Teaching and learning cycles in a constructivist approach to instruction

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Teaching and Teacher Education 24 (2008) 1613–1634

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Teaching and learning cycles in a constructivistapproach to instruction

Florence Mihaela Singera,�, Hedy Moscovicib

aInstitute for Educational Sciences, Bucharest, RomaniabCollege of Education, California State University, Dominguez Hills, USA

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 teacher-

training 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 andBauersfeld (1995), Resnick and Klopfer (1989),Sierpinska (1998), Singer (1999), and, probablymost significant, the new curriculum guidelinesin North American and European countries pro-mote new missions for the teacher and thelearner. These bring new roles into schools, whichfocus on:

�
the learner as an autonomous thinker and explorerwho expresses his/her own point of view, asks
ee front matter r 2007 Elsevier Ltd. All rights reserved.

te.2007.12.002

ing author. Tel.: +40723 542 900;

9 642.

ess: [email protected] (F.M. Singer).

questions for understanding, builds arguments,exchanges ideas and cooperates with others inproblem solving—rather than a passive recipient

of information that reproduces listened/writtenideas and works in isolation;
� the teacher as a facilitator of learning, a coach as
well as a partner who helps the student tounderstand and explain—rather than a ‘knowl-

edgeable authority’ who gives lectures and im-poses standard points of view;
� classroom learning that aims at developing
competences and is based on collaboration—instead of developing factual knowledge focusedon only validated examples and based oncompetition in order to establish hierarchies

among students.

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mailto:[email protected]

ARTICLE IN PRESSF.M. Singer, H. Moscovici / Teaching and Teacher Education 24 (2008) 1613–16341614

However, compared with the effective results, allthese new roles seem to pertain to a new ideology(e.g., Sierpinska, 1998). They require differentprocesses in order to transform the new aims fromideal targets into outcomes of current teaching–learning practice. Another challenge arises fromthese new processes: the need to make the classroominteractions generate new mental frames for theones involved, both the teacher and the student. Thestudent’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 theseare not real questions. Therefore, it is about myplaying a role. And this role should not be takenvery seriously; otherwise I risk being ridiculous.’’(Goffman, 1974). The teacher, on her part, toooften sees the ‘didactical contract’ (Brousseau, 1980)in a limited formal way. The hidden understandingof the school as a stage, and of learning andteaching as formal role playing affects in-depthlearning and the collaborative climate in the class-room. The questions are: How could these roles bemade more realistic and exciting? How couldteacher and students become partners in knowledgeconstruction?

The model for the teaching and learning cycle wepresent in this article was developed as a follow-upof teacher-training sessions and revised as it wasimplemented at a school site. This paper presents ananalytic description of the model. We start byproviding an overview of the literature on learningcycles in various knowledge areas, with examplesfor sciences, mathematics, and interdisciplinarycurricula. To make the basis for the constructionmore explicit, a teaching–learning experience inmathematics in 9th grade introduces the model andallows ‘bootstrapping’ into its description. Thetable-based presentation that follows can be usedas a functional tool for teaching. A case studyinvolving a set of science lessons in 3rd grade showshow the model can be applied, and is an example ofthe model’s flexibility. Some possibilities for extend-ing the model implementation are discussed in theend of the article.

2. Learning cycles in various areas

For a long time researchers tried to understandthe different steps that one takes when solving aproblem in an attempt to comprehend how the mindworks and how to best educate the next generation.Are there set steps, or is it a conceptual template

that allows for individual modifications accordingto the question under scrutiny, the researcher’sbackground and the available resources? Variousanswers have been proposed to this question. Welist some of them below.

2.1. The sciences

In the area of the sciences, there is an abundanceof learning cycle models perhaps because science isperceived by the general public as the only bias-freeand objective way of knowing. Generalizability andreliability studies suggest that results from one studycan be easily duplicated if one follows proceduresand uses the same materials as in the original study.These statistical entities also suggest that differencesbetween the experimental group and control(s) canbe correlated to the intervention rather than torandom coincidence/mishap.

Looking at the role of the student in a problem-solving situation, Lawson, Abraham, and Renner(1989) reviewed several years of research on studentreasoning and concluded that appropriate teachingcan lead to generalizable and significant improve-ments. They identified three required stages meantto improve students’ reasoning skills: exploration,term/concept introduction, and concept application.During exploration, students are encouraged toexplore a phenomenon/phenomena and identify apattern. The initial pattern gets reinforced, mod-ified, or changed by using appropriate terminologyand by exploring concepts during the second stage,which is term/concept introduction. The third stage,concept application, ensures that students are ableto translate concept(s) learned and use them in newsituations. Later, Lawson (2002) identified anincreased difficulty in the problem-solving processthat amplified the potential for reaching faultyconclusions when students explore science-relatedphenomena that involve unobservable entities.Lawson’s conclusion is more disturbing as thestudents in his study are preservice biology teacherswith at least a bachelor degree in the sciences. Wenoticed a similarity between the model proposed byLawson et al. (1989), and the three levels of learningproposed by Wolfe (2001): concrete experience,representational or symbolic learning, and abstractlearning.

In an attempt to enhance and clarify theengagement stage and integrate an evaluativedimension to the inquiry process, the BiologicalScience Curriculum Study (BSCS) developed a

ARTICLE IN PRESSF.M. Singer, H. Moscovici / Teaching and Teacher Education 24 (2008) 1613–1634 1615

5-step model, also known as the 5 Es: Engage,Explore, Explain, Elaborate, and Evaluate (Layman,Ochoa, & Heikkinen, 1996). This model definesteacher’s role during the five steps of inquiry. Themodel has similarities with the Lawson et al. (1989)model. The first two BSCS steps, Engage andExplore, correlate to the first stage in the model ofLawson et al. (1989), the third BSCS step, Explain,parallels the second stage, and the fourth BSCS stepis similar to the third stage. BSCS introduces a newdimension to the inquiry process: ‘Evaluation’ ofwhat the students discovered during the process andwhat they have yet to figure out. It is important toemphasize the fact that this evaluation does notimply finality, just the opposite. It encourages thelearner to think forward to other linked avenues forresearch. The 5 Es model resembles the modelproposed by the National Research Council (2000)in which five features of inquiry are defined from thelearner’s perspective as: (1) Engagement in scienti-fically oriented questions; (2) Giving priority toevidence in responding to questions; (3) Formulat-ing explanations from evidence; (4) Connectingexplanations to scientific knowledge; and (5) Com-municating and justifying explanations (p. 29).

