d13 development of pedagogical methodology€¦ · modellingpsace project deliverable d13...

INFORMATION SOCIETY TECHNOLOGIESEDUCATION AND TRAINING

SCHOOL OF TOMORROW

Project: IST-2000-25385

MODELLINGSPACESpace of ideas' expression, modelling and collaboration

for the development of imagination, reasoning and learning

Development of Pedagogical Methodology

Deliverable: D13Contractual Delivery Date: September 2003

Version: FinalType: Public

Responsible Partner: FCTUNL

Faculdade de Ciências e Tecnologia (UNL), March 2004

IST - School of Tomorrow MODELLINGSPACE : IST-2000-25385

MODELLINGPSACE Project Deliverable D04 Development of Pedagogical Methodology 2

Development of PedagogicalMethodology

Contact: Vitor Duarte Teodoro, FCTUNL, Campus da Caparica, 2829-516 Caparica,[email protected], tel. +351 917509939.

Main Authors:

Contributors:

Vitor Duarte TeodoroMônica Mesquita

ModellingSpace partners

SUMMARY: This report builds on work done on the ModellingSpace project butalso on the educational and psychological literature. A pedagogicalmethodology is a set of procedures that a teacher candevelop in order to help all students learn. A methodology isseen as something one cannot receive from others. On thecontrary, it is the complex result of instruction, personalexperience and reflection.

The report establish a framework for the procedures, based onsix tenets (Commitment to teaching, to students and to theirlearning; knowledge of science and mathematics; knowledge ofstudents; knowledge of the art of teaching; science as a way ofthinking; and reflection and professional growth) and makesthirteen proposals for a Modelling Methodology: (1) makeclear goals and plan how concepts and ideas evolve during the activities, anticipating learning difficulties; (2) elicit andverbalize students’ conceptions; (3) promote interaction,collaboration, and group cohesion; (4) give prompt feedback; (5)induce self and group formative assessment; (6) proceed fromconcrete to abstract; (7) verbalize mathematical procedures; (8)promote schematic drawing and writing as “tools-to-think-with”;(9) scaffold the transition from direct computations to algebraicreasoning, from number sense to symbol sense; (10) exploremultiple representations; (11) make abstract objects as concreteas possible but spot the differences between the “real thing” andthe representation; (12) balance exploratory learning with guidedlearning; (13) anticipate, check, and revise the coherence of themodel and data.

KEYWORDS: Pedagogical/teaching methodology; modelling in science andmathematics; exploratory learning; computers in teaching andlearning; collaborative work; teacher education; teacher training.



TABLE OF CONTENTS

What is a Pedagogical Methodology? . . . . . . . . . . . . . . . . . . . . . . .4

Effective teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Six Tenets of the Pedagogical Methodology . . . . . . . . . . . . . . . . .6

Commitment to Teaching, to Students and to Their Learning . . . . . .7

Knowledge of Science and Mathematics . . . . . . . . . . . . . . . . . . . . .8

Knowledge of Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Knowledge of the Art of Teaching . . . . . . . . . . . . . . . . . . . . . . . . . .10

Science as a Way of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

Reflection and Professional Growth . . . . . . . . . . . . . . . . . . . . . . . . .12

Thirteen Proposals for a Modelling Methodology, with Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Make clear goals and plan how concepts and ideas evolve during the activities, anticipating learning difficulties . . . . . . . . . . . . . . . . .14

Elicit and verbalize students’ conceptions . . . . . . . . . . . . . . . . . . . .15

Promote interaction, collaboration, and group cohesion . . . . . . . . . .16

Give prompt feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Induce self and group formative assessment . . . . . . . . . . . . . . . . . .18

Proceed from concrete to abstract . . . . . . . . . . . . . . . . . . . . . . . . .19

Verbalize mathematical procedures . . . . . . . . . . . . . . . . . . . . . . . .20

Promote schematic drawing and writing as “tools-to-think-with” . . . .21

Scaffold the transition from direct computations toalgebraic reasoning, from number sense to symbol sense. . . . . . . . .22

Explore multiple representations . . . . . . . . . . . . . . . . . . . . . . . . . .23

Make abstract objects as concrete as possible but spot the differences between the “real thing” and the representation . . . . . .24

Balance exploratory learning with guided learning . . . . . . . . . . . . . .25

Anticipate, check, and revise the coherence of the model and data . .26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27



It is well know that most people assume thatanyone who has been a student can teach (“justremember your best teacher, and do like him...”).

This is probably true... for teaching, not forlearning. As a matter of fact, it is now also wellknown that learning is not necessarily theoutcome of teaching. For example, in thereport Science for All (AAAS, 1989), the authorswrote:

Cognitive research is revealing that evenwith what is taken to be good instruction,many students, including academicallytalented ones, understand less than we thinkthey do. With determination, students takingan examination are commonly able to identifywhat they have been told or what they haveread; careful probing, however, often showsthat their understanding is limited ordistorted, if not altogether wrong.

