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Exploring the Computational Medium

Chapter 1. Introduction

Huub de Beer

May 11, 2016

Ever since the invention of the computer in the first half of the 20th century, computerization has had an increasing impact on our society. Some speak of an information revolution, arguing that the effects of ubiquitous computing and computer networks are fundamentally altering our self-image and the way we live our lives (Illari, 2013). Collins & Halverson (2009) compare information and communication technology (ICT) to technologies of the past that fundamentally altered our world, arguing that:

‘No one will be able to solve complex problems or think effectively in the coming world without using digital technologies. (…) Just as reading was made necessary by the printing press and arithmetic by the introduction of money, so computer technologies are changing the very ways we think and make sense of the world.’ (Collins & Halverson, 2009, p. 11)

Although the transformative powers of computer technology on our society have been recognized since the conception of the computer, that recognition was never widespread. For decades, despite their size—or perhaps precisely because of their size—, computers were almost invisible in daily life. Only after the advent of the personal computer in the 1980s and, in particular, the 1990s, the effect computers have had on society became more and more visible. While this process sped up in the 2000s, from a computer in almost every home at the start of the decade to almost everyone carrying around a personal mobile computing device at the end, society at large transformed from a (post) industrial society into an information society.

In education too, ‘sophisticated computers and telecommunications are on the verge of reshaping the mission, objectives, content, and processes of schooling (…) to meet the challenge of making pupils ready for a future quite different than the immediate past.’ (Dede, 2000, p. 281) In the global interconnected information society, our children have to compete with others around the world, while their skills and knowledge are becoming obsolete faster (Gravemeijer, 2009; Molnar, 1997) and more occupations are in danger of being automated (Frey & Osborne, 2013). In anticipation of these changes, ever since the late 20th century, there has been a movement to teach so-called 21st century skills, which includes skills like problem solving, critical thinking, ICT competency, computational thinking, and innovation skills (Cobo, 2013; Dede, 2010; Ferrari, Punie, & Redecker, 2012; Rotherham & Willingham, 2010; Wing, 2010).

The need for new STEM education in primary school

Furthermore, in recognition of the historical and cultural important role that science, technology, and mathematics (STEM) play in the high-tech information society (Millar & Osborne, 1998), many a researcher voiced the need for a new STEM education in primary school (Gravemeijer, 2013; Léna, 2006; Millar & Osborne, 1998) because the reality in our primary schools does not reflect this need. STEM in primary education, particularly in the Netherlands, is often limited to a small number of hands-on activities a month (Keulen, 2009). According to the Dutch Inspectorate for Education, only 19% of Dutch primary schools did achieve a satisfactory level of STEM education in 2009 (Inspectie van het Onderwijs, 2010, p. 45). Although this was a huge improvement over the percentage of primary schools performing satisfactory in 2004 (2%) (Inspectie van het Onderwijs, 2005), STEM in Dutch primary schools is still underdeveloped.

Since the 1960s, when STEM education meant “science for all”, STEM education switched to a process-oriented approach in the 1980s and, now, to a literacy approach (Millar & Osborne, 1998). Although STEM education has become more important in the primary curriculum, its content has stayed largely the same since the 1960s, retaining

‘its past, mid-twentieth-century emphasis, presenting science as a body of knowledge which is value-free, objective and detached—a succession of “facts” to be learnt, with insufficient indication of any overarching coherence and a lack of contextual relevance to the future needs of young people. The result is a growing tension between school science and contemporary science as portrayed in the media, between the needs of future specialists and the needs of young people in the workplace and as informed citizens.’ (Millar & Osborne, 1998, p. 4)

Léna (2006) concurs, arguing that: ‘Science curricula seem unable to convey anything but the “old” physics, and its associated, outdated representations.’(Léna, 2006, p. 5) Instead, he continues, science literacy is important for all children because it allows them to explore the world around them, to use scientific knowledge, and its results. He attributes the main problems of current STEM education in primary school to the teachers, who do not teach STEM often enough, fear its complexity, fear doing experiments, fear students’ questions they cannot answer, and have a poor understanding of STEM. Because teachers do not feel safe teaching STEM, they resort to what they know from their own experiences with STEM education. Consequently, classes labeled STEM are often workshop classes (Keulen, 2009; Léna, 2006) where pupils conduct traditional science experiments or tinker with familiar materials and tools, such as wood, nails, a hammer, cardboard, glue, and scissors.

