Guess my Garden

Submitted By MICHELE CERULLI on 03 April, 2006

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mathematical content	GmG_MathematicalContent.060329.png	GmG_MathematicalContent.060329.mm
learning and instuction	GmG_LearningAndInstruction060320.png	GmG_LearningAndInstruction060320.mm
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Abstract

The Guess my Garden game is part of the activities of the WebLabs project, and concerns randomness and probability  (Cerulli et al. 2005). In this game, proposers fill a “garden” with objects such as trees and flowers. A component provided by us extracts random samples from this garden. The proposer then publishes these extractions as a challenge. The task of the responder is to construct a garden with the same numbers of objects of each kind. Speaking mathematically the responder has to reconstruct the sample space defined by the proposer.

Introduction

The Guess my Garden game was developed and experimented within the context of the “Models Systems and Randomness” activity sequence of the WebLabs project. The activity aims to introduce pupils to randomness and probability and consists of a set of activities based on ToonTalk and/or on the employment of LEGO RCX robots.

The designed approach to randomness relies on the exploration of some key concepts (e.g. predictability, unpredictability, fairness, unfairness, determinism, indeterminism, etc.), and of some key properties of random phenomena (e.g. the properties of random walks, the independence of events from their history). The selected concepts and aspects of randomness are explored in three main phases:

Randomness Small Talks: a collection and analysis of sentences, talks, and previous experiences of the students, directly or indirectly, where the random concept emerges in some way.

Phenomenological approach to randomness: based on the manipulation and reflection on the nature and functioning of ad hoc designed RCX LEGO robots.

Toward mathematization: some ad hoc designed computer microworlds, based on ToonTalk, are used to introduce a formal language and mathematical formalization, leading to a first approach to probability and some of the key concepts such as frequency, relative frequency, equivalence of sample spaces, etc. 

In each phase, pupils are required to write individual and collective reports on the activities. In particular the class is engaged in the joint enterprise of building a shared Encyclopaedia of randomness. The items of the produced encyclopaedia (and their contents) are derived from the class experiences and from individual and group reports, and are meant to represent the shared culture of the class (Cerulli & Mariotti, 2003). The general methodology is that of negotiating the contents of the encyclopaedia by means of class mathematical discussions (Bartolini Bussi, 1996). Items in the Encyclopaedia are thought of as evolving entities, and in practice they are revised and updated periodically by the class over the course of the experiments.

It is in the third phase, “toward mathematization” that the Guess my Garden game is situated, as a means to introduce pupils to probability, after they had been working on related concepts during the previous phases, focused on the properties of random events. A detailed discussion of the first two phases is out of the scope of this paper, however it can be found in a dedicated paper (Cerulli et al. 2005).

The Random Garden Tools

The Guess my Garden game is based on a ToonTalk tool, the Random Garden, which we developed ad hoc for representing generic random extraction processes. The tool consists of a sample space, called the “Garden”, a bird and a nest: when the user gives a number to the bird, a corresponding number of objects is extracted from the garden and deposited in the nest[1] (see Figure 1). The user can modifying the garden by adding or removing objects which can be numbers, text, or images of any kind, which implies that with this simple device can be used as a means for representing any kind of random phenomenon. In particular it is called a garden because in the version presented to pupils the objects of the sample space were flowers and trees.

Figure 1: A number “1” is given to the N bird that takes it to the garden (the green square); a new bird comes out of the garden holding an object extracted at random; the object (a yellow flower in this case) is dropped in the output nest. In this example the objects contained in the sample space are a number “1”, a text “A”, a violet flower and two yellow flowers.

The elements extracted from the Random Garden are collected in a box containing a nest (see Figure 1 and Figure 10) but, in order to visualize the whole sequence of extractions it is possible to convert it to a box with as many holes as the number of extracted objects (see Figure 10).

