Edited by: Ramona Maile Cutri, Brigham Young University, United States
Reviewed by: Elisa Kupers, University of Groningen, Netherlands; Tom Porta, Flinders University, Australia
This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Providing differentiated instruction (DI) is a complex teacher task that many secondary school teachers do not master well. In the current study, a cognitive task analysis of this teacher task was conducted by analyzing how expert teachers do this and why, resulting in an inventory of the necessary teacher skills and knowledge for providing DI, and a description of the factors that influence the complexity of DI. The results of this analysis show what providing DI in secondary education entails, which is valuable for designing teacher professional development programs for DI at that level.
Teachers are increasingly expected to adapt their teaching to their students’ needs, since this can have a positive effect on students’ achievement (
DI is a teaching approach “
Teachers use various strategies to provide DI, such as ability grouping (e.g., to create a group of lowachieving students who receive more explanations), or the use of a computerized system to support DI (
The effectiveness of providing DI, however, mainly depends on what the teacher actually does. The quality of DI depends on the degree to which a teacher deliberately, proactively, and successfully adapts instruction to their students’ needs (
Research has shown that secondary school teachers find it hard to provide DI.
A first explanation for why Dutch secondary school teachers provide less DI than their colleagues in primary education might be a difference in their initial teacher training. While 85% of beginning teachers in primary education indicated that they were sufficiently prepared for providing DI, this was the case for only 55% of beginning secondary school teachers (
As mentioned earlier, however, providing DI is a complex teacher task. Hence, TPD to support teachers in the context of secondary education in mastering DI is necessary (
Research question 1: What skills are required from teachers in secondary education in order to provide DI?
Research question 2: What knowledge is required from teachers in secondary education in order to provide DI?
To answer research questions 1 and 2, the current study describes what expert teachers do to provide highquality DI, what their reasoning behind their DI activities is, and what knowledge DI requires from teachers. It was chosen to study mathematics teachers specifically, so the results can be compared to the study of
Research question 3: What factors make providing DI in secondary education more or less complex?
As the required skills and knowledge for teachers to provide DI in secondary education were not yet known, a cognitive task analysis (CTA) was conducted in the current study, to identify, analyze, and structure both the skills and knowledge that experts (in our case, teachers who differentiate well) use while carrying out a complex task (
The steps followed for conducting the CTA stem from
CTA activities in the present study.
CTA Steps according to 
Activity in this study  

Step 1: Gathering of information that is necessary for carrying out the CTA  a.  Literature review 
b.  Lesson observations to map out reallife tasks and class situations that require DI skills  
Step 2: Identify knowledge representations  This was chosen based on 4C/ID Skill hierarchy Overview of underlying required knowledge List of complexityrelated factors 

Step 3: Use of elicitation methods  a.  Lesson observations followed by semistructured interviews (cued recall) 
b.  Expert meeting with teachers  
Step 4: Analyze/verify data acquired  a.  Iterative qualitative analysis of data from observations, interviews and expert meetings with expert teachers 
b.  Expert meeting with content experts  
Step 5: Shaping the results for the intended end product  In a later study, the results will be used to design and develop a TPD program for DI. 
For the lesson observations and expert meetings, 11 mathematics teachers who differentiate well and 10 content experts (mathematics or DI) were selected. In this section, the selection procedure and participants’ characteristics are described.
This study involves cooperation between the University of Twente and a Dutch school board that governs about 50 secondary schools. School leaders, mathematics department heads, and other colleagues within said school board were asked to identify mathematics teachers who were above average at providing DI. This resulted in a group of 11 teachers with a wide variety of years of experience, the educational levels they teach and what years they teach. More information about the participating mathematics teachers can be found in
Characteristics of participating expert teachers.
