Preservice mathematics teachers’ reasoning about their instructional design for using technology to teach mathematics

ABSTRACT This study explored mathematics teachers’ reasoning regarding their instructional design for technology use in teaching mathematics. Various types of qualitative data were obtained and analysed from a sample of 19 secondary and 28 primary preservice teachers. The findings showed that the purposes of technology use, teachers’ conceptions of mathematics and its learning, their learning experience and knowledge of student thinking as well as mathematical cognition, and empirical inquiry were the main sources of teacher reasoning. Three levels of teacher reasoning were identified: descriptive, explanatory, and justifying. Our findings contributed to research on teacher professional development by identifying four types of technology use with a variety of pedagogical purposes, and formulating two main dimensions to characterise the three levels of teacher reasoning.


Introduction
Our lives and society are changing rapidly in response to substantial advances in technology and growth in big data (Brynjolfsson et al., 2014). Reflecting on the role of mathematics in the digital age, Gravemeijer et al. (2017) acknowledged that the effort of teachers to enhance students' mathematical competencies so as to complement the computer-based work that can characterise the digital society of the future was significant. To meet such challenges, mathematics teachers require continuous professional development to design instruction for technology use that can deepen students' understanding and application of mathematics.
Facilitating preservice teachers' learning, particularly in the area of design instruction, is challenging for both teachers and teacher educators (Ball & Forzani, 2009). Pepin et al. (2008) conceptualised the design capacity of a mathematics teacher; it comprises three components: the formation of an orientation for the design, a set of design principles, and reflection-in-action. Teachers' decision-making in the design and implementation of instructions is influenced by their beliefs about mathematics and its teaching (Munby, 1982). Nonetheless, little is known about teachers' reasoning about instructional design for technology use in teaching mathematics. Shulman's model of pedagogical reasoning and action was proposed to recognise pedagogical content knowledge required in teachers' actions during the processes of developing comprehension of content knowledge, transforming content knowledge into pedagogical representations, conducting instruction, evaluating students' learning, reflecting on teaching, and forming new comprehension (Shulman, 1987). Starkey (2010) further applied Shulman's model of pedagogical reasoning and action to explore beginning teachers' pedagogical decisions when utilising digital technologies based on the connectivist view of learning. Then, Starkey changed Shulman's transformation stage into one of "enabling connections," and emphasised the pedagogical aspect of "students creating knowledge in the digital era through connections in an open and flexible curriculum" (Starkey, 2010, p. 241). The change to student-centred teaching renders instructional design for technology use more challenging.
To prepare teachers to teach in technology-rich learning environments, teacher educators have to understand teachers' reasons underlying their pedagogical decisions. The aspect of teachers as designers and the need of extending the focus from teacher knowledge to teacher reasoning to understand teacher decision making in using technology to teach mathematics laid a foundation of this study. Particularly, the present study focused on preservice mathematics teachers' reasoning about their own instructional design for technology use in mathematical teaching by addressing the following research questions: (1) What technologies are used in preservice mathematics teachers' instructional design, and what are the pedagogical purposes of using such technology?
(2) What conceptions of mathematics and its learning underlie their instructional design? and (3) How do they reason with regard to their instructional design?

Reasoning as the basis of teaching competence
While Shulman's model focused pedagogical reasoning on what is required for pedagogical actions, we adopted Blömeke et al.'s (2015) holistic view of competence to investigate teachers' reasoning which underpins teaching competence. In this view, teaching competence is defined as a horizontal and vertical continuum comprising the cognitive and affective traits underlying teachers' perceptions, interpretations, and decision-making skills. This view can extend teacher reasoning from a focus on teacher knowledge to the inclusion of their perceptions, interpretations, and decision-making skills. That is, teacher reasoning, as the basis of teaching competence, generates teachers' pedagogical decisions and actions.
Teacher reasoning is required to meaningfully perceive the connection between the content and the learner, to substantially predict and interpret what may happen, and to flexibly make decisions leading to pedagogical actions. Such actions, drawing on teaching competence, include the comprehension of subject knowledge, the connections of subject knowledge within and between learners which enables them to develop and create knowledge, teaching with formative and summative evaluations, reflection on teaching decisions based on evidence, and new comprehension of the subject, students, and teaching (Shulman, 1987;Starkey, 2010).

