Open Research OnlineThe Open University’s repository of research publicationsand other research outputs

Thinking in progressJournal ItemHow to cite:

Mor, Yishay; Hoyles, Celia; Kahn, Ken; Noss, Richard and Simpson, Gordon (2004). Thinking in progress.Micromath, 20(2) pp. 17–23.

For guidance on citations see FAQs.

c© 2004 Micromath

Version: Version of Record

Link(s) to article on publisher’s website:http://www.telearn.org/open-archive/browse?resource=58

Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.

oro.open.ac.uk

1716

This article tells the story of 11-14 year- o l dstudents using a rather new and re l a t i v e l yuntested programming system to represent anddiscuss some deep mathematical ideas. There isone caveat: we have chosen the episodes toindicate the possibilities that can emerge whenchildren are given new ways to talk and thinkabout mathematics. We cannot lay claim to anygenerality: we would invite the reader to lookwhere we are pointing, not at our fingers!

We are programming with ToonTalk, a languagewe have used in the past with younger children toc o n s t ruct video games, and which we havewritten about previously in Micromath. ToonTalk isa computer game, programming environment andprogramming language in one. In most languages,p rogramming means writing text (code) in ahighly stru c t u red syntax. Some enviro n m e n t sallow the user to replace pieces of this text withiconic representations. But ToonTalk takes this onestage further, in that programs take the form ofanimated cartoon robots (ToonTalk is so namedbecause one is "talking" in (car)toons).Programming is done by training these robots,leading them through the task they are meant toperform. After training, programs are generalisedby "erasing" superfluous detail from ro b o t s '"minds" (to train a robot, you just go in andprogram an example in its thought bubble!). Sothe process of what it means to program is very

Train therobot to takea number 1from thetoolbox anddrop it onthe input, toincrement it.

Generalisethe programby erasingthe value ofthe inputfrom therobotsmemory.

Give therobot itsinput box.The robot willcontinuouslyrepeat theactions it hasbeen taught.

Figure 1: Training a robot to count

Yishay Mor, Celia Hoyles, Ken Kahn, Richard Noss and Gordon Simpson

From these, we were able to identify five basiccontexts for the use of IWB.

● Teacher as demonstrator ● Teacher as modeller ● Teacher in control - inviting the pupils

(shared)● Pupils in control with the "teacher" advising

(guided)● Pupils working independently

Demonstration was used to describe teacher inputthat simply demonstrated the mathematicalprocess e.g. how to add two 3 digit numbers, ors o f t w a re features. Modelling suggests themodelling of mathematical thinking with the aid ofthe IWB software (metacognitive modelling).Sharing occurs when the teacher has control of theIWB but invites pupils to participate in a task.Guided use refers to pupils leading with guidancefrom the teacher concerning content, direction ortechnical issues. Pupils can work independently,either in small groups or alone on IWB. The size ofthe image and the interactivity means small groupscan share ideas with greater ease than if they weresitting around a single computer screen.

Implications for the teachersinvolved in the projectThe teachers adopted models for the use of thet e c h n o l o g y, at least initially, in a way in which theyfelt most comfortable. Although the IWB was feltto change the flow, content and pace of the lessons,to begin to use it, teachers did not feel that theyhad to make large shifts in their classroom practice.

Whilst the teachers recognised the first twoteaching contexts, demonstrating and modelling,as part of their general classroom practice theycould offer little anecdotal evidence of pupilsparticipating in any form beyond the "come upand show us" model in the whole class activities atthe start and end of the numeracy lesson. Theidentification of the lack of pupil involvement ledto discussion about teachers’ everyday IWBpractice. As a consequence, teachers consideredthe need to expand the sphere of influence of theIWB into parts of the lesson where they were nottaking the lead.

C e rtain software encouraged the adoption ofp a rticular approaches to the use of the IWB by theteachers. The majority of the software being used

was for whole class- teacher centred work. Wherethe software had the potential for pupils workingi n d e p e n d e n t l y, it re q u i red additional impetus forchange. Even when the software had the potentialto support pupils working independently thisfacility was not utilised. The following examples inF i g u re 1 are those that the teachers came up within the session. They are not necessarily pre s e n t e das the best practice in these areas.

Although the purpose of this article has not beento discuss the role of alternative means ofachieving similar practices with other technologies– it does appear to be the case that the"interactiveness" of the IWB comes more into itsown as you move "down" the contexts that appearin Figure 1. That is, models of teaching andlearning that are more open and social are morelikely to result in the IWB offering a uniquefeature to the classroom.

