report

Teacher training program - 3 daagse workshop - Wiskundige denkactiviteiten’ s report

Maria Solange da Silva1

Table of Contents

1 PhD student at Institute of Education of Lisbon University - Promotor: Prof. Dr. Henrique Guimarães.

Teacher training program - 3 daagse workshop - Wiskundige denkactiviteiten report 1

1. Foreword……………………………………………………………….……………. 31.1 Acknowledgments……………………………………………….………….…….31.2 Summary………………………………………………………….………….…...3

2. Introduction………………………………………………………….…………….….42.1 Study Overview………………………………………………………..………….4

3. The Purpose of the Research and Research question…………….….…….………….53.1 The study’s purpose and research question………………….…………..………..5

4. Theoretical framework………………………………………………………..………64.1 Teaching practice when working with WDA…………………………….………64.2 Recognizing a WDA task………………………………………….……………..64.3 Processes of teaching and learning in an inquiry environment – What to

observe?..................................................................................................................75. Methods and Data collection………………………………………..….……………..9

5.1 The sample…………………………………………………….………………...106. Learning from a PD course……………………………………………………….….10

6.1 Creating an environment for teacher learning……………………………….…..106.2 PD course proceedings……………………………………………………….….11

7. Teachers cultures and beliefs from an instructional environments – Some results7.1 First meeting – Understanding WDA…………………………………….……...12

7.1.1 Teachers’ understood about WDA……………………….……………...137.1.2 Classroom observation 1 – Teachers promoting learning…………….…157.1.3 The context of the classroom…………………………….…...……….…157.1.4 The WDA task……………………………………………………..…….167.1.5 Managing the Resources to Support Learning….…………………….…17

7.2 Second meeting – Designing WDA………………………………….……….…217.2.1 Classroom observation 2 – Teachers promoting learning…….………....227.2.2 The WDA task………………………………………………….…….….227.2.3 Managing the Resources to Support Learning………………….……….22

7.3 Third meeting – Teachers working with WDA……………………….………....267.3.1 Classroom observation 3 – Teachers promoting learning…….…………297.3.2 Task of the teacher 1 (M)…………………………………….………….297.3.3 Task of the teacher 2 (F)……………………………………..…………..32

8. What the research show – final conclusions………………………….….…………..349. References…………………………………………………………………….……...37

Annex I……………………………………………………………………………….……..38

Annex II……………………………………………………………………………….…….43


1.Foreword

1.1 Acknowledgments

This study was prepared with support from Erasmus Plus program of Lisbon University and Utrecht University – FIsme. My special thanks to prof. dr. Paul Drijvers who gently received me at Freudenthal Institute providing me the opportunity to develop this study. Also, my special thanks to dr. Michiel Doorman who promptly guided me during all my stayed at FIsme teaching me on designing this report and, finally, my special thanks to the school Coornhert-Gymnasium (Gouda) and to the two teachers Mirtijn Koper and Regine van Dodewaard who promptly offered to participate in this survey. The author gratefully acknowledge their support. The research was conducted independently by the author and does not represent the views of the sponsors.

1.2 Summary

Research shows that professional learning can have a powerful effect on teacher skills and knowledge and on how well students learn. This report describe some results of an investigation that concerns the influences of a Professional Development

(PD) course for mathematics teachers on teachers’ pedagogical skills and what changes are supposed to arise in their teaching in classroom. The report intend to be a key to Freudenthal Institute’s experts involved in the implementation of the PD course to understand aspects related with teachers’ actions in two different environments of learning: the PD sessions and in the teacher’s classroom when applying the contents discussed at the course. The report is based on a set of data that has been compiled during the course and during lessons’ observations of two-participant voluntary teachers. The activities developed by the experts included to promote opportunities to teachers on working collaboratively during the face-to-face sessions and to equip teachers with resources that possibility the preparation of a series of tasks to be applied to students in classroom, enabling teachers to achieve the kind of teaching that has substantial impact on student learning to the new curricula.


2. Introduction

Since 2014, The Netherlands is under an implementation-oriented reform of Mathematics curricula of the secondary school. To help the government in stablishing the reform the Committee on Mathematics Education cTWO has worked for designing new programs and final exams in Mathematics for secondary schools of general education. The new reforms will require significant changes in classroom, requiring a style of teaching that fosters critical thinking to students. To assist teachers to understand the curricula’s changes and to guide them to the program’s implementation a team of experts of Freudenthal Institute (FIsme) developed, from September to November of 2015, a PD course in order to involve HAVO and VWO secondary education teachers into the process of the changes. Aligning with the concepts that teachers have central role in how the new approaches must to be interpreted and developed in classroom, experts of FIsme work with the perspective of the necessity to improve teacher’s education.

To the implementation of a subject or to a methodology be effective, studies about professional development indicate that teachers must to need a good understanding of how to teach in classroom. Therefore, different visions of what teachers are supposed to know to their work as a teacher have driven researchers to examine different teachers’ education outcomes. In thinking about teacher learning, it can occur in many different aspects of their practice, including their classroom and professional development courses (Borko, H. 2004). According to Doorman (2013, pg. 890) the term ‘‘teacher professional development’’ relates to changes in the teachers’ professional knowledge and competence.

This report is focus on teacher, their teaching and their professional development. The purpose is to provide a research-based answer to how a PD course influence teacher’s pedagogical competences. The report will address the many facets of the PD course considering the structure of professional development that truly changes teachers’ work through “Wiskundige Denkactiviteiten” (WDA) conceptual aspects in light of the new reform. Next, the report will examine what changes arise in teachers’ practice that enable teachers to create an inquiry based learning environment (IBL) in classroom that can support them on working with thinking activity.

2.1 Study Overview

This study is organized into three sections. The first section, “The Purpose of the Research and Research question”, provides an explanation of the study’s purpose, the sample, data collection, and methods. The second section, “Learning from a PD course”, provides a discussion of the themes that emerged from the data analysis. Finally, the third section, “Teachers cultures and beliefs from an instructional environment”, provides miniature case studies of two teachers that participated on the PD course.


3. The Purpose of the Research and

Research question

3.1 The study’s purpose and research question

Focusing on teacher environment learning on a PD course and teacher’s classroom work, the aim of the study is to find links between what teachers learn on a PD course with their work in classroom. Therefore, the study seeks to identify similarities, changes or empowerment on teacher’s pedagogical content knowledge (knowledge about how to teach a subject) and teacher’s professional competence (person’s ability to transfer knowledge into action) (Maass, K.; Doorman, M.; pg. 890; 2013).

In order to identify these subjects the study addresses the following research questions:

In terms of changes in teachers' practices when working with a WDA activity, how are the empowerment of teaching approaches? Which changes emerge on these approaches?

To what extent does a WDA course contribute to the empowerment of teachers’ competence in working with inquiry learning?

The investigation can be resume as following:

Research question Data Theoretical model

1. In terms of changes in teachers' practices when working with a WDA activity, how are the empowerment of teaching approaches? Which changes emerge on these approaches?

- Video recording lessons of teachers on an environment of learning (PD course).

- Video recording lessons on a teaching environment (classroom).

- Teacher’s interviews.- Questioner.

- WDA

2. To what extent does a WDA course contribute to the empowerment of teachers’ competence in working with inquiry learning?

- Video recording lessons of teachers on an environment of learning (PD course).

- Video recording lessons on a teaching environment.

- Teacher’s interviews.- Questioner.