To overcome these problems, new STEM education in primary school has been proposed (Gravemeijer, 2009; Keulen, 2009; Léna, 2006; Millar & Osborne, 1998). This new STEM education is based on the observation that children are curious about the world around them. STEM education can build on that curiosity by giving students new means to explore their environment:

‘It endows them with a rich understanding of our complex world, helps them practice an intelligent approach to dealing with the environment and develops their creativity and critical mind, their understanding of reality, compared to virtuality and teaches them the techniques and tools that societies have used to improve the human condition.’ (Léna, 2006, p. 8)

To accomplish this goal, the following characteristics for innovative STEM education are formulated: inquiry-based, close to students’ world-view, ICT-rich, and integrated into the primary curriculum. STEM education should be inquiry-based. Inquiry-based learning supports students’ learning by involving them in real-life situations with an emphasis on questioning, hypothesizing, and experimenting by the students themselves (Léna, 2006; Osborne & Dillon, 2008; Rocard et al., 2007). At the same time STEM education should also be close to the students’ world-view. Their environment has almost no relation to the STEM subculture. STEM education should try to minimize this gap between students’ realities and the reality propagated by the scientific world-view (Osborne & Dillon, 2008). It should try to ‘produce a populace who are comfortable, competent and confident with scientific and technical matters and artifacts.’(Millar & Osborne, 1998, p. 9) Furthermore, STEM education should use ICT. Not only will students grow up in a society where ICT is ubiquitous, the use of ICT has also an enormous potential in education. It enables new ways of teaching and learning (Bingimlas, 2009; Murphy, 2003; Osborne & Hennessy, 2003; Woodgate, Fraser, & Crellin, 2007). Finally, STEM education should be integrated into other subjects in the primary school curriculum. As it is, that curriculum is already overflowing. It will be difficult to try to fit in STEM as a separate subject. Furthermore, because STEM is an integral part of our information society, that should also be reflected in the curriculum by making it and integral part throughout the entire curriculum.

In the Netherlands, the government started the so-called VTB-Pro project in 2007, a large program to stimulate interest, knowledge, and skills of primary school teachers in the domain of science and technology. To implement this program in the South of the Netherlands, a knowledge center named KWTZ was erected by a consortium of teacher education institutes and the Eindhoven School of Education, which is part of the Eindhoven University of Technology. The main task of this knowledge center was to develop and offer in-service and pre-service teacher education. In addition, various research activities were started—among which the research that is reported in this dissertation. This thesis centers around exploring how these characteristics of innovative STEM education can be put into practice in primary school. Of course, as STEM is an extensive domain, this thesis can only focus on a small sub-domain of STEM, for which the concept of instantaneous rate of change is chosen. This choice is explained next.

Instantaneous speed in primary school?

In our information society the interpretation, representation, and manipulation of dynamic phenomena are becoming key activities. Conventionally, monitoring and controlling dynamic phenomena, in particular in real-time, was firmly rooted in the realm of STEM. With the advent of ubiquitous networked computing devices connected to sensors, however, awareness of dynamic phenomena in every-day life is growing. With smart phones, we can track our movements. With smart watches and other wearable technology we can monitor our heart rate, temperature, perspiration, and more. Government and utility companies want to introduce smart meters in every home. Combined with a smart thermostat we are able to monitor and adapt our energy usage patterns very precisely. On the horizon is the Internet of Things, promising for the future the ability to continuously monitor almost every aspect of our lives.