Meaning and levels of teacher reasoning
Based on the perspective of teaching competence (Blömeke et al., 2015), we extend Shulman's pedagogical reasoning to the explanations of decision-making about their pedagogical decisions and actions. Teacher reasoning not only involves making sense of what has been noticed by linking observed information and situations to subject and pedagogical knowledge, but also encompasses that which has been designed and reflected on for learning and teaching by transforming knowledge and beliefs. That is, teacher reasoning, as a basis of perceptions, interpretations, and decision-making skills, can transform their knowledge and beliefs into their pedagogical decisions and actions (Rowland et al., 2003). This view on teacher reasoning can differentiate it from teacher noticing and be helpful for contrasting novice and expert teachers' interpretations because teacher noticing may be based on little reasoning (Seidel & Stürmer, 2014;Vanes & Sherin, 2002).
Qualitative research by Seidel and Sturmer (2014) identified three aspects of reasoning that together form a key component of professional vision: describing, explaining, and predicting classroom situations. Teachers expanded their vision from (i) observing and recognising something in terms of learning and teaching without making any further judgements, (ii) linking the observed events to professional knowledge in order to make judgements, and (iii) predicting the consequences of the observed events by drawing on professional knowledge about teaching and learning. We would extend the application of these three levels to an analysis of teachers' reasoning about instructional design, and adapted it where necessary.

Instructional design for using technology to teach mathematics
Teachers' knowledge, beliefs, skills and learning experiences have been justified as significant predictors of technology use (Farjon et al., 2019;Knezek & Christensen, 2008). Learning technology by design has been identified as one key to prepare preservice teachers' technology integration in education (Tondeur et al., 2012). These studies imply that it is critical to investigate how teachers reason about instructional design based on their knowledge, beliefs, experiences, and skills.
Research on integrating technology in mathematics education has recognised that technology has various and specific capacities (Clark-Wilson et al., 2015), such as software and applications (Stein et al., 2020). Software can be used to provide instructions that tell a computer what to do and when to execute the instructions, while applications or websites are embedded into a computer or software program developed for users to accomplish a specific task, and mostly can be downloaded onto a mobile device (Stone et al., 2005).
Based on the theoretical framework of an instrumental orchestration, composed of two processes: (1) instrumentation which highlights how users are constrained by technology and (2) instrumentalization which emphasises how the function of technology is selected and reframed by users (Trouche, 2004), Drijvers et al. (2010) investigated three mathematics teachers' use of technology to teach mathematics and identified six orchestration types. These were: demonstration, explanation, connections of representations, collective discussion, spot-and-show, and students' use of technology. Herein, we focus on types of technology as part of instrumentation, and on teachers' pedagogical purposes as well as conceptions of mathematics and its learning as part of instrumentalization to investigate preservice mathematics teachers' instrumental orchestration underlying their instructional design.

Context of primary and secondary teacher education programs
Teacher education programs in Taiwan can be classified into four types: primary, secondary, kindergarten, and special education. A nation-wide examination has been required since 2005. In 2013, the examination for primary school teachers added the subject of mathematics and its teaching, which is not required for secondary teacher education because it is assumed that content knowledge of secondary teachers should be well prepared by their subject.
The subject of mathematics and its teaching have been required courses in primary school teacher education programs. The difference in the teaching of mathematics between primary and secondary schools in Taiwan is that most primary school teachers must teach many subjects, including mathematics, for the same class students; however, most secondary teachers teach a single subject (e.g., mathematics) to different classes of students.