Teaching can be easy, but helping students learnis surely a not so easy task. Students carry withthem many “learning obstacles”, ranging fromcommon science misconceptions toepistemological naïve thinking.

A pedagogical methodology is seen, in thisreport, simply as a set of procedures that ateacher can develop in order to help allstudents learn, not just those who learn almostspontaneously. Note the important verb“develop” in this statement: a methodology isnot something one can receive from others. It isthe complex result of instruction, personalexperience and reflection.

Pedagogical methodologies can vary and change.For example, some decades ago, reinforcing(the relationship between the incidence ofbehaviour, the occurrence of a consequence, andthe increased or decreased likelihood of thatbehaviour occurring in the future) was seen asthe essential aspect of a good methodology.Nowadays reinforcement is still considered animportant aspect but others are considered morerelevant to learning. E.g., exploring multiplerepresentations (verbal, graphical, analytical,etc., particularly in science and mathematics)and concrete experience of abstractconcepts are two of current essential aspects toconsider on an effective methodology.

What is aPedagogicalMethodology?



Effectiveteaching

Effective teaching is seen on the educational andpsychological literature as having multiplecomponents, such as:

• personal traits of the teacher;

• teacher competencies;

• teaching methods;

• classroom atmosphere;

• teacher decision making-skills;

• students previous knowledge and skills;

• students characteristics.

The interaction between all these factors and thecomplexity of each make difficult (orimpossible?) to assert which one is the singlemost important factor.

Some authors, such as Ausubel (1968) postulatethat students' previous knowledge is the singlemost important factor:

If I had to reduce all of educationalpsychology to one principle, I would say this:the most important single factor influencinglearning is what the learner already knows.Ascertain this and teach him accordingly.

Ausubel also introduced relevant ideas andconcepts, such as:

• the distinction meaningful and rotelearning;

• the most general ideas of a subjectshould be presented first and themprogressively differentiated;

• instructional materials must integrate newmaterial with previously presentedinformation;

• learning materials should be logicallyorganized and potentially meaningful tolearners.

• anchoring new concepts into the learner’salready existing cognitive structure makenew concepts recallable.

In this report, we are particularly interested inthose factors teachers can manage bythemselves. For example, it is not possible for ateacher to have influence on most students'characteristics (e.g., personal characteristics andfamily background).



Six Tenets ofthePedagogicalMethodology

Commitment to Teaching, to Studentsand to Their Learning

Teachers acknowledge and value the individualityand worth of each student, believe that all stu-dents can learn and demonstrate these beliefsin their practice.

Knowledge of Science and Mathematics

Teachers study continuously to have a broadand deep knowledge of the concepts, princi-ples, techniques, and reasoning methods ofmathematics and science (and the connectionsbetween them and with other fields of knowl-edge), and they use this knowledge to establishcurricular goals and shape their instruction andassessment.

Knowledge of Students

Teachers know and care about their students,know how they learn and develop, understandthe impact of home life and cultural background,and use this knowledge to guide their curricularand instructional decisions.

Knowledge of the Art of Teaching

Teachers have an extensive base of pedagogi-cal knowledge to stimulate, motivate and facili-tate student learning, using a wide range of for-mats and procedures to create environments inwhich students are active learners, show will-ingness to take intellectual risks, develop confi-dence and self-esteem, and valueknowledge.

Science as a Way of Thinking

Teachers develop students’ abilities to reasonand think alone or with support from others,to investigate and explore patterns, to discoverstructures and establish relationships, to formu-late and solve problems, to justify and communi-cate their conclusions, and to question andextend those conclusions.

Reflection and Professional Growth

Teachers reflect on what and how they teachand collaborate with others to strengthen thelearning community.

Inspired on the Standardsdeveloped by the National Boardfor Professional TeachingStandards, 2001.



Commitment toTeaching, toStudents and toTheir Learning

Teachers acknowledge and value theindividuality and worth of each student, believethat all students can learn and demonstratethese beliefs in their practice.

In recent years, international studies such as theTrends in International Mathematics and ScienceStudy (TIMSS, http://timss.bc.edu) and theProgramme for International Student Assessment(PISA, http://www.pisa.oecd.org) have shownthat most countries face complex problemswith student learning in Science andMathematics. These studies are being used bygovernments and schools to promote changes inteaching and learning, not only in Europe butalso in many other countries, including the US(see, e.g., http://nces.ed.gov/timss).

Science and Mathematics in schools has alwaysbeen considered a difficult subject by mostpeople, in probably all countries. And it is...particularly when teachers assume that only “thebest” students can learn it. The famousPygmalion Effect is a uniquely humanphenomenon: a persistently held belief becomesa reality.