Figure 10: On the left 8 extractions are collected in a box containing a nest; only the first element is clearly visible. On the right the nest has been converted into a box with 8 holes showing the whole sequence of extracted elements starting from the first (the leftmost) to the last (the rightmost).

 

In this case it is possible have a rough qualitative view of the sequence and of its properties, however, if the number of extractions is very high, and/or if one needs a more detailed qualitative and quantitative analysis of the data, some more tools are required. For this reason we developed tools such as the bar graph and counters (see Figure 3). The counters show how many times each object has been extracted, while the bar graph gives a visual representation of the percentages, in terms of the proportional heights of the bars.   

Figure 3:  The bar graph on the left shows the proportions of the elements extracted for each kind of object in the sample space; the counters on the right show the exact numbers of elements extracted for each kind of object.

The game

If one is given a nest or a box of extraction (see Figure 10) it is possible to address the question of what was the original composition of the Random Garden that produced it. To put it in mathematical terms, given a set of data produced by a random phenomenon, one could try to guess the sample space that produced such set of data. This key question is at the core of the Guess my Garden game which is conducted as follows. One team, namely a small group of pupils, creates a random garden by defining its sample space[2], then produces a set of boxes containing increasing numbers of extractions: 2 boxes with 100 extractions and 2 boxes with 1000 extractions.

 

Figure 4: 3 different pages of the notebook containing the challenge of Lollo, Molly and Teo. The boxes in pages 4 and 10 contains respectively a little box with 100 extractions and a little box with 1000 extractions.

The boxes are all included in a ToonTalk notebook identified by the name of the team, and the notebook is published on the WebReports system, as a challenge for other players (see Figure 4). Another team can then download the notebook and analyse the data it contains in order to try to guess the makeup of the original Random Garden produced by the challenging team. The team can either simply observe the sequences of extractions, or study them using the bar graph and counters tools. Once they make a conjecture concerning the garden to be guessed, they can produce a new corresponding garden and use it to produce a number of extractions that may be compared to those provided by the challenging team. Once the team is satisfied with the conjectured garden, they can publish it on the WebReports system and wait for their counterparts to validate or invalidate their answer. Thus, finally the challenging team checks the published answer and posts a comment to inform the other team whether they have correctly guessed their garden or not. If the garden has not been correctly guessed then the exchange between the pupils can continue until an agreement is reached.

Main educational goals

The Guess my Garden game has been designed in order to address the following key educational goals:

Introduction to the law of large numbers. This is addressed in terms of the representations offered by the bar graphs (Figure 3), and in terms of the relationship between the number of extractions analysed and the chances to guess the garden.

Introduction to a formal language to express random or random-related phenomena. We consider a language derived from the interaction with ToonTalk and participation with the game.

Introduction to key concepts of probability such as sample space, frequency, relative frequency, and definition of probability. In particular we consider Random Gardens   as incorporating the concept of sample space, while frequency and relative frequency are represented by the counters and the bar graphs.

Within this paper we are going to address mainly the educational goal concerning the concept of sample space, a key idea which is at the core of the classical definition of probability. We are going to elaborate on this issue by means of commenting on our pupils protocols.

The experiment

The entire activity sequence including the Guess my Garden game was developed across three countries: Italy, Sweden and Portugal. In Italy one class of 21 pupils, in Milan, participated in the whole activity sequence, thus playing the Guess my Garden game after the first two phases of the activity sequence. This means in particular that pupils were very familiar with the concept of randomness and were about to be introduced to probability. The pupils published their challenges and received answers from their Swedish and Portuguese counterparts, discussing in class the answers they received. In the episode that we will analyse, an Italian team of 3 pupils built a garden following a peculiar strategy based on a special ambiguity between different gardens. This led a Swedish team to answer by publishing a garden that did not correspond to the original one, but that was somehow compatible with it. As a consequence the Italian class set up a discussion in order to decide whether or not to accept the Swedish garden as a correct answer. We describe the episode in detail in the following.