Participants
Whereas most studies have revealed that teachers lacked sufficient knowledge for teaching and the skills of integrating technology into the teaching of mathematics, the aim of this study was to identify to what extent Taiwanese preservice primary and secondary teachers can reason about their instructional design for technology use in teaching mathematics. The participants in this study consisted of 19 preservice secondary teachers and 28 preservice primary teachers (as shown in Table 2). All 19 preservice secondary teachers were taking a teaching practicum course. They had taken or were taking the other required pedagogical courses for they would conduct a 6-week teaching practicum in a school in the second semester of the course. The preservice primary teachers came from two different universities. Six of them were undergraduates from one university in the capital city and taking a teaching practicum course. Instructional design and microteaching were requirements of the practicum courses before the preservice primary and secondary teachers really conducted a teaching practicum in a school. The other 22 primary preservice teachers, who were taking a teaching method course, were postgraduates from one university in a city of central Taiwan. All of the participants were asked to individually design one lesson that involved technology use in teaching mathematics. It was informed that their instructional design would be evaluated from the facets of content and structure, learning purposes, technology use and purposes, and teaching methods when they presented it in the courses.

Data collection
We collected data from the teaching materials designed and written by the participants, and their presentations of the instructional design in the courses. Semi-structured interviews were conducted by authors and research assistants after their presentations to collect data on the reasoning behind their design. The interview questions encompassed what was considered, why it was considered, the purpose of using technology for the learning and teaching of mathematics, why they decided to do this, whether they had considered an alternative approach, and why it was effective. The interview took between 10 and 25 minutes. Verbal informed consent was obtained from all the participants, and they could refuse to be interviewed at any time. In addition to the three sources of data, we took field-notes to record the feelings and perceptions of the teachers regarding both their presentations and the interviews.

Data analysis
Content analysis was applied to encode the data and categorise it into meaningful qualitative findings that can be presented to readers. This research developed qualitative codes and categories relating to the teachers' reasons for integrating technology into the teaching of mathematics. Before analysis, the data were transcribed into written format in accordance with the method employed in grounded theory.
A descriptive-interpretive analysis (Elliott & Timulak, 2005) was used in this study. This meant that the first two research questions were investigated in a more descriptive manner, while the third research question was investigated in a more interpretive manner. This difference was based on the connectedness of the research questions and the data collected as well as the nature of the research questions. A descriptive analysis was used to describe and understand the pedagogical purposes of preservice teachers' technology use and their underlying conceptions of mathematics and its teaching. Taking the analysis of pedagogical purposes as an example, each author recognised the purpose of technology use from their own students' teaching materials and presentations. Then, the categories which emerged from each author were discussed with our own students' examples, and the six categories were finalised for further coding.
An interpretative analysis characterised preservice teachers' reasoning about instructional design based on the researchers' knowledge of previous findings from other studies (Quinn & Clare, 2003). For the interpretive analysis, we first read and re-read each interview several times to familiarise ourselves with the content. Key words were noted and compiled into several categories. For instance, from the presentations of their instructional design and the interviews, key words related to mathematics and its learning and teaching could be identified and then clustered together so that categories began to emerge.
Each individual instructional design and interview were then re-read under the categories of purposes, mathematics, and its teaching to extract the primary reasoning each teacher gave for instructional design with reference to the three levels of description, explanation, and prediction. After analysing our data, the levels were revised as description, explanation, and justification. Detailed illustrations of this were provided in the findings. When analysing and reporting the data, participants' privacy is protected.