At least since late 1960s, the mastery learningand the formative assessment movements and,later, research-based teaching has shown thatmost if not all students can learn, moreconcretely or more formally, the habits ofmind, the concepts and the ideas of science andmathematics.

Theories, like Howard Gardner’s Theory ofMultiple Intelligences (1983), recognize thatall human beings have different intelligences,connected to core operations (e.g., logical-mathematical, connected to number,categorization, and relations; spatial, connectedto accurate mental visualization, mentaltransformation of images). Different humanbeings have different degrees of eachintelligence, but all have some degree of allintelligences.

Learning how to make models, in science inmathematics, can be done by all students, withdifferent degrees of success. For example,modelling with tables, as shown below, is easilygrasped by all, but modelling with differentialequations can only be done by formal thinkers,with a long training path.



Knowledge ofScience andMathematics

Teachers can be defined as professionallearners. What they learnt in the University is avery small fraction of what they need to learnduring their professional life, particularly insubjects like science and mathematics, which arein the front run of intellectual and technologicalchange and progress.

For example, for a teacher that finished his orher professional training in early 1980s, thefollowing is completely or partially new:cellular phones, personal computers, universalnetworks, graphical based software, digitalmedia, genetic manipulation, many materialsused in fabrics and sports, etc. Thesedevelopments where only possible due to scienceand technology and most are now part of theeveryday life of students.

It can be said that foundational knowledgestays more or less the same for decades. That’sprobably true. The current scientific paradigms(in the Kuhnian sense, Kuhn, 1962)–quantummechanics, relativity, plate tectonics, big-bangtheory, evolution, genetics, etc., were developedbetween 1900 and 1980. But who can learndeeply even a very specific subject in theUniversity? And students, particularly the youngones, are always eager for specificknowledge–only a teacher who is always alearner can give “food for thought” to hisstudents.

Experienced teacher’s supervisors know thatthose teachers who do not regularly study newand old things tend to have problemsappreciating student learning difficulties. Thatcan be easily understood: if someone repeatsmany times what he teaches, it becomes trivialand completely familiar to him. But if he isalways studying, learning difficulties areconstantly present and he can understand ofhow difficult it can be for students to learnsomething they are not familiar with.

Teachers study continuously to have a broadand deep knowledge of the concepts,principles, techniques, and reasoning methodsof mathematics and science (and theconnections between them and with otherfields of knowledge), and they use thisknowledge to establish curricular goals andshape their instruction and assessment.



Knowledge ofStudents

Traditional teachings methodologies were basedon teaching the same, in the same conditions,with the same approach to all students. On thesecond half on the 20th century, “teaching thesame to all” was discarded as a feasiblemethodology due to multiple factors(generalization of secondary studies, results ofeducational and psychological research,multiculturalism in schools, etc.).

The now dominant current practices recognizeeach student as a different learner, withdifferent personal knowledge and skills. A goodmetaphor (suggested by Bruner, 1960) for thestudents learning path is a spiral line.Different students can be at different places onthe spiral line, on each class. The spiral formsuggests that learning progress is not linear andit happens with cycles and steps forward.

According to Bruner, the spiral learning approachcan be characterised by a continuum with threereference levels: enactive, iconic and symbolic(these levels were inspired on the developmentalpsychology of Jean Piaget). On the enactivelevel, the child manipulate materials directly.On the iconic level, deals with mental imagesof objects but does not manipulate them directly.On the symbolic level, can manipulate symbolsand no longer needs mental images or objects.

To understand how students learn, Bruner alsoproposed an important principle: cognitiveprocesses precede perceptions rather thanthe other way around. The relevance of thisprinciple was reinforced by science andmathematics constructivist authors. For example,Driver (1983), on her seminal book The Pupil asScientist, wrote:

'Looking at' is not a passive recording of animage like a photograph being produced by acamera, but it is an active process in whichthe observer is checking his perception againsthis expectations (pp. 11-12).

Teachers know and care about their students,know how they learn and develop,understand the impact of home life and culturalbackground, and use this knowledge to guidetheir curricular and instructional decisions.



Knowledge ofthe Art ofTeaching

Active or practical learning methods are not apanacea for learning. Practical activities don’tmean anything if students don’t have anchoringconcepts on their cognitive structures. Thebalance between practical and discoverymethods is probably the most important issue ateacher must manage when planning andpromoting learning activities. As Driver (1983)pointed out:

If we wish children to develop andunderstanding of the conventional conceptsand principles of science, more is requiredthan simply providing practical experiences.The theoretical models and scientificconventions will not be 'discovered' bychildren through their practical work. Theyneed to be presented. Guidance is thenneeded to help children assimilate theirpractical experiences into what is possibly anew way of thinking about them (p. 9).

Quoting a famous proverb, («I do and Iunderstand»), Driver changed it to betterillustrate how relevant students’ knowledgestructures are: «I do and I am even moreconfused».