- WDA- IBL

Tab. 1 Research questions and theoretical framework


4. Theoretical Framework

4.1 Teaching practice when working with WDA

The WDA focus is on teaching by thinking activities. Therefore, training programs with emphasis on teachers’ professional development needs to emphasize practices that will turn students into critical thinkers with teachers looking for a new teaching in order to improve students learning. Specialists from FIsme agree that working with thinking activity can be an effective way to promote the needed changes. They underline that not always students memorize and reproduce knowledge but also develop a certain skill in a productive way. In the case of activities that help teachers to turn students into critical thinkers, WDA intend to move away from rote memorization tasks to an active way of thinking developing students’ critical thought, enabling students to apply the mathematical knowledge on modelling and reasoning, allowing a network between knowledge and skills. The procedure for working with WDA can be correlated with new methods of learning. On this study, WDA specification is related with an inquiry – based way of teaching and learning.

4.2 Recognizing a WDA task

Fig. 1 Main features that involve WDA tasks

Not routine problem solving that rely on analytical thinking

Thinking

Strategy development

Sharing information

Flexibility and Reflection

Changing information


4.3 Processes of teaching and learning in an inquiry environment – What to observe?

To observe teachers… Tips for dealing with WDA tasks

Being able to provoke students to let them think about main questions.

Give students time to understand the problem, to think about the problem and to get involved in the issue. Take good care with rapid judgments about pupils’ reactions because it can inhibit the process of thinking. Do not give answers, but set (open) questions. Create a safe and stimulating working environment.

Starting for solving a question. If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

Keep asking. Provide strategic notes rather than technical notes. Try the thinking of students in their contributions (e.g. in-group work or class) to explicit by naming what steps they take. Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

Knowing how to ask questions such that everybody is involved to the lesson.

Encourage students to look for alternative solution. Encourage students to explain, even to each other. Challenge your students not be satisfied with particularities, but to search for the general approach.

Moving from a structured (textbook) task to a task supporting WDA.

Provide inspiration and variety by offering attractive problems and different methods.

Preparing such a lesson …thinking of a lesson plan, designing or redesigning a task.

Encourage students to examine the state of the progress of the solution’s process. Provide guidelines about the final evaluation.

To analyze to what extent a WDA course contribute to the empowerment of teachers’ professional competences (that includes “a person’s ability to transfer knowledge into actions”; Maass,K.; Doorman,M.; pg.890;2013) the teachers were observed under the following competences.

Tab. 2 Table of teachers’ professional competences


Observe if teachers… Tips for dealing with WDA tasks

Allow students time to understand the problem and engage with it.

Give students time to understand the problem, to think about the problem and to get involved in the issue. Take good care with rapid judgments about pupils’ reactions because it can inhibit the process of thinking. Do not give answers, but set (open) questions. Create a safe and stimulating working environment.

Offer strategy to solve the problem rather than technical hints.

If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

Keep asking. Provide strategic notes rather than technical notes. Try the thinking of students in their contributions (e.g. in-group work or class) to explicit by naming what steps they take. Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

Encourage explanation. Encourage students to look for alternative solution. Encourage students to explain, even to each other. Challenge your students not be satisfied with particularities, but to search for the general approach.

Move from a structured (textbook) task to a task supporting WDA.

Provide inspiration and variety by offering attractive problems and different methods.

Are able to prepare such a lesson …thinking of a lesson plan, designing or redesigning a task.

Encourage students to examine the state of the progress of the solution’s process. Provide guidelines about the final evaluation.

To understand changes in teacher’s practice this table summarize which teachers’ pedagogical content knowledge (“knowledge about how to teach the subject”; Maass,K.; Doorman,M.; pg.890;2013) were observed when teachers work in classroom with a WDA task.

Tab. 3 Teachers’pedagogical content knowledge


5. Methods and Data collection

The study is framed in a qualitative interpretive perspective, focusing on teachers’ participation on the PD course and its specific rules. The research data is relate to teachers’ work in two different environments of learning: (i) in the sessions of the PD course (learning off-job) and (ii) in teachers’ classroom (learning by-job). Based on these theoretical approaches the study aims to identify links between what teachers learn on PD course with their work in classroom in order to describe what knowledge, changes or empowerment arise on teacher’s pedagogical skills.

The main source for data collect were: (i) a post-course semi-structured interview; (ii) video recordings of teachers working on the three WDA meetings; (iii) video recordings of teachers’ lessons at school; (iv) a post-course questionnaire (v) notes of observations. The qualitative data analyzes will carry out with help of a software2 making use of video registration data and on in-depth analysis of teachers approaches in classroom, a questionnaire and an interview.

Using a semi-structured interview each teachers were asked about their level of experience, the nature of collaboration with the design of the tasks applied in classroom, their beliefs regards the literature and resources worked on the PD course, their beliefs regards the collaborative work during the PD sessions. The data collect takes into account not only the processes of the PD course environment but also the classroom atmosphere, the teachers’ beliefs, their aims and the tools they use in teaching. Thus, after documenting each PD session’s experiences it was conducted a case analysis, which revealed the actions of teachers in their classroom. As part of the data analysis, the report presents a description of how the course was structured and how it was developed.

Thus, the report of findings’ results will follow this structure:

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5.1 The sample

This study’s sample was drawn from two volunteered teachers, which work at the same secondary school in the neighborhood of Gouda and participated on the WDA course from September to November of 2015 at Utrecht University/Freudenthal Institute. The two teachers were selected based on the following criteria:

(i) Both teachers must teach to the same level and work together on the face-to-face sessions of the PD course,

(ii) Both teachers agree to be video recorded during the course and on one or more mathematics lessons at school,

(iii) Both teachers agree in working at least once with the same activity with their students.

The teachers that participated on the study have at least two years of experience teaching at the secondary school and they teach at a novice level of VWO.

Tab. 2 Participant characteristics (N=2)

Characteristics

Subject matter Level Gender Experience (in years)

Mathematics Novice VWO M(1) F(1) M (5) F (3)

M = male, F = female

6. Learning from a PD course

6.1 Creating an environment for teacher learning

The objective of the PD course was to support teachers knowledge to work with “Thinking Activities” developing a training plan with particular reference to design (or re-design) tasks and inquiry learning. Three modules have been designed. They were organized in the context of the current needs of Mathematics Education and specifically on “Denkactivitait” philosophy. Pointing out an overview of WDA, the trends and how does it can be worked in classroom. The three sessions consisted of a plenary session followed by small groups’ activities and at the last meeting, a gallery walk with teachers presenting some productions.

The nowadays-learning meetings were structured in three four-hour sessions over a period of three months followed by a teacher network environment of learning where teachers and experts connect each other using all sorts of ICT tools in order to extend teachers’ education out of the meetings. One of the key structural supports from the PD course for teacher learning was the allocation of time between sessions to teachers share experiences, working with colleagues on developing or re-designing thinking activity tasks, preparing and analyzing


lessons and participating of the same activities on their classrooms. The outcomes enabled an enrichment of face-to-face sessions. A self-evaluation form was sent to the teachers after the second meeting and a short presentation of teachers’ productions took place at the end of the third meeting as a final evaluation. The model of educating teachers follows the model described by Muller model (2003)(Doorman, M. 2012): “Learning-off-job, learning-by-job and learning-on-job”.