Name 
Teaching experience (years)  Number of classes taught  Mean number of students per class  Educational level of observed class  Grade 

Amy  3  5  29  Prevocational  7 
Anna 
3  5  30  Prevocational/senior general  7 
David  9  8  29  Senior general  10 
Emily 
10  6  31  Preuniversity  8 
Jennifer 
18  5  29  Prevocational  9 
Kelly 
25  2  27  Preuniversity  9 
Robert 
25  5  23  Preuniversity  11 
Sandra  16  3  25  Preuniversity  12 
Sharon 
10  9  28  Preuniversity  10 
Steven  3  7  29  Preuniversity  7 
Thomas  4  4  27  Preuniversity  10 
All names are pseudonyms to preserve anonymity.
Also participated in the expert meeting with teachers.
After the expert meeting with teachers, an expert meeting for content experts with expertise on DI and/or secondary mathematics was organized (see step 4b in
Data collection started with lesson observations followed by semistructured interviews (as described in step 3a in
To map out what skills are necessary to provide DI, expert teachers were observed for two consecutive lessons with the same class of students to analyze the coherence between the lessons and those lesson’s goals. Permission for collecting data from both observations and interviews was granted by the ethical committee of the University of Twente. Prior to data collection, students (or their parents, depending on the student’s age) had to give active informed consent. The observed lessons were video recorded for eight of the 11 participants (three teachers did not want to be recorded for either personal or school organizational reasons, in which case the researcher took notes). Video recordings of the observed lessons were used only as input for the interviews.
Every observation was directly followed by a semistructured interview, to gain insight into the reasoning of teachers when implementing DI. Additionally, more information was gathered about the necessary skills for providing DI, which are not visible in lesson observations. These interviews included three parts. The first part included questions about the teacher’s background (e.g., years of teaching experience). In the second part, the researcher selected specific situations from the recordings or notes from the observations using an overview of classroom situations that (might) call on DI skills, based on
After all classroom observations and interviews were conducted, all of the 11 expert teachers were invited to the expert teacher meeting to verify the outcomes from the observations and interviews. Six of the participating teachers (as indicated in
The second goal of the meeting was to map out the teacher’s steps, actions, and decision points when differentiating and what knowledge it requires to do that. The teachers were asked to design a standard approach for how teachers ideally differentiate for each of the four phases (lesson series preparation, lesson preparation, teaching during the lesson, and lesson evaluation). This was followed by a plenary discussion, to synthesize all procedures into a single joint procedure. Further discussion concerned what a teacher’s DI actions and decision points are, what knowledge a teacher needs for carrying out each of the steps, and when a teacher executes a step correctly. During this meeting, the research team guided the teachers by asking questions and requested examples or clarifications if necessary. The research team took notes and the meeting was recorded.
During this meeting, the results of the lesson observations, interviews, and the expert meeting with teachers (i.e., the preliminary skill hierarchy (see section 2.3 Data Analysis), the ranking of complexityrelated factors, and overview of required knowledge) were presented to the content experts. The first goal was to reach consensus on the necessary skills for providing DI by presenting the alreadycollected data to participants in the study (i.e., memberchecking;
The data for this cognitive analysis included the transcripts of the interviews (for which the recordings of the lesson observations were used as input) and the notes and recordings from both expert meetings. First, the data from the interviews and the expert meeting with teachers were analyzed. Codes were assigned to the actions and the reasoning of teachers, the knowledge they used and the factors that make providing DI more or less complex. These codes were subsequently clustered into categories. The first four categories were the four phases of DI: the preparation of the lesson series, preparation of the lesson, teaching during the lesson and evaluation of the lesson (
Our respondents confirmed that for DI in secondary education, the four closely interrelated phases (preparing a lesson series, preparing a lesson, teaching during the lesson, and evaluating a lesson) also play an important role. All of these phases can be decomposed into constituent skills (see
Skill hierarchy of necessary skills for providing DI in secondary education.
In the rest of this section, the constituent skills will be described per phase. For each of the skills, the number of teachers who were observed doing or who mentioned the skill will be stated. While most skills were used by all teachers, at least nine of the 11 teachers performed all the skills. An exception is ‘to determine the lesson goal’ during lesson preparation, which was mentioned by seven teachers. As the content experts emphasized the importance of this skill, it was decided to include this skill as well. In the rest of the paragraphs, examples will be given of how teachers enacted the specific skill.