Even more recently, the Museum of Science inMiami, Florida, has developed a learning cycle thatbuilds upon the BSCS’ 5Es model. The first stagechanges from Engage to Excite, Elaborate is dividedinto Expand—applying the pattern/concept in asimilar situation—and Extend—applying pattern/concept in a different subject area. The Exchange

stage is introduced in order to encourage students totake advantage of Internet and share their informa-tion with individuals around the world, andEvaluate is changed into Examine, in order todiminish the tendency to see evaluation as a finalprocess, and to encourage a variety of evaluativeprocesses to be considered as evidence for students’learning (alternative assessment strategies).

In an attempt to synthesize the inquiry process asexperienced by research scientists in the sciencelaboratory, Reiff, Harwood, and Phillipson (2002)proposed the ‘inquiry wheel.’ From the practi-tioners’ point of view, the research questions lie inconcentric circles connected with two-way arrows tothe following elements/processes of inquiry: obser-ving, defining the problem, forming the question,investigating the known, articulating the expecta-tion, carrying out the study, interpreting the results,reflecting on the findings, and communicating theresults to the scientific community and to society.

The immediate connection among the elementsthrough questions allows for the necessary fluidityof research. However, the ‘inquiry wheel’ model isquite rigid as it does not emulate research sciencewhere results are interpreted based on the elementsof the specific study, the questions under scrutiny,and the studies addressed during the ‘investigatingthe known’ stage. Should the wheel be more like aweb? And how does this model translate into theK-12 or K-16 classroom? We will discuss thisfurther, when talking about the proposed model.

2.2. Mathematics

Mathematicians have been constantly preoccu-pied with conceptualizing a model for mathematicalproblem solving. Thus, Polya (1957) focused on thefollowing stages of a solving cycle: (1) under-standing; (2) devising a plan; (3) carrying out theplan; (4) looking back. Schoenfeld (1985) set thefollowing basic steps: (1) analysis; (2) exploration;(3) verification. In a more cognitive-oriented de-scription, Verschaffel et al. (1999) highlighted thefollowing inventory: (1) build a mental representa-tion of the problem; (2) decide how to solve theproblem; (3) execute the necessary calculations;(4) interpret the outcome and formulate an answer;(5) evaluate the solution. The process through whichmathematics concepts are internalized supposes aprogressive formalization, described by Dienes(1963) as the psychodynamic process of artificialgenesis of mathematical concepts. This model ofprogressive formalization identifies six stages: struc-tured play or game, isomorphic games, abstraction,schematization, formulation and symbolization,axiomatization, generalization. Studying how pri-mary school students approach problem solving,Marcou and Lerman (2006) propose a model forself-regulated mathematical problem solving thatconsists in the following phases: (1) forethoughtphase: analyzing the text; (2) performance phase:carrying out the plans; and (3) self-reflection phase:looking back.

These taxonomies mainly refer to the processcarried out by individuals when solving problems.However, the variety of implicit and explicitinteractions that take place in a classroom contextshould be considered when speaking about schoollearning. The ‘60 s New Math program (Brailly,1968–9) promoted the ideal of the learner as aresearcher in mathematics and the whole class as aresearch team. More recently, an interactionist


perspective proposed viewing the classroom as amicro-culture in which meanings emerge throughshared activities. The interpretation of these activ-ities is negotiated and eventually standardizedduring the interactions between the teacher andthe students. This view is inspired by Bruner’stheory of interaction formats in language acquisi-tion (1985). The interactionist research methodol-ogy draws upon the qualitative, naturalisticapproaches that are based on exploration andinspection of phenomena, where a theoretical modelis constructed and continually reconstructed duringthe study. The interactionist research programs leadnot to recommendations for action but to descrip-tions and discussions of various possibilities. Themicro-sociological approach developed throughinteractionism had a strong influence in France inbuilding the concept of the didactical contract(‘contrat didactique’), developed by Brousseau(1980). In Germany and Great Britain, its impactled to the research program started by Bauersfeld,focusing on micro-ethnographical studies of tea-cher–student interactions (see Cobb & Bauersfeld,1995).

The constructivist program overshadowed thistrend for a while, but recently, its scope hasbroadened to encompass the study of classroomculture and the mechanisms of their emergence andstability. As Sierpinska (1998) pointed out, interac-tionist ideas lead to a much more down-to-earthview of the teaching–learning process. In Dienes’vision, to move from a teacher-driven thinkingframe to a more research-based one, games can bethe starting point. Another starting point might bereality. The daily life problems are increasinglyemphasized in recent mathematics curricula invarious countries. For example, the algebra strandusing Mathematics in context in the Netherlands is agood illustration of the idea of progressive for-malization for middle school students (NationalCentre for Research in Mathematical SciencesEducation and Freudenthal Institute, 1997). Itbegins by having students use their own words,pictures, or diagrams to describe mathematicalsituations, to organize their own knowledge andwork, and to explain their strategies. In later units,students gradually begin to use symbols to describesituations, to organize their mathematical work, orexpress their strategies. At this level, students devisetheir own symbols or learn some conventionalnotations. Their representations of problem situa-tions and the explanations of their work are a

mixture of words and symbols. Later, students learnand use standard conventional algebraic notation.In a context of freedom, students might move backand forth among levels of formality depending onthe problem, situation, and the mathematics in-volved.

2.3. Interdisciplinary curricula

Whitehead (1929) describes the inquiry process asa course of action revolving around freedom anddiscipline. Learners begin their inquiry in a stage offreedom or romance, when they are getting im-mersed in their question and search for possiblesolutions and patterns. This stage, (parallelingLawson et al. first part of the Engagement stage)is perceived as unsystematic to the outside observer.With the tentative development of the pattern, thelearning process enters into a disciplined or preci-sion stage. During this stage, patterns are devel-oped, verified, and modified constantly. The thirdstage of Whitehead’s model brings the learner to adifferent kind of freedom or generalization thatencourages the learner to explore other situationsusing what was learned. This third stage in White-head’s model parallels Lawson et al.’s ‘ConceptApplication’ stage, and fourth stage in the BSCScurriculum—‘Elaborate on the Concept.’

Similar to Whitehead, Tchudi and Lafer (1996)concentrate on structure (parallel to disciplinein Whitehead’s model) and spontaneity (parallelto freedom in Whitehead’s model) and developan interdisciplinary model called S2 (Structure�Spontaneity ¼ S

2). Central concepts for the inter-disciplinary units are the results of the intersectionamong mandated standards, teacher’s interest, andstudents’ interests. In a rich-text and rich-resourceenvironment, participants get involved in researchabout their chosen theme/issue/problem, researchthat results in knowledge production. Variouspresentation strategies as well as group formationsmediate learning. In contrast to Whitehead’s model,the S2 shows a successful conversation betweenfreedom and discipline all along the inquiry process.

