as much as she had hoped that her students would demonstrate an understanding of the importance of spending less, saving more, and looking for ways to make money, she reported that this was not the case. Specifically, many of her fifth graders used the simulation as an opportunity to pretend to spend money and run out of money as quickly as possible. When she provided suggestions and challenged them to see how much money they could have by the end of the simulation, some students responded and saved more, but most of her students kept trying to spend a lot of money.

Mrs. Herrera had her students write a reflection about what they learned after the simulation, and all of the students wrote about the challenge of wanting to buy things but also knowing that eventually money would either start to run out or be gone. This reflection showed more evidence that students learned about budgeting than their actual behavior during the simulation when they spent their imaginary money very frivolously.

2.4. Vignette 3: Exploring Multiplication with an Open-Ended Tool in Grade 3

2.4.1. Description of Multiplication Activity

Mrs. Wilson taught third grade in a suburban school where 85 percent of her students were performing at or above grade level in mathematics. Mrs. Wilson strongly believed in teaching mathematics through letting students explore word problems while she supported them by asking questions. Mrs. Wilson began her multiplication lesson by giving her students the following word problem: Pencils come in packs of 3. If you have less than 31 pencils and all of your packs are full, how many pencils could you have?

Mrs. Wilson gave her students whiteboards and markers to explore the problem in groups of three or four. The students worked on the task as Mrs. Wilson checked in with various tables and asked questions. After the students had found a few possible answers, Mrs. Wilson had the students open up their Chromebooks and pull up an interactive hundreds board (https://toytheater.com/hundreds-chart/, accessed on 1 June 2024, Figure 5). In Mrs. Wilson’s planning meeting with other third-grade teachers, the website was shared by another teacher as an activity where students could just practice skip counting by making patterns on the website. Mrs. Wilson decided to use the website to support her students’ work on the word problem.

Mrs. Wilson gave these instructions to her students: “Find as many answers as possible for the total number of pencils. Use the hundreds board to explore and keep track of answers.” The students worked in their original groups to explore the task. While the students worked, Mrs. Wilson asked them, “what do you notice on the hundreds chart?”

2.4.2. Role of Learning Technology in Multiplication Activity

The website that was used was an open-ended tool with no specific activity embedded in it. When the students were invited to use their Chromebooks and go to the website, they were asked to use the website to find and keep track of possible answers to the word problem. The students solved the problems using a whiteboard and markers and then simply used the website to keep track of their answers. Some students, though, discovered that the possible answers were 3, 6, and 9, and they noticed a pattern on the website. As a result, they immediately extended the pattern and then tested to see if the next few numbers in the pattern (e.g., 12, 15, and 18) could be correct answers.

Since the website was open-ended, there was variation among groups on how the website was used. In its most helpful way, students used the existing pattern to find other correct answers, like it is described above. In its least helpful way, students solved the problems using whiteboard and markers and simply used the website to help keep track of their answers.

2.4.3. Enactment of TPACK in Multiplication Activity

During the activity, there was evidence of Mrs. Wilson’s knowledge of PCK as she posed a word problem for students to solve that had multiple correct answers and supported students by asking questions about their thoughts and strategies. Her selection of the website with the intent to have students keep track of the answers and hopefully discover the pattern of the correct answers (3, 6, 9, 12, 15, 18, 21, 24, 27, and 30) shows evidence of TCK and TPACK since she was planning to use the technology to support her students’ understanding of multiplication in the context of the word problem. One noteworthy part of this activity related to TPACK resulted since Mrs. Wilson was very open-ended and gave students autonomy on how to use the website. Specifically, how the website was used varied, as some students used it to explore the pattern and find correct answers while others just used it to keep track of their answers.

2.4.4. Student Learning Outcomes in Multiplication Activity

From a conversation with Mrs. Wilson after the lesson she was happy with the way her students had explored and solved that task. She noted that while some students did only use the website as a way to record the answers, some of her students noticed the pattern that all of the answers were multiples of three.

3. Discussion and Implications

This article provides insight into three elementary school teachers’ enactment of TPACK and their students’ use of learning technologies during their mathematics classrooms. This article contributes to the field as it provides vignettes and detailed descriptions of teachers’ enactment of TPACK with variance in the levels of thinking that students engaged in. This section includes connections between the vignettes and the current literature related to TPACK and technology use in mathematics classrooms. This section is organized in these parts: (1) the role of learning technologies, (2) the enactment of TPACK, and (3) school contexts.

3.1. Role of Learning Technologies

In the three vignettes in this article, there was variance in the types of learning technologies used by students as well as the amount of higher-level thinking associated with each technology. In the first vignette, students used MathFactLab, an online program that supports students’ learning of basic multiplication and division facts. The second vignette allowed students the opportunity to engage in a simulation about budgeting and financial literacy. Students received information through an internet-based simulation and kept track of their expenses and earnings. In the third vignette, students had access to an open-ended internet-based tool to keep track of their answers and look for relationships between the answers to a multiplication word problem.