When preparing the lesson series (generally lasting 4 to 8 weeks), teachers lay the foundation for providing DI. In this phase, teachers analyze student characteristics and performance and they make a plan for the lesson series, wherein they determine both the curriculum and the homework for the lessons. The relationships between the necessary skills are shown in
Part of the skill hierarchy about the preparation of a lesson series.
In this study, all teachers analyze student characteristics and performance when preparing a lesson series for a class. Teachers analyze their students’ achievement level and preferences to see whether students have achieved earlier stated learning goals or not. They do so by combining different resources including tests, observations, and information from colleagues.
One source of information is a previous or earlier lesson series with related goals, to paint a picture of where the students stand regarding the goals of the lessons in the upcoming lesson series. Because inadequate performance can have various causes (e.g., some students may have difficulties grasping a certain topic, while other students may have made a lot of mistakes in how they wrote down the solutions), teachers try to explain those performances by talking to students and observing student behavior. Teachers analyze how well the students, in general, did on the test they took at the end of a previous lesson series by checking both the grades and whether there were any frequently made mistakes. They also map out what topics the students still find hard and take these as points of attention for (the planning of) the upcoming lesson series. Besides that, teachers look at the results and mistakes of those students who did not achieve the goals of the earlier lesson series, or who performed below expectations. Teachers have diagnostic conversations with students to analyze why the student performed differently from what was expected and to discuss with the student how both the student and the teacher can ensure they will do better next time.
Another source of information for the teachers are colleagues: other mathematics teachers within the school or teachers who teach different subjects to the same students. Teachers indicated that it is important to know the students themselves as well, and to combine the information from colleagues with their own experiences. When starting a new school year with classes with students they have never taught before, some expert teachers find it important to look at students’ results from previous school years, while others find it important to get to know the students themselves without bias. The content experts agreed with the latter, stating that looking at students’ data while not yet knowing the students themselves (well) could lead to a selffulfilling prophecy; therefore, they advised being careful with it. There are, of course, exceptions where information should be shared between colleagues, for example, when a student has a visual impairment and should sit in the front of the class to see the teacher well.
Combining the information from all of the different available resources, teachers assess students’ achievement levels. Especially for lowperforming students, teachers check if they have mastered the earlier goals that are necessary for new, upcoming goals. The teachers specified that they find it important to know upfront whether the students have sufficient prerequisite knowledge, because if they do not, the teachers can take this into account in their planning. Teachers deem it important to know what part/topic (of mathematics) highperforming students are good at. Teachers keep in mind that those students might not need as much instruction for that subject, and often give highperforming students more freedom in choosing what assignments to complete.
All teachers mentioned that they make a plan for a lesson series. They first need to consider the yearly timetable in which the subjects to be taught and exams and other summative tests are specified and established by all the teachers of the mathematics department. Although these timetables are drawn up for all grades, there is more room for flexibility in the lower grades, due to the absence of exams. In the plan for the lesson series, teachers develop a timetable for when to attend to what content and in how much time. A teacher might need to develop multiple timetables as they can differ for (groups of) students, depending on their individual learning needs. To develop a plan for a lesson series, teachers need to master two constituent skills: determining the subject matter for the lessons, and determining the homework for the lessons.
All teachers mentioned that they
Along with determining the subject matter, 10 teachers stated that at this point they
When preparing a lesson, the teacher extends the foundation laid when preparing the lesson series by mapping out their students’ starting point and determining the goal of the lesson, instruction and lesson plan. For teachers to determine the instruction(s), they must determine what explanations and assignments are going to be used during the lesson. The necessary skills for lesson preparation are depicted in
Part of the skill hierarchy about preparing a lesson.