The Australian Planning Model proposed byPeter Forrestal (1986) has five stages for thedevelopment of the interdisciplinary inquiry afterthe establishment of the ‘cue’ event: input (teacherlocates core information), exploration (students usethe ‘input’ materials and make sense of them interms of the problem), reshaping (students areengaged in the synthesis of the information found),


presentation (students present new knowledge to theaudience), and reflection (teacher and studentsevaluate the work that has been done in the lightof the findings).

In an attempt to build a learning cycle that usescollected information and results in improvingstudents’ achievement, Love (2003) discusses waysin which data collected at the district or site levelcan 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 toeducational practices change and enhancing stu-dents’ achievement in their own classrooms.

To summarize, in an attempt to establish stages/elements of educational research, questions ofcontext are raised and challenged. Actions andevents need to be analyzed using the participants’explanations and ‘thick descriptions’ (Geertz, 1973).In many ways educational research reminds us ofthe model of Whitehead (1929) for interdisciplinaryinquiries. Researchers and participants enter thesituation trying to understand what is going on(freedom), they develop and refine patterns thatillustrate their interests and learning (disciplined orstructured stage), and then, by applying what waslearned in a new situation, they get into the thirdstage of relative freedom.

3. Why a new model?

It is necessary to point out that learning cyclesmodels are very dynamic entities and they tend toevolve and build on previous work as well as on thechange in the context. Students, teachers, commu-nities, as well as materials and equipment availableall changed over time. Learning cycles got adjustedto fit the new learning environments. Therefore,there are elements that successfully resisted anychange over time (e.g., the need to engage/excite thelearner), while others underwent dramatic changes(e.g., the use of resources, the availability of Internetto collect information and communicate results). Onthe other hand, in a classroom context, there is aninevitable interaction between the teacher and thestudent’s actions, as well as a fluid relationshipbetween the various stages of inquiry. Both bringimperceptible changes to the learning process. Wehave focused our attention on four main problemsof the teaching/learning models/cycles previouslydescribed, which we further address within the newproposed model.

First, most models presented above shift from therole of the teacher to that of the students whentalking about different stages. For example, in themodel of Lawson et al. (1989), the students explore,but the teacher is the one in charge with the secondstage—term introduction and pattern reinforce-ment; in Forrestal’s model (1986), the teacher is incharge during the input stage—the stage of collect-ing pertinent resources to support students’ inquiry,while the students’ activities are less emphasized.For most of the models, it is unclear how the teacherand the student(s) coexist during these learningcycles and how the shift in responsibility affects theoutcome of the inquiry process.

Second, there is a tendency to introduce (maybeunintentionally) a certain level of rigidity throughthe stages or steps that lead to the right answer. Theimmediate consequence of this linearity is the‘scientific method’ or the prescribed set of linearsteps that supposedly lead to scientific discoveries inthe world and to student’s discoveries in the day-by-day classroom. Reiff (2003) summarized research on40 science textbooks, 35 at the college level, thatshowed that the ‘scientific method’ is still describedas a set of linear, predetermined steps leading to atheory/discovery. The subject of the scientificmethod is restricted to the first two chapters of thescience textbook, it usually lacks a figure, and whena figure is provided it is still linear with few, or, inmost cases, no feedback loops (allowing/encoura-ging jumps—meaning going back to the previousstage or jumping to another stage according to theresearcher’s need). Even if the authors mention thatresearchers employ a variety of research methods,the linear step-by-step description or figure contra-dicts the text. From the science text to the scienceclassroom, teachers perpetuate the linearity of ‘the’scientific method, sometimes even checking thesequence of the steps, their correct spelling, andtheir definitions rather than their expressionthrough students’ explorations. Needless to say thatthis linear model differs tremendously from themodel described by practicing researchers; thepractitioner researchers’ model shows the questionunder scrutiny in the middle of an inquiry wheelthat encourages flexibility and loop jumping possi-bilities according to questions asked, informationneeded, and background of the researcher (Reiffet al., 2002).

The third issue involves the teaching–learningpractice and is derived directly from the second.New approaches to curriculum development support


learning with understanding and encourage sensemaking; despite this generous idea, a closer look atthe current practices shows that the student is not somuch discovering; she is more or less guessing theanswers targeted by the teacher. The students’problem becomes, to a large extent, finding thehidden rules of the game that takes place in theclassroom, unraveling the expectations of theteacher, the constraints in which the teacher acts,and adapting to these rules, expectations andconstraints (Moscovici, 2001, 2003). Being awareof all the subtle patterns that govern the life of aclassroom is indispensable in planning and design-ing teaching–learning activities. In this respect, themodel needs to be flexible enough to assureadaptation to very diverse social contexts andinteractions, without letting the context dominatestudents’ construction of knowledge.

The fourth considered aspect is that an effectivemodel should prevent at least some of the risks ofusual behaviors. Thus, the reality of the classroommay be different for the teacher and for the researcherin education. For example, a cognitively orientedresearcher will see the classroom as a collection ofindividual minds. For the common teacher, theclassroom is mostly seen as a unit that behaves in aspecific way, beyond the individuals; the result of thisvision is that the teacher practically addresses hermessage to only a few students and their reactions areextrapolated to the whole class (e.g., Bruner, 1985;Sierpinska, 1998). In this approach, the teacher is anexternal body who does not really interact with theclassroom community. We believe that the classroomactivities should become learning experiences for bothsides: teacher and students; keeping in mind hereducational aims, the teacher has to get off theknowledge barricade and construct meaning togetherwith students. The teacher’s role in genuine inquiryprojects is even more complex because it is aboutshortening the cultural history of human behaviorinto authentic inquiry pieces.

4. The proposed model

A constructivist multidimensional approachstarts with the informal ideas that students bringto school and develops ways to help them see howthese ideas can be transformed and formalized.Movements along this continuum are not necessa-rily smooth, or all in one direction. Patternedtechniques are needed to encourage students tobuild on their informal ideas in a gradual but

structured manner so that they re-build domain-specific concepts and procedures (e.g., Bailin, Case,Coombs, & Daniels, 1999; Shulman & Sherin, 2004;Singer, 2003). Students need to be challenged tomake sense of what they are doing. However,because of time constraints, the approach cannotprovide absolute freedom to search for meaning andsolutions. The teacher is in the position to build onstudents’ understandings and carefully orchestratethe students’ contributions to knowledge construc-tion (Singer, 2001). To accomplish such result, theteacher should know what is significant from thedomain-specific perspective in terms of concepts,procedures, and historical evolution; and, simulta-neously, what methods are most efficient for thelearner’s progress (see concept of pedagogicalcontent knowledge in Gess-Newsome & Lederman,1999; Shulman, 1986; Singer, 2002, 2007).