Papert (1980), a pioneer of computers ineducation, wrote about the importance ofstudents being “computer off”, reflecting anddiscussing ideas and procedures. Technologicaldevices, such as computers, can reinforce the“button syndrome” (users tend to press themaximum number of buttons on the shortesttime...) if practical learning activities are done fortheir own sake, without previous preparation,reflection and discussion.

In a simple short statement, the art of teachingis the art of balancing “minds-on” activitieswith “hands-on” activities, giving studentssufficient time to internalise concepts andbuild a coherent cognitive structure.

Teachers have an extensive base ofpedagogical knowledge to stimulate, motivateand facilitate student learning, using a widerange of formats and procedures to createenvironments in which students are activelearners, show willingness to take intellectualrisks, develop confidence and self-esteem,and value knowledge.



Science as a Wayof Thinking

Science is not just about facts. It is bestdescribed as a “way of thinking” about theworld, as the Nobel winner Leon Ledermanwrote (1998).

The fist amazing idea of science as a way ofthinking is that the “world is understandable”(as Einstein pointed out), i.e., the world isaccessible to human thought. Humanity took along time to adopt this point of view. We knownow that our understanding of the world islimited–science makes and test models abouthow the world works, but models are not the“real thing” (see, e.g., Giere, 1989)–but theessence of Einstein’s idea is a breakthrough ofscientific thought.

A second fundamental way of thinking in sciencecan be described as informed scepticism.Popper (1989) has shown that any statement canonly be a scientific statement if it “can be testedby systematic attempts to refute them”. Contraryto common sense (including students’ commonsense!), scientific ideas are not dogmas and theyare always subject to test and refutation. Thisdoesn’t mean that it is easy to accept a newscientific view: for example, Planck wrote, in hisautobiography (1950), how difficult it is for mostscientists.

Before formal schooling was invented, family,religion and apprenticeship were the availablemodes to transmit and share ways of thinking,values, attitudes, and skills from one generationto the next. They still have today a relevantplace, together with media and peers. Butscience requires a way of thinking that cannot betransmitted by these modes and need a moreformal setting to develop. For example, carefulcontrolled observation, based on theory andexpectations, is a typical process/way of thinkingof science not used in everyday contexts. Thesame is true for most other scientific processes.

Teachers develop students’ abilities to reasonand think alone or with support from others,to investigate and explore patterns, to discoverstructures and establish relationships, toformulate and solve problems, to justify andcommunicate their conclusions, and to questionand extend those conclusions.



Reflection andProfessionalGrowth

A traditional classroom is an isolated place forthe teacher: he is “alone”... with his students.Working together with other teachers, usingteam teaching or just preparing worksheets, issomething relatively new in most schools and itis not a current practice in many.

Seeing themselves as partners with otherteachers is an essential issue, particularly whenthe use of technology tools is embedded in thecurriculum. Teachers have many opportunitiesand motivations to collaborate, learn, and worktogether with computer tools, either becausethese tools facilitate interaction (e.g., email andchat) or because technological difficultiescannot most of the time be solved by only oneteacher.

Through reflection and collaboration, ateacher can develop the art and science of goodteaching practice. Reflection requires thoughtfuland careful reporting and analysis of teachingpractice, philosophy, and experience.Understanding why an activity or practice wasproductive or non-productive in the classroom isa key element in professional development.

Teacher training has been a recurrent topic ofeducational research and educational policy. Butthe concept of “teacher training” is acontroversial one because “training” is a narrowconcept if one envisions teaching as much morethan using a repertoire of teaching techniques.“Professional development” or “professionalgrowth”, particularly for those who are certifiedteachers, is a much more inclusive concept toname what is desirable for the teachingprofession.

Schools that have a technology embeddedcurriculum need to share a vision of themselvesas collaborative learning communities andlearning organizations (i.e., a group of peoplewho are continually enhancing their capabilitiesto create what they want to create, Senge,2000). The pace of change in society and intechnology doesn’t offer any opportunity for aless demanding vision.

Teachers reflect on what and how they teachand collaborate with others to strengthen thelearning community.



ThirteenProposals fora ModellingMethodology,withIllustrations

As discussed above, a pedagogical methodologyis seen, in this report, as a set of proceduresthat a teacher can develop in order to helpall students learn.

A methodology is something a teacher develops,based on his or her own experience andknowledge and on proposals made by others(scientists, peers, teachers’ educators, etc.). Amethodology is, then, a complex result ofinstruction, personal experience andreflection.

The following thirteen proposals highlightrelevant procedures to help teachers build acoherent methodology. Most proposals areillustrated with specific Modelling Space examplesin the following pages.

