TOC The Autopoiesis of Social Systems Constructionism and its Criticisms Introduction The Autopoiesis of Social Systems From self-reference to autopoiesis of social systems Applications of autopoiesis to social systems Building Bridges to Algebra through Biological criticisms Sociological criticisms Conclusion a Constructionist Learning Environment Open Peer Commentaries What Is Sociology? Eirini Geraniou • University College London, UK • e.geraniou/at/ioe.ac.uk Towards a Consistent Constructivist General Systems Theory Manolis Mavrikis • University College London, UK • m.mavrikis/at/ioe.ac.uk Does Social Systems Theory Need Autopoiesis? Does Social Systems Theory Need a General Theory of Autopoiesis? Missing: The Socio-Political Dimension > Context • In the digital era, it is important to investigate the potential impact of digital technologies in education Autopoiesis Applies to Social Systems Only and how such tools can be successfully integrated into the mathematics classroom. Similarly to many others in the Explaining Social Systems without Humans constructionism community, we have been inspired by the idea set out originally by Papert of providing students Communication is Meaning-based Autopoiesis with appropriate “vehicles” for developing “Mathematical Ways of Thinking.” > Problem • A crucial issue regarding the The Concept of Autopoiesis: Its Relevance and Consequences for Sociology design of digital tools as vehicles is that of “transfer” or “bridging” i.e., what mathematical knowledge is transferred The Concept of Autopoiesis from students’ interactions with such tools to other activities such as when they are doing “paper-and-pencil” Authors’ Response mathematics, undertaking traditional exam papers or in other formal and informal settings. > Method • Through the On the Criticisms against the Autopoiesis of Social Systems lens of a framework for algebraic ways of thinking, this article analyses data gathered as part of the MiGen project Authors’ Response from studies aiming at investigating ways to build bridges to formal algebra. > Results • The analysis supports the Combined References need for and benefit of bridging activities that make the connections to algebra explicit and for frequent reflection and consolidation tasks. > Implications • Task and digital environment designers should consider designing bridging activities that consolidate, support and sustain students’ mathematical ways of thinking beyond their digital experience. > Constructivist content • Our more general aim is to support the implementation of digital technologies, especially constructionist learning environments, in the mathematics classroom. > Key words • Algebraic generalisation and language, transition, exploratory learning, microworlds, bridging tasks.

Introduction designed to help them with. Therefore the ment (eXpresser) specially designed to sup- tool cannot immediately become an efficient port and address students’ difficulties with « 1 » There is a growing concern that mathematical instrument for them (Artigue learning algebra (see Mavrikis et al. 2013) despite the increased availability of dig- 2002). Consequently, teachers can be hesi- – to paper-and-pencil (PaP) activities. ital technologies designed for mathematics tant to use such tools as it is hard for them « 2 » We are not the only ones con- learning, students rarely use ideas, concepts to be convinced of the tools’ value and they cerned with the transition to formal algebra or strategies that they could acquire through are reluctant to dedicate time to learning (e.g., Radford 2014), and the literature is their interaction with such technologies in how to use them effectively and incorporate replete with examples of student difficulties other contexts (cf. EACEA Eurydice Report them into mathematics teaching practice (e.g., Stacey & Macgregor 2002). Our view 321 2011). As such, the full or intended po- (Clark-Wilson, Robutti & Sinclair 2014). is similar to that of Luis Radford (2014), tential of such technologies is diminished. However, a growing awareness of digital who claimed that there is a need for spe- One of the earliest (and most articulate) technologies’ potential and limitations can cially designed classroom activity to support examples of these concerns is discussed in support teachers using these technologies students’ developmental path to formal al- Jean-Luc Gurtner (1992). Referring to the in the classroom (Abboud-Blanchard 2014). gebra, and to Gurtner (1992) and Stephen Logo environment, Gurtner demonstrated In this article, we discuss our approach to Godwin and Rob Beswetherick (2003), who that the tool’s features, which are designed supporting students’ transition in moving suggested presenting structured tasks, using to support students when faced with com- back and forth from paper-and-pencil to appropriate digital tools and making explicit plex mathematical problems, may impede interacting with digital tools. We therefore interventions during students’ interactions. them from making connections between consider ways of facilitating the integration We claim that a digital tool specially de- their work in Logo and any mathematical of digital technologies into the mathematics signed to support the development of alge- or geometrical ideas they are already famil- classroom in an effort to shed some light on braic ways of thinking (AWOT), together iar with and use when problems seem less this issue that did not have the attention it with carefully designed bridging activities, complex. One of the reasons behind this deserves, as yet. In particular, our focus is should scaffold the transition to formal al- is that students might know how to use a on the transition to formal algebra and how gebra. Besides “learning” the tool and devel- digital tool procedurally, but can fail to un- students “transfer” their knowledge from oping expertise in using it, students should derstand conceptually the mathematical their interactions with an algebraic micro- be supported in making the connections to concepts and procedures that the tool was world – a constructionist learning environ- the maths.

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 322 Educational Research Experiments in Constructionism org http://www.migen. ES/J02077X/1), no: (Award ESRC by funded is project follow-on MiGen the and RES-139-25-0381) no: (Award Programme Research Learning and Teaching ESRC/EPSRC e project Gen Mi the in out carried research the above, utf gnrlt cn e osdrd “the considered be can generality justify and express to represent.”need the Instead, to anything without systems representation 546) put it, students are to “learn routinely asked (1992: Kaput James As expressions. algebraic form to symbols and letters use to struggle “every” or “always” as such words of use the through rule general a expressing of capable students Even 1981). (Hart ality gener express using to vocabulary mathematical with inexperienced are and 2007), ters to represent let using behind idea the understand to gle strug Students 2002). Macgregor & Stacey (e.g., algebra learning in difficulties student of examples many revealed has literature education mathematics the and re searched extensively been has transition latter ( algebra to arithmetic from 1990), (Fuson from arithmetic to 1987), number (Cobb number to counting from transition the studied: much are life tools, into the mathematics classroom. mathematics intothe tools, digital similar of integration successful the as well as eXpressermicroworld, the of tion integra successful the regarding shared are outcomes research Some general. in gebra al to as well as tasks, PaP to transition this on initial discussed are analysis data our of the stages on based remarks concluding General tasks. generalisation algebraic PaP traditional to eXpresser, namely ronment, envi learning constructionist MiGen the with interacting from transition their port on bridging activities carefully designed to sup worked who students 11–14-year-old from gathered data present we article, this In environments. learning exploratory with interaction their through AWOT develop of ment students’ of understanding the