By identifying a framework for "the use of theinteractive whiteboard" the teachers were alertedto a wider variety of teaching and learn i n gcontexts and possibilities. This has begun toinfluence their practice and enabled them toengage their established understanding of the flowof pupil/teacher activity in a lesson with theirdeveloping use of the IWB. The IWB thus becomesan instrument of change as it becomes more fullyadopted and teachers adopt new models ofc l a s s room practice that more fully utilise thepotential of the whiteboard.

The mapping of a possible framework for the use ofthe interactive whiteboard meant that teachers nowhad a way to refine their re s e a rch questions so thatthe reader and writer could have a greater chanceof having the same understanding as to how theIWB was being used, rather than relying onassumptions as discussed in our introduction. Theneed for such a framework is clearly evident, theone presented here is intended to stimulate debateas to the precise nature of such a framework.

ReferenceBECTA,2004,Getting the most from your interactive whiteboard;a guide for Primaryschools.

Note:Easiteach Maths is an RM product used in many Primary Schools.The latestversion is now part of the Easiteach Studio package. More information can be foundon the RM website http://www.rm.com/rmcomhome.asp

Penny Knight and Jennie Pennant work with the BracknellForest LEA and Jennifer Piggott works at the Faculty ofEducation,University of Cambridge.

1918

d i ff e rent from any existing pro g r a m m i n glanguage, and because programming objects areanimated, it is very difficult to conjure up the feelof the process without trying it. Fortunately, this isstraightforward. The important point is that theprocess is made concrete in a robot, which can bepointed at, named, picked up, and moved about.

Figure 1 shows three snapshots of what it meansto write a program (train a robot) to count throughthe natural numbers. In fact, we only have to trainthe robot to "add 1" to a number and thengeneralise it to any number. The robot iterates theactions it was trained to do, for as long as theconditions it expects hold.

Alongside the programming system, and a set ofappropriate tools developed within it, we havec o n s t ructed a web-based system we callWebReports, designed to allow students to shareand discuss the models of mathematical objectsand processes they have built in ToonTalk. Thestudents use a visual on-line editor to composereports detailing their work, can comment on and

annotate each others' re p o rts, and – mostimportantly – can publish working ToonTalk modelsof their ideas as they develop. This is a simpleprocess: students can grab any object in theirToonTalk environment, and include it in their on-line report. The object is shown as an image, but itis also a hyperlink, which, when clicked, causesthe object to open in the ToonTalk environment –which could be in another classroom or anothercountry. This last point is crucial: rather thansimply discussing what each other thinks, studentscan share what they have built, and rebuild eachothers’ attempts to model any given task or object.F i g u re 2 shows the welcome page of a"WebReport", seen after logging into the system. Itincludes a personal navigation sidebar, tabs toaccess general areas (such as topics and groups),and a quick view of the most recent updates.

After about a year of work, we are beginning tonotice how students are starting to use theirp rogrammed models as elements of theirdiscourse. By expressing themselves through theirmodels, they are beginning to find ways to talk

Figure 2: WebReports welcome page Figure 3: "Guess My Robot" WebReport pages

about deep mathematical ideas without thenecessity f i r s t to become fluent in algebraicsymbolism, and are becoming accustomed toposting their developing ideas on the web,commenting on other students' re p o rts, andchallenging them to extend their mathematicalideas.

Guess My Robot: Rita's

challenge and Nasko's

response

We now turn to the story. First, we shouldintroduce the setting. We focus on the interactionsbetween two groups of students, one in Sofia andthe other near Lisbon. The Sofia group consists of6 boys and girls, aged 11-12, working withWebLabs researchers. They have been workingwith ToonTalk for several months, approximatelyonce a week for a couple of hours. The secondgroup is from a village south of Lisbon. Paula, ateacher and re s e a rcher in the We b L a b s t e a m ,worked there with a school group (aged 12-13)during the first project year. Researchers in bothg roups act as teachers, guiding the studentsthrough the mathematical ideas as well as throughthe programming skills. At the same time, there s e a rchers facilitate interactions, by pointing

children to interesting peer reports and helpingthem to add a few words in English to their ownreports.

The activity we designed was based on the well-known "Guess my rule" game, employed by us inthe context of Logo and spreadsheets, and bymany teachers and researchers as a well-knownactivity. It has also been used in many classroomsin the UK over many years to provoke children todiscuss and compare the formulation of rules, andin particular the equivalence (or not) of theiralgebraic symbolism. In its classical form, it hasbeen used as an introduction to functions and toformal algebraic notation. As Carraher and Earnest(2003) have recently reported, even children inyounger grades enjoy participating in this game,and can be drawn into a discussion of algebraicnature through using it. It is an excellent contextfor students to come to understand that differentarticulations of their constructions can indeed yieldthe same results and it does this as children feelsome ownership of their construction and arewilling therefore to engage with others to compareand contrast solutions.