All teachers look back at the previous lessons for the class to map out the starting point. The teachers’ goal is to know about the students’ prior knowledge and learning needs. Teachers look back at the evaluation of the previous lesson to check whether the learning material was transmitted as planned and whether there were any problems. For example, there might still be a lot of questions, and students could experience difficulty grasping the theory. Teachers also check on for whom this was the case. They map out the starting situation so that they can take this into account during the rest of the lesson preparation.
Along with this, seven teachers determine the goal of the lesson, which is the second constituent lesson preparation skill. The content experts emphasized that it is very important to determine the goal of the lesson when providing DI, and that the goal should be connected to the bigger picture of what the class has already discussed and what are they working toward. Teachers decide what they want the students to be able to do or to understand at the end of the lesson. For example: “
All teachers determine instruction(s) when preparing a lesson. To do this, all teachers first
Along with determining the explanations, all teachers
The explanations and assignments together make up the instruction. The teachers decide what they want to do and how they want to do this. For the teachers, instruction thus means not only the explanation of subject matter, but also the use of learning activities and assignments during the lesson, with the aim being for the students to reach the goal of the lesson.
Finally, during lesson preparation, the teachers determine the lesson plan. This was done by nine of the teachers. Teachers decide what they are going to do, for how long, what they will do with the whole class, and what is obligatory for all, or only for a specific group of students. This is also where teachers decide if and when they want to discuss the homework from the previous lesson. Teachers also indicated that they find it important to make time during the lesson for any individual questions students may have.
When teaching a lesson, teachers make use of four constituent skills. A teacher starts by introducing the lesson. Next, teachers give instruction(s) aligned to the learning needs of the students, while simultaneously stimulating the students’ selfregulation. This phase ends with teachers wrapping up the lesson. As can be seen in
Part of the skill hierarchy about teaching during a lesson.
This phase starts by introducing the lesson. All teachers explain the subject of the lesson and four teachers explicitly pay attention to the goal of the lesson. It was stressed by the content experts that introducing and explaining the goal of the lesson is very important. They encouraged teachers to explain what the students will learn, why that content is important and what it will yield for the students. Teachers also often state what the schedule for the lesson is going to be and what is compulsory for all or only for specific students.
When introducing the lesson, teachers check what the prior knowledge of the students is, for example, through discussing homework, asking and answering questions, or wholeclass discussion of an assignment. In this way, the teacher not only monitors but also activates students’ prior knowledge, so that students can relate the new lesson material to what they have already learned.
All teachers provide adapted instruction(s). According to content experts, this is the most crucial constituent skill for providing DI. Instruction includes explanations, assignments, and all other instruments to help students achieve the goal of the lesson. Teachers adapt this instruction to the learning needs of their students, based on their constant monitoring. For example, when teachers notice that there are many similar questions, they might choose to give an extra explanation to a selected group of students or even the class as a whole. The content experts indicated that a teacher should give all students guidance that is adapted to their learning needs.
The observed lessons frequently followed the same pattern. After introducing the lesson, teachers started with general class instruction about that lesson’s theory. Teachers specified that they try to keep the general class instruction short to ensure that it is not too long for the students who do not need it. Some teachers give highperforming students the choice of whether or not to follow this instruction or to work independently. After this general class instruction, teachers in the observed lessons often gave extended instruction to students who had not fully grasped the subject matter yet. The goal is to make sure all students reach the goal. Depending on the number of participating students, teachers choose to do this in front of the class or in a small group. Teachers often do an assignment together with the students to provide guided practice, but they can also give more elaborate explanations using simpler words or visualizations.
Three teachers mentioned that they offer a more indepth explanation to explain underlying theories or a higherlevel assignment to challenge the students. Sometimes students may choose which one they prefer to do and other times the teacher decides what is most suitable based on what they have seen in their monitoring. Highperforming students can work on these instead of following the general class instruction or can start after the instruction. During the meeting with content experts, it was specified that a teacher has provided adapted instruction well when students at all levels feel challenged.