We monitor dynamic phenomena to gain a better understanding of them and, ultimately, to increase our control over them somehow. Key to a better understanding of dynamic phenomena is the concept of instantaneous rate of change. Instantaneous rate of change is defined mathematically as follows: Given a dynamic phenomenon described by a function \(f\), the instantaneous rate of change at moment \(m\) is defined as

\[\lim_{h \rightarrow 0}\frac{f(m+h) - f(m)}{h}\]

This definition suggests that understanding of instantaneous rate of change builds on average rate of change, function, algebra, and the limit concept. In line with these suggestions, conventionally, students first encounter instantaneous speed mathematically in a calculus course, which is placed at the end of the secondary school mathematics curriculum. Due to its formal mathematical nature, however, only about 10% of all students in the USA would enroll in a calculus course (Kaput & Roschelle, 1998); in the Netherlands, the percentage of students learning calculus in high school is about three times as large1. Furthermore, many of those students have trouble learning calculus. According to Tall (1993), students’ difficulties with calculus include the limit concept, trouble connecting calculus to real world applications, their preference for procedural knowledge over deeper mathematical understanding, and a lack of understanding of and skill with algebra and function. Clearly, calculus is unsuited for primary education.

At the same time, originating from a widely held dissatisfaction with calculus courses and the growing availability of computer technology came a push for calculus reform (Tall, 1993; Tall, Smith, & Piez, 2008). Reform initiatives followed technological developments: the microcomputer of the late 1970s brought on approaches based on numerical algorithms, improved graphics capabilities in the workstations of the 1980s resulted in approaches based on visualization, enactive approaches followed the introduction of new interactive input devices, and once computers became powerful enough to run computer algebra systems approaches based on those gained traction as well (Tall, 1997; Tall et al., 2008). Although most calculus reform initiatives focused either on improving traditional calculus courses or changing the mathematics curriculum in service of teaching and learning calculus in the formal mathematical sense (Tall, 1997; Tall et al., 2008), other researchers went beyond the educational implications outlined by traditional calculus.

For example, a group of researchers led by James Kaput saw in ICT an opportunity to democratize access to calculus (Kaput, 1994, 1997; Kaput & Roschelle, 1998): with the advent of affordable computer technology, every child could and should be enabled to learn the mathematics of change and variation (Roschelle, Kaput, & Stroup, 2000). Therefore, in 1993, the SimCalc project was started. After showing that the average student could indeed successfully learn the mathematics of change and variation during the try-out phase, the SimCalc project was scaled up to find out how to integrate it into the curriculum of middle school and high school (Kaput & Schorr, 2007). Beyond this ‘Kaputian program’, as Tall (2013) called it, other researchers too focused on learning of calculus-like concepts already in elementary or middle school (Boyd & Rubin, 1996; Ebersbach & Wilkening, 2007; Galen & Gravemeijer, 2010; Nemirovsky, 1993; Nemirovsky, Tierney, & Wright, 1998; Noble, Nemirovsky, Wright, & Tierney, 2001; Stroup, 2002; Thompson, 1994). These initiatives suggest that calculus-like topics can be suited for primary education after all.

In a way, the computerization of our society led to a convergence of a need for a better understanding of instantaneous rate of change and new possibilities to teaching and learning calculus-like topics offered by computer technology. Because primary school students will not have developed the level of abstraction associated with the use of “rate of change” in the literature, the term “speed” is preferred. Currently, instantaneous speed is not part of the primary school curriculum. To explore teaching topics in earlier grade levels than they are usually taught, design research is well-suited (Kelly, 2013). Therefore, to explore the coming together of the need to better understand instantaneous speed from an early age on and the possibilities for innovative ways of learning offered by computer technology, a design research project was started on: How can we teach instantaneous speed in grade five?

Design research

In the Netherlands ideas similar to design research have been explored since the 1970s by Freudenthal, Streefland, and Gravemeijer (Eerde, 2013; Gravemeijer & Cobb, 2013). Internationally, however, design research gained traction in the 1990s—many point to the seminal work of Brown (1992) and Collins (1992) as the starting point of this development—as a reaction to a perceived gap between educational practice and research (Eerde, 2013; Reeves, McKenney, & Herrington, 2010; The Design-Based Research Collective, 2003). To bridge this gap, design research seeks to address:

In addressing these needs, design research builds on educational design. It uses an interventionist process of iterative refinement to create some instructional artifact that takes real-world classroom practice into account (Bakker & van Eerde, 2013; Barab & Squire, 2004; Cobb, 2003; Plomp, 2013; Reimann, 2011; The Design-Based Research Collective, 2003). In contrast to educational design, however, which is product-driven, design research aims at developing theory where ‘the design is conceived not just to meet local needs, but to advance a theoretical agenda, to uncover, explore, and confirm theoretical relationships.’ (Barab & Squire, 2004, p. 5)