1 Make clear goals and plan how conceptsand ideas evolve during the activities,anticipating learning difficulties.

2 Elicit and verbalize students’ conceptions.

3 Promote interaction, collaboration, andgroup cohesion.

4 Give prompt feedback.

5 Induce self and group formativeassessment.

6 Proceed from concrete to abstract.

7 Verbalize mathematical procedures.

8 Promote schematic drawing and writingas “tools-to-think-with”.

9 Scaffold the transition from directcomputations to algebraic reasoning,from number sense to symbol sense.

10 Explore multiple representations.

11 Make abstract objects as concrete aspossible but spot the differences betweenthe “real thing” and the representation.

12 Balance discovery and exploratory learningwith guided learning.

13 Anticipate, check, and revise thecoherence of models and data.



Make clear goals andplan how conceptsand ideas evolveduring the activities,anticipating learningdifficulties

1Establishing goals and objectives is a commonpractice in education since late 1950s, after thepublication of Bloom's Taxonomy of EducationalObjectives (1956). Objectives and goals offereasily understandable guidelines forsystematic planning and evaluation, coveringthe multiple cognitive processes (and not onlythe lower mental processes, such asmemorization). Goals must put a biggeremphasis on processes rather than on contentmatter.

For example, most students have problemsunderstanding ratios (e.g., "an object is fivetimes larger than other"). Then, an activity onthe direct proportionality relation can bepreceded by a familiar context, as far aspossible, where learners can use ratios. The goalis direct proportionality but before reaching thisgoal, understanding ratios is a fundamental step.Most learning difficulties can and should beanticipated, giving students less demandingactivities, in order to help them make smoothtransitions, from familiar contexts to more formalones.

Images and sizes. Usinggrids, students discuss howmany times the height of animage is bigger that other.

This discussionis an essentialpre-requisiteto understandproportionaly.

The goal is a “trip to theMoon”... and someelementary mathematicsabout proportionality(direct and inverse).



Elicit and verbalizestudents’ conceptions

2Eliciting and verbalizing is a powerful didactictool. Teachers must give many opportunities tostudents talk about what they see, do, andthink. Typical teacher encouragements can be:

• What are you seeing on the screen (or onthe paper)?

• Can you describe what you have done?

• What do you think it will happen?

• What do you think of this?

• What does this word/statement mean toyou?

• What are the limits of ...?

• What does this image mean to you?

• What is the relevant information you cangather from this ...?

It is possible to include these type of elicitingactivities as part of written worksheets but abalance must be made be between “asking”and “recording”.

Some examples of eliciting questions:

What does “0” meansin this image?

What happens when timestarts? What does it mean tosay “time starts running”?

How does the height of thewater increases as time runs?

What will happen if timecontinues to run?

What do you “read” onthese two arrows?

Can you plan and make a table of values fromthis animation?

What is the maximum valueof this variable or entity?



Promote interaction,collaboration, andgroup cohesion

3Most if not all modelling activities that can bedone with Modelling Space should be made bygroups of 2 to 4 students. Teachers can givestudents the chance to choose their partners, orcan put mixed ability students in the samegroup.

A typical activity starts “computer-off”:students read and discuss the goal of theactivity, as stated on the worksheet. Teacherssupervise students reading and make clear howimportant is to understand what is the goal ofthe activity. Students should also be encouragedto “browse” through all the pages of theworksheet to have a global idea of its contentand purpose, actions and records required, andto have a feeling of the time required tocomplete it. Certain activities can only be doneon more than one class period.

Worksheets should encourage group interaction.E.g., instead of asking “what do you think of...”it can say “what does your group think of...”.

For younger students, it can be interesting toallow students to choose a name for theirgroup. Students usually choose “strong andoriginal” names, which helps to create groupcohesion and identity. It can also be useful toassign roles in each group (e.g., keyboardmanager, worksheet record coordinator,spokesperson for the group). These roles canrotate by group members in different activities.

Modelling Space allows collaborative activities,locally and remotely. Collaborative environmentsdemand extra-effort from the teacher tosupervise the discussions among the students, aswell as their work on the shared or the individualworkspaces, but it usually pay-off sincestudents become more aware of their ownreasoning and teachers know explicitly howstudents reason. Three main strategies can beused: “tutoring” between groups (one groupexplains the other how to make a model);“coaching” between groups (one groupobserves the work done by the other andintervenes to give advices and makesuggestions); “confrontation” betweengroups (both groups discuss the work done bythe other and both intervene to support actions).In certain cases, activity logs can provideteachers with rich information about studentsreasoning and knowledge construction.



Give prompt feedback

4Feedback is the information provided to alearner or to a group concerning the correctness,appropriateness, or accuracy of actions.Feedback occurs only after an action, isobservable and describes the effects of theaction. Feedback is, essentially, informationabout performance.

Feedback is pervasive in teacher-classinteractions but it less common on an individualbasis since most classroom activities involves theclass as whole.