6.2 PD course proceedings

Day 1

The first afternoon is devoted to “think about activities”. On this

session will be discussed backgrounds of WDA. In addition, tasks of “Active Think” will be reviewed by characteristics on the model of Van Streun Wiskundige denkactiviteiten, hoofdstuk 11

Day 2

The second meeting is about “designing think active commands”.

Thinking on modifying existing tasks, for example from the book, and on the design of new tasks, for example, for a test.

Day 3

The third session goes on “performing active thinking math classes”.

In what way can you ensure that there is actually, what happens in the minds of the students?

Each session was going on into three rounds:

To the first round, in a big plenary, teachers were invited to discuss backgrounds of WDA, stressing aspects related to Active Thinking, its main concepts and the curricular innovations that will be implemented in Netherlands schools from 2015. The facilitator provided a summary of the presentations and discussions of the meeting before and the agenda of the day work was introduced to the teachers; To the second round, teachers were invited to work the task of the day in small groups under the supervision of an expert from Freudenthal Institute; To the third round, a general meeting took place, with the purpose to make reflections relate with the workday and to report to teachers their homework


http://www.fi.uu.nl/professionalisering/wda/bijeenkomst1/VanStreunKop2012Hoofdstuk-11.pdf

Fig. 1 Teachers in a WDA session

7. Teachers cultures and beliefs from an

instructional environment – Some results

After each session of the PD course is expected a certain basic level of application in classroom of the WDA philosophy under inquiry methods attitudes. This study shows the routine of two teachers during WDA course and a small routine of the application of WDA tasks by them in their classroom.

7.1 First meeting – Understanding WDA

The first afternoon is devoted to “Think about Activities”. On this session were

discussed backgrounds of WDA. In addition, tasks of “Thinking Active” were reviewed by characteristics on the model of Van Streun. Wiskundige denkactiviteiten, hoofdstuk 11

Assignment 1: Teachers were asked to reflect and discuss about a meaning to ‘Thinking activity” in order to compose a common understanding.

Assignment 2: Teachers were asked to make an analysis of a mathematical activity from current program of Netherlands schools in order to recognize thinking activities’ characteristics.




Tab. 4 Table of resources to teachers’ work to this session

Several information by experts to allow teachers to build up an understanding of WDA concepts.

Literature about WDA (Drijvers, 2011) and (van Streun). Literature with tasks to teachers’ analysis, in order to learn how recognize a WDA

task. Moments of collaborative work with teachers working group. A webpage was created for teachers to follow the meetings. A website with models of WDA activities for teachers works in classroom. An evaluation sheet to teachers sent to experts before the second meeting.

Because WDA is not a defined term in mathematics teaching, on the first meeting teachers works with the question: What do you mean by WDA?

On this session, teachers were asked for analyzing mathematical activities by a learning trajectory under experts’ guidance in order to recognize a WDA activity. The mathematical activities partly reflected the new curricula and partly reflected the current program of Netherlands schools. From the material, teachers select and analyses a task applying their understanding about ‘thinking activity’ developed previously by the group.

Fig. 2 The specialist of Freudenthal Institute guiding teachers’ reflections

7.1.1 Teachers’ understood about WDA

1. What is WDA?

According to Paul Drijvers (2011) a WDA is a mathematical thinking activity if presents the following characteristics: non-routine problem solving that rely on analytical thinking, creativity, strategy development, flexibility and reflection. Because of the deliberate way in which facilitators went about building an understanding of WDA, teachers expressed their views in a confidence attitude enabling to identify some criteria that emerged from teachers’ answers and that are in accordance with WDA principles.


As a result of the first session some criteria of WDA have emerged. This was used as a start to teachers’ homework and next meeting work.

Thus, what WDA is?

On an informal interview, the two teachers express their feelings about their understanding of WDA.

Teacher 1 (M)

Teacher 2 (F)

To understand and identify a WDA task is a first step to teachers to know how to design and how to work with this kind of task. To this meeting the discussion of teachers’ work were structured around a short explanation of their understanding followed by peer feedback. The teachers of this case study conclude that:

Tab. 5 Teachers’ arguments

Teacher 1 (M) Teacher 2 (F)

To a WDA task.

“The exercise must have many steps to stimulate student’s thinking”.“It is possible to (re) write a task and its solution’s procedure in order to have a new elaboration to the task that permit students focus more to the task”.“A task does not need to be difficult to stimulate student’s thinking”.

“When you design a task, you must to be precise in what you want to know”.

To a pedagogical work with a WDA task

“Students must to search the answer by themselves or ask for help to the teacher”.

“You must to find a limit of information for students, and then some tips on how to arrive at the answer. The student must to start to think by himself and not the teacher doing everything for him. Students are not stimulate when you gave to them a lot of information”.

[…]”Well, we don’t know…but that’s no problem, because that’s good…because that is a start point

and you start thinking what it is”[…]

[…] "I think that it is something new, things that I have to learn, specially

the things that they don’t know anything about, and they have to

think how they can solve it […]


Teachers’ homework

7.1.2 Classroom observation 1 – Teachers promoting learning

Assignments 1 – Read the article of Paul Drijvers and the chapter 11 of Handboek wiskunde of Anne van Streum and Peter Kop and then try a task that encourages WDA in classroom. Do the report through an evaluation and/or a movie.Assignment 2 – Try a task that encourages WDA in classroom

7.1.3 The context of the classroom.

In Netherlands most of the schools provide teachers with comfortable classrooms fitted with interactive whiteboard and computers to teachers. In addition, in this school, teachers worked with a digital Mathematics book and all of the students working with their on laptop.

Teacher 1 (M) classroom

In the Teacher’s (1) classroom students are placed in an informal way and working in pairs or in groups of three or four. Tables are around the wall with some tables in the middle of the room. Students can walk between each other and they are confident with teacher and friends when they have doubts. According with the teacher this environment work was more comfortable to walk among students.

Fig. 3 – In the classroom of teacher (2)


Teacher 2 (F) classroom

In the classroom of the teacher 2 students are placed in a formal way and working in groups of three or four. Tables are one besides the other but, as with teacher 1, students can walk around the class and they are confident with teacher and friends when they have doubts.

Fig. 4

7.1.4 The WDA task

Different from the original picture and commands, the new task was introduced to students only with the picture of the tower projected on the interactive board followed by the question: “What is the high of the tower?”

The introductory way to the task can be resumed as follow on this picture.

Fig. 5 The task (Doorman, 2015)


With the task, teachers aimed to provide students with opportunities for thinking on different ways and for discussing about different strategy to the solution. Therefore, In order to help students’ thinking, this practical lesson is carried out with teacher inquiring students while guide them to the right solution. The table below refers a tool of a researcher understanding of a teacher’s lesson plan that was used to guide the researcher all teachers’ lessons observation.

Time Teacher’s actions

Introduce the problem.

Allow students think about the problem alone.

Allow students work in groups of two or three identifying further information.

Stimulate whole class to discuss the problem bringing up questions and different ideas to the solution.

Stimulate students to work with different solutions.

Conclude the problem.

Reflect about the solution.

Tab. 6 Researcher sample less guide

7.1.5 Managing the Resources to Support Learning

Teacher 1(f) – The teacher introduce the task projecting the picture of the tower followed by the question: “What is the high of the tower?” He explains that the task is an introductory situation to a new theme emphasizing to students that the answer is not important to him but, the way that the students will find the answer. The teacher start the task with an exciting dialogue analyzing with students in the picture what the tower is.

Tab. 7 what happened in the classroom?