The aims of "Guess my robot" are to encouragestudents: to build sequences with robots and tochallenge others to program the robot that madethe sequence; to compare the robots used and todiscuss the different methods; and to take a robot

The "Guess my Robot" game presented in a WebReportthat explains the game rules.

Each student creates her own game page, and the pagesare listed at the bottom of the main game page.

2120

that "encapsulates the process" and use it togenerate new sequences by using new inputs.

So, in our game, proposers (students) invent a ruleand program it so that it generates a numericalsequence, and publish the first few terms itgenerates in a WebReport. Responders then have tobuild a robot that will produce this sequence, andthus work out the underlying rule. The newelement in our variant of the game is that "rules"have to be encoded as robots: one responds to achallenge sequence by posting a robot thatproduces "the same" sequence. So the encoding ofthe rule takes the form of a process-description inthe form of a ‘program’. Managing to reproducesomeone else's sequence by training a robot, is away to show that you have grasped how thesequence may have been originally generated. Asone girl said:

"So, like, the robot is my proof that I got it?"

Let's go back to our story. Figure 3 shows the main"guess my robot" game page. Students enter thegame via this page, where they are challenged togenerate a "smart sequence" and post it on theweb.

Rita is a 14 year old girl from Lisbon, who has beenparticipating in WebLabs since February 2003. Shelikes maths, but has not yet learnt much aboutsequences in school: this topic is not highlydeveloped in the Portuguese curriculum. In fact,most of her experience in this topic comes fromher involvement with WebLabs.

Rita found the ‘guess my robot’ activity, anddecided to pose her own challenge. The sequenceshe posted (see Figure 4) was:

2, 16, 72, 296, 1192 …

A few days after she posted it, the Sofia WebLabsgroup held a session, and some of the studentstried solving Rita’s challenge. Nasko, a 12 year-old

Figure 4: Rita's "Guess My Robot" challenge

Figure 5: Nasko's response to Rita and his twofold newchallenge

boy, posted his response. He had built a robot thatproduced Rita's five terms, but also realised thatthe same robot could be used to generate othersequences by changing its initial inputs. So, heposed a two-part challenge for Rita:

● Could she use his robot to generate a new sequence of five terms?

● Could she use her robot to generate the same sequence?

Nasko’s response and challenge are shown inFigure 5.

Ivan, who is also a member of the Sofia group,could not participate in that particular WebLabssession. Still, that did not stop him from trying hishand at Rita’s challenge, and posting his ownresponse (Figure 6). Ivan succeeded in building arobot, but was especially proud of what hethought was a clever solution as it was ‘simpler’:("I only use two holes in the box"). He adds: "I’mcurious to see other solutions".

A few days later Rita came to her next session. Shewas very excited to find comments on her page –and from children on the other side of Europe!She immediately clicked on the ToonTalk robots inthe responses, and watched them step through thep rocess of rule-generation. She was totallysurprised: Nasko and Ivan had solved herchallenge, but their robots seemed completelydifferent from hers (and one from the other)!

Figure 6: Ivan's response to Rita

Comparing solutions

We will now look briefly at two of the robots (Ritaand Nasko’s) and try to clarify, from an algebraicperspective, what mathematical ideas are put intoplay in this story. To o n Ta l k robots operaterepeatedly on the data in their boxes. This makesrecursive computations very intuitive: just like aspreadsheet, the values in use at step n are used to

2322

derive the value for step n+1. Snapshots of Rita’srobot as it runs through its first cycle, are shown inFigure 7.

When this cycle completes, the number in the left-most box is then passed to the "bird", as a secondelement of the sequence, starting the second cycle.A "bird" is the way that To o n Ta l k uses tocommunicate between objects. In this case, it takesthe number of the sequence in turn, and collectsthem (on its nest!).

Rita’s robot therefore computes:

An+1 = (An +2) *4A1 = 2

Nasko’s robot (Figure 8) differs from Rita’s in tworespects. Firstly, its programming style is different;while Rita uses a bird to carry the sequenceelements out, Nasko displays them in the robot’sbox. The second disparity is more interesting – therobot appears at first sight to be computing acompletely different sequence!

Nasko’s robot computes:

An= An-1+14) *4n-2

A1 = 2

Are these two robots equivalent? And what, inany case, does "equivalent" mean (this in itself isan interesting talking point for the two childrenconcerned). This may be an interesting exercise inalgebra for the reader.

Copy the value of An to the bird, tosend it out, so the first term is thenumber in the left-most box.

Copy the value of i (14) over the current sequence term(2), adding them up.

Copy the value of j (*4) over i, multiplying it by 4.

Copy i (2) and place on top of An,which adds 2 to it.

