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This article tells the story of 11-14 year- o l d
students using a rather new and re l a t i v e l y
untested programming system to represent and
discuss some deep mathematical ideas. There is
one caveat: we have chosen the episodes to
indicate the possibilities that can emerge when
children are given new ways to talk and think
about mathematics. We cannot lay claim to any
generality: we would invite the reader to look
where we are pointing, not at our fingers!
We are programming with ToonTalk, a language
we have used in the past with younger children to
c o n s t ruct video games, and which we have
written about previously in Micromath. ToonTalk is
a computer game, programming environment and
programming language in one. In most languages,
p rogramming means writing text (code) in a
highly stru c t u red syntax. Some enviro n m e n t s
allow the user to replace pieces of this text with
iconic representations. But ToonTalk takes this one
stage further, in that programs take the form of
animated cartoon robots (ToonTalk is so named
because one is "talking" in (car)toons).
Programming is done by training these robots,
leading them through the task they are meant to
perform. After training, programs are generalised
by "erasing" superfluous detail from ro b o t s '
"minds" (to train a robot, you just go in and
program an example in its thought bubble!). So
the process of what it means to program is very
Train the
robot to take
a number 1
from the
toolbox and
drop it on
the input, to
increment it.
Generalise
the program
by erasing
the value of
the input
from the
robots
memory.
Give the
robot its
input box.
The robot will
continuously
repeat the
actions it has
been 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 basic
contexts for the use of IWB.
l Teacher as demonstrator
l Teacher as modeller
l Teacher in control - inviting the pupils
(shared)
l Pupils in control with the "teacher" advising
(guided)
l Pupils working independently
Demonstration was used to describe teacher input
that simply demonstrated the mathematical
process e.g. how to add two 3 digit numbers, or
s o f t w a re features. Modelling suggests the
modelling of mathematical thinking with the aid of
the IWB software (metacognitive modelling).
Sharing occurs when the teacher has control of the
IWB but invites pupils to participate in a task.
Guided use refers to pupils leading with guidance
from the teacher concerning content, direction or
technical issues. Pupils can work independently,
either in small groups or alone on IWB. The size of
the image and the interactivity means small groups
can share ideas with greater ease than if they were
sitting around a single computer screen.
Implications for the teachers
involved in the project
The teachers adopted models for the use of the
t e c h n o l o g y, at least initially, in a way in which they
felt most comfortable. Although the IWB was felt
to change the flow, content and pace of the lessons,
to begin to use it, teachers did not feel that they
had to make large shifts in their classroom practice.
Whilst the teachers recognised the first two
teaching contexts, demonstrating and modelling,
as part of their general classroom practice they
could offer little anecdotal evidence of pupils
participating in any form beyond the "come up
and show us" model in the whole class activities at
the start and end of the numeracy lesson. The
identification of the lack of pupil involvement led
to discussion about teachers’ everyday IWB
practice. As a consequence, teachers considered
the need to expand the sphere of influence of the
IWB into parts of the lesson where they were not
taking the lead.
C e rtain software encouraged the adoption of
p a rticular approaches to the use of the IWB by the
teachers. The majority of the software being used
was for whole class- teacher centred work. Where
the software had the potential for pupils working
i n d e p e n d e n t l y, it re q u i red additional impetus for
change. Even when the software had the potential
to support pupils working independently this
facility was not utilised. The following examples in
F i g u re 1 are those that the teachers came up with
in the session. They are not necessarily pre s e n t e d
as the best practice in these areas.
Although the purpose of this article has not been
to discuss the role of alternative means of
achieving similar practices with other technologies
– it does appear to be the case that the
"interactiveness" of the IWB comes more into its
own as you move "down" the contexts that appear
in Figure 1. That is, models of teaching and
learning that are more open and social are more
likely to result in the IWB offering a unique
feature to the classroom.
By identifying a framework for "the use of the
interactive whiteboard" the teachers were alerted
to a wider variety of teaching and learn i n g
contexts and possibilities. This has begun to
influence their practice and enabled them to
engage their established understanding of the flow
of pupil/teacher activity in a lesson with their
developing use of the IWB. The IWB thus becomes
an instrument of change as it becomes more fully
adopted and teachers adopt new models of
c l a s s room practice that more fully utilise the
potential of the whiteboard.
The mapping of a possible framework for the use of
the interactive whiteboard meant that teachers now
had a way to refine their re s e a rch questions so that
the reader and writer could have a greater chance
of having the same understanding as to how the
IWB was being used, rather than relying on
assumptions as discussed in our introduction. The
need for such a framework is clearly evident, the
one presented here is intended to stimulate debate
as to the precise nature of such a framework.
Reference
BECTA,2004,Getting the most from your interactive whiteboard;a guide for Primary
schools.
Note:Easiteach Maths is an RM product used in many Primary Schools.The latest
version is now part of the Easiteach Studio package. More information can be found
on the RM website http://www.rm.com/rmcomhome.asp
Penny Knight and Jennie Pennant work with the Bracknell
Forest LEA and Jennifer Piggott works at the Faculty of
Education,University of Cambridge.

1918
d i ff e rent from any existing pro g r a m m i n g
language, and because programming objects are
animated, it is very difficult to conjure up the feel
of the process without trying it. Fortunately, this is
straightforward. The important point is that the
process is made concrete in a robot, which can be
pointed at, named, picked up, and moved about.
Figure 1 shows three snapshots of what it means
to write a program (train a robot) to count through
the natural numbers. In fact, we only have to train
the robot to "add 1" to a number and then
generalise it to any number. The robot iterates the
actions it was trained to do, for as long as the
conditions it expects hold.