1:20 min Introduce the problem.

_ Students thinking about the problem alone.

_ Students working in groups of two or three identifying further information.

4:00 min Whole class discussing the problem bringing up questions and different ideas to the solution.

1: 45 min Working with a different solution.

1:05 min Conclusion of the problem.

0:20 min Reflection about the solution.


Teacher 2 (M) – The teacher introduce the task projecting the picture of the tower and explains to students that the task is an introductory situation to a new theme. She asks students to turn off the laptops and stay in silence, paying attention to her explanation. In addition, the teacher guide the students to think about mathematics in different situations: “In what do you think when I say the word ‘mathematics’?” After a short debate where students work collaboratively, “finding mathematics” on the picture the teacher introduced the task with the question: “Can you in pairs, speaking quietly, to think a way how to find the high of the tower?” “How high is the tower?”




_ Students thinking about the problem alone.

1:40 min Students working in groups of two or three identifying further information.

3: 05 min Whole class discussing the problem bringing up questions and different ideas to the solution.

_ Working with a different solution.


_ Reflection about the solution.


Tab. 9 Analyzing teachers’ practices (Annex I)

Observe if teachers… Tips for dealing with WDA tasks Teacher 1 (M) Teacher 2(F)


1. Give students time to understand the problem, to think about the problem and to get involved in the issue.

2. Take good care with rapid judgments about pupils’ reactions because it can inhibit the process of thinking.

3. Do not give answers, but set (open) questions.4. Create a safe and stimulating working environment.

1. No2. No3. Yes4. Yes

1. Yes2. No3. Yes4. Yes


1. If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

2. Keep asking.3. Provide strategic notes rather than technical notes.4. Try the thinking of students in their contributions (e.g.

in-group work or class) to explicit by naming what steps they take.

5. Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

6. Encourage students to examine the state of the progress of the solution’s process.

1. Yes2. Yes3. Yes4. Yes5. Yes6. No

1. Yes2. Yes3. No4. Yes5. No6. No

Encourage explanation. 1. Encourage students to look for alternative solution.2. Encourage students to explain, even to each other.3. Challenge your students not be satisfied with

particularities, but to search for the general approach.

1. Yes2. Yes3. No

1. Yes2. Yes3. No


Are able to prepare such a lesson: thinking of a lesson plan, designing or redesigning a task.

1. Provide inspiration and variety by offering attractive problems and different methods.

2. Provide guidelines about the final evaluation.

1. Yes2. No information



Tab. 10 Analysing teachers’ professional competences (Annex I)

To observe teachers… Tips for dealing with WDA tasks Teacher 1 (F) Teacher 2 (M)





1. No2. No3. Yes4. Yes


Starting for solving a question. 1. If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

2. Keep asking.3. Provide strategic notes rather than technical notes.4. Try the thinking of students in their contributions (e.g. in-group work or

class) to explicit by naming what steps they take.5. Demonstrate how you address a problem, think aloud and thus make

your own problem-solving strategies negotiable.

1. Yes2. Yes3. Yes4. Yes5. Yes

1. Yes2. Yes3. No4. Yes5. No


1. Encourage students to look for alternative solution.2. Encourage students to explain, even to each other.3. Challenge your students not be satisfied with particularities, but to search

for the general approach.4. Encourage students to examine the state of the progress of the solution’s

process.

1. Yes2. Yes3. No4. No

1. Yes2. Yes3. No4. No








7.2 Second meeting – Designing WDA

The second meeting is about designing “think active commands”. Thinking on

modifying existing tasks, for example from the book, and on the design of new tasks, for example, for a test.

Assignment 1: Teachers working in group getting started editing existing material exchanging experience, thinking on WDA tasks and discussing on how it will be applied in the classroom.

To the second meeting facilitators made sure that teachers understood WDA principles getting at the beginning of the session a resume about WDA philosophy, aiming establish a shared understanding of purpose. In addition, the expert worked with the evaluation sheet that teachers answered as homework. The session carried with experts drawing on a detailed comment, linking the tasks explored on the first meeting with WDA structure. The expert commentary was important because it suggests ways in which teachers might be moved forward the development of their own tasks. To this session, teachers’ work were structured around WDA “tips”. At the end of the session were expected all the teacher being familiar with WDA.

Tab. 11 Table of resources to teachers’ work

A review of previous meeting about WDA concepts. A list of “tips” to help teachers to recognize and work with WDA. Moments of collaborative work with teachers being grouped by level of teaching. A list of informal definitions about WDA (teachers’ production). A list of “handles” for designing WDA.

After the theoretical introduction, teachers started working in small groups. The plan for the meeting was to expert of FIsme start working together with teachers in terms of development a WDA tasks. Therefore, the meeting proceeded with teachers discussing and preparing a lesson to be applied on their classrooms. The expert that works with the group explain to them that tasks in which students need to create formulas from graphic data, might require a greater degree of difficulty to students. Also to situations that involves equality in an expression, or involves a graphic or a formula is more difficult for students. The teachers decide to work with a problem that involves “percent”.


Classroom observation

Assignment 1: - Try alone or with a colleague a WDA task in class.

7.2.2 The WDA task

During the face-to-face meeting, the teachers decided to use the task to the second lesson observation. The two teachers shared with colleagues the task and a teacher of the group says that she had a good experience in classroom with a same type of task, a task with different words but with the same idea and she had a nice time in classroom. On that moment, teacher 1 says to himself “ok, I am going to do it!” To this teacher the group at face-to-face meeting has a positive influence to his decision.

The task: There are two groups. An amount of people is moving from one group to the other group. So, how many percent of the first group is higher and how many percent of the other group is down?

The goal is: How students will start.

7.2.3 Managing the Resources to Support Learning

Teacher 1(f)

Again, the teacher start his lesson with an introductory explanation about their work of the day. He explained that they were going to work with a thinking activity and that the most important on this kind of activity is on how they go towards the task and not the result. The teacher invited students to inquiry themselves during the lesson.

The table below refers of an understanding by the researcher of a teacher’s lesson plan to the second lesson.


Tab 12 what happened in the classroom?


3:20 min Students thinking about the problem alone.

_ Students working in groups of two or three identifying further information.

3: 40 min Whole class discussing the problem, bringing up questions and different ideas to the solution.


_ Conclusion of the problem.

_ Reflection of the solution.

Teacher 2 (M)

The teacher introduce the task inviting students to the work with the task on board. She explained that they have five minutes to think about the problem and they must to start working alone.

The table below refers of an understanding by the researcher of a teacher’s lesson plan to the second lesson.



4:00 min Students thinking about the problem alone.

3:20 min Students working in groups of two or three identifying further information.

5: 40 min Whole class discussing the problem bringing up questions and different ideas to the solution.



1:35min Reflection about the solution (because a misunderstanding).


Tab. 14 Analyzing teachers’ practices (Annex II)

Observe if teachers… Tips for dealing with WDA tasks Teacher 1 (M) Teacher 2(F)





1. Yes2. Yes3. Yes4. Yes



1. If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

2. Keep asking.3. Provide strategic notes rather than technical notes.4. Try the thinking of students in their contributions (e.g.

in-group work or class) to explicit by naming what steps they take.

5. Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

6. Encourage students to examine the state of the progress of the solution’s process.

1. Yes2. Yes3. Yes4. Yes5. No6. Yes

1. Yes2. Yes3. Yes4. Yes5. No6. Yes

Encourage explanation. 1. Encourage students to look for alternative solution.2. Encourage students to explain, even to each other.3. Challenge your students not be satisfied with

particularities, but to search for the general approach.

