Teaching and learning cycles in a constructivist approach to instruction

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Teaching and Teacher Education 24 (2008) 1613–1634 www.elsevier.com/locate/tate

Teaching and learning cycles in a constructivist approach to instruction Florence Mihaela Singera,, Hedy Moscovicib a

Institute for Educational Sciences, Bucharest, Romania College of Education, California State University, Dominguez Hills, USA

b

Received 14 January 2007; received in revised form 4 December 2007; accepted 11 December 2007

Abstract This study attempts to analyze and synthesize the knowledge collected in the area of conceptual models used in teaching and learning during inquiry-based projects, and to propose a new frame for organizing the classroom interactions within a constructivist approach. The IMSTRA model consists in three general phases: Immersion, Structuring, Applying, each with two sub-phases that highlight specific roles for the teacher and the students. Two case studies, one for mathematics in grade 9 and another for science in grade 3, show how the model can be implemented in school, making inquiry realistic in regular classes. Beyond its initial purpose, the IMSTRA model proved to be a powerful tool in curriculum development, being used in producing mathematics textbooks, as well as in developing teaching courses for a long-distance teachertraining program. r 2007 Elsevier Ltd. All rights reserved. Keywords: Inquiry; Learning activities; Teaching and learning cycle; Teaching models; Teacher’s role

1. Setting the problem A large body of literature, such as Cobb and Bauersfeld (1995), Resnick and Klopfer (1989), Sierpinska (1998), Singer (1999), and, probably most significant, the new curriculum guidelines in North American and European countries promote new missions for the teacher and the learner. These bring new roles into schools, which focus on:



the learner as an autonomous thinker and explorer who expresses his/her own point of view, asks





Corresponding author. Tel.: +40 723 542 900; fax: +40 213 139 642. E-mail address: [email protected] (F.M. Singer).

0742-051X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tate.2007.12.002

questions for understanding, builds arguments, exchanges ideas and cooperates with others in problem solving—rather than a passive recipient of information that reproduces listened/written ideas and works in isolation; the teacher as a facilitator of learning, a coach as well as a partner who helps the student to understand and explain—rather than a ‘knowledgeable authority’ who gives lectures and imposes standard points of view; classroom learning that aims at developing competences and is based on collaboration— instead of developing factual knowledge focused on only validated examples and based on competition in order to establish hierarchies among students.

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However, compared with the effective results, all these new roles seem to pertain to a new ideology (e.g., Sierpinska, 1998). They require different processes in order to transform the new aims from ideal targets into outcomes of current teaching– learning practice. Another challenge arises from these new processes: the need to make the classroom interactions generate new mental frames for the ones involved, both the teacher and the student. The student’s frame might be presumed as follows: ‘‘I am the student and I have to answer questions; but the teacher already knows the answers, so these are not real questions. Therefore, it is about my playing a role. And this role should not be taken very seriously; otherwise I risk being ridiculous.’’ (Goffman, 1974). The teacher, on her part, too often sees the ‘didactical contract’ (Brousseau, 1980) in a limited formal way. The hidden understanding of the school as a stage, and of learning and teaching as formal role playing affects in-depth learning and the collaborative climate in the classroom. The questions are: How could these roles be made more realistic and exciting? How could teacher and students become partners in knowledge construction? The model for the teaching and learning cycle we present in this article was developed as a follow-up of teacher-training sessions and revised as it was implemented at a school site. This paper presents an analytic description of the model. We start by providing an overview of the literature on learning cycles in various knowledge areas, with examples for sciences, mathematics, and interdisciplinary curricula. To make the basis for the construction more explicit, a teaching–learning experience in mathematics in 9th grade introduces the model and allows ‘bootstrapping’ into its description. The table-based presentation that follows can be used as a functional tool for teaching. A case study involving a set of science lessons in 3rd grade shows how the model can be applied, and is an example of the model’s flexibility. Some possibilities for extending the model implementation are discussed in the end of the article. 2. Learning cycles in various areas For a long time researchers tried to understand the different steps that one takes when solving a problem in an attempt to comprehend how the mind works and how to best educate the next generation. Are there set steps, or is it a conceptual template

that allows for individual modifications according to the question under scrutiny, the researcher’s background and the available resources? Various answers have been proposed to this question. We list some of them below. 2.1. The sciences In the area of the sciences, there is an abundance of learning cycle models perhaps because science is perceived by the general public as the only bias-free and objective way of knowing. Generalizability and reliability studies suggest that results from one study can be easily duplicated if one follows procedures and uses the same materials as in the original study. These statistical entities also suggest that differences between the experimental group and control(s) can be correlated to the intervention rather than to random coincidence/mishap. Looking at the role of the student in a problemsolving situation, Lawson, Abraham, and Renner (1989) reviewed several years of research on student reasoning and concluded that appropriate teaching can lead to generalizable and significant improvements. They identified three required stages meant to improve students’ reasoning skills: exploration, term/concept introduction, and concept application. During exploration, students are encouraged to explore a phenomenon/phenomena and identify a pattern. The initial pattern gets reinforced, modified, or changed by using appropriate terminology and by exploring concepts during the second stage, which is term/concept introduction. The third stage, concept application, ensures that students are able to translate concept(s) learned and use them in new situations. Later, Lawson (2002) identified an increased difficulty in the problem-solving process that amplified the potential for reaching faulty conclusions when students explore science-related phenomena that involve unobservable entities. Lawson’s conclusion is more disturbing as the students in his study are preservice biology teachers with at least a bachelor degree in the sciences. We noticed a similarity between the model proposed by Lawson et al. (1989), and the three levels of learning proposed by Wolfe (2001): concrete experience, representational or symbolic learning, and abstract learning. In an attempt to enhance and clarify the engagement stage and integrate an evaluative dimension to the inquiry process, the Biological Science Curriculum Study (BSCS) developed a

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5-step model, also known as the 5 Es: Engage, Explore, Explain, Elaborate, and Evaluate (Layman, Ochoa, & Heikkinen, 1996). This model defines teacher’s role during the five steps of inquiry. The model has similarities with the Lawson et al. (1989) model. The first two BSCS steps, Engage and Explore, correlate to the first stage in the model of Lawson et al. (1989), the third BSCS step, Explain, parallels the second stage, and the fourth BSCS step is similar to the third stage. BSCS introduces a new dimension to the inquiry process: ‘Evaluation’ of what the students discovered during the process and what they have yet to figure out. It is important to emphasize the fact that this evaluation does not imply finality, just the opposite. It encourages the learner to think forward to other linked avenues for research. The 5 Es model resembles the model proposed by the National Research Council (2000) in which five features of inquiry are defined from the learner’s perspective as: (1) Engagement in scientifically oriented questions; (2) Giving priority to evidence in responding to questions; (3) Formulating explanations from evidence; (4) Connecting explanations to scientific knowledge; and (5) Communicating and justifying explanations (p. 29). Even more recently, the Museum of Science in Miami, Florida, has developed a learning cycle that builds upon the BSCS’ 5Es model. The first stage changes from Engage to Excite, Elaborate is divided into Expand—applying the pattern/concept in a similar situation—and Extend—applying pattern/ concept in a different subject area. The Exchange stage is introduced in order to encourage students to take advantage of Internet and share their information with individuals around the world, and Evaluate is changed into Examine, in order to diminish the tendency to see evaluation as a final process, and to encourage a variety of evaluative processes to be considered as evidence for students’ learning (alternative assessment strategies). In an attempt to synthesize the inquiry process as experienced by research scientists in the science laboratory, Reiff, Harwood, and Phillipson (2002) proposed the ‘inquiry wheel.’ From the practitioners’ point of view, the research questions lie in concentric circles connected with two-way arrows to the following elements/processes of inquiry: observing, defining the problem, forming the question, investigating the known, articulating the expectation, carrying out the study, interpreting the results, reflecting on the findings, and communicating the results to the scientific community and to society.