The three vignettes provide scholars who research TPACK and technology integration with a set of purposefully selected technology-rich activities from school contexts where teachers are supported and encouraged to teach with technology. Additionally, the three vignettes were purposefully selected since the learning technologies used varied so much, but they still reflect the types of technologies that are commonplace in current classrooms: basic skills, simulations or games, and open-ended tools. Each of these three of these activities embodied research-aligned practices, and the learning technologies supported mathematics learning in each of these cases.

The connections to student learning for all three vignettes was anecdotal for the most part. However, Vignette 1, which focused on multiplication fluency, had definitive data that students had become more fluent and were showing growth based on the data in the MathFactLab.com internet-based program. Past studies have found empirical associations between student learning and basic-skill practice activities such as Vignette 1 [12], simulations such as Vignette 2 [12,27], and problem-solving activities like Vignette 3 and student learning [4,12].

These vignettes include limitations since they are limited to three classroom teachers who all come from supportive school contexts that encourage technology use. Future research should include more intentional designs that examine how learning technologies in mathematics influence student learning outcomes using processes such as pre- and post-assessments, in-depth analysis of student work samples, or other evidence of students’ mathematics learning and understanding. Specifically, there needs to be further research that examines mathematics learning technologies that focus on lower-level thinking skills compared to the use of learning technologies that focus on higher-level thinking skills to more closely examine potential differences between the extent of higher-level thinking and student learning outcomes.

3.2. Enactment of TPACK

Teachers’ enactment of TPACK varied across the three vignettes. Since the first vignette with the MathFactLab fluency activity did not require teacher involvement, all evidence of TPACK was in Mrs. Paleo’s planning and selection of the tool. There was evidence of her TPACK as she had her students use a learning technology that aligned with research on developing students’ fluency. In the second vignette, Mrs. Herrara demonstrated evidence of TPACK during the activity when she posed questions to her students about the math and budgeting concepts as they were working through the internet-based budgeting simulation. In the third vignette, Mrs. Wilson provided evidence of TPACK when she provided students with the opportunity to use the internet-based hundreds chart to keep track of answers to their word problem and use the pattern to find more correct answers.

As stated earlier, each of these three vignettes was purposefully included to show how TPACK can be enacted with various types of learning technologies. In each case, teachers were already willing to use technology in their mathematics classroom and were convinced that it supported teaching and learning. This affirms previous work citing that teachers need to have positive beliefs about teaching with technology as well as positive beliefs that technology will support student learning [18,19,24].

As evident in these vignettes, these three teachers did not just have knowledge of pedagogy and content, but they showed enactment of Technological Knowledge that was specific to mathematics (TCK) with the use of research-based pedagogies to support the activities (TPACK). These three vignettes, though, appeared very different in terms of the pedagogies that were used. Vignettes 2 and 3 aligned more closely to the recommendations from mathematics education scholars to allow students to learn by exploring mathematics concepts that are embedded in non-routine, complex word problems or scenarios [9,10]. A limitation to this article that should be addressed in follow-up research studies includes examining teachers’ enactment of TPACK while teaching in a more direct instruction process, as advocated by special education scholars [11].

Further, more research is needed to examine ways to further describe or quantify the amount of TPACK needed to effectively use specific learning technologies. Specifically, in Vignette 1, Mrs. Paleo needed to set up accounts for the students and provide opportunities for students to log in and complete the activity. This use of technology is rather straightforward pedagogically compared to Vignettes 2 and 3 where the teacher was required to facilitate students’ progress through the activities by introducing the activity and posing questions to support students during mathematics class. Scholars may question, and they may be right in doing so, whether much TPACK is needed to use the fluency website (Vignette 1) compared to the other vignettes. Future studies should closely examine what specific aspects of TPACK are needed to enact specific technologies, especially those internet-based technologies that are primarily skills-based practice.

3.3. School Contexts

The role of school context and culture continues to heavily influence whether and how teachers integrate learning technologies into their teaching [17,25]. In the case of all three vignettes, teachers had complete autonomy to use whatever activities that they wanted as long as it aligned to the standards. Mrs. Wilson, in the third vignette, decided to use the website that a colleague showed her but used it as a tool to support the solving of a word problem and discovering the pattern of correct answers instead of the lower-level skills commonly developed using that website.

As educational technologists, instructional designers, and scholars continue to examine how to best support teachers’ decisions on whether and how they use technology in their teaching, there is a need to more closely examine the context of the school and whether or not teachers feel supported in integrating technology in their teaching. Additionally, research studies should continue to look for ways to collect trustworthy data about school contexts and culture. Logically, it is intuitive that we cannot expect teachers to effectively enact TPACK and use technology in ways that support teaching and learning if they do not have a supportive context and culture that allows teachers to make decisions about how they will teach their state’s standards.