1. Yes2. Yes3. Yes

1. Yes2. Yes3. Yes





1. Yes2. Yes

1. Yes3. Yes


Tab. 15 Analysing teachers’ professional competences (Annex II)

To observe teachers… Tips for dealing with WDA tasks Teacher 1 (F) Teacher 2 (M)







Starting for solving a question. 1. If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

2. Keep asking.3. Provide strategic notes rather than technical notes.4. Try the thinking of students in their contributions (e.g. in-group work or

class) to explicit by naming what steps they take.5. Demonstrate how you address a problem, think aloud and thus make

your own problem-solving strategies negotiable.

1. Yes2. Yes3. Yes4. Yes5. No

1. Yes2. Yes3. Yes4. Yes5. No


1. Encourage students to look for alternative solution.2. Encourage students to explain, even to each other.3. Challenge your students not be satisfied with particularities, but to search

for the general approach.4. Encourage students to examine the state of the progress of the solution’s

process.







1. Yes2. Yes

1. Yes2. Yes


7.3 Third meeting – Teachers working with WDA

The third session goes on performing active thinking math classes. In what way can

you ensure that there is actually, what happens in the minds of the students?

Assignment 1: Discussion of a video lesson related with inquiry learning.

As part of this session, teachers analyzed a video lesson as an assignment in order to identify and interpreting a situation involving inquiry learning. The proposal of the task was to teachers’ carefully analysis the pedagogical features presented on the video in order to help them to develop and/or understand pedagogical content knowledge for teaching with inquiry learning. Teachers were asked to reflect on the video situation and identify following teaching procedures as “To think about commands”, “to establish a sample lesson” and “to consider some tips”.

Tab. 16 Table of resources to teachers work

A review of previous meeting with information about WDA concepts. A video with the “Spirolaterals lesson” as a model of WDA. A final presentation of teachers’ WDA task to each other.

After teachers watch the video, a list of commands arise and it was decided which procedures and commands they consider important to apply in classroom (fig.1). Firstly, the facilitator invited the group of teachers share what they observed on the video providing among them an opening for discussion. This initiative were to facilitator validate their ideas writing their conclusions on the white board.


Fig. 6 Conclusions of teachers related with the video

What the Students Teachers

- Calculate with Numbers

- First try

- Time for exploration

- Own question / problem formulate

- Inventory questions

- Without judgment

- Stimulate

- Play, create, predict

- What, how, why

- Reflection

- Structured read

Secondly, teachers were invited to reflect about the elaboration of a “sample lesson plan”, if they want to promote a task involving inquiry learning. According to Lin, F. and Chin, E (p.2, 2013) “training teachers to design a detailed lesson plan of a new teaching strategy step-by-step is very helpful for them in successfully implementing the new teaching strategy into their classroom teaching” and “directed and tool mediated activity are fundamental to this process” (Doorman, M. 2013).


Fig 7. Sample less plan free translation of fig. 7

Finally, the facilitator indicates to particular “tips” that fits with working in an inquiry-learning environment.

Fig. 8 Tips Free translation of fig. 8

The steps above offer an overview of features that characterize a PD course that aims to transform teacher’s beliefs, knowledge and habits of practice. The bottom line to this process seems to be teachers planning attitudes that must to be incorporated on their practices.

With respect to the Teacher 2 (F), she selected instructional commands that she believes were able to support her attitudes in classroom (fig. 6) and which the teacher comply with.

Sample less plan

Introduction (5 min) Exploration (5 min) Discussion for focus ( 10

min) Research ( 10 min) Discussing the process (and

feedback) (10 min) Discussing the results Reflection on the purpose

connecting the activity with its sequence

Tips

Give time for exploration of a problem situation

Structure the problem (not too fast)

Give students an opportunity to master a question

Give feedback process (and later feedback product)


Fig. 9 Teacher’s notes

In addition, a particular tip explored on the video situation – “Time to explore”- seems to make part of the teacher’s beliefs and consequently they are supposed to arise on teacher’s practice.

After the last meeting of the course, the researcher asked for a last observation lesson to two teachers. To this lesson, it was asked to teachers to prepare, individually, a WDA task and to apply this task in two different classes with the same level. The main goal was to verify how teachers could manage with WDA tasks and what could be observed in terms of changes or improvement on teachers’ pedagogical knowledge and competences between classrooms, after the end of the course. This observation lesson was followed by an interview with each teacher. The interview also gave the opportunity to researcher support data.

7.3.1 Classroom observation 3- Teachers promoting learning

Assignment 1: Prepare, individually, a WDA task and apply this task in two different classes.

7.3.2 Task of Teacher 1 (M) – “The mosaic in the left is from Persia (now Iran). It is comprised of hexagons, four rectangles, and triangles. Here, you see again a very small piece of the whole.

a) What can you say about all the triangles, quadrilaterals and hexagons?”


Fig. 10

Het mozaïek hiernaast is afkomstig uit Perzië (het huidige Iran). Het bestaat uit zeshoeken, vierhoeken (rechthoeken) en driehoeken. Je ziet weer een heel klein stukje van het geheel.

a) Wat kun je zeggen over de aantallen drie-, vier- en zeshoeken?


Task Teacher’s actions

First group Second group

2:35 min 4:00 min Introduce the problem.

- 5:30 min Students thinking about the problem alone.

2:45 min 1:50 min Students working in groups of two or three identifying further information.

2:23 min 6:20 min Whole class discussing the problem bringing up questions and different ideas to the solution.

2:40 min 4:50 min Working with a different solution.

1:20 min 1:00min Conclusion of the problem.

- - Reflection about the solution (because a misunderstanding).

Total

11:23 min 23:00 min

To this lesson, the teacher decided a different introductory way between classes. To the first group the teacher used the picture followed by the question to introduce the problem. To the second group the teacher just decided to project the picture without the question. Consequently, he spent more time to introduce the problem to this group. In addition, an initial discuss started related with the goal of the problem before the students becomes to work by themselves. Teacher says to researcher that he was trying to apply the same task without explanation because he believes children will focus better to the problem situation.


This belief is closely related with teacher’s point of view that emerged at the beginning of the course.

Task

First group Second group

Fig. 11 One task, two introductory way

Tab. 18 Analyzing teachers’ practices Observe if teachers… Yes/No


Yes – Teacher asked students to work alone and in-group in part of the lesson.


Yes – Teacher asked students for helping during the lesson. He had students’ whole class contribution to solve the problem. Students share their ideas with friends.

Encourage explanation. Yes – Teacher started the dialogue following the thinking of a student and then, he becomes to get answers from others.



Yes – Teacher developed a task from a picture. Teacher changes his methodology to work with the same task redesigning the task to his next class.

Tab. 19 Analysing teachers’ professional competences

To observe teachers… Yes/No


Yes – Teacher works with open questions creating an environment of learning with students contributing with their inquiring.

Starting for solving a question. Yes – Teacher started the solution of the problem following the thinking of a student.


Yes – Teacher guided students to the answer during the dialogue enabling them to develop their own strategy of solution.



Yes – Teacher designed a task that give opportunity to engage students in a dynamic discussion.


Task of the Teacher 2 (F) – To this observation lesson the teacher prepared a 45 minutes lesson with four different tasks.

Task one Task two

What do you think here?