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The immediate connection among the elements through questions allows for the necessary fluidity of research. However, the ‘inquiry wheel’ model is quite rigid as it does not emulate research science where results are interpreted based on the elements of the specific study, the questions under scrutiny, and the studies addressed during the ‘investigating the known’ stage. Should the wheel be more like a web? And how does this model translate into the K-12 or K-16 classroom? We will discuss this further, when talking about the proposed model. 2.2. Mathematics Mathematicians have been constantly preoccupied with conceptualizing a model for mathematical problem solving. Thus, Polya (1957) focused on the following stages of a solving cycle: (1) understanding; (2) devising a plan; (3) carrying out the plan; (4) looking back. Schoenfeld (1985) set the following basic steps: (1) analysis; (2) exploration; (3) verification. In a more cognitive-oriented description, Verschaffel et al. (1999) highlighted the following inventory: (1) build a mental representation of the problem; (2) decide how to solve the problem; (3) execute the necessary calculations; (4) interpret the outcome and formulate an answer; (5) evaluate the solution. The process through which mathematics concepts are internalized supposes a progressive formalization, described by Dienes (1963) as the psychodynamic process of artificial genesis of mathematical concepts. This model of progressive formalization identifies six stages: structured play or game, isomorphic games, abstraction, schematization, formulation and symbolization, axiomatization, generalization. Studying how primary school students approach problem solving, Marcou and Lerman (2006) propose a model for self-regulated mathematical problem solving that consists in the following phases: (1) forethought phase: analyzing the text; (2) performance phase: carrying out the plans; and (3) self-reflection phase: looking back. These taxonomies mainly refer to the process carried out by individuals when solving problems. However, the variety of implicit and explicit interactions that take place in a classroom context should be considered when speaking about school learning. The ‘60 s New Math program (Brailly, 1968–9) promoted the ideal of the learner as a researcher in mathematics and the whole class as a research team. More recently, an interactionist

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perspective proposed viewing the classroom as a micro-culture in which meanings emerge through shared activities. The interpretation of these activities is negotiated and eventually standardized during the interactions between the teacher and the students. This view is inspired by Bruner’s theory of interaction formats in language acquisition (1985). The interactionist research methodology draws upon the qualitative, naturalistic approaches that are based on exploration and inspection of phenomena, where a theoretical model is constructed and continually reconstructed during the study. The interactionist research programs lead not to recommendations for action but to descriptions and discussions of various possibilities. The micro-sociological approach developed through interactionism had a strong influence in France in building the concept of the didactical contract (‘contrat didactique’), developed by Brousseau (1980). In Germany and Great Britain, its impact led to the research program started by Bauersfeld, focusing on micro-ethnographical studies of teacher–student interactions (see Cobb & Bauersfeld, 1995). The constructivist program overshadowed this trend for a while, but recently, its scope has broadened to encompass the study of classroom culture and the mechanisms of their emergence and stability. As Sierpinska (1998) pointed out, interactionist ideas lead to a much more down-to-earth view of the teaching–learning process. In Dienes’ vision, to move from a teacher-driven thinking frame to a more research-based one, games can be the starting point. Another starting point might be reality. The daily life problems are increasingly emphasized in recent mathematics curricula in various countries. For example, the algebra strand using Mathematics in context in the Netherlands is a good illustration of the idea of progressive formalization for middle school students (National Centre for Research in Mathematical Sciences Education and Freudenthal Institute, 1997). It begins by having students use their own words, pictures, or diagrams to describe mathematical situations, to organize their own knowledge and work, and to explain their strategies. In later units, students gradually begin to use symbols to describe situations, to organize their mathematical work, or express their strategies. At this level, students devise their own symbols or learn some conventional notations. Their representations of problem situations and the explanations of their work are a

mixture of words and symbols. Later, students learn and use standard conventional algebraic notation. In a context of freedom, students might move back and forth among levels of formality depending on the problem, situation, and the mathematics involved. 2.3. Interdisciplinary curricula Whitehead (1929) describes the inquiry process as a course of action revolving around freedom and discipline. Learners begin their inquiry in a stage of freedom or romance, when they are getting immersed in their question and search for possible solutions and patterns. This stage, (paralleling Lawson et al. first part of the Engagement stage) is perceived as unsystematic to the outside observer. With the tentative development of the pattern, the learning process enters into a disciplined or precision stage. During this stage, patterns are developed, verified, and modified constantly. The third stage of Whitehead’s model brings the learner to a different kind of freedom or generalization that encourages the learner to explore other situations using what was learned. This third stage in Whitehead’s model parallels Lawson et al.’s ‘Concept Application’ stage, and fourth stage in the BSCS curriculum—‘Elaborate on the Concept.’ Similar to Whitehead, Tchudi and Lafer (1996) concentrate on structure (parallel to discipline in Whitehead’s model) and spontaneity (parallel to freedom in Whitehead’s model) and develop an interdisciplinary model called S2 (Structure  Spontaneity ¼ S2). Central concepts for the interdisciplinary units are the results of the intersection among mandated standards, teacher’s interest, and students’ interests. In a rich-text and rich-resource environment, participants get involved in research about their chosen theme/issue/problem, research that results in knowledge production. Various presentation strategies as well as group formations mediate learning. In contrast to Whitehead’s model, the S2 shows a successful conversation between freedom and discipline all along the inquiry process. The Australian Planning Model proposed by Peter Forrestal (1986) has five stages for the development of the interdisciplinary inquiry after the establishment of the ‘cue’ event: input (teacher locates core information), exploration (students use the ‘input’ materials and make sense of them in terms of the problem), reshaping (students are engaged in the synthesis of the information found),

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presentation (students present new knowledge to the audience), and reflection (teacher and students evaluate the work that has been done in the light of the findings). In an attempt to build a learning cycle that uses collected information and results in improving students’ achievement, Love (2003) discusses ways in which data collected at the district or site level can be used to plan and promote student learning. In the same issue, Mayer (2003) and Greco (2003) report on the importance of teacher’s involvement (as a ‘stakeholder’) in the Learning Cycle leading to educational practices change and enhancing students’ achievement in their own classrooms. To summarize, in an attempt to establish stages/ elements of educational research, questions of context are raised and challenged. Actions and events need to be analyzed using the participants’ explanations and ‘thick descriptions’ (Geertz, 1973). In many ways educational research reminds us of the model of Whitehead (1929) for interdisciplinary inquiries. Researchers and participants enter the situation trying to understand what is going on (freedom), they develop and refine patterns that illustrate their interests and learning (disciplined or structured stage), and then, by applying what was learned in a new situation, they get into the third stage of relative freedom. 3. Why a new model? It is necessary to point out that learning cycles models are very dynamic entities and they tend to evolve and build on previous work as well as on the change in the context. Students, teachers, communities, as well as materials and equipment available all changed over time. Learning cycles got adjusted to fit the new learning environments. Therefore, there are elements that successfully resisted any change over time (e.g., the need to engage/excite the learner), while others underwent dramatic changes (e.g., the use of resources, the availability of Internet to collect information and communicate results). On the other hand, in a classroom context, there is an inevitable interaction between the teacher and the student’s actions, as well as a fluid relationship between the various stages of inquiry. Both bring imperceptible changes to the learning process. We have focused our attention on four main problems of the teaching/learning models/cycles previously described, which we further address within the new proposed model.