Types and purposes of technology use
The participants' instructional design demonstrated four main types of technology for teaching mathematics, including Applications (Apps)/Websites for general or mathematics education as well as Software for general or mathematics education. The function of Apps/Websites for a general purpose (e.g., Kahoot and search engines) is to make it easy for students to engage with mobile devices. For mathematics education, its function is to motivate students to achieve the specific learning goal simply by using mobile devices (e.g., mathematical games or simulations). The software for general education is intended to be used in education, but not specifically for mathematics (e.g., PowerPoint and Flash). The software for mathematics education is designed to provide mathematical functions (e.g., GeoGebra and Desmos).
In addition to the types of technology, we further analysed the participants' instructional designs with respect to their purposes of using technology, including demonstrating information, describing mathematics, representing mathematical thinking, enhancing students' understanding, increasing students' affect, and facilitating students' mathematical inquiry.
For each type of technology, the participants presented various purposes of using technology in their instructional design. Table 1 displayed the frequencies of purposes of using technology for each type of technology with brief descriptions. Because the participants were not representative, we compared the types and purposes of technology use only within rather than between preservice secondary and primary teachers. The types of technology most frequently used by preservice secondary mathematics teachers were software for mathematics education (41), and by the preservice primary teachers they were apps/websites for mathematics education (7 + 33). This indicates that the participants were more likely to adopt mathematics education-related technology to teach mathematics for both primary and secondary students. No participants used Apps/Websites for general education with the purposes of describing mathematics and representing mathematical thinking. This may imply that the two purposes can be better accomplished by using Apps/Websites and Software for mathematics education.
The purpose most frequently stated by the preservice secondary mathematics teachers was to facilitate students' mathematical inquiry. This purpose extends beyond the orchestration types of technology use (Drijvers et al., 2010) to the exploratory dimension. This indicates that our secondary mathematics teacher education program can cultivate preservice teachers' skills of using technology to meet different purposes of teaching mathematics, in particular, weighing more on mathematical inquiry. Few preservice primary teachers used technology for mathematical inquiry to teach, although learning theories (e.g., Piaget's theory of cognitive development) have been instructed to emphasise making thinking operational through inquiry in the primary teacher education programs. The purposes of technology use stated by the preservice primary teachers were mainly spread across the four types: describing mathematics, representing mathematical thinking, increasing students' affect, and enhancing students' understanding. Comparatively, the preservice secondary mathematics teachers paid less attention to  To ask students to explore mathematical relationships.
the purpose of increasing students' affect when using technology to teach mathematics. These purposes can be viewed as pedagogical purposes. Although some of the participating teachers used the same kind of technology, they might have different purposes. For example, using PowerPoint included saving time spent writing on the blackboard, outlining knowledge structure or concept maps, and presenting information step by step without adding any unnecessary information. The last two purposes echo the "two main reasons for why students prefer PowerPoint as a teaching supplement: (1) organisation and (2) sustained attention" (Baker et al., 2018, p. 377). The first purpose may neglect students' concerns about fast-paced instruction in relation to PowerPoint (Hill et al., 2012).

Conceptions of mathematics and its learning that underlie technology use
We found that preservice secondary and primary mathematics teachers were able to make various assertions about mathematics in their instructional design for technology use. They recognised mathematics as involving rigour, dynamic process, metaphor, reasoning, and thinking. For instance, "the four operations of matrices are complicated and accurate" (rigour); "graphs of functions are dynamically constructed" (dynamic process); "to multiply an inequality by a negative number is like an upside-down world" (metaphor); "mathematics is to reason why it happens" (reasoning); "to find mathematical properties and then apply them is thinking mathematically" (thinking). An international study showed that Taiwanese mathematics teachers performed very well in mathematical content knowledge (Schmidt et al., 2011). This may be one reason why our participants can present a variety of conceptions of mathematics.
Our participants perceived the learning of mathematics as involving challenging, understanding, questioning, solving problems, manipulating, exploring, exercising, and memorising. For instance, some preservice teachers recognised that "Students have difficulties in learning some mathematical concepts. . . . Technology can be used to ease the difficulty by displaying mathematical procedures step by step or by visualising dynamic processes." This expression concerns challenge and understanding. One preservice secondary mathematics teacher emphasised the importance of asking questions for understanding and clarification in private because of shyness. He opined that "Students can ask questions in Zuvio [https://www.zuvio.com.tw/], . . . , questioning is critical when students do not understand." This expression concerns understanding and questioning.
Some participants provided students with an opportunity to manipulate something (e.g., "Students can move or drag it on the screen"). Although it implies the importance of manipulating for learning, few participants further asked students to conjecture or justify. Further action following manipulation implies a perspective on learning mathematics as exploring and problem solving. The manipulating function of technology recognised by our preservice teachers differs from the doing function recognised by Drijvers (2015), where manipulating refers to pressing buttons on the mouse or screen to see what happens, while doing refers to outsourcing work (e.g., calculation). As well as asking students to explore the underlying mathematics using technology, some participants highlighted the importance of exercising and memorising when learning mathematics (e.g., "Deliberative practice is also necessary to learn mathematics."). This finding is not only consistent with the Confucian heritage culture of learning through constant perseverance and application (Bryan et al., 2007), but also implies a mixed constructive-conservative teaching method used by the participants.
The participants' explanations referred less to their conceptions of teaching, but instead mainly focused on the purposes of teaching mathematics in relation to their instructional design, that is, the purposes of technology use in Table 1. These purposes can be further classified into three dimensions: cognition, affect, and metacognition. The cognitive purposes involved were that technology can "impress students with a dynamic process," "give students a sense of infinity," and "let students see a visual representation." The affective purposes involved were that technology can be used to "interest students," "engage students in learning by doing," and "attract students' attention." The metacognitive purposes involved were that "the exploratory activity can facilitate students' thinking about the ratios of width and length," and "I do not just give students fish but a fishing rod." The former is related to active thinking which may lead to self-awareness of thinking, whereas the latter is related to reflective thinking which may lead to self-regulation of thinking. Although the participants did not explicitly relate the pedagogical actions to the development of students' metacognition, teacher educators could make the connection obvious and guide them to predict students' thinking as well as illustrate examples of how to fish mathematically.