Plomp (2013) distinguishes two main categories of design research, each with a different aim (Plomp, 2013): “developmental studies” aim at developing research-based design theories for use in educational design projects, while “validation studies” aim at exploring innovative learning ecologies to develop a local instruction theory (LIT). The latter category fits the design research project described in this thesis. In particular, the design research approach outlined in Gravemeijer & Cobb (2013)2 is applied to answer the research question, how can we teach instantaneous speed in grade five, by developing a local instruction theory on teaching instantaneous speed in 5th grade.

A LIT pays attention to the intended learning goals, the instructional starting points implied by students’ prior instruction and their intuitive understanding, and it

‘consists of conjectures about a possible learning process, together with conjectures about possible means of supporting that learning process. The means of support encompass potentially productive instructional activities and (computer) tools as well as an envisioned classroom culture and the proactive role of the teacher.’ (Gravemeijer & Cobb, 2013, p. 78)

To start off the design research process, an initial LIT is formulated based on the literature and any other source that might contribute to the researcher’s understanding of students’ prior conceptions (Figure below, starting up). After this starting-up phase, the LIT is elaborated, adapted, and refined in one or more design experiments (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003) or macro design-cycles (Figure below, three design experiments were performed). A design experiment consists of three phases. First, in the preparatory phase, the LIT is (re)formulated and, based on that LIT, an instructional design is developed. Then, in the second phase, that design is tried in a teaching experiment consisting of a sequence of micro design-cycles of redeveloping, testing, and evaluating instructional activities and materials. Finally, in the retrospective analysis phase, an analysis of what happened during the teaching experiments informs the refinement of the design, and an analysis why this happened informs the refinement of the LIT. This LIT is the starting point of the next design experiment.

This design research process yields both a working prototype and a theory on how that prototype works. In line with the theory-driven nature of design research the focus in this thesis is on the LIT rather than on the prototype. Nevertheless, parts of the prototype and its development will be discussed while elaborating on the development of the LIT. For a more thorough exploration of the prototype, in particular the educational simulation software that was developed and tested in the various teaching experiments, the website https://heerdebeer.org/DR/ is set up.

Overview of the design research project

Overview of the design research project: after a starting-up phase three successive design experiments were conducted. (Taken from Beer, Gravemeijer, & Eijck (2015))
Overview of the design research project: after a starting-up phase three successive design experiments were conducted. (Taken from Beer, Gravemeijer, & Eijck (2015))

Before detailing the organization and content of this dissertation, an overview is given in terms of the four partial projects which together comprise the design research project (see Figure above): after the starting-up phase, three subsequent design experiments followed. Given the iterative nature of design research, the experiences gained and results found in earlier design experiments feed into the next. As result of this iterative and cumulative character, the findings of the starting-up phase and the first two design experiments are reflected and apparent in design experiment 3. In this thesis most attention is therefore paid to that last design experiment (Chapter 3) and the substantive relationships between that design experiment and the three partial projects that preceded it (Chapters 4 and 5). In addition, when construing the proposed LIT in Chapter 5 the data of the earlier design experiments are included in order to create a broader empirical basis. To offer a sound background for those data an overview will be given of all four partial projects in terms of their research context, data collection, and procedure of analyzing that collected data.

Data collected during the four partial projects of the design research project.
1) when the author acted as a teacher in the one-on-one teaching experiments and during design experiment 2, no observations were made.
2) only the teacher’s computer session was captured during the lessons.