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First, most models presented above shift from the role of the teacher to that of the students when talking about different stages. For example, in the model of Lawson et al. (1989), the students explore, but the teacher is the one in charge with the second stage—term introduction and pattern reinforcement; in Forrestal’s model (1986), the teacher is in charge during the input stage—the stage of collecting pertinent resources to support students’ inquiry, while the students’ activities are less emphasized. For most of the models, it is unclear how the teacher and the student(s) coexist during these learning cycles and how the shift in responsibility affects the outcome of the inquiry process. Second, there is a tendency to introduce (maybe unintentionally) a certain level of rigidity through the stages or steps that lead to the right answer. The immediate consequence of this linearity is the ‘scientific method’ or the prescribed set of linear steps that supposedly lead to scientific discoveries in the world and to student’s discoveries in the day-byday classroom. Reiff (2003) summarized research on 40 science textbooks, 35 at the college level, that showed that the ‘scientific method’ is still described as a set of linear, predetermined steps leading to a theory/discovery. The subject of the scientific method is restricted to the first two chapters of the science textbook, it usually lacks a figure, and when a figure is provided it is still linear with few, or, in most cases, no feedback loops (allowing/encouraging jumps—meaning going back to the previous stage or jumping to another stage according to the researcher’s need). Even if the authors mention that researchers employ a variety of research methods, the linear step-by-step description or figure contradicts the text. From the science text to the science classroom, teachers perpetuate the linearity of ‘the’ scientific method, sometimes even checking the sequence of the steps, their correct spelling, and their definitions rather than their expression through students’ explorations. Needless to say that this linear model differs tremendously from the model described by practicing researchers; the practitioner researchers’ model shows the question under scrutiny in the middle of an inquiry wheel that encourages flexibility and loop jumping possibilities according to questions asked, information needed, and background of the researcher (Reiff et al., 2002). The third issue involves the teaching–learning practice and is derived directly from the second. New approaches to curriculum development support

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learning with understanding and encourage sense making; despite this generous idea, a closer look at the current practices shows that the student is not so much discovering; she is more or less guessing the answers targeted by the teacher. The students’ problem becomes, to a large extent, finding the hidden rules of the game that takes place in the classroom, unraveling the expectations of the teacher, the constraints in which the teacher acts, and adapting to these rules, expectations and constraints (Moscovici, 2001, 2003). Being aware of all the subtle patterns that govern the life of a classroom is indispensable in planning and designing teaching–learning activities. In this respect, the model needs to be flexible enough to assure adaptation to very diverse social contexts and interactions, without letting the context dominate students’ construction of knowledge. The fourth considered aspect is that an effective model should prevent at least some of the risks of usual behaviors. Thus, the reality of the classroom may be different for the teacher and for the researcher in education. For example, a cognitively oriented researcher will see the classroom as a collection of individual minds. For the common teacher, the classroom is mostly seen as a unit that behaves in a specific way, beyond the individuals; the result of this vision is that the teacher practically addresses her message to only a few students and their reactions are extrapolated to the whole class (e.g., Bruner, 1985; Sierpinska, 1998). In this approach, the teacher is an external body who does not really interact with the classroom community. We believe that the classroom activities should become learning experiences for both sides: teacher and students; keeping in mind her educational aims, the teacher has to get off the knowledge barricade and construct meaning together with students. The teacher’s role in genuine inquiry projects is even more complex because it is about shortening the cultural history of human behavior into authentic inquiry pieces. 4. The proposed model A constructivist multidimensional approach starts with the informal ideas that students bring to school and develops ways to help them see how these ideas can be transformed and formalized. Movements along this continuum are not necessarily smooth, or all in one direction. Patterned techniques are needed to encourage students to build on their informal ideas in a gradual but

structured manner so that they re-build domainspecific concepts and procedures (e.g., Bailin, Case, Coombs, & Daniels, 1999; Shulman & Sherin, 2004; Singer, 2003). Students need to be challenged to make sense of what they are doing. However, because of time constraints, the approach cannot provide absolute freedom to search for meaning and solutions. The teacher is in the position to build on students’ understandings and carefully orchestrate the students’ contributions to knowledge construction (Singer, 2001). To accomplish such result, the teacher should know what is significant from the domain-specific perspective in terms of concepts, procedures, and historical evolution; and, simultaneously, what methods are most efficient for the learner’s progress (see concept of pedagogical content knowledge in Gess-Newsome & Lederman, 1999; Shulman, 1986; Singer, 2002, 2007). 4.1. Starting from an example Inevitably, we need to describe the continuum of classroom learning through discrete stages in order to ensure its replication in various contexts. What phases could be more productive and effective? The answer is situated at the interface of micro-developmental analyses, cognitive research, and curriculum construction. To illustrate our approach, we have chosen as the starting point, from our empirical database, a lesson in mathematics with students in grade 9 (15–16 years old). Because our focus is to synthesize at a micro-level the main phases of the model, we detail only the aspects of the lesson that focus on the model construction. The lesson started with the problem presented below. Two taxi companies have the following offers: QUICK TAXI: Start: 6 h +1.3 h per 1 km

SPEED TAXI: Start: 3 h +1.9 h per 1 km

Which is the best choice for a trip of: (a) 3 km? (b) 10 km? We present below the main classroom activities. The teacher invited students to work in pairs/ groups for the purpose of: (1) discussing solving the problem (brainstorming, without any suggestions from the teacher), (2) identifying the mathematical objects involved in the task,

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(3) exploring ways to solve the problem by experiencing particular cases and analogous situations. Three types of students’ approaches have been identified. One group of students gave values on a table, noticing this way that for 3 km, SPEED TAXI is cheaper, while for 10 km, QUICK TAXI is a better choice. Another group that gave values on a table first anticipated this variation and then discussed it, based on the values. A third group started by discussing how to express the travel costs as functions that vary with the distance. What have students actually done during this phase in terms of generalizable experience? They have:

 

All these activities can be concentrated around the keyword Evoking. They also have:

      



checked the results obtained in the previous phases, analyzed and described the behaviors of the two functions, y

identified a specific difficulty and have decided to solve it by performing a specific task, searched for means/methods, gathered information from the group members, anticipated targets (answers to questions) through analogy with known situations, communicated pertinent experiences to student peers, recorded, compared, classified data using modalities adequate to the targets, processed partial results, trying to build own understanding of the concept, begun to correlate between intervention and results in terms of variables.

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6 4 3

1.

All these activities can be concentrated around the keyword Exploring. Evoking and Exploring are meant to immerse students in the problem-solving experience. Immersion—with its two components, Evoking and Exploring—is the state that the teacher might construct

3x

+

6





3



x+



identified terms, relationships, methods already known that might be connected to the topic, and shared these with others, selected personal experiences that might prove pertinent to the situation at hand, planned and performed a first trial to solve the problem, checked the trial, patterned experiences and defined variables.

in order to get the students involved in creative activities. Further, a group considered x the distance, and defined the trip costs as functions of x; they found: q(x) ¼ 1.3x+6 and s(x) ¼ 1.9x+3. Thus, the students transposed the given practical problem into an algebraic task—i.e., solving a linear inequation: q(x)ps(x). The teacher noticed that this group has proposed a mathematical model for the problem, and decided to share this with the whole class. Through discussions, the model discovered by one of the groups in the classroom was adopted by some other groups and rejected by others, without teacher’s direct intervention. Some groups started to solve the inequation. In the meantime, a group preferred to continue to guess and check the solutions. Another group had the idea that, instead of solving the inequation, it might be better to represent the graphs of the two functions (see Fig. 1). Using students’ observations, the teacher then initiated a discussion about the degree of generalizability of each solution. Students reviewed their results and adjusted their solutions. Then, using the graph from Fig. 1, some students:

1.9



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O

1

5

x

Fig. 1. A graphic representation for the two functions involved in the problem.

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expressed the connections between the two functions in mathematical terms, explained why the answers for the two questions were different.