Copy k (*4) on to An, multiplying itby 4.

Figure 7: Rita’s robot in action

Figure 8: Nasko’s robot in action

The next step

Now that something unexpected has happened,the next step is to challenge the students toexplain it. Both groups are scheduled to discuss

the different solutions, compare them and explainhow they appear to generate the same sequence.The questions they will discuss include:

● Explain how you worked out what the sequence was, and how you ended up with therobot you built.

● We think your robots will generate the same sequence forever, but how can we be sure?

● Discuss the different robots, and explain why they seem to generate the same sequence.

● Describe what the robots are doing on the whiteboard or on paper, in a way that will make it easier to compare them.

As at the previous stage, we are planning forsurprises. We designed the original challenge inthe belief that responders would solve the sametask differently from proposers, and that both sideswould find this interesting enough to have amathematical discussion. This time, we’re preparedto be as surprised as the students. Will they basetheir explanations on the ToonTalk representation,or will they ask for alternative notations? In eithercase they will be exploring new mathematicalterrain. The objects of discussion have emergedfrom their own activities. This sense of ownershipallows, we believe, children to access structuresand ideas which may normally be considered tooadvanced for them. If this is the case, although theToonTalk representation cannot help furnish theproof, it may be a useful tool in motivating aninteresting and non-trivial piece of mathematicalp roblem-posing (the situation is somewhatanalogous to that of Dynamic Geometry in thisrespect).

We do not mean to restrict the role ofprogramming to a matter of motivation. We wrote"the robot computes the sequence…". In fact, weshould have written "The robot IS the sequence…" inthe same way as we talk of "the sequenceAn= a + b * n ". The same sequence can berepresented as "An= An-1+b; A1= a" or by the robotwith a 3-hole box. All three notations are valid,precise and well-defined. Just as a formal equationdefines a unique sequence, or class of sequences,so does a ToonTalk robot.

As mathematically-educated people, we may takefor granted the meaning of a statement likeAn= a+b*n. In much the same way, participants in

the WebLabs community are coming to share themeaning of ToonTalk robots as representations ofmathematical entities. Each representation has itss t rengths. While the algebraic form is moreconducive to formal manipulations and proof, theTo o n Ta l k re p resentation is situated within afamiliar activity, and is easier to tweak in anexploratory manner. Most important, what you"write" (i.e. do on the screen) has a real effect interms of what the program produces (how do youknow when you've got an algebraic calculationwrong?).

To illustrate this point, we focus on Rita's responseto Nasko's challenge. Nasko had challenged Rita tothe use the robot he had constructed for hersequence 2, 16, 72, 296, 1192,... to generate anothersequence: 9.5, 14, 16.25, 17.375, 17.9375

Rita responds to Nasko by comparing each step ofthe process that the robot acted out, and imaginingwhat numbers the robot must have worked on, inorder to obtain the required number:

"In Nasko's box for my sequence he used 2 in the first hole, and added 14 from the second hole to get the second term (16), and multiplies that by 4. Then I think to get the first term of Nasko's sequence I need 9.5 in the first hole. The number in the second hole has to be a number that you add to 9.5 to get 14 (the second term), this number is 4.5. In my sequence he uses x4 to get the third term (16 + 14 x 4). Then for his sequence I think like this: 14 + 4.5 "something" should give me 16.25, or 4.5 "something" should give me 2.25. But 2.25 is half of4.5, then in third hole of the box I need to put /2".

Does this count as a mathematical solution? Weleave it to the reader to decide.

ReferenceCarraher, D. & Earnest, D. (2003) Guess My Rule Revisited in Proceedings of 27thInternational Conference for the Psychology of Mathematics Education,2003:HonoluluNotes:ToonTalk is a commercial product.Free trial and Beta versions are availablefrom http://www.ToonTalk.com.We have designed a set of tools to assist in theprogramming tasks.These are available on the web reports system under the Toolssection.See weblabs.eu.com To read more about the WebReports system,and to seeit in action,visit:http://www.WebLabs.org.uk/wlplone/Help/about_index_html.Themain "Guess my rule" page can be found onhttp://www.WebLabs.org.uk/wlplone/Members/yish/my_reports/Report.2004-01-06.5353

Acknowledgement:We acknowledge the support of the European Union,Grant # IST-2001-32200. WebLabs is a three-year European research project on the use ofprogramming and web-based collaboration in mathematics and science education.Ourfocus is on communities of young learners (10-14 years),engaged in collaborativemodelling of mathematical and scientific phenomena,across five EU countries.

Yishay Mor , Celia Hoyles, Ken Kahn,Richard Noss andGordon Simpson are involved in The WebLabs Project ,Instituteof Education,University of London