Results

Our analytical approach attempts to interleave the socio-cultural theories with the framework of communities of practice while maintaining a focus of the epistemological observations arising from the specific knowledge domain of number sequences.

Our analysis is informed by the notion of ‘community of practice’ as it is used within the situated approach to learning (Lave and Wenger, 1991; Wenger 1998). The insights we gain from this analysis are fed into the next iteration of the activity design. Thus, we have built on our initial observations of communities to actively cultivate their existence.

Wenger proposes three dimensions of practice as the property of a community:

• Mutual engagement: a sense of “working together”. Sharing ideas and artefacts, with a common commitment to the interactions between members of the community.
• Joint enterprise: having some object as an agreed common goal, defined by the participants in the very process of pursuing it, not just a stated agenda but something that creates among participants relations of mutual accountability; that become an integral part of the practice.
• Shared repertoire: agreed resources for negotiating meanings. This includes routines, words, tools, procedures, stories, gestures, symbols, and so on. Artefacts that the community has produced or adapted in the course of its existence and have become part of its practice. The repertoire combines both reificative and participative aspects. It includes the discourse members use to create meaningful statements about the world as well as the styles in which they express their forms of membership and their identities as members.

We wish to set these elements within an epistemological context, in that we intend to encourage the formation of mathematical communities. That is, we are trying to generate communities of practice – both physically and virtually – in which there are agreed socio-mathematical norms, where it is natural to make conjectures, test hypotheses, offer counter-examples and so on. By restricting our attention to a specific domain of mathematical activity, we commit ourselves to make specific and concrete claims. Our focus on design provides us with a unique opportunity to go beyond explanatory observations. We can verify our claims by changing the activity system and monitoring predicted change.

Guess my Garden

The first protagonists of the episode are Jeka, Jè and Rossana, they have to build their Garden and to publish a challenge[1]:

Jeka: we could do…the same number…of flowers and threes

Jeka: …I don’t know, maybe thre times… four eight twelve

Rossana: for instance this one (pointing at one flower on the screen) three times

Jeka and Rossana: three times this one, three times this one, three times this one and three times this one (pointing at the flowers and the tree in the random garden)

The girls build their garden and ask the software to produce 100 extractions from the garden, then comment on the results:

Jk: yellow flower 25 extracted times…

R: …but they are all the same?! (looks surprised)…more or less…25, 24, 25 and 26 …ah, yes, of course, we put (in the garden) all the same numbers  (of flowers and tree)…(she looks around to stress that she is stating something obvious and her pals nod).

The girls take note of the obtained result and a researcher (Michele) intervene asking them what they are doing:

Michele: so what is the garden that…?

R: we multiplied each object of the garden by 3 (pointing to the monitor)…we tripled

M: Why do you think this is difficult to be guessed? (reads one of the written questions the pupils are supposed to answer in order to accomplish their task)

R: no, it is not difficult, we just tried…

M: but are you going to publish this one or another one?

R: no, we won’t publish this…(she looks at Jeka who agrees but seems to be doubting) 

M: but it is not easy to guess this …(he is interrupted by Jeka who intervenes promptly)

Jk: exactly! Because…one may think of two (objects) maybe… 

R: …yes…. (thoughtful)

Jk: I would think of two (objects of each kind in the garden) 

The original idea of Jeka begins to be clearer and becomes more explicit when the teacher (Annalisa) asks them an explanation:

Annalisa: why do you think this garden is difficult…

Jk: maybe because with the resulting numbers (after the extraction) one may…one may get confused

Annalisa: why? What answer could you get?

Jk: maybe two

M: you mean two…

Jk: I mean, if one sees 20

A: 2, 2, 2, 2? (meaning a garden with 2 objects of each kind)

Jk, Jè + R nod

After some more reflections and discussions the girls decide to publish the garden they had produced, made of 3 objects of each kind (see Figure 5).