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Considering the issues discussed discussed issues the Considering ao tastos n learner’s a in transitions Major v ist 1 a ofrd ao gis in gains major offered has F ound any value ( atio n D s vol. 10,N°3 avis 1985). The The 1985). avis D uke uke & Graham ------2005b: 2). 2005b: algebra”of (Mason purpose and root heart, edge, edge, demonstrated and knowl of“entails transfer re-use that claim authors Other learning. or knowledge not and move who ones the are people that ers in fact a process of transition and he consid is another to location one from boundaries crossing as transfer, of instead transition as viewed has be should metaphor the that argued who (2003) Beach King with aligned is view Our Wagner2005). & diSessa (e.g., “transfer” on research of lot a There course of is communities. different for con structs different to refers it because quotes in transfer put We activities. PaP digital to tools with interactions their from knowl edge their “transfer” students whether and how investigate to im therefore, is perative, It tools. digital with PaP interacting from to forth and back moving of that transition, another with faced are students classroom, mathematics the in appearance their making increasingly are technologies learning learning to both similar and new situations.” that of application the and new learning something when learning past of learning use our of is “Transfer claims xiii) (2001: Haskell Robert Similarly, 2005). Wagner & (diSessa situations)” of class (or situation ‘new’ a in situations), of class (or situation nor verify their answers (Hieler, Gurtner Gurtner (Hieler, answers their verify actions nor their upon reflect necessarily but not tool the from get they feedback diate imme the on rely can “works” and tool the how discover they as tools, digital with act with students is observed who usually inter that one the is knowledge of type latter The procedure. the in actions successive tween only be apparentthe are those connections where actions, of series a of learning the as defined is knowledge Procedural tionships. rela in rich is and networks of connection a of consists knowledge Conceptual edge. knowl procedural and conceptual troduce in to (schemas) networks internal between connections of lack or existence the used (1986) Lefeivre Patricia and Hiebert James 1999; & diSessa Wagner 2005). For example, (Beach knowledge of types different revealed many has literature transferred, educational being the as is knowledge of type what consider to needs one knowledge, of « 5 » 5 « « 6 » 6 «

In the digital era, where digital digital where era, digital the In hn osdrn te “transfer” the considering When

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or acquired or in acquired one ------students’learning. on impact greater a have to hopefully and mathematics and tool digital the of main do the between possible as early as made and identified to be connections it as allows transfer) of notion the to opposed (as phor meta our bridge work the with and aligned We life. valued everyday the and life school the of aspects or domain same the within topics or domains different between to build try educators or the students connections describe to metaphor “bridging” the used also He 1987).” Perkins & (Salomon occur can mathematics to transfer before necessary is reflection) in rich is that (one practice Logo of period long rather “a that as mathematics” and practice Logo between observed, dom sel very and expected, con generally nections of type “the considered 247) (1992: teractions. Even though a lot of research research of lot a though Even teractions. in students’ during interventions making explicit and tools digital appropriate ing us tasks, structured presenting through is maths formal to bridges build to way One classrooms. mathematics their in nologies acquired have tech digital with interactions their through to seem they strategies or concepts ideas, use rarely students that gest sug observations anecdotal our and 1992) tions. connec these make to students support to isThe al tochallenge interactions. find ways connec digit their tions to behind mathematics the the make to able be then should students it, using in experts becoming and tool the “learning” Besides algebra. of bank for possible them to reach the mathematical im it rendering without algebra formal to transition the scaffold should example, for algebra, of topic the on misconceptions ble possi and difficulties students’ support to designed specially are that tools digital ing otherwise. approached have not would can rich visit areas mathematically that they that (1980) students who interact with Logo Papert Seymour in arguments the re visiting worth is it Beswetherick though, that Saying & 2003). Godwin 1992; Gurtner tools (e.g., and with digital more experience more gaining and interacting while ematics math on perspectives interesting ignore to tend students Moreover, 1988). Kieran & « 8 » 8 « « 7 » 7 «

In the case of Logo, Gurtner Gurtner Logo, of case the In Relevant research (e.g., Gurtner Gurtner (e.g., research Relevant transfer and suggested suggested and U s ------Constructionism Building Bridges to Algebra Eirini Geraniou & Manolis Mavrikis

has been carried out on how to design such dents’ interactions in the form of a database tools to address students’ difficulties with and interviews of teachers, and transcripts mathematical concepts (e.g., Hoyles & Noss and bridging activities specially designed to 1996; Noss et al. 2012), the issue of the in- gauge students’ knowledge. tegration of the tools in question needs to « 12 » We have presented results from be investigated. This involves looking into our data analysis of our various studies in a what happens after students interact with a number of papers (e.g., Mavrikis et al. 2013; digital environment and what resources can Noss et al. 2012; Geraniou et al. 2011). In render the transition to formal mathematics this article, however, we focus on the data successful. collected from the bridging activities stu- « 9 » Despite the advances in the tech- dents worked on during, but mostly after nological infrastructure in schools and the their final interaction with the eXpresser plethora of digital tools specially designed tool. We present here our initial analysis Figure 1 • A model for 7 red tiles surrounded to support the teaching and learning of offering examples of students’ typical re- by grey tiles. Students must construct a mathematics, the integration of digital tech- sponses. We plan to do more in depth anal- general model and find the general rule. nologies in mathematics education has not ysis for each type of bridging activity, as we always met expectations (e.g., Drijvers et started to do in the case of the collaborative al. 2010). One of the reasons seems to be activity presented later in the article (cf. « 13 » In the results section, we share teachers’ usual practices (Clark-Wilson, Geraniou et al. 2011). Since the eXpresser our insights gained from the initial analy- Robutti & Sinclair 2014). The teachers’ per- tool has been specially designed to support sis of this data on bridging activities and spectives and their abilities to develop a new students dealing with some well-known students’ strategies to solve figural pattern repertoire of appropriate teaching practices and researched misconceptions on algebra generalisation tasks without the support or for technology-rich classrooms can play a (Noss et al. 2012), the goal of this prelimi- immediate feedback of eXpresser. crucial role in identifying the best strate- nary analysis was to identify the impact gies in supporting the successful integration of those design decisions on students’ un- of digital technologies in the mathematics derstanding and reasoning about algebraic The eXpresser microworld classrooms (Ruthven 2007). generalisation, and whether students use and bridging activities « 10 » Considering all the above issues, any of the strategies they were encouraged in this paper the focus is on bridging activi- to employ while interacting with eXpresser « 14 » The MiGen system is a pedagogi- ties specially designed to support students’ on the PaP tasks and are successful in solv- cal and technical environment that is de- transition from their interactions with the ing figural pattern generalisation tasks. We signed to improve 11–14-year-old students’ MiGen constructionist learning environ- focused on two AWOT, as described in our learning of algebraic generalisation, a spe- ment to PaP activities outside the tool. We previous work (Mavrikis et al. 2013): cific and fundamental mathematical way share some research outcomes in an effort ƒƒ Perceiving structure and exploiting its of thinking. Its core component consists of to support students, and consequently their power, which is about noticing what a microworld, eXpresser, which is designed teachers, towards a successful integration of stays the same and what is repeated in to support students in their reasoning and the MiGen tool and other similar tools in a figural sequence so as to understand problem-solving of a class of generalisation 323 general into the mathematics classroom. how the sequence is “structured,” sup- tasks. Previous work (e.g., Küchemann 2010; porting therefore “the development Mason 2005b) has demonstrated in particu- of structural reasoning” and the hab- lar the potential of designing activities that Methodology its of “breaking things into parts” by can help students focus on the structure of identifying “the building blocks of a the figural pattern by providing generic ex- « 11 » Using a Design-Based Research structure” (Cuoco, Goldenberg & Mark amples and challenging them to identify the methodology (Design-Based Research Col- 1996); and rules that underlie them. In eXpresser, stu- lective 2003), over the past seven years we ƒƒ Recognising and articulating gener- dents construct figural patterns by express- carried out a number of studies in six dif- alisations, including expressing them ing their structure through repeated build- ferent schools in London, worked with 11 symbolically, which is the process of ing blocks of square tiles, and articulating mathematics teachers and collected data translating the observed structure in an the rules that underpin the calculation of the from 553 students aged 11–14 years old. Our algebraic expression, using formal al- number of tiles in the patterns. A typical eX- data comprises interviews and transcripts gebraic notation to write general rules presser activity asks the student to reproduce that are one-to-one and with small groups for numerical sequences. Students’ an- a dynamic model presented in a window that of students, video (mostly screen record- swers were viewed several times and appears on the side of the activity screen. ings) and audio files from interviews, obser- analysed using those two AWOT as an « 15 » Figure 1 shows a model where a vations that are one-to-one, of small groups analytical framework for interpreting row of red tiles is surrounded by grey tiles. and of classrooms, detailed logs from stu- students’ strategies. Students are asked to construct a model that