4.1. Starting from an example

Inevitably, we need to describe the continuum ofclassroom learning through discrete stages in orderto ensure its replication in various contexts. Whatphases could be more productive and effective? Theanswer is situated at the interface of micro-devel-opmental analyses, cognitive research, and curricu-lum construction. To illustrate our approach, wehave chosen as the starting point, from ourempirical database, a lesson in mathematics withstudents in grade 9 (15–16 years old). Because ourfocus is to synthesize at a micro-level the mainphases of the model, we detail only the aspects ofthe lesson that focus on the model construction.

The lesson started with the problem presentedbelow.

Two taxi companies have the following offers:

QUICK TAXI:
SPEED TAXI:
Start: 6 h
Start: 3 h +1.3 h per 1 km +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 inthe task,


(3)
exploring ways to solve the problem by experi-encing particular cases and analogous situa-tions.
Three types of students’ approaches have beenidentified. One group of students gave values on atable, noticing this way that for 3 km, SPEED TAXIis cheaper, while for 10 km, QUICK TAXI is abetter choice. Another group that gave values on atable first anticipated this variation and thendiscussed it, based on the values. A third groupstarted by discussing how to express the travel costsas functions that vary with the distance.

What have students actually done during thisphase in terms of generalizable experience?

They have:

�
identified terms, relationships, methods alreadyknown that might be connected to the topic, andshared 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.
All these activities can be concentrated aroundthe keyword Evoking.

They also have:

y17
� 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,
�
+ 3

recorded, compared, classified data using mod-alities adequate to the targets,

9x

�

1.6
processed partial results, trying to build ownunderstanding of the concept, �
xO 1

34

5

1.3x +

6

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

the problem.

begun to correlate between intervention andresults in terms of variables.

All these activities can be concentrated aroundthe keyword Exploring.

Evoking and Exploring are meant to immersestudents in the problem-solving experience. Immer-

sion—with its two components, Evoking and Ex-

ploring—is the state that the teacher might construct

in order to get the students involved in creativeactivities.

Further, a group considered x the distance, anddefined the trip costs as functions of x; they found:q(x) ¼ 1.3x+6 and s(x) ¼ 1.9x+3. Thus, the stu-dents transposed the given practical problem into analgebraic task—i.e., solving a linear inequation:q(x)ps(x). The teacher noticed that this group hasproposed a mathematical model for the problem, anddecided to share this with the whole class. Throughdiscussions, the model discovered by one of thegroups in the classroom was adopted by some othergroups and rejected by others, without teacher’sdirect intervention. Some groups started to solve theinequation. In the meantime, a group preferred tocontinue to guess and check the solutions. Anothergroup had the idea that, instead of solving theinequation, it might be better to represent the graphsof the two functions (see Fig. 1).

Using students’ observations, the teacher theninitiated a discussion about the degree of general-izability of each solution. Students reviewed theirresults and adjusted their solutions.

Then, using the graph from Fig. 1, some students:

�
checked the results obtained in the previousphases, � analyzed and described the behaviors of the two
functions,


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

What have students actually done during thisphase 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 aroundthe keyword Synthesizing.

They also have:

�
reflected upon some examples, while analyzingthe 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 knowl-
edge and searched for other information,
� described/explained the observed patterns.
All these activities can be concentrated aroundthe keyword Explaining.

For the simplicity of the model, Synthesizing andExplaining can be seen as two facets of Structuring.

The inquiry process continued even after gettinganswers to the problem. The teacher used the initialproblem to devise other tasks in order to analyze thevariation 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 studentsto use the pattern found in the previous phases tosolve the following problem:

The ‘‘Micron’’ store sells CD video gamesrecorded in its own studio. Each CD costs 6 h

to produce and it is sold for 10 h. The initialinvestment is of 5000 h. How many CDs must besold 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 thisphase in terms of generalizable experience?

They have:

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

All these activities can be concentrated aroundthe keyword Practicing.

They also have:

�
explored applications of the solving schema innew 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 aroundthe keyword Extending.

Practicing and Extending bring the inquiryprocess into the Applying phase.

This analysis led to three major phases of theteaching and learning cycle, with each phase dividedinto two sub-phases. The three phases will be called:Immersion, Structuring, and Applying, albeit theselabels do not cover the complexity of actions to betaken under this framework. They are describedbelow from a more general perspective, highlightingthe main characteristics of the model.

The model construction is derived from observedgood practice, further elaborated from an expertperspective. Its working versions were then im-proved during a teacher-training program thatinvolved 200 teachers of Mathematics and Sciences


from primary to upper secondary. The programconsisted in two sessions: a 4-day institute inOctober, 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 assessmentwas based on the projects they developed during thesessions. The timetable of the course alternateddisciplinary work that grouped teachers of the samesubject matter for a specific key stage (Mathematics,Physics, Chemistry, Biology; respectively, primary,lower secondary and upper secondary), and inter-disciplinary work, where the trainees interacted withone another within the curricular area. The team oftrainers (among them, one of the authors) preparedin detail each training session. Each class of teacherswas taught by a team of at least two trainers, usuallythree. We used this training opportunity to test andrefine the model. In the following sections, weprovide a description of the model and an exampleof how this model was applied in a real class.Although the phases of the model are the same forthe teacher and the student, the sub-phases aredifferent, in order to capture the different goals,roles, and views.

4.2. Immersion phase

During this phase, students get immersed into theproblem—address and use previous knowledge,seek more information, plan and perform experi-ments, and, based on all these resources andprocesses, identify tentative pattern(s). The studentsalso explore their own knowledge and anticipate theknowledge development through planning personalprojects. With respect to teacher’s role during thisphase, the model proposes questions that theteacher might ask or be asked, and actions that willlead to the expected action performed by thestudents. Thus, during the first sub-phase of thefirst phase (Anticipation), the teacher is preoccupiedwith developing students’ curiosity; helping themformulate learning targets, identifying and selectingthe chunks of information and knowledge (includ-ing previous knowledge) that are pertinent to thespecific problem. During the second sub-phase,Problem construction, the teacher scaffolds students’research by providing necessary hints, encouragesstudents’ explorations, and helps students to recorddata. As students acknowledge safety rules andremain focused on solving the problem, the teachershould abstain from leading them to her ownsolution(s).