What have students actually done during this phase in terms of generalizable experience? They have:

    

remarked invariants, identified constants and patterns, elaborated on a first claim concerning the possible solution, elaborated on rules that can express some general inferences, synthesized the statement obtained.

All these activities can be concentrated around the keyword Synthesizing. They also have:

     

reflected upon some examples, while analyzing the results of their explorations, proposed examples and counter-examples of their claims, adjusted the version of their claims in correlation with new examples and counter-examples, discussed the underlying causes and effects, accepted limitations of personal and peer knowledge and searched for other information, described/explained the observed patterns.

All these activities can be concentrated around the keyword Explaining. For the simplicity of the model, Synthesizing and Explaining can be seen as two facets of Structuring. The inquiry process continued even after getting answers to the problem. The teacher used the initial problem to devise other tasks in order to analyze the variation of a linear function, as follows: What would happen if the two companies:

 

were doubling the starting costs? were increasing with 0.5 h (or 1 h) the costs per kilometer?

A more complex transfer task invited the students to use the pattern found in the previous phases to solve the following problem: The ‘‘Micron’’ store sells CD video games recorded in its own studio. Each CD costs 6 h

to produce and it is sold for 10 h. The initial investment is of 5000 h. How many CDs must be sold to have this business profitable? The students were then asked:

 

to vary some data and to analyze the result, to devise their own new problems that can be solved using similar patterns.

What have the students actually done during this phase in terms of generalizable experience? They have:

  

imposed the already known pattern to a similar situation, transferred through analogy the model they developed during previous activities, proposed particular cases and checked the transfer.

All these activities can be concentrated around the keyword Practicing. They also have:

    

explored applications of the solving schema in new situations, set criteria to assess the final solution, taken decisions concerning the learned strategies to solve this category of problems, extended the pattern found in the previous phases, found ways and means to shortcut further research procedures.

All these activities can be concentrated around the keyword Extending. Practicing and Extending bring the inquiry process into the Applying phase. This analysis led to three major phases of the teaching and learning cycle, with each phase divided into two sub-phases. The three phases will be called: Immersion, Structuring, and Applying, albeit these labels do not cover the complexity of actions to be taken under this framework. They are described below from a more general perspective, highlighting the main characteristics of the model. The model construction is derived from observed good practice, further elaborated from an expert perspective. Its working versions were then improved during a teacher-training program that involved 200 teachers of Mathematics and Sciences

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from primary to upper secondary. The program consisted in two sessions: a 4-day institute in October, and a 3-day follow-up, 6 months later. The format of the program was an interactive one, the teachers worked in groups, and the assessment was based on the projects they developed during the sessions. The timetable of the course alternated disciplinary work that grouped teachers of the same subject matter for a specific key stage (Mathematics, Physics, Chemistry, Biology; respectively, primary, lower secondary and upper secondary), and interdisciplinary work, where the trainees interacted with one another within the curricular area. The team of trainers (among them, one of the authors) prepared in detail each training session. Each class of teachers was taught by a team of at least two trainers, usually three. We used this training opportunity to test and refine the model. In the following sections, we provide a description of the model and an example of how this model was applied in a real class. Although the phases of the model are the same for the teacher and the student, the sub-phases are different, in order to capture the different goals, roles, and views. 4.2. Immersion phase During this phase, students get immersed into the problem—address and use previous knowledge, seek more information, plan and perform experiments, and, based on all these resources and processes, identify tentative pattern(s). The students also explore their own knowledge and anticipate the knowledge development through planning personal projects. With respect to teacher’s role during this phase, the model proposes questions that the teacher might ask or be asked, and actions that will lead to the expected action performed by the students. Thus, during the first sub-phase of the first phase (Anticipation), the teacher is preoccupied with developing students’ curiosity; helping them formulate learning targets, identifying and selecting the chunks of information and knowledge (including previous knowledge) that are pertinent to the specific problem. During the second sub-phase, Problem construction, the teacher scaffolds students’ research by providing necessary hints, encourages students’ explorations, and helps students to record data. As students acknowledge safety rules and remain focused on solving the problem, the teacher should abstain from leading them to her own solution(s).

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In terms of students’ learning activities, the Immersion phase can be divided into two interconnected sub-phases: Evoking, and Exploring. During the Evoking sub-phase, students bend their previous knowledge to the problem, discuss and challenge their ideas with peers. Some students might be looking for resources to document/enrich their knowledge, and that leads to exploring. During the Exploring sub-phase, students plan, perform, and analyze their investigations while always addressing the problem that they have to solve. During the Immersion phase, the students learn to select pertinent knowledge from what they know realizing that personal knowledge might prove insufficient and deciding to look for resources (including library and Internet), and to judge resources in terms of reliability of information. Students also learn to correlate between variables and experimental results (hypothetical-deductive/ hypothetical-predictive, ifythen type), understand limitation of experiments (e.g., number of specimens/groups, variability of specimens within the same group), and become familiar with the use of higher-order thinking skills such as synthesis, analysis, evaluation, and the creation and expression of a complex solution to a problem. Moreover, students constantly shift between concrete (e.g., personal experiences relevant to the problem under scrutiny) and abstract (e.g., understanding patterns underlying peer’s concrete examples or described in texts). Students learn to move from concrete (personal experiences) and semiconcrete (other students’ experiences) to abstract (patterning) and again to concrete (planning experimentation and make first trials to solve the problem). These constant shifts will help them during the patterning process when they need to begin explaining the found pattern and trying to generalize it, whilst detaching the abstract piece of knowledge from the concrete experimental/trial phases (Singer, 1995, 2004). 4.3. Structuring phase During this phase, students move to another level of understanding when they interpret their concrete experiential results from the Immersion phase and adjust the pattern. They explain the claim developed during the previous phase in terms of examples and counter-examples, and create new situations in order to challenge their own claim and to add to the generalizability of the knowledge they produced.

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In the role of facilitator, the teacher supports students by helping them synthesize observations, summarize findings, and explore inferences during the Systematization sub-phase. During Conceptualization sub-phase, the teacher helps students use the new terminology, generalize conclusions, and expand their findings beyond the specific problem that they researched, into related issues. Synthesizing sub-phase involves the students in the process of identifying and contrasting patterns, helping to extend their findings into more generalizable statements. Explaining sub-phase requires students to connect the concrete exploration to a more abstract model that describes the results of explorations and challenges findings through concrete and hypothetical examples and counterexamples. The students define the concepts through interpreting the results of their activity and reinforce them through connection with other activities. In terms of skills, during the Structuring phase, students learn how to differentiate between opinion and fact, about the limitations of experiments, and about the use of appropriate language when sharing findings. They also learn about validity of claims and the role of the constructive criticism provided by peers and teacher during the dynamic and fluid process of knowledge construction. Students move during this phase from the concrete aspects of experimentation into the complex and multifaceted conditions of real life problems. They also move from concrete into abstract during the generalization process, when they shift from a specific solution for a specific problem to finding solution(s) for classes of problems. 4.4. Applying phase During this phase, students learn to use the abstract pattern that they developed into related and unrelated situations, they modify/adjust their pattern to be more generalizable and applicable in a wider range of situations. They apply learned concepts and patterns to new situations by trying to solve existing problems, and by creating/describing new hypothetical or realistic situations that need solving. These processes lead to a more generalized pattern that identifies constraining elements. The teacher during the Applying phase is concerned with assessing students’ understanding of the concepts developed and with the process of inquiry and its limitations. Sometimes the teacher might