Stimulating students’ selfregulation is a skill that all teachers mentioned. Teachers involve students during the lesson by giving them responsibility for their learning process. They can do this, for example, by giving them the choice to not to follow the general class instruction and/or extended instruction, or by giving choices about what assignments to do as their homework. Teachers stimulate selfregulation by helping students to form an image of where they stand in the learning process and what they still need to do to attain the goal of the lesson. For example, teachers can indicate that if students find a certain assignment hard to do, they should do more practice using an easier version of that assignment. When stimulating students’ selfregulation, teachers take on the role of a coach, continually monitoring whether the students are making the right choices by asking the students questions about their learning process.
Wrapping up the lesson was identified as constituent skill based on observations and explanations of ten teachers. The teachers in the expert meeting stated that this should be part of every lesson. In practice, teachers were sometimes still answering questions in class, which left them with no time to explicitly wrap up the lesson. When teachers do wrap up the lesson, they do so by discussing a difficult assignment, giving an extra tip, taking stock of any unanswered questions, and/or mentioning the homework, based on their monitoring during the lesson. While teachers look back at the lesson, they often summarize how the lesson went, how the work ethic of the students was, and by asking students their opinion about the lesson. Content experts found it important that teachers check during the wrapup whether or not the goal of the lesson has been achieved. Teachers do so by asking the students questions about the lesson or by giving an assignment to check if the students can put into practice what they have learned.
After the lesson, teachers look back at the lesson, which, as can be seen in
Part of the skill hierarchy about evaluating a lesson.
All teachers evaluate for the short term. Teachers reflect on how the lesson went, what kind of questions were asked, and whether or not all students achieved the goal of the lesson. First of all, the teacher evaluates the group of students who received the lesson. A teacher might find out that the explanation during the extended instruction was too hard for some of the lowperforming students. Teachers think about what they can do differently in the next lesson so the students can learn the content and achieve the goal of the lesson. Some teachers make notes of this, while others think about it and take it into account when preparing the next lesson. Content experts stressed that it is important that the teacher not only looks at the class as a whole, but also zooms in on individual students. Second, teachers can evaluate a group or groups of students who are going to experience the same lesson, namely, a parallel class. If this lesson did not go the way the teacher planned, they might change some things so that the parallel class will not run into the same problems and will achieve the goal of the lesson, while the first group of students maybe did not.
Along with evaluating for the short term, all teachers also evaluate for the long term. They reflect on what they could do differently next school year. Teachers do not just look at one lesson, but evaluate multiple lessons at once, or even all lessons within a lesson series. For example, when teachers notice that they are going through the subject matter too quickly, they take this into account when preparing the lesson series in the next school year. They can make notes in their curriculum material manual and plan to take more time for this topic. Furthermore, all teachers check how well the explanation of the subject matter worked; if it did not work very well, they will consider changing things for next year. The teachers often make notes about this in their own textbook.
To provide DI by deploying all of the constituent skills, teachers also need specific knowledge. In this CTA, three types of knowledge emerged: knowledge about students, general didacticpedagogical knowledge, and subjectmatter knowledge.
All teachers indicated that it is important to have knowledge about the students. First, a teacher must know the level of all individual students per class. A teacher in secondary education sees a lot of different students per week, which makes it hard to always have all this knowledge about every specific student available. Getting to know the students takes time. Teachers mainly gather information by continually monitoring during the lessons. They acknowledged that it is important to know which students find mathematics hard, which students are good at math, and which students are in between those two groups. It is also important that a teacher knows how motivated and independent students are.
Next, teachers need to have general didacticpedagogical knowledge: knowledge about how students learn and what activities can help them learn, which teachers mostly gather when monitoring during a lesson. Teachers use this knowledge, for example, to decide how they can help an insecure student or a student who is good at math.