In terms of students’ learning activities, theImmersion phase can be divided into two inter-connected sub-phases: Evoking, and Exploring.During the Evoking sub-phase, students bend theirprevious knowledge to the problem, discuss andchallenge their ideas with peers. Some studentsmight be looking for resources to document/enrichtheir knowledge, and that leads to exploring. Duringthe Exploring sub-phase, students plan, perform,and analyze their investigations while alwaysaddressing the problem that they have to solve.

During the Immersion phase, the students learn toselect pertinent knowledge from what they knowrealizing that personal knowledge might proveinsufficient and deciding to look for resources(including library and Internet), and to judgeresources in terms of reliability of information.Students also learn to correlate between variablesand experimental results (hypothetical-deductive/hypothetical-predictive, ifythen type), understandlimitation of experiments (e.g., number of speci-mens/groups, variability of specimens within thesame group), and become familiar with the use ofhigher-order thinking skills such as synthesis,analysis, evaluation, and the creation and expres-sion of a complex solution to a problem.

Moreover, students constantly shift betweenconcrete (e.g., personal experiences relevant to theproblem under scrutiny) and abstract (e.g., under-standing patterns underlying peer’s concrete exam-ples or described in texts). Students learn to movefrom concrete (personal experiences) and semi-concrete (other students’ experiences) to abstract(patterning) and again to concrete (planning experi-mentation and make first trials to solve theproblem). These constant shifts will help themduring the patterning process when they need tobegin explaining the found pattern and trying togeneralize it, whilst detaching the abstract piece ofknowledge from the concrete experimental/trialphases (Singer, 1995, 2004).

4.3. Structuring phase

During this phase, students move to another levelof understanding when they interpret their concreteexperiential results from the Immersion phase andadjust the pattern. They explain the claim developedduring the previous phase in terms of examples andcounter-examples, and create new situations inorder to challenge their own claim and to add tothe generalizability of the knowledge they produced.


In the role of facilitator, the teacher supportsstudents by helping them synthesize observations,summarize findings, and explore inferences duringthe Systematization sub-phase. During Conceptuali-

zation sub-phase, the teacher helps students use thenew terminology, generalize conclusions, and ex-pand their findings beyond the specific problem thatthey researched, into related issues.

Synthesizing sub-phase involves the students inthe process of identifying and contrasting patterns,helping to extend their findings into more general-izable statements. Explaining sub-phase requiresstudents to connect the concrete exploration to amore abstract model that describes the results ofexplorations and challenges findings through con-crete and hypothetical examples and counter-examples. The students define the concepts throughinterpreting the results of their activity and reinforcethem through connection with other activities. Interms of skills, during the Structuring phase,students learn how to differentiate between opinionand fact, about the limitations of experiments, andabout the use of appropriate language when sharingfindings. They also learn about validity of claimsand the role of the constructive criticism providedby peers and teacher during the dynamic and fluidprocess of knowledge construction.

Students move during this phase from theconcrete aspects of experimentation into the com-plex and multifaceted conditions of real lifeproblems. They also move from concrete intoabstract during the generalization process, whenthey shift from a specific solution for a specificproblem to finding solution(s) for classes ofproblems.

4.4. Applying phase

During this phase, students learn to use theabstract pattern that they developed into relatedand unrelated situations, they modify/adjust theirpattern to be more generalizable and applicable in awider range of situations. They apply learnedconcepts and patterns to new situations by tryingto solve existing problems, and by creating/describ-ing new hypothetical or realistic situations that needsolving. These processes lead to a more generalizedpattern that identifies constraining elements.

The teacher during the Applying phase is con-cerned with assessing students’ understanding of theconcepts developed and with the process of inquiryand its limitations. Sometimes the teacher might

choose another concrete example in the samedomain (for Reinforcement sub-phase) or in arelated/unrelated domain (for Transfer sub-phase)to illustrate relationships, or to increase complexity(e.g., number of possible solutions). Teachers mayalso explicitly prompt students to think aboutaspects of their everyday life that are potentiallyrelevant for further learning.

4.5. Sinthesizing the features of the model

The model described above presents a teachingand learning cycle focused on IMmersion, STRuc-

turing, Applying. It will be called on short IMSTRA.Tables 1–3 contain a functional presentation ofthe teacher’s role and students’ learning activitieswithin the IMSTRA framework. The descriptionfacilitates the model’s translation into the teachingpractice and encourages its adaptation to varioustopics.

To have a clearer view of this construction, theschema in Fig. 2 gives a synthetic presentation ofthe IMSTRA cycle. This schema emphasizes theclose relationship between the teacher’s targets andthe students’ activities during the inquiries. Westress that the purpose of the whole cycle is to fosterstudents’ learning. Within the IMSTRA framework,the students are involved in a multitude of inductiveand deductive pathways that help them move withease among concrete or semi-concrete experiences,and abstract patterns. Furthermore, during thisprocess, students are forced to use higher-orderthinking skills and metacognition.

The schema in Fig. 2 emphasizes the qualities thatmake this model reliable and effective: generality,extensiveness, flexibility, functionality, and inquiry-

focused dimensionality.Generality shows that this model can be used

across a few disciplines, at least for mathematicsand natural sciences. We hypothesize that its usecan be extended to teaching and learning socialsciences, but evidence for this claim is to beobtained through specific studies.

Extensiveness consists in reflecting the parallelroles of the student and the teacher whilst theyattempt to reach their goals. Although the activitiesare different, their goals are convergent: buildingknowledge in an interactive process. By makingexplicit the difference in roles and actions of the twoactors during the didactical process, the modelmoves the approach from the ideology of thedidactical 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 Student’s learning activities

Key questions Targets

Immersion Actualization– anticipation Evoking

� 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)?

� Offer problem-pretext (real-

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)

� 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

Problem construction Exploring

� 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?

� 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

� 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

F.M. Singer, H. Moscovici / Teaching and Teacher Education 24 (2008) 1613–1634 1623

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Table 2

Key roles of the teacher and the students during the IMSTRA cycle: Structuring phase

Structuring

Systematization Synthesizing

� 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

� 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.)

Conceptualization Explaining

� 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

� 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

F.M. Singer, H. Moscovici / Teaching and Teacher Education 24 (2008) 1613–16341624

on descriptive approaches, to the reality of theclassroom’s interactions.