Besides the data collected during the lessons and the one-on-one teaching experiments, the meetings with the teacher in design experiments 1 and 3 to prepare and evaluate the lessons were captured on audio or video as well. This data has not been used in this dissertation, however.
video/audio transcript computer session student products observations
Starting up
one-on-one teaching experiments × × × × 1
Design experiment 1
lesson 1 × × × ×
lesson 2 × × × ×
lesson 3 × × × ×
lesson 4 × × × × ×
one-on-one teaching experiments × × × × 1
lesson 5 × × × ×
Design experiment 2 (at university)
lesson 1 × × × × 1
lesson 2 × × × × 1
lesson 3 × × × × 1
Design experiment 3 (2 classrooms)
lesson 1 × × ×2 × ×
lesson 2 × × ×2 × ×
lesson 3 × × ×2 × ×
lesson 4 × × ×2 × ×
evaluation by students ×

Because the starting-up phase is discussed in detail in Chapter 2, this section focuses on the three design experiments. Each design experiment started by designing an instructional sequence based on the ideas put forward in the LIT. Subsequently, that instructional sequence was tried in a teaching experiment while a multitude of data was collected. Before elaborating on the procedure of analysis used during the design experiments, the research context of all four partial projects is sketched using Table above detailing the background of the participants and Table below detailing the origin and type of the data collected.

classroom with teacher students gifted students grade mixed 5th/6th grade mixed 4th/5th/6th grade boys girls
starting up 9 × 4 5
design experiment 1 × 25 21 × 15 10
design experiment 2 4 × 2 2
design experiment 3, classroom C1 × 24 24 × 17 7
design experiment 3, classroom C2 × 24 24 × 18 6
total 86 69 56 30

Research context and data collection

Starting up

While starting up the design research project, it became apparent that a literature review did not offer enough support to formulate an initial LIT. In addition, 8 one-on-one teaching experiments (Steffe & Thompson, 2000) were performed with 93 5th grade students to explore their prior understanding of speed in situations with two co-varying quantities using computer simulations and Cartesian graphs. Spread over three separate days, these teaching experiments were performed during school hours at the students’ school; one of their teachers was present also and supported the researcher in having the students think aloud. This teacher discussed the experiments with the researcher before and after they took place.

These meetings with the teacher, as well as the experiments were captured on video. The video recordings of the teaching experiments were transcribed. During the experiments, the computer session was captured as well, from which the student products were collected.

Design experiment 1

What was learned during the starting up phase allowed for the formulation of the initial LIT, which, together with the experiences gained during the starting-up phase of the design research project formed the basis for the development of a five-lesson instructional sequence on instantaneous speed. This instructional sequence was tested in a gifted mixed 5th/6th grade classroom with 21 students and their experienced teacher. During the test, 4 students from the regular mixed 5th/6th grade classroom participated as well. These four students performed above average in mathematics and were regularly invited to participate in mathematical activities in the gifted classroom. There were 15 boys and 10 girls.

The five lessons in the instructional sequence were each spaced by four through seven days. Not all students took part in all five lessons due to illness, standard test taking, and other activities. Over all, however, there were always more than 15 students in class during the lessons. Two days before the last lesson, three one-on-one teaching experiments were performed with three pairs of students: one pair from the regular 5th/6th grade and two pairs from the gifted program. The intent was to have a preparatory and evaluational meeting with the teacher before and after each lesson. Unfortunately, there were no such meetings before the 2nd and 5th lesson, nor a separate meeting to evaluate the 4th. Nevertheless, each lesson was discussed briefly during and at the end of each lesson.

The teacher did have many years of experience teaching and ever since the introduction of the gifted program three years prior, she had been teaching the gifted 5th/6th grade. In the gifted program, students spend less time on the core curriculum, making time for other subjects and more challenging projects. The gifted program was set up to pay attention to the special needs of gifted children. In the program, students work on their social-emotional development, learn meta-cognitive skills, and improve their overall happiness about going to school. Nevertheless, the class behaved like any other class.

A multitude of data was collected during the teaching experiments (see Table above). All lessons and meetings with the teacher were captured on video, of which the class discussions were transcribed. During the lessons the students worked on tasks via web-based interactive worksheets. Their answers were gathered, tabulated, and used to enrich the transcripts. During the lessons, the researcher present made observations. At the end of the 5th lesson, the students were given an open ended evaluation worksheet, which was filled in when the researcher was not present. The one-on-one teaching experiments were also captured on video, the computer sessions recorded, and the students’ work on paper collected.

Design experiment 2

After reflecting on the results of design experiment 1, it became clear that the LIT had to be revised significantly. To try out these new ideas, a three-lesson instructional sequence was developed and tried in a small-scale teaching experiment. Four above average performing 5th grade students (2 boys and 2 girls) were invited to visit the university and participate in the teaching experiment. The three lessons were split up in two sessions, one before and one after lunch. The author acted as teacher.