Three levels and four types of reasons
Based on the analysis of classroom discussion and interview data, preservice teachers' reasoning regarding their instructional design can be classified into three levels: descriptive, explanatory, and justifying. Table 2 shows the distribution of the reasoning level of preservice primary and secondary mathematics teachers. Participants in both groups most frequently belong to the descriptive level. Preservice primary teachers in postgraduate education (n = 22) were more likely presented the explanatory level than those in graduate education (n = 6). The reasoning of only a few participants reached the justifying level. This indicates that our preservice mathematics teacher programs may cultivate preservice teachers' vision of using technology to increase students' affect or facilitate their mathematical inquiry, but may fail to coordinate their knowledge and practice, including design and instruction, by developing their pedagogical reasoning. Furthermore, four types of reasons emerging from the three levels of reasoning included a variety of cognitive, affective and metacognitive purposes, personal experience or conceptions of learning and teaching mathematics, student thinking or mathematical cognition, as well as ways of justification.

Descriptive level focusing mainly on pedagogical purposes
In the descriptive level, their reasoning mainly focuses on the cognitive purposes of technology use and how this is connected to affective purposes or personal experiences and conceptions of learning mathematics. Que's interview transcript provides an example of connecting one pedagogical purpose of technology use (daring to ask questions) with his learning experience (his questions being laughed at). His reasoning was mainly based on his own learning experience and did not really explain why questioning is critical.

Explanatory level connecting pedagogical purposes to thinking
In the explanatory level, preservice teachers were able to clarify their various purposes of technology use by relating them to student thinking and mathematical cognition, although their reasoning may also be based on personal experiences and conceptions of learning mathematics. Jay: Students can use GGB to explore the attributes of trigonometric functions and see how basic trigonometric functions can generate all other trigonometric functions. (cognitive purpose, conceptions of learning mathematics) Instructor: Why is it important?
Jay: My teacher just taught me how to calculate the values of trigonometric functions and showed me the graphs of basic trigonometric functions. I did not know how other trigonometric functions were generated. I want students to explore this by themselves.
(personal experience, conceptions of learning mathematics) Instructor: How can students explore this using GGB?
Jay: I do not give them domains, ranges, periods, amplitudes. After drawing and comparing, you (students) can induce the same feature. For instance, the range of sine functions. . . .