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 324 Educational Research Experiments in Constructionism corner) havebeenaccomplished. allthegoalsshownin“ActivityThe systemhasrecognisedthat window” (lowerleft-hand the left, andMyModelontheright), ruleforthetotalnumberofsurroundingtiles. andacorrect Figure 2 of the variable the of representationof a animatedsimultaneousthefeedbackon and eXpresser capitalises of on design The dom. ranattiles rednumberputerof change the matingthe model can test generalityThey tiles.red the by rule for the total number of tiles surrounding worksfor example)automaticallychosenis randomat

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n u suis suet ae pre are students studies, our In correct for the different val different the for correct Rule forthe number of blue tile Computer’sModel only Rule forthetotal surrounding the number oftiles s red tile coloured if the rule paper tasks exam-like Textbook s and Rule forthe number of green tile model s indi ------1 types of bridging activities: We4 tasks.designedPaPtransition their to ties (Figure 3), which are designed to support bridgingactivi ofnumber presenteda with are students afterwards, immediately and Throughout their interactions with eXpresser out of the total 553 students we worked with. students175 with used were and tasks tion at the end. These we referred to as consolida their interactions with eXpresser and not just used they strategieslem-solving prob the and mathematical concepts upon reflections students’ promote that activities for need the recognised we teachers, with studies,though, and after close collaboration impactanstrategiesthoseon not.laterorIn had eXpresser whether and paper on tasks similar solving for strategies their reveal to ately after the final eXpresser task, in an effort off-computerpresentedwith immedi tasks, two models and rules (see Geraniou et al. 2011). value for their Model andquite often they were observed using the same Model the for values different trial and explore to encouraged are students Instead ber. Model same the for presented necessarily etd n iue , r aiae i eXpresser in shownwhen to students. animated are 4, Figure in sented pre- model train-track grey the e.g., models, all therefore and activities eXpresser accompany to http://expresser.lkl.ac.uk any Modelany rule for the number of square tiles needed for general a derivemodel’s andstructure,task perceptionsterns,theirdependingonthe of pat combinations of and patterns different presser. eX in on work can students tasks harder 3 2 | | | tasks; eXpresser of sequence a throughout er eXpresswith interactions - their on flect re- studentsencourageto and intervene to used are which tasks, consolidation 5 4 3 are presented and on paper; that tasks generalisation pattern figural are which tasks, paper eXpresser-like ofalence derived their rules; regarding the equivalence or non-equiv- strategies justification students’ on cus fo- and task eXpresser an of end the at presentedare which tasks, collaborative

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Consolidation Task – Train-track

1. How many tiles are needed for Models 4, 8, 1 and 100? 2. If we use “M” to stand for the Model Number, how many tiles are needed for Model M? 3. Use the space below to explain the different parts of your rule – use the diagrams left or your own if it helps. (Note: Task models for these tasks are presented animated in eXpresser)

Collaborative Task – Equivalent expressions

5 Model number + 1 + 2 Model number 7 Model number + 5

5(n+1)+ 2n 7n+5

1. Convince each other that your model and rules are correct. 2. Can you explain to each other why the rules look different but are equivalent? Discuss and write your explanations. eXpresser-like Paper Task – Bridges

Model number 4 1. Find the rule for the number of tiles for any Model Number. 2. Find the number of tiles for Model Numbers 5, 10 and 100.

325 Model number 7

Text-book Paper Task – Tables and Chairs

4 tables 1. Find the general rule for the number of chairs for any number of tables. 2. Use your rule to find the number of chairs for 20 tables and for 200 tables. 3. If I have 26 chairs, how many tables do I need?

10 tables

Figure 4 • Examples of the 4 types of Bridging Activities.

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C d o D B A building block model’s 1 buildingblock | n tdns rm okn fr h term-to- the for looking from students prevent to designed carefully were ties activi bridging All structure.patterns’ the perceiving of terms students’ in approaches assess to as so sequentially, and involvepatternspresented,non- paperusually on students to given tasks generalisation traditional the to similar are which tasks,exam-like or text-book str u cti first v 1st modelonly extra 5tilesforthe a model+the There are7tiles ist F ound Figure 5 atio on the Train-track consolidationbridgingactivity. n s vol. 10,N°3 • Examples of13-year-old students’ answers 1 columnof take away building block repeating model there isone1ofthem shaded butbecause but notaextra5fromthe an extra7tilesareadded with everynewmodel 5 - / E C sented in Figure 4. preactivitiesbridgingare of typesfour Model any open-ended is(“Find the rule for questionthe number of tiles for task the and sequentially(eXpresser-like tasks)paper non- or (consolidationeXpresser tasks) eithermodelsarepresented animatedin term rule in a sequence. For example, the Model Model 2 1 N me”. xmls f the of Examplesumber”). same block staysthe repeats theblack The greenblock - though though even and Blocks), Building for stands (BBs BBs” red of “number as variable pendent inde her named Janet example, For tiles. five of one “L”-shaped an of that and tiles square three of column a of that eXpresser, in task this they solving were that they if use blocks could building two the drew 7 Figures in ample, ex For models. or blocks building of ber num e.g., terminology, eXpresser the ing us were they as eXpresser’sfeatures the by influenced as be to seemed they and terns pat task the visualise to ways demonstrated of variety a Students blocks. different building these identify to pens coloured used eXpresser, of feature colouring the by influenced perhaps them, of Some the pattern. of instances different the create time, every repeated which, parts, the and tern pat the of instance any in same the main clearly re would that Students parts different the marked respectively. task like paper textbook- Chairs” and “Tables the and task eXpresser-like “Bridges” the task, collaborative expressions” task, “Equivalent the consolidation “Train-track” answers the students’ on of examples pres some we ent 8, and 7 6, 5, Figures In given model. the of structure the visualised they how paper on presented figures model the on demonstrated students 175 the of most its outcome is presented in Figure 6. in presented is outcome its An and collaboration students’ two of example equivalence. their about conclude to them compared and rules and models their in blocks building their students identified verbally most task, collaborative the for Especially Figure 6). (see Janet with sions in her discus eXpresser’sused terminology initial resultsinitial two headings. under those our present we activities, bridging the ing - undertak when students’strategies preting interfor framework analytical an as earlier of three. of groups or pairs in worked they which for one, collaborative the bridging for except of activities, types the all on dependently « 18 » 18 « power its exploiting and structure Perceiving « 17» Results N U

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« 19 » In Figure 7 J, the student has de- rived two “theories” as they describe the two ways in which they were able to construct the Bridges model. In both these cases, they clearly identified the building blocks that are repeated to form the task model and used different colours to indicate which blocks, for example, need to be removed Properties Properties from the constructed model so that the task model is produced (two grey tiles for “The- = 4 = 4 = 4 = 4 = 4 ory 1”) and which building block would be added as an extra one to complete the task 4 Number of red BBs ++ 3 Number of red BBs 2 Nancy 7 + 5 1 model in “Theory 2” (the green building block of three tiles). For their “Theory 2,” they also indicated clearly that the blue tiles 5n+ 3(n+1)+ 2 7n+5 are used for the “repeating model,” whereas the green tiles are used for the “not repeated Nancy: Yeah. It’s one red building block plus one blue building block so that would actually model.” kind of make the… « 20 » In all these examples, it is evident Janet: yeah, it would make the same shape… how many different ways students have visu- Nancy: because one red building block added to one blue building block… alised the task models. There were also a few Janet: and that’s the same as one of my green building blocks. students who managed to derive a correct general rule, but they did not demonstrate Figure 6 • An example of two 12-year-old students’ discussion on paper the structure in which they possi- on the collaborative bridging activity. bly visualised the pattern. Such an example is given in Figure 7 F.