Figure 5. The garden used by Jeka, Jè and Rossana to publish their challenge: they believe that their opponents may think that the garden is made of 1 object (or 2 objects) of each kind.

The following week the class goes back in the computer laboratory and each team finds an answer from a Swedish team, in particulare Jeka, Rossana and Jè find the answer published by Amelie, Sonja and Johan (see Figure 6).

Figure 6. A response from three Swedish students, the proposed garden contains 1 object of each kind, instead of 3, as expected by the Italian team.

They now have to check their opponents’ answer to the challenge (Figure 6) and decide if to answer that they guessed or not:

R: which we believe to be wrong

Michele: which you believe to be wrong, in fact I remember that your garden was different.

 R+Jk: It is wrong!

…

Michele: thus they didn’t guess, why? How many objects did they put?

Jk: they put one for each object ….which we thought it was going to happen

Michele: exactly, they fall in your trap…

Jk: exactly, because we said they could put 2 flowers for each kind, or 3, or any number

Michele: thus now we have to be kind and tell them that they were wrong, but without publishing the solution…you choose what to say and I can help you in translating it in english …

…

Michele: to answer we can send a personal message to one of them, ok?

The girls write in English this time (helped by Michele): “We are sorry because you did not guess try again”.

After answering to their Swedish pals the pupils were required to answer some challenges provided by the researchers, in particular here we will consider the challenge proposed by Michele which was responded by two teams of pupils and whose graph is represented in Figure 7. We recall that a challenge consists in a set of elements extracted from the garden to be guessed, and that responders can use the graphing tool, and the counters tool as means for guessing what the original garden was.

Figure 7. The graph obtained from 1200 extractions published by Michele for his challenge. Notice that the 4^th column, the one representing the quantity of trees, shows clearly that there are no trees in the garden used by Michele.

The two teams that responded to Michele’s challenge gave two different answers, one of the teams was that of R, Je and Jeka whose guess was a garden constituted of 2 red flowers, 4 pink flowers and 6 yellow flowers; on the other hand Lollo’s team’s guess was 1 red flower, 2 pink flowers and 3 yellow flowers. Both teams guessed that there were no trees, however, the key question is, did any of the two teams guess Michele’s Garden? This question was addressed a couple of weeks later within a class discussion which was set up by the teacher in order to discuss several issues which were raised by the Guess my Garden game and by some related homeworks. As it was usual in this class, during the class discussion a poster was produced, representing the main findings of the discussions or agreement found among the class. The discussion took place in class, so pupils couldn’t access to the computer and to the Random Garden tools of ToonTalk, however the teacher provided a set of cards representing the main gardens she intended to discuss about.

Figure 8. The guesses of Lollo’s (“Giardino A”, i.e. “Garden A”) and Jeka’s (“Giardino B”, i.e. “Garden B”) teams as they were reported in the poster.

What is interesting in Michele’s challenge is that speaking mathematically the two answers proposed represent equivalent sample spaces, and actually it was one of the aims of the whole Guess my Garden activity to introduce the idea of equivalence of sample spaces.

The discussion concerning Michele’s challenge begins with the teacher recalling the different answers given to the challenge and asking which of them is right or wrong or if they are both wrong or right, causing C1’s reaction:

C1: we cannot know because we didn’t see the graph 

Je: but who responded to Michele’s challenge saw it

C1: in fact, we responded Augusto’s challenge, and couldn’t see the graph, so we cannot say if one is right or wrong.

The implicit hope in the teacher’s question was that pupils could come out with some ideas of equivalence of sample based on the fact that different gardens (thus different sample spaces) may produce the same kind of graph. Pupils obviously see a relationship between gardens and graphs, but they tend to interpret it only in terms of guessing what garden corresponds to a given sample space.  We may argue that they are interpreting the graph basically as a product of the process represented by the garden.