choose another concrete example in the same domain (for Reinforcement sub-phase) or in a related/unrelated domain (for Transfer sub-phase) to illustrate relationships, or to increase complexity (e.g., number of possible solutions). Teachers may also explicitly prompt students to think about aspects of their everyday life that are potentially relevant for further learning. 4.5. Sinthesizing the features of the model The model described above presents a teaching and learning cycle focused on IMmersion, STRucturing, Applying. It will be called on short IMSTRA. Tables 1–3 contain a functional presentation of the teacher’s role and students’ learning activities within the IMSTRA framework. The description facilitates the model’s translation into the teaching practice and encourages its adaptation to various topics. To have a clearer view of this construction, the schema in Fig. 2 gives a synthetic presentation of the IMSTRA cycle. This schema emphasizes the close relationship between the teacher’s targets and the students’ activities during the inquiries. We stress that the purpose of the whole cycle is to foster students’ learning. Within the IMSTRA framework, the students are involved in a multitude of inductive and deductive pathways that help them move with ease among concrete or semi-concrete experiences, and abstract patterns. Furthermore, during this process, students are forced to use higher-order thinking skills and metacognition. The schema in Fig. 2 emphasizes the qualities that make this model reliable and effective: generality, extensiveness, flexibility, functionality, and inquiryfocused dimensionality. Generality shows that this model can be used across a few disciplines, at least for mathematics and natural sciences. We hypothesize that its use can be extended to teaching and learning social sciences, but evidence for this claim is to be obtained through specific studies. Extensiveness consists in reflecting the parallel roles of the student and the teacher whilst they attempt to reach their goals. Although the activities are different, their goals are convergent: building knowledge in an interactive process. By making explicit the difference in roles and actions of the two actors during the didactical process, the model moves the approach from the ideology of the didactical research, where various models still rely

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Table 1 Key roles of the teacher and the students during the IMSTRA cycle: Immersion phase Phase

Teacher’s role Key questions

Immersion

Actualization– anticipation  How shall I develop students’ curiosity?  How shall I support student questions, and help them formulate learning targets?  How will the students analyze what they already know (their previous knowledge and experiences)?

Student’s learning activities Targets

 Offer problem-pretext (real-





Problem construction  What content should I present (as a teacher), and what content would be explored by the students?  What exploring activities will allow students to understand the concept?  What kind of observations should students perform?  What questions should I ask to encourage student exploration?

life problems) to create cognitive conflicts which motivate the students to engage in the task Create learning situations that generate the recall of the notions, operations, and behaviors necessary to understand the new concept (topic) Identify the students’ knowledge and understanding about the topic (through testing, interviews, or simply by questioning during the first trials/experiments)

 Expose students to a variety  



of resources connected to the topic Give hints and cues to keep the exploration going Avoid defining terms/ explaining evidence until the students have made enough trials to orient to the solution Facilitate the student’s searching, without shortening this search by teacher’s own adult projection

Evoking  Search for means to solve the problem situation (the cognitive conflict): identify notions, terms, relationships, phenomena, methods already known, connected to the topic; share opinions with others about these things  Select concrete experiences from memory, experiences that are pertinent to the situation at hand  Plan and perform a first trial to solve the problem, completing or adjusting the searching steps  Survey/do a practical verification of the trial, observe phenomena, gather data from various sources that help to think about the concept/topic  Use imagery and try to understand how other students’ experiences relate to the problem/situation at hand  Pattern experiences and define variables Exploring  Identify/challenge a specific difficulty and decide to solve it through performing a specific task  Search for means/methods, and eventually redo the experience/ experiment using other means/method, if the previous was not efficient  Gather and record own information  Redo, depending on the given/discovered criteria, the set of the initial searches/ means/methods  Through analogy with previous situations, anticipate targets (answers to questions) and search for means to reach them (construct investigations to test the hypotheses), and check them, hazardously  Share pertinent experiences with student peers and be aware of the limitations of one’s experiences  Process (record, compare, classify, represent) data using modalities adequate to the targets, compute partial results, trying to build his/her own understanding of the concept  Begin to correlate between intervention and results in terms of variables

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Table 2 Key roles of the teacher and the students during the IMSTRA cycle: Structuring phase Structuring Systematization  How will the students evaluate the explorations?  How will the students check their own concept’s understanding?

 Help students to express their observations/conclusions/inferences

 Help students to summarize their findings

 Synthesize the students’ observations concerning the new concept

Conceptualization  How will the students make use of concept understanding?  How will the students be challenged to search for supplementary information and to find answers to still existing questions?

 Introduce new terminology  Help the students idealize through



models the objects explored and generalize to other objects the conclusions emerged during explorations Help the students describe/define/ explain the new notions, concepts

on descriptive approaches, to the reality of the classroom’s interactions. Flexibility is manifested in two ways: by the fluid transition from one phase (sub-phase) to another or by the possibility of merging the sub-phases into a single phase. This is in close connection with the next characteristic, functionality. Thus, depending on the topic, and the students’ age and abilities, it is possible to consider three, four, five or six sub-phases. The cycle still remains consistent within this variability if connections among the phases are highlighted. The amplitude and depth of the learning cycle is the teacher’s decision. Many studies report on the students’ natural focus on exploring while learning in a natural context (e.g., Lawson, 2003). However, elements from the other

Synthesizing  Remark invariants, identify constants, patterns  Generalize the features, pattern concrete results of experiments  Elaborate a first claim concerning the concept/solution  Synthesize statements obtained, elaborate rules/definitions/laws that express the results/conclusions  Address other sources of information (textbooks, articles, internet, contact experts, etc.) Explaining  Reflect upon examples/cases, analyze the results of exploration  Describe systems, stages, etc. observed; describe and/or define new notions/concepts  Connect knowledge and understanding to express new ideas about the entity under study  Device examples and counterexamples for the statements  Argue/proof/demonstrate the general assumptions  Adjust the version of his/her claim (definition, rule, theorem, etc.) in correlation with the new examples and counterexamples  Understand and accept limitations of personal and peer knowledge and search for other information resources

phases: structuring and applying might be developed taking into account the constraints of the school system, otherwise the learning remains incomplete and the knowledge achieved by the student is very rarely transferable in a stable acquisition (e.g., Bransford, Brown, & Cocking, 2000). To put it briefly, the IMSTRA framework to put it requires that, in order to construct student’s knowledge, a learning cycle should touch each of the three main phases, even when there is a focus on only one of the phases. The teacher might assume this concentration in its premises and consequences. Inquiry-focused dimensionality simply means that, in each phase, student and teacher equally address questions to themselves and to others. By setting the

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Table 3 Key roles of the teacher and the students during the IMSTRA cycle: Applying phase Applying Reinforcement  How shall I assess the students’ understanding?  What aspects should be reinforced?  What limits/conditions could be stressed when the learned concept is applied?  How will the students use the new competencies?

Transfer  How might the students identify relationships between the studied concept and other concepts?  How will the students identify other contexts in which they can apply the techniques they have already learned?  How could the gained experience be used in other situations/domains?

 Propose and orient the activity in applicative tasks

 Propose supplementary reinforcement activities

 Offer opportunities to students to independently apply the learned concept/product/theorem, to independently develop their ideas in applications

 Systematize further connections: the

  

relationships of the studied concept with other concepts within the domain or with concepts from other domains Make connections to other disciplines Propose new contexts to apply the models Extend learning outside the classroom

reflective attitude in the core, the model centers on students’ learning with understanding. We should also stress that even the formulation of the basic research question could evolve within a constructivist learning environment—sometimes it becomes more specific, while other times it changes to match the researcher’s (student’s) new interest. In this frame, the teacher is orchestrating the inquiry process. That means that the teacher continuously assesses students’ involvement, improvises to let the inquiry process go on, and harmonizes students’ participation in knowledge building.