Recognising and articulating els, it should be (7 × M) + 5” (Figure 5 A) able and used these examples to justify fur- generalisations, including or “there are always 2 chairs to the ends of ther their derived rules. For example, in Fig- expressing them symbolically the single tables, then 2 chairs on the end ure 7 J, the student calculated the number of « 21 » Students seemed to rely on the of all tables put together” (Figure 8 K). But blocks for Model Number 50 and found that structure of the given task model in order the crucial step was their ability to translate there are 253 blocks. As they had derived a to articulate a general rule. Most of them that generalisation in parallel to their visu- second rule, i.e., 5 (m + 1) – 2 (top corner of provided clear explanations to justify their alised structures into general rules, as well the Figure 7 J), they could see that for the derived rules and share their solution and as to argue about similarities (or differences) same Model Number 50, both rules give the revealed some fluency in using the formal between their models and derived general same number of building blocks. In Figures algebraic language. They identified what rules when discussing rule equivalence (e.g., 8 K and 8 L, even though both students justi- 327 stayed the same and translated that into Figure 6). Most students used the eXpresser fied in words how they derived their rules, a constant in their rule. For example, in language and terms such as “model num- they even used numerical examples, choos- Figure 7 J, the student had even annotated ber” to represent the variable in their rule ing 7 chairs and 50 chairs respectively. his / her rule (5 × M) + 3. He / she showed (e.g., “5 × whatever model number n is +3,” « 24 » Of course challenges remain, and that the coefficient 5 is the number of build- Figure 7 F, as an intermediate step before even though the presentation of each of the ing blocks in his/her second building block, expressing their derived rules in a formal bridging activities was carefully designed to which, as he/she claimed, is repeated. The algebraic expression (e.g., “(5 × M) + 3,” Fig- prevent students from looking for the term- constant 3 is the number of tiles in their ure 7 J). eXpresser seems to have influenced to-term rule in a sequence, there were some first building block, which is not repeated, this outcome, as it encourages students to cases of students, especially in the text-book and “M is the “model number.” Similarly, the name their variables (“unlocked” numbers) like bridging activities, who reverted to student in Figure 7i successfully identified based on what its various values represent their past experiences and worked out the two building blocks that can produce the and therefore allows students to give mean- answers for each consecutive term in a se- task model, and indicated which building ing to that variable. This step eased students’ quence. For example, in Figure 8 N, the stu- block stays the same and which is repeated. transition to the formal algebraic language dent calculates the number of chairs when « 22 » Students’ answers revealed their and seems to have given meaning to the use having 1 table, then 2 tables, then 3 tables, ability to articulate general statements, of letters to represent “unknown” values. etc. He / she focused on the term-to-term such as “with every new model, another 7 « 23 » Some students evaluated their rule and managed to spot the correct gen- is added and if there’s “M” amount of mod- rules by using specific values for their vari- eral rule and wrote “Chairs = tables × 2 + 4.”

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 328 Educational Research Experiments in Constructionism

C J H F o Separate buildingblock This oneneverchanges,itmoves. n str 5 xwhatevermodelnumberitis+ u cti v ist it isapatternandcall This onedoesmovebecause F ound atio Figure 7 on the eXpresser-like Bridges bridging activity. n s vol. 10,N°3

•

Examples of 12-year-old students’ work M 3 I G (7+1)= For examplemodel7has8strands number +1 The numberofstrandsinthemodel where itis This blockstays 8

building block This istherepeated mathematics instruction. intotoolsICTof incorporate tanttouse the short- and long-term value and can be reluc busyconvincedworklivesbe tools’ totheof their in them for hard is it as lessons their in toolssuch usehesitant to be teacherscan Consequently, with. them help to designed cal concepts and procedures that the tool was understandingmathematiconceptuallythe in procedurally,fail maytool butthe use to how know may They interactions.their ing dur knowledge their consolidating and on reflecting necessarily without answer “an” or answer right the to get often and quote, terns as suggested by the student in the above beautifulproductions, such as colourful createpat tool, the with interact to how learn students though, often Quite mathematics. to engage them and support their learning of that developedandcarefullydesigned tools havebeen ICT of number a with interact as additional challenging tasks). challenging additional as given occasionally were which sequences, quadratic (e.g., easy been have not would strategy rule” “term-to-term a such which in task, generalisation pattern figural plex with a faced when more same strategy com the used have would student this whether investigate to interesting been have would It difficulties. researched and well-known have who students supporting to given be to emphasis greater and strategies their on constant reflections with dents’ stu interactions enrich to activities bridging through interventions more for need a is there haps Per develop. them help to designed was er do not the eXpresswhich transfer the skills students which in cases course of are There chairs.”more 2 in adds table “every writing view, to a functional butswitched with then start to approach empirical this used they approachor term-to-term a on relied solely either student the that reveals example This « 25 » Conclusion Researcher: “So, what did you learn from

Student: “I learned how to put tiles and make nice patterns in different colours” tdns oaas r akd to asked are nowadays Students interacting with the eXpresser tool?” (Excerpt from an early pilot study with eXpresser) ------Constructionism Building Bridges to Algebra Eirini Geraniou & Manolis Mavrikis