Flexibility is manifested in two ways: by the fluidtransition from one phase (sub-phase) to another orby the possibility of merging the sub-phases into asingle phase. This is in close connection with thenext characteristic, functionality. Thus, dependingon the topic, and the students’ age and abilities,it is possible to consider three, four, five or sixsub-phases. The cycle still remains consistentwithin this variability if connections among thephases are highlighted. The amplitude and depth ofthe learning cycle is the teacher’s decision. Manystudies report on the students’ natural focus onexploring while learning in a natural context (e.g.,Lawson, 2003). However, elements from the other

phases: structuring and applying might be deve-loped taking into account the constraints of theschool system, otherwise the learning remainsincomplete and the knowledge achieved by thestudent is very rarely transferable in a stableacquisition (e.g., Bransford, Brown, & Cocking,2000). To put it briefly, the IMSTRA frameworkto put it requires that, in order to constructstudent’s knowledge, a learning cycle should toucheach of the three main phases, even when there is afocus on only one of the phases. The teacher mightassume this concentration in its premises andconsequences.

Inquiry-focused dimensionality simply means that,in each phase, student and teacher equally addressquestions 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 Practicing

� 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?

� 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

� 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

Transfer Extending

� 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?

� 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

� 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


reflective attitude in the core, the model centers onstudents’ learning with understanding. We shouldalso stress that even the formulation of the basicresearch question could evolve within a constructi-vist learning environment—sometimes it becomesmore specific, while other times it changes to matchthe researcher’s (student’s) new interest. In thisframe, the teacher is orchestrating the inquiryprocess. That means that the teacher continuouslyassesses students’ involvement, improvises to let theinquiry process go on, and harmonizes students’participation in knowledge building.

5. Model application in class

In this section, we present some modalitiesnecessary to implement the model. A first stepconsists in planning the distribution of educationalobjectives and contents in instructional units. Thecriteria to set up an instructional unit are:

�
coherence, referring to the objectives on whichthe instructional unit focuses, � emphasis on the students’ competences devel-
oped during the instructional unit,

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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 summa-
tive assessment.

Some other criteria to be taken into account shouldbe: the school mission concerning students’ achieve-ments at graduation, the possibility of developing anintegrated approach, the students’ level of under-standing, the interests and abilities of students, thesocio-cultural local context, and the existent orpotential resources. Such instructional units mightbe assimilated with organizing learning throughprojects—the most frequent way of work if weexamine life out of school, as Gardner (1999) noticed.Compared with the classical planning of lessons, theconcept of the instructional unit has some advan-tages: it creates for students a coherent learningenvironment in which their expectations becomeclearer on medium and long term; it involves theteacher in anticipating the didactical meaning onmedium and long term, allowing students’ differentrhythms of learning; it offers perspectives for thelessons through a non-linear relationship amongthem, generated by their position in the teachingand learning cycle (Leahu, Singer, & Leahu, 2001).

5.1. Learning within the IMSTRA framework—an

example

We describe below an instructional unit taught ina 3rd grade class of Science, within the IMSTRA

framework. The topic was The Sun—source of

periodical changes in the environment, focusing onthe question: ‘‘Why are there days and nights onEarth?’’ The instructional unit covered eightlessons, organized around five phases of theIMSTRA cycle, which followed the stages of aninvestigation:

(i)
formulate the question and advance the hy-potheses—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 thelearning approach. The description is based on theobserver’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 observa-tions, experiences and personal events regardingnatural phenomena (such as ‘the travel’ of theSun in the sky, the motion of shadows, thesuccession day–night, thoughts about the influ-ence of the Sun on the passage from light todarkness). The children discuss and makedrawings.
(2)
Working in groups, students model interactionsSun–Earth–Moon using a lantern, geographicglobes, balls in a semi-darkened room: Earthreceives the light from the Sun; the Sun produceslight and shadows on the Earth; the Moon andthe Earth might shade on each other; Sun,Earth and Moon spin around their own; theEarth’s axis crosses the North Pole and theSouth Pole.
(3)
Working in pairs, students formulate questionsabout the natural phenomena produced by theSun. (Some of the questions devised by thestudents: ‘‘Why are there days and nights onEarth? How is the Sun moving? Where does theSun go to sleep during the night? Why, when theSun sets down, the sea water does not start toboil? What is the Sun made of? How can weknow the time observing the Sun?’’)


In the meanwhile, the teacher facilitates ex-plorations by asking questions, communicatingonly by divergent questions, and abstainingfrom giving answers. The questions were listed;from this list, the question: ‘‘Why are there daysand nights on Earth?’’ was selected as the one tobe answered.

(4)
Working in pairs, the students re-examine thechosen question, in order to clarify it: (‘‘Haveyou ever thought about it?’’ ‘‘Do you have otherquestions?’’ ‘‘How would you answer?’’), and tryto answer the question. Some students answeredsome based on what they already knew andsome accessed other resources put at theirdisposal from the beginning of the lesson—e.g., excerpts from an encyclopedia). Then, thestudents presented the versions they found asanswers. By comparing own proposals withthose of classmates, they assessed which of theanswers were testable explanations, which weresimple descriptions, or opinions, and commu-nicated their points of view. (These assessmentcriteria had been provided by the teacher.)
5.1.1.2. Explore/experiment (2 h) ‘‘How does this

information fit with what I know or think about?’’

(1)
Depending on the given answer to the question,or depending on their preferences, the studentsmake new groups in order to check thehypotheses and versions of answers (e.g., ex-planations for the day–night succession onEarth: (a) rotation of the Sun around itsaxis—as the mirror of a far); rotation of theSun around the Earth—this is visible; (b) therotation of the Earth around its axis; (c) rotationof the Earth around the Sun.).
(2)
Through discussions within the groups andsometimes with the teacher, the students:J plan the investigation (the details of the
problem, personal tasks, procurement ofmaterials, plan of activities, etc.);

J expose the ideas in front of the class (to refinetheir own project, for supplementary docu-mentation, in order to analyze other techni-ques);

J collect arguments for the answer (based onconnections/analogies with own experiences,from various sources).

(3)
Within these working groups, the students uselanterns, balls, geographic globes or their own
body to model experimentally the hypothesispromoted to explain the day–night succession ina certain place on Earth:i. ‘‘The Sun spins around its axis’’ (Does itlight the Earth from afar?).

ii. ‘‘The Earth spins around the Sun’’ (In 1year!).

iii. ‘‘The Sun spins around the Earth’’ (becausethe globe kept fixed is lighted by the lanternwhich rotates around the globe-the place wasmarked with a spot on the globe).

iv. ‘‘The Earth spins around its axis’’ (thelantern is fixed and lights the globe whichrotates around its own axis-the place wasmarked with a spot on the globe).