Practicing  Apply the already known pattern on concrete similar situations  Transfer through analogy the properties and the models developed during the project  Propose particular cases and explain (through reasoning) if they are or not satisfying the features of the new product/concept/theorem/theory  Discuss about the limits of applicability of the new concept/ theorem/theory  Relate various categories of representations of the product/ concept and assess these relationships Extending  Explore applications of the concept/ product in new situations  Make trials of the results, observe and analyze the results, set up new criteria to assess the product (the final solution)  Extend the pattern found in the previous phase to new situations  React to the context: integrate, optimize, negotiate contexts connected to the product, react to the possible problems/limits of the new knowledge  Systematize the process: establish to what extent answers to questions have been found  Take decisions concerning how to use the learned strategies to solve various types of problems  Anticipate ways and means that allow a shortcut to further research procedures

5. Model application in class In this section, we present some modalities necessary to implement the model. A first step consists in planning the distribution of educational objectives and contents in instructional units. The criteria to set up an instructional unit are:

 

coherence, referring to the objectives on which the instructional unit focuses, emphasis on the students’ competences developed during the instructional unit,

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framework. The topic was The Sun—source of periodical changes in the environment, focusing on the question: ‘‘Why are there days and nights on Earth?’’ The instructional unit covered eight lessons, organized around five phases of the IMSTRA cycle, which followed the stages of an investigation:

Fig. 2. The IMSTRA framework for the teaching and learning cycle.

   

topic relevance throughout the unit, continuity over a certain period of time, feasibility, referring to the teaching and learning cycle, minimal length that allows a significant summative assessment.

Some other criteria to be taken into account should be: the school mission concerning students’ achievements at graduation, the possibility of developing an integrated approach, the students’ level of understanding, the interests and abilities of students, the socio-cultural local context, and the existent or potential resources. Such instructional units might be assimilated with organizing learning through projects—the most frequent way of work if we examine life out of school, as Gardner (1999) noticed. Compared with the classical planning of lessons, the concept of the instructional unit has some advantages: it creates for students a coherent learning environment in which their expectations become clearer on medium and long term; it involves the teacher in anticipating the didactical meaning on medium and long term, allowing students’ different rhythms of learning; it offers perspectives for the lessons through a non-linear relationship among them, generated by their position in the teaching and learning cycle (Leahu, Singer, & Leahu, 2001). 5.1. Learning within the IMSTRA framework—an example We describe below an instructional unit taught in a 3rd grade class of Science, within the IMSTRA

(i) formulate the question and advance the hypotheses—evoke and anticipate, (ii) test the hypotheses—explore and experiment, (iii) find an explanation—synthesize and explain, (iv) test the explanation through including other cases; report the results—apply, (v) devise an impact of new knowledge in various domains—transfer. For reasons of space, we focus on the students’ activities, as they are the most relevant for the learning approach. The description is based on the observer’s notes (Sarivan & Leahu, 2005). 5.1.1. Immersion phase 5.1.1.1. Evoke/anticipate (1 h) ‘‘What I know or think about?’’ (1) The students work in pairs and suggest observations, experiences and personal events regarding natural phenomena (such as ‘the travel’ of the Sun in the sky, the motion of shadows, the succession day–night, thoughts about the influence of the Sun on the passage from light to darkness). The children discuss and make drawings. (2) Working in groups, students model interactions Sun–Earth–Moon using a lantern, geographic globes, balls in a semi-darkened room: Earth receives the light from the Sun; the Sun produces light and shadows on the Earth; the Moon and the Earth might shade on each other; Sun, Earth and Moon spin around their own; the Earth’s axis crosses the North Pole and the South Pole. (3) Working in pairs, students formulate questions about the natural phenomena produced by the Sun. (Some of the questions devised by the students: ‘‘Why are there days and nights on Earth? How is the Sun moving? Where does the Sun go to sleep during the night? Why, when the Sun sets down, the sea water does not start to boil? What is the Sun made of? How can we know the time observing the Sun?’’)

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In the meanwhile, the teacher facilitates explorations by asking questions, communicating only by divergent questions, and abstaining from giving answers. The questions were listed; from this list, the question: ‘‘Why are there days and nights on Earth?’’ was selected as the one to be answered. (4) Working in pairs, the students re-examine the chosen question, in order to clarify it: (‘‘Have you ever thought about it?’’ ‘‘Do you have other questions?’’ ‘‘How would you answer?’’), and try to answer the question. Some students answered some based on what they already knew and some accessed other resources put at their disposal from the beginning of the lesson— e.g., excerpts from an encyclopedia). Then, the students presented the versions they found as answers. By comparing own proposals with those of classmates, they assessed which of the answers were testable explanations, which were simple descriptions, or opinions, and communicated their points of view. (These assessment criteria had been provided by the teacher.)

body to model experimentally the hypothesis promoted to explain the day–night succession in a certain place on Earth: i. ‘‘The Sun spins around its axis’’ (Does it light the Earth from afar?). ii. ‘‘The Earth spins around the Sun’’ (In 1 year!). iii. ‘‘The Sun spins around the Earth’’ (because the globe kept fixed is lighted by the lantern which rotates around the globe-the place was marked with a spot on the globe). iv. ‘‘The Earth spins around its axis’’ (the lantern is fixed and lights the globe which rotates around its own axis-the place was marked with a spot on the globe). (4) Continuing their work in groups, the students organize the data in various ways (drawings, tables, schemes, somebody wrote a short essay), and formulate new questions, before making other investigations. Some change the plan and try to collect other data; when they finish their own activity, they move to other groups which are still working on their investigations.

5.1.1.2. Explore/experiment (2 h) ‘‘How does this information fit with what I know or think about?’’

5.1.2. Structuring phase 5.1.2.1. Synthesize/explain (2 h) ‘‘How are my beliefs influenced by these ideas?

(1) Depending on the given answer to the question, or depending on their preferences, the students make new groups in order to check the hypotheses and versions of answers (e.g., explanations for the day–night succession on Earth: (a) rotation of the Sun around its axis—as the mirror of a far); rotation of the Sun around the Earth—this is visible; (b) the rotation of the Earth around its axis; (c) rotation of the Earth around the Sun.). (2) Through discussions within the groups and sometimes with the teacher, the students: J plan the investigation (the details of the problem, personal tasks, procurement of materials, plan of activities, etc.); J expose the ideas in front of the class (to refine their own project, for supplementary documentation, in order to analyze other techniques); J collect arguments for the answer (based on connections/analogies with own experiences, from various sources). (3) Within these working groups, the students use lanterns, balls, geographic globes or their own

(1) Discussing in groups and with the teacher, the students synthesize the gathered data, model the rotation of the Earth around its axis using a top, distinguish rules/patterns in collected data, then expose to peers the data, the products (experimental models, tables, drawings, copies of documents, posters), and their conclusions. (2) Through discussions within their working groups and with the teacher, the students analyze the plausible data (What data are we to keep, which to eliminate?) and assess their own results and procedures (What conclusions should be kept and what eliminated? Is this model appropriate? What explanations are sustained by evidence? Which explanation is better?). (3) Through discussions within the groups and with the teacher, the students compare alternative explanations (terrestrial rotation around the axis and the Sun’s revolution around the Earth) and search for arguments. For example, some children experimented how, when shaking hands in pairs and rotating one around the other, the

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heavier individual can ‘control’ the motion of the other (Does the heavier student spin around the lighter or vice versa? Do the heavier bodies spin around the lighter or vice versa? Does the revolution of the Sun around the Earth really happen?) (4) Through discussions within the groups and with the teacher, the students revise prior knowledge and formulate the explanation in the light of the new experiences: the succession day–night is due to the Earth’s rotation around its own axis.