« 26 » In the case of the eXpresser tool, K L and as has been revealed in the few ex- amples presented in the previous section, most students seemed to have successfully “transferred” their gained knowledge or crossed the “bridge” from the eXpresser algebra to formal algebra. They have dem- onstrated their conceptual understanding of deriving a general rule and allowed us to claim they can generalise and adopt AWOT when solving paper and pencil figural pat- there is always 2 chairs to the ends of for any number of Chair tern generalisation tasks. Our experience the single table then 2 chairs on the And tables At the End each tble there will from the various studies for the MiGen end of all the tables put together be the Same amount there will always project so far has supported the need for For example: be two at each End bridging activities, whose objective is to RULE make the connections to algebra explicit. Tables x 2 + 4 The need and value of such activities have been claimed by Gurtner (1992: 253) too, M N who claimed that “the do-math-without- noticing-it philosophy of Logo can be abandoned in favour of techniques that ex- plicitly present looking for connections be- tween Logo and mathematics as an objec- tive of a task.” We also recognized the need for constant reflections by students when interacting with the eXpresser tool, which was achieved through the consolidation There always going to be 2 every table adds in 2 more chairs tasks. Similarly to our research outcomes, tables at each end of the table End table – 4 chairs Godwin and Beswetherick (2003), when and times the number Tables – 2 chairs investigating the use of Omnigraph in the of tables and add 4 more chairs Chairs = tables x 2 + 4 classroom, claimed that there is a need for Rule: Tables x 2 + 4 tasks that encourage more focused interac- tions by students in an effort to help them formalise and concretise their generalisa- Figure 8 • Examples of 12-year-old students’ work tions, notice relevant properties and devel- on the Tables and Chairs textbook-like bridging activity. op mathematical ways of thinking. There also seems to be a need for a long period 329 of practice with the eXpresser tool, and any mathematics digital tool, rich in reflection with notable exceptions of concentrated re- them to co-design activities around eX- and consolidation, before transfer to math- search efforts, e.g., Küchemann 2010). This presser and to use them for teaching. We ematics can be deemed possible. This view is the case for other areas of mathematical have seen several creative approaches di- has been supported by other researchers learning as well, and although digital tools rectly or indirectly aiming towards the ob- in the past (e.g., Noss et al. 2012; Gurtner like eXpresser provide part of the answer, jective of making links between students’ 1992; Godwin & Beswetherick 2003). the article demonstrated how carefully de- experience with eXpresser and algebra. For « 27 » Finally, we do not claim that our signed bridging activities may be of value. example, teachers have designed activities approach to bridging activities is the only « 28 » In a series of projects related to that invite students themselves to design way to encourage transfer, neither that the eXpresser tool6 we have engaged with eXpresser tasks and challenge their peers eXpresser is the only environment to help a number of teachers, trainee teachers or or create posters to share their views of eX- students develop AWOT. However, as we mathematics educators, and encouraged presser, its activities and what they believe have elaborated in Mavrikis et al. (2013), they have learned during their interactions the interaction with eXpresser provides a 6 | These projects include the ESRC funded and to identify similarities to traditional substrate of activity and experience for the “follow-on” project (ES/J02077X/1), the EU- algebra. There are several examples of this teaching and learning of algebraic gener- funded METAFORA project, http://www.meta- in the MiGen follow-on package resource alisations that is difficult to achieve with fora-project.org, and the M C Squared project, (http://link.lkl.ac.uk/migen-package), and traditional paper-based activities (perhaps http://mc2-project.eu. the M C Squared project (http://www.mc2-

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 330 Educational Research Experiments in Constructionism ideas. these around c-book a designing is that est inter of community a involves project.org) gets noticed by the interaction with such a such with interaction the by noticed gets that knowledge residual the is what above: mentioned as same the is daily) with teract in we teachers (and us for concern the But artefacts. share and create or puzzles to solve order in skills develop or use to tunity oppor the given are students where tools, programming or production elaborate games to mathematical from ranging tools,

C o n « 29 » 29 « str u cti

There are of course several several course of are There v ist F ound atio n s vol. 10,N°3 { { - - - i rmis o netgt frhr h is the further investigate to remains aim our vision, this consideration into Taking knowledge. such of sustainability the itly explic encourage to ways identifying and activities, interactions PaP to technologies digital students’ with from be transferred potentially can that knowledge the of awareness the to tool inter digital a from with acting transition the involves view constructionist learning environments? learning constructionist in of mathematics learning the support and explicit more it make we can how and tool students aswellteachersthroughouttheirinteractionswithinnovativedigital media. implements mathematicseducationresearchinactualclassroomsettings,supporting of materialandaccompanyingpedagogicalscenariosactivitiesformathematics. She Assessment, UCLInstituteofEducation.Herresearchworkfocusesontheeducational design is aLecturerinMathematicsEducationattheDepartmentofCurriculum,Pedagogy and Eirini Geraou increase teachers’awarenessandresearchers’understandingoftheprocessesinvolved inlearning. designing andevaluatingintelligenttechnologiesthatprovidedirectfeedbacktolearners and mathematics education,human-computerinteractionandartificialintelligence.He hasbeen His researchinterestsandexperiencelieattheintersectionoftechnology-enhanced learning, is aSeniorResearchFellowattheLondonKnowledgeLab(LKL),UCLInstituteofEducation. Manolis « 30 » 30 «

A successful integration in our our in integration successful A vrikis - - - mathematical ways of thinking. of ways mathematical that students’ sustain activities and support consolidate, bridging innovative and designed carefully through classroom the in tools digital of implementation the port sup and “bridging” and “transfer” of sues Received: 22FebruaryReceived: 2015 Accepted: 24April 2015 - Constructionism Proposing a Framework for Exploring “Bridging” Nicole Panorkou

Open Peer Commentaries on Eirini Geraniou & Manolis Mavrikis’s “Building Bridges to Algebra through a Constructionist Learning Environment”

Proposing a Framework « 2 » The authors study “transfer” ƒƒ Consolidation tasks for Exploring “Bridging” through a series of what they refer to as ƒƒ Collaborative tasks “bridging activities” to assist the “transition” ƒƒ eXpresser-like paper tasks Nicole Panorkou to formal algebra. Unlike Harry Broudy’s ƒƒ Textbook paper tasks. (1977) notion of “applicative knowing” or Since context plays such a vital role in this Montclair State University, USA the ability of students to apply their prior study, in this commentary I raise some de- panorkoun/at/mail.montclair.edu knowledge in order to solve new problems, sign issues that can be considered while Geraniou and Mavrikis’s goal is not just studying the notion of bridging. First, I > Upshot • Geraniou and Mavrikis raise about replicating or applying knowledge would like to challenge the authors to de- the important issue of “transfer,” when but rather about placing students “on a fine what they mean by “context.” If the goal students transition from activity in tech- trajectory towards expertise” (Bransford & of the design-based research is to develop nological tools to paper-and-pencil tasks. Schwartz 1999: 68) by scaffolding the transi- a contextual variety of bridging activities In this commentary, I contribute to the tion to formal algebra. In order to make this aimed at broadening the scope of the math- conversation by focusing on the relation- distinction, they take a view of transfer that ematical abstractions in the eXpresser activ- ship between task design and students’ is aligned with King Beach’s conception of ities, then the choices of context in the four development of knowledge. transfer as “transition,” since “people are the PaP tasks need to be made explicit. ones who move and not knowledge or learn- « 4 » The choice of context is influenced « 1 » Education researchers have tried ing” (Beach 2003: 3). Although the authors both by the initial eXpresser context and by to describe the situated nature of knowl- describe the notions of “transfer,” “bridging” the two algebraic ways of thinking (AWOT) edge that students construct as they interact and “transition” as separate but inter-relat- that the authors aim to explore, namely with contextual problems and digital tools. ed, at points throughout the article these are ƒƒ Perceiving structure and exploiting its Examples include the notions of “abstrac- used interchangeably. In my view, Geraniou power; and 331 tion in context” (Hershkowitz, Schwarz & and Mavrikis’s goal of transfer aligns with ƒƒ Recognizing and articulating gener- Dreyfus 2001), “situated generalizations” the metaphor of “bridging” as “a process alizations, including expressing them (Nemirovsky 2002) and “situated abstrac- of abstraction and connection making,” as symbolically. tions” (Hoyles & Noss 1992). What is com- described by David Perkins and Gavriel Design studies involve “‘engineering’ par- mon among these notions is the idea of Salomon (1988: 28). Consequently, bridg- ticular forms of learning and systematically constructing knowledge that is situated in ing can be seen as expanding what Dave studying those forms of learning within the the specific context in which that construc- Pratt and Richard Noss (2002, 2010) refer context defined by the means of support- tion takes place. Although powerful for to as students’ contextual neighbourhood, or ing them” (Cobb et al. 2003: 9). Likewise, developing advanced mathematical ideas, the range of contexts and variety of circum- the process of “bridging” can be “system- “there is, however, little evidence that stu- stances in which the students’ knowledge is atically” studied by making the iterations dents can abstract beyond the modeling made relevant and accessible. of the task design process explicit and context” (Doerr & Pratt 2008: 272). Eirini « 3 » By using a design-based research presenting how students’ development of Geraniou and Manolis Mavrikis make a methodology, Geraniou and Mavrikis de- AWOT informed the design of those itera- significant contribution to the field by rais- signed a series of “bridging activities” to ex- tions. In other words, it can be studied by ing the issue of the “transfer” of students’ plore how students use the situated knowl- exposing the study’s iterative nature, where constructed knowledge from the eXpresser edge they constructed through their activity researchers formulate conjectures about digital environment to paper-and-pencil in eXpresser to solve similar paper-and-pen- student learning and then revisit and re- activities. cil (PaP) tasks, namely: fine them throughout the study (Confrey