However we witness a shift of focus when the teacher on purpose introduces the idea of comparing sample spaces by asking them to guess from which of the two gardens it is more likely to pick a red flower[2]. In this case gardens are no more regarded as processes (or inputs of processes) but as objects to be compared. A brief discussion follows where pupils agree that one has the same chances to get a red flower from garden A or B, as expressed by C1:

C1: they are the same, in fact they are all doubled (meaning the the numbers of flowers in garden B are the doubles of the numbers of flowers in garden A)

The gardens are now regarded as objects and their constitutive properties are compared, and garden B is now regarded as a sort of “double” of garden A. This idea reminds Jeka of the challenge they proposed to their Swedish pals:

Jeka: teacher, we sent this challenge to the Swedish

Clearly Michele’s challenge is different from the one proposed by Jeka’s team (Figure 5) which would produce a graph where the bars have all the same heights, but she refers to the idea of exploiting as a winning strategy the fact that different, but equivalent, gardens will produce the same kind of graph. However, now the terrain seems to be ready for planting the see of equivalence and the teacher takes her occasion introducing explicitly the word “equivalent”:

Annalisa: […] thus these two gardens, in theory, are equivalent?

C1: yes, they are equiv…

Annalisa: (or) Are they equal?

C2: they are equal

Annalisa: (or) Are they identical?

C1: they are equal

Jeka: they are equivalent

C1: they are equiv…

Follows a noise of chat among pupils follows which end with C1 stating:

C1: equivalent!

Jeka: they are equivalent!

Someone in the background says “equal”

C1: equiva…equivalent…seems to be doubting

Annalisa: will you explain me what you mean with equivalent?

Jè: [they are] not equal because there are not the same elements in the two gardens

Annalisa: Thus Michele surely had one or the other (meaning that Michele couldn’t have both gardens but only one of the two proposed gardens)

Jè: [they are] equivalent because they have the same values….in practice…we can say so!

The excerpt shows that questioning on the equivalence of gardens is not obvious for the class, however, after a while the position expressed by Jè (when she states that the two gardens are not equal because they have different elements) opens the space for discussing new criterions (different from pure “equality”) for comparing gardens. She introduces her criterion of equivalence, based on the “values” of the gardens. At this point it is not clear what is meant with the word “values”[3],  but from the context we deduce that Jeka refers to some kind of result produced by the gardens, could it be the graphs, the numbers in the counters, or the sets of extractions. On the other hand some other pupils propose their interpretation or alternative “definition” of equivalence, like that of Bo3:

Bo3: the percentages are the same…I believe it is because the percentage is more or less the same.

Annalisa: the same of what?

Bo3: of…of twelve…of all the flowers…for instance

Annalisa: give me an example

Bo3: in the first garden, the garden A, the percentage is 6…in the first garden the percentage is 6, six is …it is …

Annalisa: the percentage? The total?

Bo3: yes, the total, the 6 is like the 100, and the red is 1, thus it is ….the red is 1 over 6…oh god!

Annalisa: 1 over 6

Bo3: 1 over 6 and the second (garden) is 2 over 12 which like 1 over 6

Annalisa: uhm, thus you say “they are equivalent as the two fractions” that you said?

Bo3: yes

Annalisa: right?

Annalisa: but they are not the same, thus we can say that these two answers are equivalent […] but without knowing Michele’s garden we cannot give a definite verdict, ok?

Bo3: because even if we could make the extraction….surely the numbers would not be the same, they would be almost the same!

Annalisa: almost the same, and the colums [of the graphs]?

Bo3: equ….with the same heights I think

A: in the two gardens?

Bo3: in the two gardens the heights of the columns would be the same but the values different

Bo3’s idea of equivalence, differently from Jeka, is clearly based on an analysis of the constitutive elements of the garden, and he tries to formalize it associating fractions to the garden, these fractions are only by chance coherent with the classical definition of probability which at this stage is not known to these pupils[4]. However also Bo3 agrees clearly that the two gardens would produce the same graph.