5.1.3. Applying phase 5.1.3.1. Apply/practice (2 h) ‘‘What are the beliefs behind this information? (1) Working in groups, the students test the explanation on other particular cases (A student remarked: ‘‘If this is the explanation, then it has to be well documented!’’). For example: J Some investigate previous questions with their own means: ‘‘Why the New Year is celebrated in the world at different hours?’’, ‘‘Does the Earth completely spin around its axis in 1 day?’’ or another initial question: ‘‘Why, when the Sun sets down, the sea water does not start to boil?’’ J Others formulate questions about other results. J Others evoke (through drawings or short stories) own experiences in the light of new knowledge (such as about their own program that depends of the Sun position in the sky, or in a cloudy day). J A group built watches that measure spans smaller than a day (‘shadows watch’—using a stick pushed to the ground, and denoting the hours on different directions of the shade, anticipating next hours; a sand glass, observing the flow of the sand in a sand glass— Egyptians were using sand glasses during the night. Why? What could they use during the day?). (2) Then, through discussions within the groups and with the teacher, the students: J Make a short report concerning the results of their own investigation (‘‘Now you know why the Earth has days and nights. Because it spins around its axis like a top!’’) J Present the realized products and the working reports.

During this phase, the teacher has moderated the discussions among the students, abstaining from giving immediate answers to questions. As the students presented their reports in front of the class, they learned that some samples do not explain—or explain only partially—the initial observations. 5.1.3.2. Transfer/extend (1 h) ‘‘What can I do differently now, when I have gathered this information?’’ (1) Through discussions within the groups and with the teacher, the students investigate the consequences of the explanations found: J Some compared their own arguments (heliocentric) with other conceptions about the Solar System (geocentric), with other beliefs (stories) or TV information. J Some built patterns or toys (‘the compass’ of the hand watch; a watch ‘made of the Sun rays’—the positions of the sun rays on the walls of the room associated with the activities along the day; a watch made of a candle with pushing pins—(‘‘Why the pins should be pushed in the candle at equal distances? What kind of candle is more appropriate to measure a longer period of time?’’). J Others experiment with orientation and time measurement with the help of the Sun. J Others record observations of some natural phenomena that depend of the positions of the Sun in the sky, or of the day–night succession (timetables during a day—‘‘The day moments help you to organize your time. Make a list of activities you are doing each day, and mention the hours’’). (2) Individually or in groups, the students: J Expose the products (drawing, patterns, schemes, etc.). J Conclude with comments about phenomena that might influence their health (activities during daytime, best hours for sunbathe, etc.). An instructional unit as it is seen in this article targets to help students develop personal or group projects that are presented and assessed at the end of that unit. In this way, the student explores her own level of knowing and anticipates the knowledge development through perfectible planning; she experiments with some phenomena during learning activities, and constructs personal meaning, which is

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then compared and contrasted with peers’ meanings; she develops metacognitive skills that help to base future learnings on the gained experiences. Within the IMSTRA framework, the students arrive at defining concepts based on analyzing the results of their own explorations, reinforce them through connections with peers and resources, synthesize the knowledge built, apply it to novel cases, and extend it to the world around them.

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approach leads to a spiral model (see Fig. 4). In a spiral development, the use of resources is optimized at both informational and procedural levels. 6. Model application in curriculum development and teacher training Surprisingly, the most visible application of the model happened not in teaching, but in curriculum

5.2. The IMSTRA cycle: models of implementation The IMSTRA model might be used in teaching during the school year: regularly, frequently, or just from time to time. Two distinct options can be emphasized: the cylinder and the spiral as representing two different implementation frameworks leading to two different approaches. The cylinder model represented in Fig. 3 gives an image for integrating relatively independent instructional units. Classical instructional units (based on lessons or book chapters) could alternate with projects in which the IMSTRA teaching–learning cycle is systematically followed. Fig. 3 emphasizes multiple uses of the IMSTRA model during the same year in different subject areas or in the same subject area. Lack of cohesiveness and communication among the different implementations of the model are represented as definite slices of the cylinder. Given its flexibility, the IMSTRA methodology could inoculate classical lessons as the teacher gains experience and confidence with the process. A more effective way of using the IMSTRA model is to structure the didactical process as a part of long-term planning, in which the acquisitions of one cycle are to be valued in the next cycles. This

Fig. 4. Integrating IMSTRA model into the teaching practice: the spiral development.

Fig. 3. Integrating IMSTRA model into the teaching practice: the cylinder development.

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development. In the last 5 years, teams of authors (including the authors of the model) have developed textbooks structured in instructional units that follow the IMSTRA structure. The three main phases: immersion–structuring–applying allow constructing the chapters of a textbook that highlights the child’s participation in learning. Thus, for instance, in a mathematics textbook for the 1st grade, as well as for the 2nd grade, the mathematical content of each instructional unit was doubled by a tale with characters who follow the phases of the model, making the assigned mathematics more realistic and contextual. In grades 3 and 4, the textbooks in the same series continue this construction strategy, giving more space to students’ personal projects and investigations; moreover, the products to be made by children within these projects are proposed throughout the books. This extension of the IMSTRA model in curriculum development, which was not anticipated from the beginning, shows that the model can be adapted for various educational purposes. Another application arose with the occasion of a professional development program for rural areas. The program for rural education addressed teachers

who graduated university in a specific subject matter, but they were to teach a discipline for which they were not qualified. To gain the qualification, they had to take specific courses within a 3-years long-distance learning program. From the perspective of this article, it is significant that the didactical courses in the domains of: mathematics, biology, chemistry, physics, and ICT were developed within the IMSTRA model. It is about 23 courses produced by 21 authors, covering topics such as, for example in teaching chemistry: Didactics of Chemistry, Methodology of teaching, learning and assessing chemistry, Didactics of learning chemistry through devising and solving problems, The history of chemistry and its applications. Beyond these disciplinary courses, an interdisciplinary one, taken in all the strands of the program, was ‘Didactics of the curricular areas Mathematics and Natural Sciences and Technologies’. To get a feeling about how these courses have been developed within the IMSTRA framework, the next table lists the titles and subtitles of the chapters of two of the courses. Beyond the formal distribution of content, presented in Table 4, the didactical approach of these courses also highlights the conceptual framework of

Table 4 A course design based on the IMSTRA framework—two examples Phases of the Functional Contents of the course ‘Didactics of the curricular areas IMSTRA translation Mathematics and Natural Sciences and Technologies’ model within the book

Immersion

Explore and compare

Contents of the course ‘Didactics of geometry’

Instruction unit 1: Curricular area. A conceptual framework Instruction unit 1: Positional geometry 1.1. A few steps into history 1.1. The epistemological perspective 1.2. The historical perspective: From analytical programs 1.2. How do we use intuition to construe rigor in reasoning? of study to student-centered curriculum 1.2.1. How do we build intuition at 1.3. From the teaching plan to the curriculum framework the level of axioms and other 1.4. The aims and features of the curricular areas and basic notions? of the school disciplines 1.2.2. How do we build intuition at 1.4.1. The curricular area Mathematics and Natural the level of deductive systems? Sciences 1.2.3. The didactical construction of 1.4.2. The curricular area Technologies the deduced notions and claims 1.3. Types of mathematical claims

Structuring

Understand and 1.5. An objective-based curriculum for primary and lower 1.4. Helpful particular configurations in problem solving experiment secondary education 1.4.1. Square networks 1.5.1. Curriculum development in the area Mathematics 1.4.2. Parallel lines in the workbooks and Natural Sciences 1.4.3. Tiles of the plane 1.5.2. Curriculum development in the curricular area 1.4.4. The cub Technologies 1.4.5. The tetrahedron 1.5.3. The training profile of the graduate of compulsory education

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Table 4 (continued ) Phases of the Functional Contents of the course ‘Didactics of the curricular areas IMSTRA translation Mathematics and Natural Sciences and Technologies’ model within the book

1.6. A competence-based curriculum for high school 1.6.1. Taxonomies, technologies, cognitivism and constructivism 1.6.2. Developing competences for high school students 1.6.3. The pattern of developing competences 1.7. The advantages of the new competence-based curriculum for high school

Contents of the course ‘Didactics of geometry’

1.5. The analogy 1.5.1. How can we use the analogy in teaching? 1.5.2. The analogy 3D–2D 1.5.3. Dangers of the 3D–2D analogy 1.6. Modalities to describe parallelism and perpendicularity 1.6.1. Synthetic description 1.6.2. Vectorial description 1.6.3. Analytic description 1.6.4. Description based on complex numbers 1.7. Comparative problem solving

Applying

Apply and develop

1.8. TIMSS Reports 1.8.1. Knowledge facts or competences? 1.8.2. Abstract or quotidian? 1.8.3. Algorithmic or creative investigative? 1.8.4. Rigorous or estimative? 1.8.5. Structured or unstructured?