(student thinking)
Instructor: Why is it effective? Jay: After exploration, students have to integrate different parts and then practice. It can deepen their understanding of sine functions and properties. It also makes it difficult to forget. (student thinking) Instructor: Why were you not satisfied with your teacher's teaching? Jay: Calculating y by substituting numbers into x is not (mathematical) thinking. It requires students to explore the same feature. (conceptions of mathematics and its learning) Jay's interview transcript provides an example of relating one pedagogical purpose of technology use (for exploring) to student thinking (drawing, comparing, inducing, integrating) and mathematical thinking (to explore the same feature) based on his poor learning experience (only the teacher's instruction). His response implied that reasoning was not only drawn from reflection on personal (and poor) learning experiences but was also strengthened by the attention to student thinking during and after exploring the underlying mathematical invariance (the same feature). Jay's further clarification makes the explanatory level different from the descriptive level.
Chen: Students can use GGB to explore exponential functions. (cognitive purpose) Instructor: How do they do this?
Chen: Exponential functions can be classified according to their base numbers. Larger or smaller than one. Their graphs are different. I do not tell students. This property has to be found by themselves. (conceptions of mathematics and its learning) Instructor: What do you mean when you say you give students a fishing rod?
Chen: I teach them how to make a slider in GGB, rather than just using it. (metacognitive purpose) Instructor: Why is it important?
Chen: Making sliders benefits their understanding of the dynamic, changing process. It is helpful to find structures of exponential functions because making the slider helps them notice base numbers. (mathematical cognition) Like Jay, Chen's reasoning was based on discontent with his own learning experiences. The difference was that Chen related one metacognitive purpose of technology use (being able to inquire) to mathematical thinking (recognising critical conditions in order to conjecture a property). His reasoning applied a metaphor to explain the pedagogical purpose and was clarified by relating it to mathematical cognition.

Justifying level using a practice-based study
In the justifying level, preservice teachers view an instructional design as a study, justifying the potential of technology use for student thinking and mathematical cognition. This justification can be characterised as practice-based or theory-based. Practice-based justification refers to empirical testing to warrant an instructional design, while theory-based justification refers to the application of theories to design and justify instruction. No data shows a theory-based study. Fang actively tested her instructional design with textbook analysis and one case study, although the homework did not ask them to do this. This may be because Fang had been taking a research method course in the mathematics education program which gave her the idea to do it. Nonetheless, she did not refer to any theory to support her instructional design or case study.