Finally, teachers need subjectmatter knowledge: pedagogical content knowledge (PCK) for math, knowledge of the curriculum, and knowledge of the learning path. PCK is important for knowing how to explain math subjectmatter content. Knowledge about the learning path is important so a teacher can take a step back or a step forward. In this way, a teacher can respond to the needs of low and highperforming students. Knowledge of the curriculum relates to knowledge of the textbook and the assignments in the textbook. With this knowledge, teachers can use and explain subjectmatter content well in the class.
In the CTA, various factors were identified that influence the complexity of providing DI in secondary education. Multiple class characteristics influence the complexity (see
Factors influencing the complexity of DI.
Factor  Aspect  Explanation 

Characteristics of a class  Class size  Teachers indicated that they find providing DI easier when the class size is smaller, as that makes it easier to have an overview of their students. 
Variation in student levels  The higher the difference between and variation within the groups of low and highperforming students, the harder it is to differentiate. Teachers then must consider many and very different levels.  
Classroom ambience  When the ambience in a class is not good, that asks for a lot of the teacher’s attention, which makes it harder to differentiate. When the ambience is good, students will feel more comfortable answering and asking questions, providing more information for the teacher.  
Class attitude  If (a large part of) the students do not work independently when they should, providing DI is harder.  
School organization  Preparation time  Teachers mentioned often that they do not have much time to prepare a lesson. This makes it harder for them to think upfront about how they would like to differentiate during the lesson. 
Number of activities  Teachers stated that their planning of activities within a year is very tight and that they do not have much room for revisiting topics from the previous period.  
Time spent with students  The more time a teacher spends with the same students, the easier it is to get to know them and to identify their educational needs.  
Lesson duration  When lessons last longer, there is more room to provide instruction in different ways and as such to respond to the varying needs of the students.  
Physical space in classrooms  If there is more space, a teacher can use a table for instruction and sit apart with a smaller group of students. If a classroom is quite full, this is harder.  
Data regarding student achievement  Information richness  Teachers gather information about students’ progress by monitoring during lessons and in (summative) tests they administer. Having the right information gives more insight into the students’ learning needs, which makes it easier to differentiate. 
Support from the curriculum  Assignments at different levels  If the textbook has differentiated assignments according to different levels, it is easier for a teacher to align the assignments to the varying needs of the students. 
DI refers to how teachers adapt their instruction to the continually monitored needs of all learners (
The rest of this section is divided into two parts. The first part addresses the three research questions: (1) “What skills are required from teachers in secondary education to provide DI?,” (2) “What knowledge is required from teachers in secondary education to provide DI?,” and (3) What factors make providing DI in secondary education more or less complex?.” The second part of this section compares the required teacher knowledge and skills for providing DI between the contexts of primary and secondary education.
This study showed that four phases are important for providing DI in secondary education, in which teachers use various skills and different types of knowledge, and that a number of factors influence the complexity of DI. Furthermore, the indepth interviews revealed that the quality of DI depends on the deliberate adaptations a teacher makes, based on their knowledge.
In the first phase, preparation of a lesson series, teachers lay the foundation for providing DI. Teachers analyze student characteristics and performance, to add to their alreadyexisting knowledge about their students. They use their subjectmatter knowledge to make a plan for the lesson series, determining both the curriculum and the homework. Curriculum material can support teachers in preparing for DI in a lesson series; for example, it is easier to determine assignments at varying levels when the textbook material already provides assignments for different achievement levels.
In the second phase, lesson preparation, teachers map out the students’ starting point. Teachers determine the goal of the lesson and the lesson plan, and prepare the required instruction(s), for which they use their didacticpedagogical knowledge to determine both explanations and assignments so as to ensure that their students can reach the goals.
Teaching during the lesson is the third phase. Teachers introduce the lesson, provide instruction(s) aligned to the learning needs of the students and stimulate the students’ selfregulation. Finally, they wrap up the lesson. During the lesson as a whole, teachers monitor the progress and understanding of their students, and continually expand and refine their knowledge about students. Providing DI during the lesson is more complex when lessons are shorter, as that leaves less time to provide a variety of types of instruction to attend to all students’ learning needs. Enough physical space in classrooms enables teachers to sit apart with a smaller group of students, which is experienced as helpful for DI.