Feedback is generally considered a criticalcomponent in effective teaching and learning.The original behaviourist view of feedback as areinforcer automatically linked to responses hasbeen changed to new views and practices withthe raise of cognitive theories of learning.Feedback is now viewed as more linked withlearner's cognitive processes rather thensolely with answers produced by learners. Typicalinstances of feedback involves studentsdescribing their thought processes when solvinga problem, reading a text aloud and discussing it,managing the computer, listen to the thoughtprocess described by teachers, and then comparewhat they did to what the teacher did.

When using ModellingSpace, teachers have manyopportunities to give feedback to students, bothon a individual basis and on a group basis. Thepedagogical question is how to give promptfeedback. In order to succeed in giving promptand meaningful feedback, teachers must beproactive. The pattern of proactive teacherbehaviour includes:

• walking around all groups and sitting witheach group if necessary;

• reading aloud to the group what studentshave recorded on worksheets and whatthey have done on the screen and discussit;

• asking questions that can provokethinking and verbalization.

• writing comments, suggestions, andquestions directly on students’worksheets.

• making schematic drawings thatdescribe ideas and processes and discussthem with students.



Induce self and groupformative assessment

5Students’ views on assessment are still morerelated to summative assessment (related toknowledge and skills certification) than formativeassessment, a concept that arose in late 1960s(Scriven, 1967).

Formative assessment is, essentially, a way oflearning, and should be part of all learningactivities. It provides feedback for improvementof learning and instruction and occursthroughout the learning process.

The following are characteristics of formativeassessment:

• it is detailed and provide specificinformation for improvement rather thansingle composite scores in the form ofmarks or grades;

• it is nonthreatening to the student and tothe group so as not to stimulatedefensiveness and rejection but they areconscious that they are being evaluated;

• it is timeliness;

• value judgments are explicit and availableto the student, in context.

In order to promote formative assessment duringModelling Space activities teachers should:

• include questions like “write down yourgroup predictions” on worksheets;

• give preferably indirect hints aboutstudents’ work;

• promote group interaction to assessstudents’ answers (e.g., group B analyzethe answers of group A);

• show correctly filled worksheets to givestudents reference standards on how toanswer certain types of questions;

• make value judgments explicit tostudents and groups, accompanied byreflection hints (e.g.: “This is correct, canyou explain why?”. “Something is stillmissing in this in order to be correct. Canyou find what is missing?”).



Proceed from concreteto abstract

6Science and mathematics is about formalmodels of objects, not about the objectsthemselves (Giere, 1989). This is a subtle anddifficult distinction to novices. Novices'knowledge usually consists mostly of theobjects and features explicitly presented inspecific situations. For example, most novicesclassify introductory physics problems as, e.g.,inclined plane problems or projectile problems.Their solution procedures are usually syntacticand specific, translating problem statementsdirectly into “formulas”, that should have analmost immediately “solution”. In contrast,experts' knowledge tend to represent problemsby the relevant implicit physics concepts,quantities and principles, usually not mentioneddirectly in the problem description (Chi et al.1981).

As multiple authors have shown, in childdevelopment there is a shift from verbatimrepresentations to gist representations, ashift from concrete to abstract. IntroductoryModelling Space activities should be designed toscaffold this shift. For example:

• whenever adequate, start with concreteentities, like “car” or “clock”;

• discuss what are the properties of eachentity relevant to a certain problemsituation;

• define operationally the relevantproperties when analyzing a problem;

• discuss how can properties be measured,i.e., expressed by numbers that representquantities.

A “concrete” model, made with “objects”(images have a high degree of perceptualfidelity). Some questions to discuss: Whatare the relevant properties of the “clock”and the “recipient” one is interested tomodel the situation? How can the propertiesbe operationalized and mesuared?

A more formal model of thesame problem... and of manysimilar problems.



Verbalizemathematicalprocedures

7Students can learn to articulate reasoning bypresenting their thinking to themselves, to theirpeers and to adults.

Verbalization is a powerful heuristics studentscan use to learn and to become conscious of howand what they have learnt, growing into moreself-sufficient learners. Self-sufficient learners(Miller and Brewster, 1992) are those who knowwhat can affect their cognition, know how tocontrol their own cognitive endeavours, andbelieve that cognitive effort results in academicsuccess. This heuristic encourages students totalk (and write) to themselves and to their peersas they engage in learning and problem-solvingprocesses. The types of verbalizations include:

• analyzing the task and the goals to beachieved (e.g., “what do I have to do?”);

• formulating a plan of action (e.g., “howshould I do it?”);

• describing by words mathematicalrelationships (e.g., “if this increases ntimes, this decreases n times”);

• evaluating the effectiveness of each ofthe steps in the plan (e.g., “how am Idoing?”);

• giving themselves positivefeedback for succeeding witheach step (e.g., “that's finework.”);

• dealing with obstacles,employing corrective actions(e.g., “that's not completelyright.”).