Discussion
Understanding the development and ways of instructional reasoning is important as instructional reasoning influences practice and continues to be a challenge for teacher education (Loughran et al., 2016). The present study undertook the challenge by analysing preservice mathematics teachers' reasoning about their instructional design for technology use in teaching mathematics. Various purposes of technology use found in this study extended the orchestration types identified by Drijvers et al. (2010) to the explorative and affective dimensions. This extension reminds teacher educators of focusing on both cognitive and affective purposes of technology use for teachers to reason about learning mathematics.
The participants' conceptions of mathematics and its learning were varied and related to their various purposes. The finding supported that "a substantial component of action is directed by goals that reliably control and motivate the behavioural system in a dynamic world" (Aarts & Elliot, 2012, p. 7). According to the goal-driven perspective on teaching behaviour, a suggestion for teacher educators is to facilitate the needs of preservice mathematics teachers for different pedagogical purposes of technology use by developing multiple conceptions of learning mathematics.
Our interpretation of the data revealed that teachers' reasoning can be divided into three levels, namely descriptive, explanatory, and justifying, which reflect four types of reasons into the content and forms of their reasoning about their instructional design. The content of reasoning included pedagogical purposes, conceptions of mathematics and its learning, experience in learning mathematics, student thinking and mathematical cognition. The extent of the content is increasingly rich from the descriptive level to the justifying level. The forms of reasoning, including cyclic reasoning at the descriptive level, metaphorical reasoning at the explanatory level, and convincing others with a case study of an instructional design at the justifying level.
The three levels of reasoning found in the context of instructional design differed from the three levels of description, explanation, and prediction found in the context of noticing (Seidel & Stürmer, 2014). In the research on noticing, it was essential for teachers to differentiate relevant aspects of a noticed teaching and learning event, to link what is noticed to professional knowledge, and then to predict what may happen. In our study where teachers constructed teaching and learning events, they were found to set up pedagogical purposes (descriptive level), connect these with mathematical and student thinking (explanatory level), and justify the connection based on research or theoretical knowledge (justifying level).
Our study broadened the investigation of instructional reasoning to include instructional design of technology use in teaching mathematics, and thus, various pedagogical purposes of technology use emerged and connected to conceptions of mathematics and its learning, in addition to student thinking and mathematical cognition. Such broadened investigation shed light on an analytical approach to understanding teachers' instructional reasoning by referring to the three levels. Toom et al. (2015) identified two main dimensions: inductive-deductive and staticdynamic to characterise preservice teachers' reflective patterns. Inductive reflection is based on practical experiences while deductive reflection is based on theoretical aspects. Static reflection is declaratory while dynamic reflection is transformative. Differing from their findings on student teachers' patterns of reflection in teaching practice, the content and forms of reasoning, as two main dimensions, emerged to characterise preservice teachers' reasoning in a context of instructional design. Further investigation can focus on the two dimensions to assess the quality of teacher reasoning about teaching practice.
Although Taiwanese preservice mathematics teachers performed excellently on tests of mathematics content knowledge and mathematics pedagogical content knowledge in an international comparative study (Blömeke et al., 2011), few of the preservice mathematics teachers were at the justifying level. This indicated that scaffolding is necessary to develop preservice teachers' knowledge-based or theoretical-based reasoning by transforming their knowledge for teaching mathematics to the formulation of their pedagogical decisions and actions. Our findings suggested that enriching pedagogical purposes of technology use and personal experiences which can develop multiple conceptions of mathematics as well as its learning in the context of using technology to teach mathematics, and connecting them based on plausible forms of reasoning by conducting textbook analysis with the comparison of technology use, a case study on instructional design of using technology to teach mathematics or theory-based explanations of strengths and weaknesses in using technology to teach mathematics may be beneficial for improving teachers' instructional reasoning about their design of using technology to teach mathematics.

Conclusion
The main components in preservice teachers' reasoning regarding their instructional design for technology use in teaching mathematics were identified. Particularly, four types of technology and different pedagogical purposes of each type were illustrated. Three levels of reasoning and their differences were characterised by two dimensions of the content and forms. This study provided evidence to support that research on teacher reasoning regarding their instructional design could deepen research on teachers as designers where knowledge and motivation were the focuses of studies (Kali et al., 2003) by emphasising the reasoning needed to design instruction.
The results of this study also have implications for teacher education and professional development. Although the preservice teachers in this study had learned several theories relevant to learning mathematics (e.g., mathematical understanding, concept development, and intuitive theory), they did not apply these theories to support their reasoning about their instructional design. This suggests that scaffolding how to apply theories of learning mathematics is necessary for teacher education. The meaning of the three reasoning levels identified in this study can shed light on an approach to gradually advancing teacher reasoning about their pedagogical decisions. For instance, teacher educators could guide preservice teachers towards identifying pedagogical purposes, clarifying the connection between pedagogical purposes, conceptions of mathematics as well as its learning, and student thinking, justifying based on knowledge or theories of learning mathematics, and conducting a case study.
The implications of the study should be considered with caution given the following limitations. First, the number of participants was not large enough to represent the preservice teachers in various contexts. Second, this study only focused on teachers' instructional design. Thus, it would be insufficient to conclude that the participants can use technology to teach mathematics well for a variety of pedagogical purposes in real classes. Nonetheless, our study moved the focus of teacher professional development from teacher knowledge and belief to teacher reasoning. This movement contributed to the literature by suggesting an alternative factor, teacher reasoning, to explain the integration of technology (Farjon et al., 2019;Liu, 2011) and further studies on the relationship between the levels of teacher reasoning and the stages of technology integration.