Task three

An example

Task four

One more to unlearn

Jip has a number in his head, if you doubled that number and add 100 there will be as much as when you subtract 200 from the double of the number.

Fig. 12 One lesson, three tasks


Task 1 Task 2 Task 3 Task 4 Teacher’s actions

First group

Second group

First group

Second group

First group

Second group

First group

Second group

0:25 min

0:25 min

0:36 min

0:50 min

0:35

min

- 0:55 min

1:00 min

Introduce the problem.

2:30 min

1:45 min

2:35 min

1:27 min

3:05 min

1:20 min

6:00 min

3:40 min

Students thinking about the problem alone.

- - - - - - - - Students working in groups of two or three identifying further information.

5:20 min

5:20 min

4:24 min

5:50 min

1:00 min

2:05 min

1:05 min

1:30 min

Whole class discussing the problem bringing up questions and different ideas to the solution.

- - - - - - - - Working with a different solution.

0:10 min

1:10 min

3:10 min

3:55 min

1:55 min

0:40 min

0:45 min

1:20 min

Conclusion of the problem.

- - - - - - - Reflection about the solution (because a misunderstanding).

Total

8:25 min

8:40 min

10:45 min

12:02 min

6:35 min

4:05 min

8:05 min

7:30 min


Tab. 21 Analyzing teachers’ practices Observe if teachers… Yes/No


Yes – To start the lesson teacher asked students to work alone.


No – Teacher asked students for helping during the lesson but in a technical structure. Teacher asked students in an individual dialogue offering procedures to the solution.

Encourage explanation. Yes – Teacher guided students to the answer systematically asking what they think during the process.



No – Teacher used three-structured task.

Tab. 22 Analysing teachers’ professional competences

To observe teachers… Yes/No


Yes – Teacher works with structured questions creating an environment of learning where students contributed with systematic answers.

Starting for solving a question. Yes – Teacher provoke one student with a question to start de procedure of the solution.


With this lesson was not possible to analyze this competence because teacher asked to one student’s answer while the entire group waited in silence.



No – Teacher used three-structured task.


8. What does the research show – final conclusions

The main purpose of the study was to gain a better insight into the empowerment of teachers’ competences and approaches in working with a WDA task. To attend this purpose two-research questions was established.

In terms of changes in teachers' practices when working with a WDA activity, how are the empowerment of teaching approaches? Which changes emerge on these approaches?

To what extent does a WDA course contribute to the empowerment of teachers’ competence in working with inquiry learning?

To answer the research questions it was analyzed teachers’ work during the meetings at the WDA course and during their lessons in classroom. It was investigated teachers’ practice under the perspective of to find changes on their practices that were emerged from their learning on the course. This analyzes was conducted specifically relating the resources available for teachers (tab 4, tab. 11, tab. 16) which led them to understand and apply WDA tasks in the classroom, and teachers attitude in using and applying those resources.

Teachers identifying and using a WDA task, teachers pedagogical attitudes and competences

During the study, an important role that teachers assumed was managing the needed resources to select a WDA activity. To support teachers understanding about WDA philosophy the course offered theoretical tools and instructional materials that could be used in classroom. Related with the instructional materials available by the course teachers didn’t use them in classroom. They chose tasks from the e-book that is used at school and that were developed by a teacher from the school.

Trying to verify teachers understanding, after the end of the course the researcher did the following question:

Researcher: “Now, (after de course) what do you understand about WDA?”

T 1 (M): […] “It is very important for student after a kind of discomfort, where they don’t know exactly what to do, start (with this discomfort) trying to get an answer (to the task), because they don’t like the situation when they don’t know. They are not very secure and they say ‘ahhhh’”[…].[…] “It is something that we need to teach these children, we have Mathematics, and I think on the top of that is Wiskunde Denkactivitate”[…]. […] “I think, it is not for every people here.”[…]. […]” I’m not agreeing that it should be in an exam, I


really do not. Because it is very hard to test. But, what I learned from Wiskunde Denkactivitate is to see it as a different kind of material” […]. […]”and it keeps involving, it is not finished, it stays in minds”.

T: 2 (F) – […] “I think the same (of the beginning) but I think sometimes that Wiskunde Denkactivitate could be about something students must know to learn another way to solve it […]. […] So, I don’t think that it always has to be something new but it has to be thinking about the way to solve the problem: an easier way, a right way, or if there are another way, could be the same answer, a short way. I think that is the point” […]. […] “In a Wiskunde Denkactivitate, you cannot give all the steps to the solution immediately because you are killing the process for the thought” […].

After the course teachers becomes to speak more properly about WDA but we verify that they are still in a process of understanding and constructing their own definition. To teacher 1, Wiskunde Denkactivitate is related to a dynamic process of learning […] “It is very important for student to and to, after a discomfort where they don’t know exactly what to do, start trying to get the answer” […]. To teacher 2; Wiskunde Denkactivitate is related to a structural process of learning […] “It has to be thinking about the way to solve a problem” […]. However, they also agree that they are at the beginning of this understanding and that they have a lot to learn.

Teachers are consciousness about their personal outcomes, not only for the aspects of WDA but also for some difficulties from their previous teaching practice. Regarding this, after an activity teacher 1 says: […]”I think I am not so structural” […] […] “for me …I want to improve… I want to learn” […]. Teacher 2 says: […] “I am a bit satisfied. But it stayed hard to put some information some Denkactivitate always in my lesson” […].[…] “It were in a short time (the course). It is not something that you can put in right way in every lesson and say, yes, it is a thinking activity. I think we need more time to make our own and to put it always in our lesson, or to put it easier in our lesson” […].

During the research, teachers were invited to work with three different tasks on different classes and moments. The main goal was to offer an opportunity to teachers to think about to prepare a WDA task and to think on how they can work with it. Both teachers worked with the same task on their lessons for the first two observations lessons. To the last observation lesson, each teacher developed their own task.

With this experience teachers demonstrated some changes in working with WDA, distributing better the time used in classroom (tables: what happened in the classroom?) and being more careful in allow students to work individual and in group. These two teachers become to organize lessons in such a way that students got more time to think for themselves about the problem and the strategy to solve it. However, with this experience is possible to conclude that they handle this aspect better with “open” tasks (task 1 and task 3 of the teacher (M) and task 1 of teacher (F)) than with “structural” task (task 2 of both teachers and task 3 of teacher


2). The analyses of these tables suggest that teachers have difficulty in stimulating student participation to the lesson when the task has a more structural aspect. At the end of second lesson teacher 1 conclude that students were not going well on this lesson. […] “I suppose it was a little bit difficult for them” […]. In addition teacher 2 conclude: […] “I think that this activity could be a thinking activity but otherwise they cannot solve it on different kinds of ways so…in some ways I think it is a thinking activity but in somewhere I think it is not…because it is too small, too structured […].

With regard to which features a WDA task must has, teachers paid particular attention to how prepare a task with focus on students analysing, interpreting and thinking actively. To task 1 both teacher redesigned an original task (fig. 5) and the teacher 1 (M) also redesigned the task 3 among his two classes, creating a powerful environment of learning on his second class (fig 11 and tab. 17).

Regarding with these changes on the tasks teacher 1 says:

Researcher: “Why did you decide just project the picture?”

Task 1: […] “I saw the exercise and I saw a lot of words, and I can imagine that if you are in classroom, in a busy morning, and you sit down to have mathematics lesson, and you look at the board and you see all of this, you don’t know in what to focus! So, when showing this picture they only need to focus on this tower!” […].