(4)
Continuing their work in groups, the studentsorganize the data in various ways (drawings,tables, schemes, somebody wrote a short essay),and formulate new questions, before makingother investigations. Some change the plan andtry to collect other data; when they finish theirown activity, they move to other groups whichare still working on their investigations.
5.1.2. Structuring phase

5.1.2.1. Synthesize/explain (2 h) ‘‘How are my

beliefs influenced by these ideas?

(1)
Discussing in groups and with the teacher, thestudents synthesize the gathered data, model therotation of the Earth around its axis using a top,distinguish rules/patterns in collected data, thenexpose to peers the data, the products (experi-mental models, tables, drawings, copies ofdocuments, posters), and their conclusions.
(2)
Through discussions within their workinggroups and with the teacher, the studentsanalyze the plausible data (What data are weto keep, which to eliminate?) and assess theirown results and procedures (What conclusionsshould be kept and what eliminated? Is thismodel appropriate? What explanations aresustained by evidence? Which explanation isbetter?).
(3)
Through discussions within the groups and withthe teacher, the students compare alternativeexplanations (terrestrial rotation around the axisand the Sun’s revolution around the Earth) andsearch for arguments. For example, somechildren experimented how, when shaking handsin pairs and rotating one around the other, the


heavier individual can ‘control’ the motion ofthe other (Does the heavier student spin aroundthe lighter or vice versa? Do the heavier bodiesspin around the lighter or vice versa? Does therevolution of the Sun around the Earth reallyhappen?)

(4)
Through discussions within the groups and withthe teacher, the students revise prior knowledgeand formulate the explanation in the light of thenew experiences: the succession day–night is dueto 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 theexplanation on other particular cases (A studentremarked: ‘‘If this is the explanation, then it hasto be well documented!’’). For example:J Some investigate previous questions with
their own means: ‘‘Why the New Year iscelebrated in the world at different hours?’’,‘‘Does the Earth completely spin around itsaxis in 1 day?’’ or another initial question:‘‘Why, when the Sun sets down, the sea waterdoes not start to boil?’’

J Others formulate questions about otherresults.

J Others evoke (through drawings or shortstories) own experiences in the light of newknowledge (such as about their own programthat depends of the Sun position in the sky,or in a cloudy day).

J A group built watches that measure spanssmaller than a day (‘shadows watch’—using astick pushed to the ground, and denoting thehours on different directions of the shade,anticipating next hours; a sand glass, obser-ving the flow of the sand in a sand glass—Egyptians were using sand glasses during thenight. Why? What could they use during theday?).

(2)
Then, through discussions within the groups andwith the teacher, the students:J Make a short report concerning the results of
their own investigation (‘‘Now you knowwhy the Earth has days and nights. Because itspins around its axis like a top!’’)

J Present the realized products and the work-ing reports.

During this phase, the teacher has moderated thediscussions among the students, abstaining fromgiving immediate answers to questions. As thestudents presented their reports in front of the class,they learned that some samples do not explain—orexplain only partially—the initial observations.

5.1.3.2. Transfer/extend (1 h) ‘‘What can I do

differently now, when I have gathered this informa-

tion?’’

(1)
Through discussions within the groups and withthe teacher, the students investigate the con-sequences of the explanations found:J Some compared their own arguments (helio-
centric) with other conceptions about theSolar System (geocentric), with other beliefs(stories) or TV information.

J Some built patterns or toys (‘the compass’ ofthe hand watch; a watch ‘made of the Sunrays’—the positions of the sun rays on thewalls of the room associated with theactivities along the day; a watch made of acandle with pushing pins—(‘‘Why the pinsshould be pushed in the candle at equaldistances? What kind of candle is moreappropriate to measure a longer period oftime?’’).

J Others experiment with orientation and timemeasurement with the help of the Sun.

J Others record observations of some naturalphenomena that depend of the positions ofthe Sun in the sky, or of the day–nightsuccession (timetables during a day—‘‘Theday moments help you to organize your time.Make a list of activities you are doing eachday, 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 (activitiesduring daytime, best hours for sunbathe, etc.).

An instructional unit as it is seen in this articletargets to help students develop personal or groupprojects that are presented and assessed at the endof that unit. In this way, the student explores herown level of knowing and anticipates the knowledgedevelopment through perfectible planning; sheexperiments with some phenomena during learningactivities, and constructs personal meaning, which is

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

the spiral development.


then compared and contrasted with peers’ mean-ings; she develops metacognitive skills that help tobase future learnings on the gained experiences.Within the IMSTRA framework, the students arriveat defining concepts based on analyzing the resultsof their own explorations, reinforce them throughconnections with peers and resources, synthesize theknowledge built, apply it to novel cases, and extendit to the world around them.

5.2. The IMSTRA cycle: models of implementation

The IMSTRA model might be used in teachingduring the school year: regularly, frequently, or justfrom time to time. Two distinct options can beemphasized: the cylinder and the spiral as represent-ing two different implementation frameworks lead-ing to two different approaches.

The cylinder model represented in Fig. 3 gives animage for integrating relatively independent instruc-tional units. Classical instructional units (based onlessons or book chapters) could alternate withprojects in which the IMSTRA teaching–learningcycle is systematically followed. Fig. 3 emphasizesmultiple uses of the IMSTRA model during the sameyear in different subject areas or in the same subjectarea. Lack of cohesiveness and communicationamong the different implementations of the modelare represented as definite slices of the cylinder.Given its flexibility, the IMSTRA methodologycould inoculate classical lessons as the teacher gainsexperience and confidence with the process.

A more effective way of using the IMSTRAmodel is to structure the didactical process as a partof long-term planning, in which the acquisitions ofone cycle are to be valued in the next cycles. This

Fig. 3. Integrating IMSTRA model into the tea

approach leads to a spiral model (see Fig. 4). In aspiral development, the use of resources is optimizedat both informational and procedural levels.

6. Model application in curriculum development and

teacher training

Surprisingly, the most visible application of themodel happened not in teaching, but in curriculum

ching practice: the cylinder development.


development. In the last 5 years, teams of authors(including the authors of the model) have developedtextbooks structured in instructional units thatfollow the IMSTRA structure. The three mainphases: immersion–structuring–applying allow con-structing the chapters of a textbook that highlightsthe child’s participation in learning. Thus, forinstance, in a mathematics textbook for the 1stgrade, as well as for the 2nd grade, the mathema-tical content of each instructional unit was doubledby a tale with characters who follow the phases ofthe model, making the assigned mathematics morerealistic and contextual. In grades 3 and 4, thetextbooks in the same series continue this construc-tion strategy, giving more space to students’personal projects and investigations; moreover, theproducts to be made by children within theseprojects are proposed throughout the books. Thisextension of the IMSTRA model in curriculumdevelopment, which was not anticipated from thebeginning, shows that the model can be adapted forvarious educational purposes.