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 332 Educational Research Experiments in Constructionism - activi the of contribution the of analysis and schemes, those in changes of nations expla - schemes, initial students’ of nation expla - an of consists that AWOT of ment develop - students’ of trajectory a portray will models These 1982). Thompson 1983; developed thinking through the research process (Cobb & Steffe students’ how of els - mod constructing by described be can cess “bridging” process. “bridging” the studying for framework stronger a vide pro could tasks those with working while students’ about had ofthinking progression authors the any,conjectures if the and sign, choices of context, of iterative the de cycles the explicit makes that sequence) nature, (e.g., tasks those designing of process the showing mechanism A 2000). Lachance &

process include: AWOTQuestionsgoals. that may this guide the reach and knowledge their developing in students assist may sequence and design task the engineering of andspecific how the thinking student about conjectures of form process learning (Simon 2013, 2014), which takes the hypothetical the is investigation further with needs What basis. the activity as eXpresser students’ having by design task their began and goals those reaching for tasks of sequence a developed goals, ing described the two AWOT clearly they have eXpresser,as - learn in thinking students’ of Geraniou and Mavrikis constructed a model 4 3 2 1 through asequence of four steps: a framework bridging offer this structure to guide may activity through learning to C ƒ ƒ ƒ o | | | | ƒ ƒ ƒ n “bridging”? explain would that tasks those in dent stu - the of thinking the be could What tions provoked be contexts? new inthe opera- and schemes similar can How eXpresser? of context initial the in provoked were What schemes and operations of AWOT « 6» « 5» related learning process. D learningthe and goal; by specified abstraction the developing ly have available that can be the basis for Specify an activity that students current- alearningArticulate goal; conceptions; mathematical relevant students’ Assess str esign a task sequence and postulate a postulate and sequence task a esign u cti io’ (03 dsg approach design (2013) Simon’s Subsequently, the “bridging” pro- “bridging” the Subsequently, v ist F ound atio n s vol. 10,N°3 - - alization. gener algebraic to up build that landmarks as steps” “intermediate process those identifying by learning the during refined even or adapted modified, developed, been has knowledge students’ how of process the as described be then would “Bridging” sion. expres- algebraic formal a in rules derived variables in the rule before they express their represent to language eXpresser the of use students’ ofstep” “intermediate the present they AWOT, where second the of opment devel- the of discussion their in examplean evolves through design. The authors provide that process a as “bridging” of perspective dynamic the show would AWOT final the to initial the thinkingfrom of changesdiate students’ of description A 2004). 2003, (Steffe changes those in involved ties provide can aframework for “bridging.” knowledge of development and students’ design task between relationship the mechanismexplainsthatversationhowa of con- a initiate to was goal My studied. and described be can process bridging the ing dur thinking students’ how of suggestions presentingsome also and design “bridging” the to essential consider I that issues some raising conversationby the to contribute to State University, USA.Herresearchinterestsincludethe validation oflearningtrajectoriesforK-8mathematics. Nicole PanorkouisanassistantprofessoratMontclair development ofstudentlearninggeometry, algebra, of mathematicalconcepts;andthedevelopment and rationalnumberreasoning;afocusontheways « 7» that technologyandmodelingcanfostertheutility She holdsadoctorateinmathematicseducation from theUCLInstituteofEducationinLondon. In this commentary, I have tried tried have I commentary, this In Accepted: 16June 2015 Received: 22MayReceived: 2015 interme- - - 26). 2014: (Hewitt equations” of solving ematics math the and notation mathematical the on “attention purely is where to fading vide pro activities bridging the and knowledge construct students help to scaffolding vides pro environment digital the That is, ure 2). (Fig tasks” exam-like or “textbook of form the taking before environment MiGen the to resemble first on at that activities move paper-based then and it, within generalised patterns describing and constructing and environment the about learning with start Students tasks. paper-based to puter com the from transition students’ aid activities to “bridging” of sequence a scribe de They environment. that of outside fer trans might MiGen, called environment, microworld a in developed knowledge how investigated Mavrikis Manolis and raniou croworlds take a lot of work, and success, success, and work, of lot a take croworlds Mi (§8). classrooms” mathematics their in technologies digital with interactions their through acquired have to seem they egies that strat or concepts ideas, use “suggest rarely students experiences own their and state that literature the and raniou Mavrikis Ge Moreover, (§26). possible” deemed be can to mathematics […] transfer before tice prac of period long “a need students that conclude and environment, the with them familiarise to designed lessons two ceived re students that report authors The time. sustained a for experienced and learned be > Upshot i.jones/at/lboro.ac.uk Loughborough University, UK Ian Jones Functional andStructural Building Bridgesthatare btat o ocee otxs should contexts suchapproaches. complement concrete to abstract from transfer that case the make texts.I con- abstract to concrete from transfer is implicit also but paper, to computer the from learning They of transfer on building. report pattern through ships to relation- algebraic explore children environment help an describe Mavrikis « 1 » 1 « « 2 » 2 «

n hi tre atce Ern Ge Eirini article, target their In The MiGen environment needs to needs environment MiGen The In their article, Geraniou and Geraniou article, their •In ------Constructionism Building Bridges that are Functional and Structural Ian Jones