During the teaching experiment, there were two kind of activities: the students worked in pairs on an assignment or the teacher led a discussion with all 4 students. These discussions as well as the students’ work in pairs was recorded on video, all of which was transcribed later. The computer sessions of both pairs and the teacher were captured as well. Students’ products on paper worksheets were collected. Following all 4 students closely during the teaching experiment allowed for developing a deeper understanding of the LIT in terms of students’ reasoning. In this sense, design experiment 2 acted as a try-out for the classroom teaching experiments in design experiment 3.

Design experiment 3

In design experiment 3 a four-lesson instructional sequence was developed and tried at a school for gifted children. Twice a week, selected students would come from all over the municipality to attend the program for half a day during normal school hours. The gifted program focused on students’ creativity and social-emotional development whilst offering an intellectual challenging environment with topics in the area of language and culture.

Two classes of 24 students from grades 4-6 participated in the teaching experiments. Classroom C1 (17 boys and 7 girls) participated on four Friday afternoons and classroom C2 (18 boys and 6 girls) participated on four Monday afternoons. Although both classrooms were taught by the same pair of teachers, only one teacher participated in our study. He was a novice teacher with less than three years of experience who also worked as a project manager and teacher of “media literacy” at the teacher training institute of a nearby college. He confessed to an affinity for science and technology.

Before the first lesson, the researcher discussed the whole instructional sequence with the teacher. Before and after each lesson, the teacher and researcher discussed the lesson and prepared for the next. These sessions were recorded. While the lessons were recorded on video, the researcher made observations and captured the teacher’s computer session. Each lesson the students’ products were collected as well.

Procedure of analysis

The procedure of analysis used during each of the three design experiments was a two-step retrospective analysis modeled after Glaser and Strauss’s (1967) comparative method. In particular, the elaboration of Cobb & Whitenack (1996) on this method was used. After formulating conjectures about What happened? during the teaching experiments and testing these conjectures against the data collected, a second round of analysis was carried out by formulating conjectures about Why did it happen?. These conjectures were also tested against the data collection.

Although all data sources contributed to develop a better understanding of students’ learning processes, the transcriptions of whole-class discussions and student products in particular formed the basis for the two rounds of formulating and testing conjectures during the retrospective analysis. The conjectures that validated the ideas put forth in the LIT in terms of what happened and why it did happen were carried over to the LIT of the next design experiment. The conjectures that were refuted were taken as indications of a mismatch between the researchers’ ideas about the students’ learning processes and their actual learning processes. In these cases, to improve the LIT, it was tried to generate new explanatory conjectures through a process of abductive reasoning (Chapter 3). These new conjectures were added to the LIT of the next design experiment. Thus refining the LIT.

Organization and content of this thesis

Again, it is emphasized that design research is a process of iterative refinement: each design experiment builds directly on the results found and experiences gained in previous design experiments and the starting-up phase. This thesis, however, is not set up as a monograph offering a continuous narrative detailing the design research project. Instead a format is chosen that consists of four separate studies that did emerge from the design research project. Each study addresses a different topic and has been developed into a self-contained article. These articles are included as such in Chapters 2–5 in this thesis. As a result, however, some repetitions in these chapters are unavoidable.

This dissertation brings together the following four studies that emerged from the design research project:

  1. The first study (Chapter 2) deals with finding starting points for the initial LIT. It starts with a review of the literature on primary school students’ conceptions of speed. However, instantaneous characteristics of speed are not part of the primary school curriculum and, therefore, are not covered in this literature. Because instantaneous speed is conventionally treated first in a calculus course, the literature review is extended towards the literature on teaching calculus-like topics early on in the mathematics curriculum. Although this review did not allow for an initial LIT on instantaneous speed to be formulated, it offered enough pointers to perform a small-scale study to explore 5th graders’ conceptions of speed. After presenting the design, conduction, and analysis of 8 one-on-one teaching experiments about speed in the context of filling glassware, the results lead to the formulation of starting points of the initial LIT for teaching instantaneous speed in grade five.