Another application arose with the occasion of aprofessional development program for rural areas.The program for rural education addressed teachers

Table 4

A course design based on the IMSTRA framework—two examples

Phases of the

IMSTRA

model

Functional

translation

within the book

Contents of the course ‘Didactics of th

Mathematics and Natural Sciences and

Instruction unit 1: Curricular area. A c

Immersion Explore and

compare

1.1. The epistemological perspective

1.2. The historical perspective: From a

of study to student-centered curri

1.3. From the teaching plan to the cu

1.4. The aims and features of the curr

of the school disciplines

1.4.1. The curricular area Mathem

Sciences

1.4.2. The curricular area Techno

Structuring Understand and

experiment

1.5. An objective-based curriculum fo

secondary education

1.5.1. Curriculum development in

and Natural Sciences

1.5.2. Curriculum development in

Technologies

1.5.3. The training profile of the

compulsory education

who graduated university in a specific subjectmatter, but they were to teach a discipline for whichthey were not qualified. To gain the qualification,they had to take specific courses within a 3-yearslong-distance learning program. From the perspec-tive of this article, it is significant that the didacticalcourses in the domains of: mathematics, biology,chemistry, physics, and ICT were developed withinthe IMSTRA model. It is about 23 coursesproduced 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 dis-ciplinary courses, an interdisciplinary one, taken inall the strands of the program, was ‘Didactics of thecurricular areas Mathematics and Natural Sciences

and Technologies’. To get a feeling about how thesecourses have been developed within the IMSTRAframework, the next table lists the titles and sub-titles of the chapters of two of the courses.

Beyond the formal distribution of content, pre-sented in Table 4, the didactical approach of thesecourses also highlights the conceptual framework of

e curricular areas

Technologies’

Contents of the course ‘Didactics of

geometry’

onceptual framework Instruction unit 1: Positional geometry

nalytical programs

culum

rriculum framework

icular areas and

atics and Natural

logies

1.1. A few steps into history

1.2. How do we use intuition to construe

rigor in reasoning?

1.2.1. How do we build intuition at

the level of axioms and other

basic notions?

1.2.2. How do we build intuition at

the level of deductive systems?

1.2.3. The didactical construction of

the deduced notions and claims

1.3. Types of mathematical claims

r primary and lower

the area Mathematics

the curricular area

graduate of

1.4. Helpful particular configurations in

problem solving

1.4.1. Square networks

1.4.2. Parallel lines in the workbooks

1.4.3. Tiles of the plane

1.4.4. The cub

1.4.5. The tetrahedron

ARTICLE IN PRESS

Table 4 (continued )

Phases of the

IMSTRA

model

Functional

translation

within the book

Contents of the course ‘Didactics of the curricular areas

Mathematics and Natural Sciences and Technologies’


geometry’

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

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?

1.9. Conclusions of TIMSS reports

Assessment test

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

Instruction unit 2: Transdisciplinary perspectives and

didactical approaches

Instruction unit 2: Metrical geometry and

trigonometry


compare

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 epoque,

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

2.2. The modern world—multiple connections

2.1. Measuring: some historical aspects

2.2. Measure

2.3. Thales’s theorem

2.4. Measuring—didactical perspectives

2.4.1. Measuring areas and volumes

2.4.2. Measuring angles


experiment

2.3. From factual learning to conceptual learning. Facts and

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


ARTICLE IN PRESS


Phases of the

IMSTRA

model

Functional

translation

within the book




geometry’

2.4.3. Conceptual maps: types, examples, uses

2.4.4. What are the macroconcepts?

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

Applying Apply and

develop

2.5. School-based curriculum

2.5.1. A few examples of integrated optional disciplines

Assessment test

2.15. Assessment through projects in

geometry

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


compare

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


experiment

3.2. The relationship inductive approach—investigation—

problematization

3.2.1. Learning through investigation

3.2.2. Problematizing and problem-based learning

3.3. Strategies for developing values and attitudes in the

curricular areas Mathematics and Natural Sciences and

Technologies

3.3.1. Values and attitudes in the new curriculum

3.3.2. Debate—didactical method with

transdisciplinary and valuing potential

3.4. Communication and action tools to make knowledge

more accessible

3.4.1. Focusing on instructional units beyond the

lessons—a modality to integrate domain specific

features

3.4.2. Communication as a learning resource

3.5. Various resources for the classroom activity

3.5.1. Paper and plastic resources

3.5.2. Educational software. Ways to use ICT in the

curricular areas Mathematics and Natural

Sciences and Technologies

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

3.3.1. Criteria for the congruence of

triangles

3.3.2. The method of congruent

triangle

3.4. The similarity relationship

3.4.1. Criteria for the similarity of

triangles

3.4.2. Similar polygons

3.5. Properties. Equal sets. Equivalent

definitions

3.5.1. How do we formulate a

definition?

3.5.2. How do we make

generalizations in geometry?

3.5.3. Some comments on the

necessary and sufficient

condition


ARTICLE IN PRESS


Phases of the

IMSTRA

model

Functional

translation

within the book




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 cliches 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?

3.7. 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

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


the model. In short, it allows each chapter to focuson the specific targets of IMSTRA phases, facilitat-ing constructive learning.

As previously shown, many studies emphasize theidea that inquiry-based instruction has the potentialto enhance the quality of student’s achievement.Our study is based on the assumption that, in orderto make this approach realistic to the student, it isnecessary to find appropriate and realistic targetsfor the teacher. This is imperative today becausesocial and economical changes compel educators tocreate new relationships between knowledge andstudents. The human knowledge extends nowadaysalmost exponentially, and school instruction shouldboth compress this informational explosion andmake it ‘realistic’ within school practice. Fromknowledge as collection of facts to knowledge asprocess that happens in school, the educational roadis long and full of challenges. The present article hastried to identify an optimal route. Yet, moreresearch is needed to see if its optimality functionsin a variety of teaching contexts.

Acknowledgements

F. M. Singer thanks the colleagues from theScience Education Seminar, at the Harvard Smith-sonian Center for Astrophysics, Cambridge, Mas-sachusetts, for their suggestions on a previous draft

of this article, and to Fulbright Commission forproviding the opportunity to develop this research.

The authors are thankful to Dr. Leni Cook,Professor Emeritus of Teacher Education at theCalifornia State University, Dominguez Hills, forher editorial suggestions on the pre-publicationversion of this paper.

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Teaching and learning cycles in a constructivist approach to instruction