This discussion on the equivalence of gardens was originated by the issue of judging which of two given responses to a challenge is correct. In other words the class found itself with the need to establish criterions to validate responses to challenges, this led them to introduce some idea of equivalence. Such idea is still fuzzy, but nevertheless it turns out to be useful in the following excerpt where the class discusses the case of Jeka, Je and Rossana and their Swedish opponents. According to Jeka the strategy used by Michele is the same used by her team, they both produced a garden whose results are “the same” as the results of other gardens, reducing the responder’s chances to guess exactly the right garden. Being this issue related to the idea of equivalence of gardens, the teacher approaches it immediately after the discussion of Michele’s challenge, and puts it in terms of validation of the Swedish anser:

Annalisa: they [Jeka, Jè and Rossana] answered to the Swedish pupils “you did not guess”, are we leaving this answer or are we going to write to them again?

C: I think it is better to re-write it and state that they did well but the values…

Jeka: they did wrong

C: they did wrong but with respect to the values…they did wrong but they were right because the original value was 3, 3, 3 and 3, the one they had to guess, and they put 1, 1, 1, 1 but it is not their fault because…

Jeka: they did wrong

C: they did right, it is only that the bars were the same, so they could put any number

Jeka: I think we should re-write it stating that they did wrong, but that the two gardens are equivalent like the other one (referring to Michele’s challenge)

According to C the swedish answer should be considered correct because their garden produce the same graph as Jeka’s team’s garden. This idea is re-formulated by Jeka in terms of equivalence relationship among the two gardens, thus the word “equivalent” (ita. “equivalente”) begins to be used by the pupils as a tool for validating responses to the challenges of the game. However the meaning associated to the word “equivalent” needs some more clarification:

Annalisa (teacher): what is the phrase you would write to explain them…

C: I would write, you did wrong…should I say also the solution?

A: suppose that what you say is going to be sent to the Swedish pupils

C: you did…you did wrong…you didn’t guess our garden but you found another one that has the same value of the one we did, apart from…

The words “values” (ita.: “valori”) and “equivalent” appears as strictly tied, and their realitionship is defenetly cleared by C in the following excerpt:

A: what do you mean with “value”

C: I mean that the graphs of the extractions are equal to the graphs of our garden 

A: perfect

C: only, our garden had different quantities of objects

A: so, if they give the same extractions they are equivalent

C: there are 3 equivalent gardens…there are 2 equivalent gardens…there is our garden, 3 red flowers, tre yellows, 3 pinks and 3 trees, your garden, one for each object, and a third one containing to elements for each object[5].

Finally the meanings of the words “values” and “equivalent” are cleared and the class agrees on the following criterion for validating answers to responses: the answer is correct if the garden proposed by the responder is equal to the original garden; the answer is almost correct if the garden of the responder is equivalent to the original garden, in the sense that they produce the same graph.

What we find interesting in this story is that the “seed” of the idea of equivalence appeared in the form of the strategy employed by Jeka’s team for winning the game. It then reappeared in the discussion of Michele’s challenge which was designed on pourpose to exploit the ambiguities related to the equivalence of gardens; in this case the idea of equivalence appeared in the form of criterion for deciding which of the two proposed answers was correct. After that the idea of equivalence appears in the form of criterion for validating answers to challenges. Each of these steps corresponded to an evolution of pupils idea of equivalence of gardens by means of reflections and class discussions that were clearly motivated and driven by the needs of the game. The two needs that drove such evolution were basically: the need for finding a principle to validate answers; the need to produce “difficult” challenges. The two need turn out to be tied, as witnessed by the fact that the discussion concerning the validation of the Swedish answer ends up with some pupils questioning the strategy used by Jeka’s team. According to such pupils their strategy was not a good one, however Jeka and her pals were confident enough to defend their strategy and obtained to include it in the list of “winning strategies” which was reported in the poster produced by the analysed class discussion.