in terms of transfer to non-digital contexts, worlds can free students from operational formal notation, a virtual and manipulable is far from guaranteed. So is it worth the patterns in order to explore algebraic ways symbol system that closely resembles that trouble? of thinking, but operations are likely to be typically seen in textbooks and classrooms. « 3 » Constructionists argue that mi- prioritised again for some students when re- Therefore, transfer from a digital to a paper- croworlds provide a powerful resource for turning to more traditional presentations of based domain might be relatively natural immersing learners in mathematics. Ab- mathematical tasks. and intuitive for many students. stract objects and concepts become tangible, « 5 » At the heart of the MiGen philoso- « 7 » We can assume that construc- allowing trial and error experimentation, phy is another important aspect of transfer, tionist approaches to introducing formal mental reflection and discussion (Papert the shift from concrete to abstract knowl- algebra naturally align with both func- 1980). Students might then discover and edge. This has been a contentious issue of tional and structural approaches. Indeed, explore ideas that are otherwise be inacces- late, with a high-profile paper by Jennifer both approaches have been shown to lend sible to them, and can be challenged in ways Kaminski, Vladimir Sloutsky and Andrew themselves to the design of microworlds not always supported by typical classroom Heckler (2008) claiming mathematical ideas that enable tangible exploration and test- activities. Some readers of this journal will should be introduced in abstract contexts ing of conjectures such that formal symbol have experienced and studied this enabling to ensure better transfer, and others chal- systems become a natural and useful me- power of microworlds. In my own research, lenging their finding (e.g.. De Bock et al. dium of mathematical learning. Ideally, we students working in the SumPuzzles envi- 2011). The use of generalised patterns to might want learners to shift flexibly between ronment interacted with formal arithmetic support algebraic ways of thinking has been thinking about concrete referents such as equations in distinctly algebraic ways, fo- termed “functional approaches” (Kirshner generalisable patterns, and thinking with cussing on structure not calculation, and 2001). Appeals are made to children’s expe- formal symbols and their transformation did so with minimal explicit instruction riences of pattern and regularity, and tasks rules. Such a fluid and dialectic mixed-ap- (Jones & Pratt 2012). However, when the are designed such that formal algebra of- proach might be expected to strengthen al- plug is pulled, is the knowledge constructed fers a powerful medium for describing and gebraic experience and understanding, and by the student switched off along with the generalising patterns. Alternatives, which so promote transfer in the broadest sense of computer? Work such as that by Geraniou are perhaps less visible in the literature, are the term. and Mavrikis is important for exploring how “structural approaches.” These start with the students might be bridged to working with abstract (that is, formal symbols and their Ian Jones is a lecturer in the Mathematics formal mathematics on paper, and helping structural relationships, with no concern Education Centre at Loughborough University. to evaluate whether the scaffolding and fad- for real-world referents) and seek to nur- Prior to this he was a Royal Society Shuttleworth ing payoff is worthwhile. ture conceptual understanding that can be Education Research Fellow and has taught « 4 » Another form of transfer, or per- transferred to new contexts, be they abstract in various schools around the world. Ian haps more accurately transition, is implied or concrete. Structural approaches perhaps is widely published in the discipline of in the research; namely, the shift from arith- have a tarnished reputation, sometimes be- mathematics education, with a particular metic to algebraic ways of thinking. The ing associated with “meaningless” arithme- interest in the assessment of mathematical authors report that many students were tic and algebraic drill. However, carefully understanding and the cognitive processes successful with the final bridging task, and designed tasks can enable interactions with of learning to think mathematically. so claim that students “can generalise and formal notation and associated transforma- 333 adopt [algebraic ways of thinking] when tion rules in a rich, meaningful and educa- Received: 4 June 2015 solving paper and pencil figural pattern tionally valuable way (Dörfler 2006). Micro- Accepted: 16 June 2015 generalisation tasks” (§26). However, there worlds that take this approach have been were exceptions in which students “reverted found to motivate engagement with algebra- to their past experiences and worked out ic ways of thinking about formal notation the answers for each consecutive term in a systems (Hewitt 2014; Jones & Pratt 2012). sequence” (§24). Researchers working in « 6 » There are two potential reasons the early algebra field will be unsurprised to consider structural approaches as com- by this. Years of learning arithmetic using plements to functional approaches. First, conventional notation has been shown to whereas functional approaches typically develop “operational patterns” (McNeil & end with the production of a formal expres- Alibali 2005), such as the expectation of a sion or equation used to describe a concrete numeric answer and a propensity to per- referent (typically a pattern), structural ap- form calculations even when they are irrel- proaches enable the exploration of how evant to the task goal. Moreover, operational formal expressions can be transformed; the patterns are stubborn and can be triggered notation becomes a medium for doing math- unhelpfully by traditional paper-based tasks ematics rather than describing mathematics. (McNeil 2008). Carefully designed micro- Second, structural microworlds start with

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 334 Educational Research Experiments in Constructionism enacted in the future. inthe enacted be to available actions effective more and more making for or one’sintofunctioning, integrating for learning, for sufficient not course, been articulated before: George San- of has, slogans these behind sentiment The “ way, another Put required. is more Something “ > Upshot [email protected] University ofOxford, UK Open University& John Mason Using NarrativeConstruction Bringing ReflectiontotheFore 1989: 275) n xeine f succession. of experience an ence alone. - experi from learn often not do we that is rience, ics. top- other with experience past to tions connec- and hold goals and actions to together narrative own their and construct effective, consider been have often, actions so what every action from withdraw to but doing, been have they what on “reflect” to simply not learners for need the against up come Mavrikis & Geraniou software, through algebra and arithmetic between bridging or tion

imagination, and with symbols. the in e-screens), (includingworld material may be visceral or virtual, taking place in the actions Such actions. integrated already on variations or actions, new functioning their into integrate to mean must which thing, powers.We ical “learn”towant them some- the use and development of their mathemat- experience and procedures, mathematical make of use mathematical concepts, employ themes, mathematical encounter to tivity, ac- subsequent their through them, want we that conjecture, I is, do to tasks ematic C o A succession of does experiences not add up to - expe from learn to seem not do we thing One n « 2» « 1» str u cti The reason we give learners math- learners give we reason The U ” n tiig o upr transi - support to striving •In v fruaey ee xeine is experience mere nfortunately (Mason 2002:8) ist F ound atio n ” s vol. 10,N°3 Msn & (Mason D avis though reflection has been worked on and on worked been has reflection though and action the fromwithdraw to the breach.”the in honoured“more suggested, munication) com - private (1984 Wilson Jim as is, back, of problem solving, the last of which, phases four delineated (1945) PólyaGeorge experience? from learn to we are then How “ that they want to express and when they they have to express when vocabulary technical the it. and express to want they that something have learners when successful only is this but vocabulary, mathematical use to students Teachers urge amples.often ex- particular more or one through seen as between concepts, or expressing a generality connections and concepts of meaning the of words own your in articulations cedure, proouta - carrying fore when the to actions of sequence a bring to incantations inner whether narrative, own your constructing and number the of that term. sequence a of term a between relationships invariant capture to generalisation, enable of symbols. Its intention is to encourage and expres- use wordsformalandsionsactions) (in and informal between gap the bridge to generality,and express and encounter to which in environment undirective but ive support- a with learners provide attemptto an is MiGen elements.symbolic and iconic, manipulative, with medium expressive an rather but task, virtual some achieving of offers not a means of doingcalculations, nor feeling of succession. and William James wrote: ated with doubt about learning from history, - associ both are Burke Edmund and tayana portant of part reflection. of Polya’s fourfold framework. This is an im- at the end as implied in most interpretations mid-action (consolidation tasks), not simply in reflection promptinglearner of necessity the discovered study, they their In enacted. be to come actions useful mathematically ing of the scope of situatedness within which situated cognition, ofevidence the broaden- oflanguage transfer, the of in evidence or is chapter their in for looking are Mavrikis lis A succession of feelings does not add up to a to up add not does feelings of succession A « 6» « 5» « 4» « 3» n motn pr o lann is learning of part important An What Eirini Geraniou and Mano- and Geraniou Eirini What Enter the MiGen project, which which project, MiGen the Enter t s bouey ia fr learners for vital absolutely is It ” (James 1950:628) - Al reflect. looking turning to James again, enacted in one context but not in others. - Re actions between gap the articulate to tempt abstraction(N of notion The readily. too all orate evap- can experiences and insights, desires, intentions, hiatus that in and medium, the outside world attention the re-entersbefore vacuum or hiatus sudden a absorbing is there an book: down putting than harder even is medium motion-colour-sound-rich a from away or Turningoff experience. the fromlearn that todifficult rememberit to is accomplishment-driven so engaging, fully so is it that in (McLuhanmessage1964) the is medium the thinking: mathematical improvementin widespread in resulted not largely why the use of digital technology has is this that is conjecture My practice. room class- in evident rarely is it listing), begin to many (too authors many by elaborated “ tions seem most appropriate,mostpro the - seem tions while educatedbeen is flexible, usingwhatever ac- has awareness whose thinker mathematical well-honed the course Of tasks. subsequent in value) term and member term between expressionratherdirect than ofrelationship (term-to-term thinking of ways established Yetmoretasks. to reverted pencil someand paper in vocabulary same the tinuinguse to hap- certainly pened with MiGen, with many subjects con- this report, Markolis and Geraniou As description. of level surface a below experiences to access provides that vocabulary technical with experience ers’ - learn enrich to teachers on incumbent is it discourseof that micro-world, is which why the vocabulary, the into people enculturate to likely is micro-world particular a in ence pres- engaged and Continued teachers. of situation,similar much consternation tothe or same the be to seems what in times ent differ differently at think and act, feel, may person same the indeed, differently; act and tion. People in the same situation think, feel, world, or even the world of mental imagina - material the from differently very inhabited our attention is.” where are we attention; our are “we as out appear to himself to inhabit. shall he universe a of sort what things, tendingto each of us literally chooses, by his way of at- of way his by chooses, literally us of each « 7» n oen alne ti comes this parlance, modern In oss & Hoyles 1996) is one at- one is 1996) Hoyles & oss V irtual e-screen worldsirtual are ” (James 1950:424) situated - Constructionism Let’s Cross that Bridge Eirini Geraniou & Manolis Mavrikis