In addition, teachers considered what questions and what procedure they might use to facilitate learner development in starting working in an inquiry environment of learning. To task 3, teacher 1 says: […] “I tried to follow the thought of this girl over here, I am not sure if what she said was totally right but then the discussion started” […].According to teacher 2 (F), her option for three tasks in one lesson was because […] “ they are really different classes. The one is a class where children understand immediately, and the other class probably children needs two times more. Then, I tried to get a way that fits with the kids. And it is different between classes”. […]

Two important features:

Teachers valued positively the course.

Teacher 1: […] “I think every experience improves” […]. […] “To see an exercise and recognize it as a Wiskunde Denkactivitate, that is the gain. Analyses an exercise and recognize it as a Wiskunde Denkactivitate. But I want to practice more”. […]

Teacher 2. […] “The course was an opportunity to learn things to teach children”. […]. […] “I want to know more about it, and how to use it. The combination of these two things I really want to know. It is about the changes that are coming with the now curriculum. I really want to read more about Denkactivitait” […].


Teachers need more time and more examples to understand and work better with

WDA. Regards with this, the result of the research suggest to experts ask teachers to apply in classroom at least one of the activities of the bank of activities offered by the course. It will gave the opportunity for teachers try a Denkactivitate before design their own.

9. References

Borko, H (2004). Professional Development and Teacher Learning: Mapping the Terrain. Education & Educational Research, November 2004. 33:3-15.

Doorman, M., & Maass, K. (2013). A model for widespread implementation of inquiry-based learning. ZDM Mathematics Education, 45:887-899.

Drijvers, P. (2011). Wat bedoelen ze toch met... denkactiviteiten? Nieuwe Wiskrant 31.

Annex I

How was teachers’ performance in working with WDA tips after the first meeting?


WDA Tips Teacher 1 (M) Teacher 2 (F)

• Give students time to understand the problem, to think about the problem and to get involved in the issue.

Teacher believes that it is important to students think by themselves but during the lesson, he didn’t stop asking.

[…]It is also important to think yourself for a moment. So, we will start with a couple of minute to think a moment. […]. […] we have already discussed, we have a strategy, now take some minutes, while you are quiet. […]

Yes Teacher asks students to work by themselves for 1:35 minutes at the beginning of the lesson.

After teacher asks for students continues working in pair.

Yes

(X)

No

(X)

No

• Take good care with rapid judgments about pupils’ reactions because it can inhibit the process of thinking.

[…]S: What is exactly the high of the tower?

T: Good question! What is exactly the high of the tower?[…]

Yes […]I've heard a lot of good answers […]

[…]Very well, does someone have another good idea how to do this? […]

[…]S: We think how big the house is.

T: That's right. In addition, how big the house is? […]

[…]S: Plus and minus?

T: No…What we found with the door? What is exactly that door relative to the whole tower?

S: Compare!

Teacher: Exactly! We will go compare […]

Yes

No

(X)

No

(X)

• Do not give answers, but set (open) questions.

[…]S: What is exactly the high of the tower?

T: Good question! What is exactly the high of the tower? […] S (1): I think it's the small piece on the top.

S (2): I think the tower is from the ceiling to the end of the small tower. T: In fact, you need to know what the tower is to calculate the high of the tower. […][…] We will stablish that the

Yes

(X)

T: […] does someone have another good idea how to do this? (The teacher invites other student) (Maaike?)

Yes

(X)

No No


tower is from the floor to the small tower […]

• If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

[…] S: Well, if you know how long the pole is…

P: OK, you are already working with a strategy, who has another idea? […]

S (1): I think the house that is on the left side of the tower. […] The average high of a house is about seven meters. It seems absurd that a normal house is so big (on the photo and compare with the high of the tower). However, the houses on the back are much smaller. You do not know how far it is from the back or from the front of the tower. T: Are you sure about what you are saying? I think that this house (on the left side) is more “to the front” of the tower but […]

Yes

(X)

Yes

No No

(X)

• Keep asking. […]S: What is exactly the high of the tower?

T: Good question! What is exactly the high of the tower? […] Let's take this high (from the floor to the small tower). How we will calculate this high? […]

[…] T: OK, you are already working with a strategy, who has another idea? If you know the high of this one (pole) then you know how to calculate the high of the tower. Do someone want to say something about this? […]

[…]T: But we do not know the high of the lamppost and it is a little far to the front of the tower. Then, you do not know exactly the high. Who has an idea? […]

[…]You said that the gateway is two meters. Let’s see if someone has something to say about it. […]

Yes

(X)

[…]Who has a good idea? (to answer the question) I've heard a lot of good ideas.

S: Well, we think that the pole is 2.5 meters, the door is 2 meters, we have to calculate how many times the object fits with the tower, and then we multiply this amount by the height […] T: […] does someone have another good idea […]? S: We were not thinking with the traffic plate but with the door. […] T: And did you do with the white part up the door or only the green part?

S: Only with the green […] T: And there, […] I see two kids saying that they thought the same? S: We use the door as well. […] […] S: Do we calculate the height of the tower?

T: Yes, we calculated the height of the tower, but […]

Yes

(X)

No No


T: There are other ways to calculate? […] S: The house. T: Oh ok, and how you made it using the house? […]

• Provide strategic notes rather than technical notes.

[…]S (1): I think it's the small piece on the top.

S (2): I think the tower is from the ceiling to the end of the small tower. T: In fact, you need to know what the tower is to calculate the high of the tower. We will stablish that the tower is from the floor to the small tower […]. Let's take this high (from the floor to the small tower). How we will calculate this high?

[…]S: Well, if you know how long the pole is…

P: OK, you are already working with a strategy, who has another idea? If you know the high of this one (pole) then you know how to calculate the high of the tower. Do someone want to say something about this?[…]

Yes

(X)

Yes

No No

(X)

• Try the thinking of students in their contributions (e.g. in-group work or class) to explicit by naming what steps they take.

[…]S (3): The pole we do not know how high it is but the gate should be about two meters. S (4): The gateway of the tower should be about two meters and besides the gateway, there is a pole. T: But we do not know the high of the pole and it is a little far to the front of the tower. Then, you do not know exactly its high. Who has an idea?

[…]S (3): The gateway has 2 meters. (The teacher draw a line representing the length on the photo). We just need to calculate how many times the door fits

Yes

(X)

[…] However, you see that if you are going to rethink how you can proceed with something so that you do a nice step plan to decide how big something is […]

Yes

(X)

No No


with the tower.

S (1): Yes, this is about two meters (the high of the gateway). Besides the gateway, something is two times its high. T: Here? Are you sure that this one is near the tower? […]

[…]T: Ok, but let me use S (3) idea. This part is 2 meters (on the photo). How do I know the whole length of the tower? […]

• Encourage students to look for alternative solution.

[…]T: Here? Are you sure that this one is near the tower? S (1): Yes, Others: No, S (1): It does not matter.

Others: Yes… T: Does it matter if something is far or near the photo? […]

Yes

(X)

[…]T: Do you now understand that all of you are doing the same together, equal to each other? […]

Yes

(

X)

No No

• Encourage students to explain, even to each other.

[…]T: Does it matter if something is far or near the photo? S (2): Yes, if you see something far from you, something that is big, it looks small. S (4): Yes, the distance is important when you want to know the high of something. […]

Yes

(X)

[…]S: We were not thinking with the pole but with the door. Because an adult man has to get through the door, and the church door is larger. Therefore, we were thinking that the church door is 2.5 meters.