  2. The second study (Chapter 3) focuses on the generation of new theoretical claims in design research by means of abductive reasoning. This process is showcased by describing the introduction of a new explanatory conjecture during the retrospective analysis of design experiment 3. To that end, the LIT and the instructional sequence instantiated by it are described, as are the teaching experiments leading up to the surprising event that triggered the abductive process: in one of the two participating classrooms wherein the instructional sequence was tried, the students themselves invented the curve as a Cartesian graph while in the other classroom the teacher had to introduce the curve. Then, by formulating conjectures about what happened during the teaching experiments and grounding these in the data collected, a second round of analysis is performed to formulate and ground a conjecture about why the students in one classroom were able to invent the curve while the others were not. This study ends with a discussion about generating new theory in design research and the generalizability of such theory.

  3. The third study (Chapter 4) is an effort to contribute to the codification of educational design practices. Indeed, design research builds on educational design. In the followed design research approach of Gravemeijer & Cobb (2013), the research part heavily leans on the virtual replicability of design experiments. This asks for a detailed account of the design research process and the researchers’ own learning process embedded in it. In this sense, design research may offer insight on how educational designers might document their practices and knowledge. In this study, the design research process is taken as an example to illuminate the researchers’ learning process by tracking in detail the development of the instructional sequence from the starting-up phase through the three subsequent design experiments. After summarizing the yield of the researchers’ own learning process in terms of a prototypical instructional sequence on instantaneous speed and the emerging LIT, the justification of the theoretical claims is discussed with respect to their origin and development. Furthermore, the utility of the LIT is discussed in light of the difference between the communities of design research and educational design.

  4. The fourth study (Chapter 5) is dedicated to the temporary endpoint and the cumulative yield of the iteration of design experiments, both in terms of insights in student thinking and in terms of a proposed LIT on teaching instantaneous speed in 5th grade. To formulate that proposed LIT the design and theoretical underpinnings of the local instruction theory are elaborated in terms of the theory of realistic mathematics education. From a methodological perspective the argumentative grammar suggested by Cobb, Jackson, & Munoz (in press) is followed. According to Cobb et al. (in press), the justification of the theoretical findings of a design experiment follows from

    1. showing that the students’ learning process is due to their participation in the design experiment,

    2. documenting that learning process, and

    3. enumerating the necessary means of support for that learning process to occur.

    Instead of documenting the learning process of the students in one go, however, first the key learning moments are described that came to the fore as patterns in the data of all teaching experiments. Then, the proposed LIT that can be construed on the basis of those learning moments is formulated by describing the students’ learning processes and the necessary means of support to bring these learning processes to fruit. Finally, the proposed LIT is placed in the literature and discussed regarding further avenues of research.

Finally, in Chapter 6, a summary of this thesis is given, followed by discussion of the proposed LIT. After evaluating the proposed LIT in light of the characteristics for new STEM education put forth in this Introduction, possible avenues for adaptation and further research on teaching instantaneous speed in 5th grade are explored. In particular, the design research methodology delineated in this thesis is discussed before envisioning a potential design research project that would disseminate the proposed LIT to a larger audience, including real-world practice.

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  1. Examenmonitor VO 2015: In the Netherlands, calculus topics are part of “Wiskunde A” and “Wiskunde B” at the havo and vwo levels. In 2015, 73% of havo students and 49.4% of vwo students selected “Wiskunde A” while 27% of havo students and 46.6% of vwo students selected “Wiskunde B”. Of all students finishing high school in 2015, 45.9% attended havo or vwo, the larger amount attending havo. The treatment of calculus topics is more complete in “Wiskunde B” than “Wiskunde A” at both levels and it is more thorough in “Wiskunde A” at the vwo level than at the havo level. The treatment of calculus in “Wiskunde B” matches that of a typical Calculus course in the USA best. This suggests that about 30% of all students in the Netherlands are taught calculus compared to the 10% in the USA as indicated by Kaput (Kaput & Roschelle, 1998).

  2. Gravemeijer & Cobb (2013) is an updated version of Gravemeijer & Cobb (2006).

  3. In the third one-on-one teaching experiment, a pair of students participated.