In the fourth and final phase, evaluating a lesson, teachers evaluate whether or not students have reached the goal of the lesson in the short term. Teachers also evaluate for the long term, where they consider if they could do anything differently in the next school year, such as changing the order of topics in the curriculum/learning path, using other activities that help students to learn better, or planning more time for a certain topic.
In the theoretical framework it was mentioned that for highquality DI teachers should adapt their instruction deliberately and proactively (
From the CTA it appeared that the core skill for providing DI is continually monitoring students’ learning and progress. In all phases, teachers identify their students’ learning level. This goes from analyzing performance in the preparation for the lesson series and mapping out starting points in the lesson preparation, to observing students’ expressions and behavior, asking them questions and checking their work during the lesson. In their evaluation, teachers monitor to what extent each student reached the goal of the lesson. Continual monitoring contributes to teachers’ knowledge about their students.
As not much was yet known about the teacher skills and knowledge required for providing DI,
Providing DI in secondary education happens in the same four interrelated phases as were found in primary education (
What knowledge is necessary for providing DI is also mostly similar between primary and secondary education. For secondary education, three types of knowledge were found: knowledge about students, subjectmatter knowledge, and didacticpedagogical knowledge. Although the last type of knowledge was not explicitly mentioned in the study by
Regarding the complexityrelated factors, in both primary and secondary education the composition of the group, school support, and available data on the students’ progress were mentioned (
In sum, the current study confirms the findings of
Although we studied teachers who are considered to be above average in terms of DI skills (within a Dutch school board) and we looked for patterns across this group (which led to a stable pattern), we cannot guarantee that the 11 participating teachers are the best in the country at providing DI in secondary education. In this study, content experts proved more normative and sometimes more ambitious than the observed teachers. However, the common patterns led to the skill hierarchy, knowledge types overview, and complexityrelated factors, which we think are a good basis for the development of a professional development trajectory, as they emerge from school practice and therefore should have high feasibility. On the other side, although the current study provides rich insight into the constituent skills in the four phases, each skill in itself could be analyzed in more detail in order to obtain an even better understanding of how teachers exactly enact these skills and which underlying knowledge and skills are required.
All expert teachers worked for the same large school board, which only has schools in the Netherlands. For a future study, the results could be verified in a broader context, such as teachers of a different Dutch secondary school board or even secondary schools in other countries. As all teachers were mathematics teachers, the results of this study cannot be generalized to teachers teaching other subjects. Hence, it would be interesting to verify the results with teachers of other subject domains, such as other STEM (e.g., physics or chemistry) subjects or languages (e.g., English or French). All participating teachers taught classes that were mostly theoretical. For a future study, it would be interesting to see if teachers who teach more practical lessons (e.g., in vocational education) use the same knowledge and skills for providing DI as the teachers in the current study. Furthermore, since the factors related to complexity (as described in
In secondary education in the Netherlands, teachers do not yet implement DI in their lessons very often and often do not feel equipped for it (
As mentioned in the introduction, secondary school teachers do not yet provide much DI. Nevertheless, the current study shows that providing DI in secondary education can be achieved, as the data are based on practice. Hence, TPD based on insights from the current study could help teachers to provide (better) DI. Although providing highquality DI is not something that happens very often yet (
The datasets presented in this article are not readily available because the data include video observations and recorded interviews that are not anonymized. Requests to access the datasets should be directed to
The studies involving humans were approved by the Ethical committee of the University of Twente. The studies were conducted in accordance with the local legislation and institutional requirements. The participants provided their written informed consent to participate in this study.
MV, MG, and AV contributed to the design of the study. MV contributed to data collection. MV and KM performed the analysis of the data and wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.
This study was funded by Stichting Carmelcollege.
We thank all teachers and other experts for their participation in our study.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The reviewer EK declared a past coauthorship with the authors MG to the handling Editor.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.