Teachers should model and displaythese steps and the verbalizations inlearning situations, in order to providestudents with information about why theverbalizations are necessary and howeffectively they work.

This introductory page isonly about promotingverbalization of linearrelationships...



Promote schematicdrawing and writingas “tools-to-think-with”

8Young students’ abilities to talk and listen areusually more advanced than their abilities to readand write. Therefore, teachers must provideexperiences that allow varied forms of writtencommunication as a natural component ofmodelling activities.

Recording and communication should alwaysbe explicit activities. Students can makedifferent types of records, ranging fromwords, simple sentences, simple drawings ormore elaborated records, like concept maps,careful schematic drawings or even full reports.

Drawing and writing are usually preferable totalk. In fact, drawing and writing demands moretime and consciousness than talk and can bemore easily revised by students.

Drawing and writing usually starts withthinking-aloud, an activity that must becommon to teachers and students alike. Studentscan learn to think aloud if the teacher usuallyillustrates that procedure and shows how it canbe used to make records and re-examinereasoning.

A good record is effective to oneself and to otherstudents. For example, the record below is aclear restatement of what’s on the text and caneasily be understood by any reader. Students

become better at listening,paraphrasing, questioning, andinterpreting others’ ideas when they canrecord them, using their own words anddrawings. With time, their recordsbecome more precise.

A drawing made by a student todescribe what happens to the distancetravelled by a car in one hour.

“This car moves at aspeed of 100 km/h.”

“1 hour after”

“This car has travelled100 km.”



Scaffold thetransition from directcomputations toalgebraic reasoning,from number sense tosymbol sense

9Different authors (e.g., Kaput, 2001) haveargued that developing algebraic reasoningsince elementary levels is critical formathematics teaching and learning. It isnecessary to reconceptualize the nature ofalgebra and algebraic thinking and how it relatesto arithmetic. The traditional but artificialseparation of arithmetic (where numbers aretreated as “real objects” that one canmanipulate) and algebra (where symbols aretreated as quantities and “real objects”–symbolsense) makes more difficult for students to learnalgebra and denies students of powerfulmathematical ideas.

Familiarity takes a long time to develop, andalgebraic thinking is no exception. But one mustbe careful because the idea is not to push thecurrent typical high school algebra curriculum tomore elementary grades. The goal is only todevelop algebraic reasoning, not skilled use ofcommon algebra procedures.

The two central algebraic procedures are (1)making generalizations and (2) usingsymbols to represent mathematical quantitiesand solve problems. A typical generalization isthat multiplying a number by a fraction of 1results on a smaller number and a commonsymbolic procedure is representing that “quantity

a is the double of quantity b” by“a = 2 × b”.

Modelling Space activities mustsmoothly support this transition fromarithmetic reasoning (e.g., makingdirect computations on tables of values)to algebraic reasoning (e.g.,transforming a table of values of twovariables into an equation).

This table requires “number sense”...

This mathematical equation illustratesthe transition from directcomputations, as on the table, tosymbolic algebraic representation.



Explore multiplerepresentations

10Multiple representations (Perkins, Schwartz,West, & Wiske, 1995) are alternativecoordinated views of a phenomenon, modelor process in a certain cognitive domain, such asa graph, an equation, and a table.

The concept of multiple representations has beena recurrent concept in exploratory softwaredesign for science and mathematics, at leastsince the publication of Making sense of thefuture (Harvard Educational Technology Center,1988). In this position paper, the authors argueabout how computers can make a difference inlearning environments: they stress the fact thatcomputers can easily present simultaneouslyrepresentations of the same formal object, suchas a function (e.g., the analytical expression, atable of values, and a graph). Multiplerepresentations, emphasizing different aspects ofthe same idea and affording different sort ofanalyses, are now a “taken for granted” issue inmost educational software for science andmathematics.

Modelling Space activities should encourageteachers to use multiple representations,particularly of simple mathematicalrelationships, including those that cannot beexpressed by algebraic expressions. Forexample, the model below shows an oscillatoryrelation between variable b (on the right) andvariable a (on the left). The relation isestablished as a “graph” but a “table” is alsoavailable. Running the model, the small squarethat represents the value of b oscillates back andforth. Another representation, the “verbal” one,should be asked on the worksheet...

The small square, representing thevalue of variable b, oscillates backand forth as variable a increases.

The oscillatory relation betweenvariable b and variable a wasestablished using the mouse but itcan also be done introducing valueson the table.



Make abstract objectsas concrete aspossible but spot thedifferences betweenthe “real thing” andthe representation

11In the early beginning of computers in education,Papert (1980) wrote:

Stated most simply, my conjecture is thatthe computer can concretize (and personalize)the formal. Seen in this light, it is not justanother powerful educational tool. It is uniquein providing us with the means for addressingwhat Piaget and many others see as theobstacle which is overcome in the passagefrom child to adult thinking. I believe that itcan allow us to shift the boundary separatingconcrete and formal. Knowledge that isaccessible only through formal processes cannow be approached concretely. And the realmagic comes from the fact that thisknowledge includes those elements one needsto become a formal thinker (p. 21).