cedure-mastering learner whose behaviour Authors’ Response: §§5f, on how students’ learning trajectories has only been trained is more hidebound, Let’s Cross that Bridge… between contexts, facilitated by bridges, more routine in the actions they enact. can be studied. We see our article as a first « 8 » Fostering and sustaining a con- but Don’t Forget to Look Back step towards this investigation. The “bridg- structive stance to learning involves more at Our Old Neighborhood ing” activities were a design-based research than providing engaging tasks, more than outcome after carrying out a number of encounters with pervasive mathemati- Eirini Geraniou studies; they therefore served a purpose in cal themes, more than experience of one’s & Manolis Mavrikis the research context rather than the object own use of natural powers in a mathemati- of investigation itself. We agree, however, cal context. It requires immersion in and > Upshot • This response addresses the that future research should be structured prompts use of a vocabulary that captures main points from the three commentar- in ways that bring out the dynamic nature those experiences and enables learners to ies, focusing particularly on additional of the bridging activities and (sticking with become aware of what has been effective and terms and concepts introduced to the the metaphor) help investigate what situated what has not, not only at the end of a piece bridging metaphor. We further clarify abstractions (Hoyles & Noss 1992) or other of work, but throughout. It is the construc- our call for future research in the area learning takes place on the two sides of the tion of a personal narrative, with on-going and conclude with reflections about the bridge. improvements and refinements, that con- practical implications emerging from « 3 » Along the same line of thought, Ian stitutes learning in the fullest sense of the our target article and the commentaries. Jones’s review first brings to our attention a word. recent paper by Dave Hewitt (2014) that can « 1 » In her commentary, Nicole Panork- also help in future research by thinking in John Mason is a professor emeritus at the Open ou explicitly reminds us, among other rel- terms of scaffolding and fading. In §2, he University and a senior research fellow at the evant literature, of the concept of contextual raises an important question that has trou- University of Oxford, Department of Education. neighbourhood from Pratt & Noss (2010), bled us and the team behind the original Mi- His interests span problem solving; mathematical which of course permeates our research due Gen project that designed the eXpresser mi- thinking; calling upon and developing learners’ to the direct and indirect influence of the coworld and its tasks: Is the time investment powers, particularly generalizing; the use and research of the authors in our work. In §3, in scaffolding students through one micro- role of mental imagery in teaching and learning Panorkou challenges us to define our context world, designed with specific algebraic ways mathematics; the nature and role of attention in explicitly. Revisiting Pratt & Noss (2010), we of thinking in mind, worth the trouble? teaching and learning mathematics; the discipline are reminded that in design-based research, « 4 » We think that an answer to this co- of noticing applied to research as well as teaching the determination of contextual neighbour- nundrum comes on the back of more than and learning. Homepage: http://www.pmtheta.com hoods is sometimes implicit, both in the 40 years of research in constructionism and case of software design, as in eXpresser, and endless debates since. Avoiding opening a Received: 29 April 2015 in the case of task design, as in our paper- can of worms in such a short response, our Accepted: 15 June 2015 and-pencil activities. The challenge with other papers on eXpresser have demonstrat- making context explicit is that it is in the eye ed its potential (e.g., Mavrikis et al. 2013), of the beholder. We therefore have at least and Jones’s eloquent summary of functional three contexts to elaborate on — researcher, and structural approaches in §§5f provides 335 student, and teacher or schooling contexts. claims towards the potential of a micro- In brief, the framework of algebraic ways of world to support flexibility. Additionally, thinking (inspired by Seymour Papert’s 1972 we rely on anecdotal teacher reports and reference to mathematical ways of thinking) our experience of the potential of using eX- gave us, as researchers, the lens through presser and other microworlds in so called which to examine students’ activities and “blended-learning” scenarios, recently pop- learning as well as a way to map those to the ularised by advocates of “flipped learning.” teacher and the schooling parlour (e.g., in We have seen first hand the potential of giv- our case, to the national curriculum). These ing students eXpresser homework or group contexts of course overlap, and perhaps the projects that can act as substrate for a teach- distinction is mainly academic, but we are er-led plenary, or subsequent engagement primarily interested in the context as per- with traditional algebra in the classroom. ceived by the students and its influence in « 5 » An additional answer to the point the knowledge or ways of thinking that they above lies between the lines of the third develop. commentary by John Mason, whose research « 2 » We see therefore the rest of Pan- on mathematics education and his contribu- orkou’s review as a call for future research, tions, in particular on algebra learning (Ma- particularly her excellent suggestions, in son 2005a; Mason et al. 1985), have heavily

http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou 336 Educational Research Experiments in Constructionism as expansion of contextual neighbourhoods. eX- the presser tasks, and achieve what throughout interactions their to become aware of their work by looking back and experiences those capture to students achieve efficiency, answer can bridging activities at appropriate time points al. et with interspersed activities eXpresser Cobb 2003) of sense the (in Engineering article. this of title the to us brings gether Bransford J. D. &Schwartz D.- L.(1999)Re K.D.Beach (2003)Consequential transi- K.D.Beach (1999)Consequential transitions: A mathematics M.(2002)Learning Artigue ina Abboud-Blanchard M.(2014)Teacher and Combined References to learn. remember and themselves immersed they have which in microworld the from back step a We take ties. studentstohelp wantto gantly frames the aim of our bridging - activi GeorgePolya’s back,”“looking ele- so which learning to conducive not is that situation a to lead paradoxically can that technology digital of potential ing engag- the torefers Mason tasks. associated and its of eXpresser design the influenced

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