1.8. Assessing in geometry 1.8.1. Perspectives of assessment 1.8.2. The matrix for structuring the competences 1.8.3. How do we use the matrix in assessment? Assessment test

1.9. Conclusions of TIMSS reports Assessment test

Immersion

Explore and compare

Instruction unit 2: Transdisciplinary perspectives and didactical approaches

Instruction unit 2: Metrical geometry and trigonometry

2.1. Short incursion into the history of science and technology 2.1.1. Period 2,400,000–600 b.c.: The beginnings of Science and Technology 2.1.2. Period 600 b.c.–530 a.c.: The Greek and Hellenistic science 2.1.3. Period 530–1452: The medieval science 2.1.4. Period 1453–1659: Renaissance and the scientific revolution 2.1.5. Period 1660–1819: The Newtonian e´poque, Illuminism, the industrial revolution 2.1.6. Period 1820–1894: The science of the 19th century 2.1.7. Period 1895–1945: Science at the beginning of the 20th century 2.1.8. Period 1946–2000: Science and Technology after World War 2

2.1. 2.2. 2.3. 2.4.

Measuring: some historical aspects Measure Thales’s theorem Measuring—didactical perspectives 2.4.1. Measuring areas and volumes 2.4.2. Measuring angles

2.2. The modern world—multiple connections Structuring

Understand and 2.3. From factual learning to conceptual learning. Facts and experiment scientific data. Scientific paradigms 2.4. Concept and macroconcept 2.4.1. Concept formation—particularities of the disciplines 2.4.2. How to check concepts understanding?

2.5. Computing formulas for areas and volumes 2.6. Exact or estimative? 2.7. Area and length of the circle; volume of round solids 2.8. Pitagora’s theorem

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Phases of the Functional Contents of the course ‘Didactics of the curricular areas IMSTRA translation Mathematics and Natural Sciences and Technologies’ model within the book 2.4.3. Conceptual maps: types, examples, uses 2.4.4. What are the macroconcepts?

Applying

Apply and develop

Contents of the course ‘Didactics of geometry’

2.9. Metric theorems in the right angled triangle 2.10. Methods to deduce trigonometric formulas 2.11. Types of problems 2.12. Classifying a problem—suggestion about its solving 2.13. Collect metric data 2.14. The didactical scenarios

2.5. School-based curriculum 2.15. Assessment through projects in 2.5.1. A few examples of integrated optional disciplines geometry Assessment test Assessment test Instruction unit 3: Strategies of scientific research— relationships to didactics of math, science and technology

Instruction unit 3: Properties of 2D and 3D figures

3.1. Scientific procedures 3.1.1. Problem solving—typologies, features, stages 3.1.2. The scientific method—typology, definition, features, stages The scientist perspective-the scientific project 3.1.3. Problem solving in technology—steps, procedures, activities 3.1.4. The self-development project 3.1.5. Perspectives on the scientific culture in school

3.1. The heuristics of problem solving in geometry 3.1.1. The dynamics of problem solving 3.1.2. Support points in exploring problem solving strategies 3.1.3. Thinking strategies to be used for efficient problem solving 3.1.4. How to think and solve a geometry problem? A strategy to be developed by students

Immersion

Explore and compare

Structuring

Understand and 3.2. The relationship inductive approach—investigation— experiment problematization 3.2.1. Learning through investigation 3.2.2. Problematizing and problem-based learning

3.2. Geometrical reasoning. Congruence and similarity 3.2.1. Logic and reasoning 3.2.2. Relationships among notions

3.3. The congruence of triangles 3.3. Strategies for developing values and attitudes in the 3.3.1. Criteria for the congruence of curricular areas Mathematics and Natural Sciences and triangles Technologies 3.3.2. The method of congruent 3.3.1. Values and attitudes in the new curriculum triangle 3.3.2. Debate—didactical method with transdisciplinary and valuing potential 3.4. The similarity relationship 3.4.1. Criteria for the similarity of 3.4. Communication and action tools to make knowledge triangles more accessible 3.4.2. Similar polygons 3.4.1. Focusing on instructional units beyond the lessons—a modality to integrate domain specific features 3.5. Properties. Equal sets. Equivalent 3.4.2. Communication as a learning resource definitions 3.5.1. How do we formulate a definition? 3.5. Various resources for the classroom activity 3.5.2. How do we make 3.5.1. Paper and plastic resources generalizations in geometry? 3.5.2. Educational software. Ways to use ICT in the 3.5.3. Some comments on the curricular areas Mathematics and Natural necessary and sufficient Sciences and Technologies condition

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Table 4 (continued ) Phases of the Functional Contents of the course ‘Didactics of the curricular areas IMSTRA translation Mathematics and Natural Sciences and Technologies’ model within the book

Contents of the course ‘Didactics of geometry’

3.6. Applications of geometry into practice 3.7. The sequences of an instructional unit Applying

Apply and develop

3.6. Monodisciplinarity perspective versus transdisciplinarity: overcoming conceptual and methodological cliche´s in teaching and learning 3.6.1. Degrees of integration: pluridisciplinarity, interdisciplinarity, transdisciplinarity 3.6.2. How does the school turn to account the new perspectives?

3.8. Assessment through problem solving 3.8.1. How do we train the students through written assessment? 3.8.2. Practical ways to improve students’ performances Assessment test

3.7. Applying the transdisciplinary view: project based learning 3.8. Transdisciplinary perspectives offered to assessment: project and portfolios 3.9. The European perspective—domains of key competences Assessment test

the model. In short, it allows each chapter to focus on the specific targets of IMSTRA phases, facilitating constructive learning. As previously shown, many studies emphasize the idea that inquiry-based instruction has the potential to enhance the quality of student’s achievement. Our study is based on the assumption that, in order to make this approach realistic to the student, it is necessary to find appropriate and realistic targets for the teacher. This is imperative today because social and economical changes compel educators to create new relationships between knowledge and students. The human knowledge extends nowadays almost exponentially, and school instruction should both compress this informational explosion and make it ‘realistic’ within school practice. From knowledge as collection of facts to knowledge as process that happens in school, the educational road is long and full of challenges. The present article has tried to identify an optimal route. Yet, more research is needed to see if its optimality functions in a variety of teaching contexts. Acknowledgements F. M. Singer thanks the colleagues from the Science Education Seminar, at the Harvard Smithsonian Center for Astrophysics, Cambridge, Massachusetts, for their suggestions on a previous draft

of this article, and to Fulbright Commission for providing the opportunity to develop this research. The authors are thankful to Dr. Leni Cook, Professor Emeritus of Teacher Education at the California State University, Dominguez Hills, for her editorial suggestions on the pre-publication version of this paper.

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