Reification (using formal objects as real objectsto think with) is at the realm of learning scienceand mathematics (Roitman, 1998). Hebenstreit(1987) coined a term that seems essential tounderstand how computers can help in thereification of knowledge. For Hebenstreit,computers allow the user to manipulate a newtype of objects, objects that he calls concrete-abstract objects. Concrete in the sense thatthey can be manipulated on the screen and reactas “real objects” and abstract because they canbe only physical or mathematical constructs suchas magnitudes, equations, fields, etc.

Modelling Space entities and relations are, inmost contexts, concrete-abstract objects,sometimes multiple levels of perceptualfidelity. Using abstract objects concretely, as inModelling Space, can be a powerful didacticheuristics. But there also pitfalls, since asignificative number of students tend to considerrepresentations as the “real thing” (Justi &Gilbert, 2002).



Balance discoveryand exploratorylearning with guidedlearning

12The inquiry curricula, associated with discoveryand exploratory learning, were inspired byPiaget and Bruner’s ideas, notably the The act ofdiscovery paper (1961), where Bruner arguesthat discovery learning is superior to learningbased on expository modes.

A very different position has been assumed byother cognitive theorists such as e.g. Ausubel(1963, 2000). Ausubel’s most common critiqueof discovery learning is that although it can beeffective in certain specific situations, for themost part it is cumbersome and overly time-consuming. Discovery learning also demandsfrom the teacher a greater contextualization inorder to have a better chance of retention thanrote memorization of a procedure. Accordingly toAusubel (2000), most meaningful learning isassociated with reception learning, not withdiscovery learning. In opposition to Bruner,Ausubel argues that discovery learning can alsobe “rote in nature because it does not conformto the conditions of meaningful learning” (p. 5)and that meaningful reception learning is anactive process, not a passive one, and requirescognitive analysis in order to define whichaspects of existing cognitive structure are mostrelevant to the new potentially meaningfulmaterial.

Balancing exploratory and guided learning is afundamental issue in the use of computer toolslike modelling software. Teachers should alwaysbear in mind that learners cannot explorewhat they don’t know already!

The balance between exploratory learning anddirect instruction must be managed bycurriculum developers and by teachers. Thisbalance can be supported via two means:

1 the student activity sheets;

2 teachers’ strategies during the sessions.

Activity sheets play a crucial role for thisbalance. When someone try to ask questions in awritten way, it is difficult to balance it , and it isdifficult to avoid saying something that finally,guide students directly. Often a question that ispresented later, reveal something that studentshave to discover previously. For this reason, oneneed to gradual delivery short and separatesheets.



Anticipate, check, andrevise the coherenceof models and data

13Models are abstract constructions of humanmind. They cannot be “true” in the sense thatthey are “like the real thing”. Models onlyrepresent some features of the “real thing”.There are two important words in the previousstatement: represent and some. Both wordsestablish the epistemological status of models.These status must be explicitly discussed andlearnt by students.

At all levels of modelling, from planning a realtrip to the Moon to learning simple models ofdirect and inverse proportions, a fundamentalissue is the anticipation of the coherence of themodel and data.

For example, a student can start making amodel of the trip to the Moon anticipating thattime of travel grows with distance anddecreases with speed... The data for thisstatement comes from common sense experiencebut it can also come from other sources, such asgraphs of experimental results.

Anticipation is absolutely fundamental to check ifthe model makes sense, once it is done. If thestudent doesn’t anticipate what he will get withthe model, he is more ready to acceptanything... and that happens frequently, usuallybased on a mislead view that “the computer isallways right”.

A possible sequence for modelling activitiesis:

1 discuss and describe the features ofwhat is going to be modelled;

2 select what properties will be used on themodel;

3 anticipate possible relations betweenthe properties (if possible, get real dataon these relations);

4 analyse and discuss the relations andmake a first model, guided by aworksheet, by the teacher, or by groupdiscussion;

5 run the model and compare the outputwith the predictions;

6 make “what-if” investigations, discuss andrevise, as necessary.



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Ausubel, D. P. (2000). The acquisition andretention of knowledge: A cognitive view.Dordrecht: Kluwer Academic Publishers.

Ausubel, D. (1968). Educational psychology: Acognitive view. N. Y.: Holt, Rinehart and Winston.

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Bruner, J. (1960). The process of education.Cambridge, MA: Harvard University Press.

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Justi, R. S., & Gilbert, J. K. (2002). Scienceteachers’ knowledge about and attitudes towardsthe use of models and modelling in learningscience. International Journal of ScienceEducation, 24(12), 1273-1292.

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