T: And did you do with the white part up the door or only the green part? […]

Yes

(X)

No No

• Challenge your students not be satisfied with particularities, but to search for the general approach.

Yes Yes

No No


(X) (X)

• Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

[…] I think that this house (on the left side) is more “to the front” of the tower but, if you consider this house (on the right side) it has more or less the same format and I think it also has about seven meters. You consider the part of the house that is 3 meters (fig 7). It fits with the tower 5 times and then you do 3 x 5 = 15 times. […]. […] T: And how we do with the high of the gateway that is two meters? […] […]You will not discuss but all of you first have to think by yourselves […]

Yes

(X)

Yes

No No

(X)

• Create a safe and stimulating working environment.

[…] T: What is important for me is not the answer, but as you arrive to the answer

Yes

(X)

[…]. Can you in pairs, speaking quietly, find a way to calculate the high of the tower? […]

Yes

(X)

No No

• Provide inspiration and variety by offering attractive problems and different methods.

Problem first session

Problem second session

Problem third session

Yes

(X) Problem first session

Problem second session

Problem third session

Yes

(x)

No No

• Encourage students to examine the state of the progress of the solution’s process.

Yes Yes

No

(X)

No

(X)

Annex II

How was teachers’ performance in working with WDA tips after the second meeting?


WDA Tips Teacher 1 (M) Teacher 2 (F)


• Give students time to understand the problem, to think about the problem and to get involved in the issue.

T:[…]”you have a few minutes to think about it”[…][…] “I think two minutes is a lot, maybe three is a good idea”[…]

T: […]”look at this for a moment” […]. […]”Do not go directly saying: I cannot do this”[…]

T: […]”ask to yourself: How do I get there?”[…]

Yes

(X)

T: […] “Silently, you have five minutes to solve the task, ok?”[…]

Yes

(X)

No No

• Take good care with rapid judgments about pupils’ reactions because it can inhibit the process of thinking.

_

Yes

(X)

S: […] “Well I'm not sure if my calculation is correct.” […] T: […]”Well, if you are at 11% then it seems to me that your calculation is strongly wrong” […]

Yes

No No

(X)

• Do not give answers, but set (open) questions.

Student: […] Well, I first looked at what could happen with 5 members of the first group, and it is 20% […] which 25 are members of the first group, that is 100%, and there are five members away” […]

Teacher: […]”But, how do you do that (to calculate)? What do you use? Do you do all in your head? […]

Yes

(X)

T: […]”what have you done? Tell me. This time quietly.”[…]

Yes

(X)

No No

• If the student has a stalemate, immerse yourself in the thinking of the student and ask a question that could have imagined the student himself.

The teacher conclude that the task was not so easy to the students and he had not time enough to keeps working with the group

Yes

(X)

S: […] “Well I'm not sure if my calculation is correct.” T: […]”what have you done? Tell me. This time quietly.”[…] S: […]”If you divide 100 by 45 than you have 1 percent and then you do it 5 times and that is 11%” […] T: […] “You said that you divide by 45?”[…]

Yes

(X)

No No

• Keep asking. T: […]”Who can say something about the question?”[…]

T: […]”How do you write this now? How do you handle with this? What is the first step?”[…]

Yes

(X)

T: […]”Is there someone who has done something different?”[…]

Yes


(X)

No No

• Provide strategic notes rather than technical notes.

T: […]”My new question is: What steps can you take?” Vincent says to use the ratio table…So, if I do…how did you go further?”[…] S: […]”I have also looked into the second groep”[…].

Yes

(X)

T:[…]”But what I am trying to say here is basically that if you write it down you make it visible to yourself”[…] I am trying to say is that this way it is easier to count”[…]

Yes

(X)

No No

(X)

• Try the thinking of students in their contributions (e.g. in-group work or class) to explicit by naming what steps they take.

T: […]”How do you write this now? How do you handle with this? What is the first step?”[…]

S: […]”Well, if one group with 25 percent have now 5 members would be something logical that …” […]

S: […] “The first group was 5 times and the second group was 4 time, and then you count those 20 and 25 with each other” […] […] “In total there is 45 members. In addition, how many five is of 45? It is 5/45 or 1/9 and then we calculate the percentage of it.”[…]

T: […] “and then you can do the percentages in the relation table…” […]

Yes

(X)

T: […]”Is there anyone who has made an example, a draw?”[…] S: […]”Well, I made a box that was 25 members and other box with 20 members. And then I draw an arrow moving 5 from one to the other” […]

Yes

(X)

No No

• Encourage students to look for alternative solution.

T: […]”Well, just try for once, try to see what happens”. […] […] “Do not go directly saying ‘I cannot do this’”.

T: […]”Do you start with a ratio

Yes

(X)

T: […]”Ok, I want you now examining the exercise with your colleague and see how you’ve done. Don’t look at to the final answer but look if you both have done the same way”[…] […]”Do

Yes

(X)

No No


table […] […] How did you get this idea? […] […] Who has a different way of thought? […] […] Try to draw also […].

we get the same answer? Where do we have done differently and where we have done the same?”[…]

• Encourage students to explain, even to each other.

T: […] “Who can say something about the task? How did it start? We will start with Vincent…” […]

S: […]”Well, I first looked at what could happen with the five members of the first group…” […]

Yes

(X)

T: […]”Do we have the same answer? Can you consult each other?”[…]

Yes

(X)

No No

• Challenge your students not be satisfied with particularities, but to search for the general approach.

T: […]”This is what you say, how did you go further?”[…]. S: […] “I have also looked into the second group” […] […] T: “Yes, but you say: ‘then I go here to 100%’…, right?”[…]

Yes

(X)

S: […]”One group is 20% smaller and there is 5 members going away. Therefore, a member is 4%. The other group will be 25% larger, so a member is 5%.”[…] T: […]”So, you have twice what percentage is a member and how much move to each other. Is there someone who has done something different?”[…] S: […]”I know that 5/45 is gone, then you do 100 divided by 45 and you will have 1% and then you do it 5 times” […]

Yes

(X)

No No

• Demonstrate how you address a problem, think aloud and thus make your own problem-solving strategies negotiable.

It did not happen in this issue.

Yes

It did not happen in this issue.

Yes

No

(X)

No

(X)

• Create a safe and stimulating working environment.

T: […]”Just look at it for a moment, read it once, try to find a solution and if you have a question, just put your finger on”. […] […]”I will run to you if your finger is up”. […]

Yes

(X)

T: […]”what have you done? Tell me. This time quietly.”[…]

Yes

No No

• Provide Yes Yes


inspiration and variety by offering attractive problems and different methods.

The task. (X) The task. (x)

No No

• Encourage students to examine the state of the progress of the solution’s process.

S: […]” the first group was 5 times, so it is 25, and in the second group, you do the same. Thus, 20 plus 25 is the total number of members, in both groups. Moreover, how much is 5 of 45? It is 5/45 that it is 1/9. Now you calculate 1/9 in percentage with the ratio table.”[…] T: […]”Ok, but…are there other approaches to get started?”…”Who has another way of thought?”…”Class, just listen to each other!”…[…]

Yes

(X)

T: […]”Well, you did it in percentages…Instead of to see how much percent is one person as Brecht did” […]

Yes

(X)

No No


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