Alongside the programming system, and a set of
appropriate tools developed within it, we have
c o n s t ructed a web-based system we call
WebReports, designed to allow students to share
and discuss the models of mathematical objects
and processes they have built in ToonTalk. The
students use a visual on-line editor to compose
reports detailing their work, can comment on and
annotate each others' re p o rts, and – most
importantly – can publish working ToonTalk models
of their ideas as they develop. This is a simple
process: students can grab any object in their
ToonTalk environment, and include it in their on-
line report. The object is shown as an image, but it
is also a hyperlink, which, when clicked, causes
the object to open in the ToonTalk environment –
which could be in another classroom or another
country. This last point is crucial: rather than
simply discussing what each other thinks, students
can share what they have built, and rebuild each
others’ 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. It
includes a personal navigation sidebar, tabs to
access 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 to
notice how students are starting to use their
p rogrammed models as elements of their
discourse. By expressing themselves through their
models, 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 the
necessity f i r s t to become fluent in algebraic
symbolism, and are becoming accustomed to
posting their developing ideas on the web,
commenting on other students' re p o rts, and
challenging them to extend their mathematical
ideas.
Guess My Robot: Rita's
challenge and Nasko's
response
We now turn to the story. First, we should
introduce the setting. We focus on the interactions
between two groups of students, one in Sofia and
the other near Lisbon. The Sofia group consists of
6 boys and girls, aged 11-12, working with
WebLabs researchers. They have been working
with ToonTalk for several months, approximately
once a week for a couple of hours. The second
group is from a village south of Lisbon. Paula, a
teacher 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 both
g roups act as teachers, guiding the students
through the mathematical ideas as well as through
the programming skills. At the same time, the
re s e a rchers facilitate interactions, by pointing
children to interesting peer reports and helping
them to add a few words in English to their own
reports.
The activity we designed was based on the well-
known "Guess my rule" game, employed by us in
the context of Logo and spreadsheets, and by
many teachers and researchers as a well-known
activity. It has also been used in many classrooms
in the UK over many years to provoke children to
discuss and compare the formulation of rules, and
in particular the equivalence (or not) of their
algebraic symbolism. In its classical form, it has
been used as an introduction to functions and to
formal algebraic notation. As Carraher and Earnest
(2003) have recently reported, even children in
younger grades enjoy participating in this game,
and can be drawn into a discussion of algebraic
nature through using it. It is an excellent context
for students to come to understand that different
articulations of their constructions can indeed yield
the same results and it does this as children feel
some ownership of their construction and are
willing therefore to engage with others to compare
and contrast solutions.
The aims of "Guess my robot" are to encourage
students: to build sequences with robots and to
challenge others to program the robot that made
the sequence; to compare the robots used and to
discuss the different methods; and to take a robot
The "Guess my Robot" game presented in a WebReport
that explains the game rules.
Each student creates her own game page, and the pages
are listed at the bottom of the main game page.

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that "encapsulates the process" and use it to
generate new sequences by using new inputs.
So, in our game, proposers (students) invent a rule
and program it so that it generates a numerical
sequence, and publish the first few terms it
generates in a WebReport. Responders then have to
build a robot that will produce this sequence, and
thus work out the underlying rule. The new
element in our variant of the game is that "rules"
have to be encoded as robots: one responds to a
challenge sequence by posting a robot that
produces "the same" sequence. So the encoding of
the rule takes the form of a process-description in
the form of a ‘program’. Managing to reproduce
someone else's sequence by training a robot, is a
way to show that you have grasped how the
sequence may have been originally generated. As
one 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 the
game via this page, where they are challenged to
generate a "smart sequence" and post it on the
web.
Rita is a 14 year old girl from Lisbon, who has been
participating in WebLabs since February 2003. She
likes maths, but has not yet learnt much about
sequences in school: this topic is not highly
developed in the Portuguese curriculum. In fact,
most of her experience in this topic comes from
her involvement with WebLabs.
Rita found the ‘guess my robot’ activity, and
decided to pose her own challenge. The sequence
she posted (see Figure 4) was:
2, 16, 72, 296, 1192 …
A few days after she posted it, the Sofia WebLabs
group held a session, and some of the students
tried 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 new
challenge
boy, posted his response. He had built a robot that
produced Rita's five terms, but also realised that
the same robot could be used to generate other
sequences by changing its initial inputs. So, he
posed a two-part challenge for Rita:
l Could she use his robot to generate a new
sequence of five terms?
l Could she use her robot to generate the same
sequence?
Nasko’s response and challenge are shown in
Figure 5.
Ivan, who is also a member of the Sofia group,
could not participate in that particular WebLabs
session. Still, that did not stop him from trying his
hand at Rita’s challenge, and posting his own
response (Figure 6). Ivan succeeded in building a
robot, but was especially proud of what he
thought was a clever solution as it was ‘simpler’:
("I only use two holes in the box"). He adds: "I’m
curious to see other solutions".
A few days later Rita came to her next session. She
was very excited to find comments on her page –
and from children on the other side of Europe!
She immediately clicked on the ToonTalk robots in
the responses, and watched them step through the
p rocess of rule-generation. She was totally
surprised: Nasko and Ivan had solved her
challenge, but their robots seemed completely
different 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 (Rita
and Nasko’s) and try to clarify, from an algebraic
perspective, what mathematical ideas are put into
play in this story. To o n Ta l k robots operate
repeatedly on the data in their boxes. This makes
recursive computations very intuitive: just like a
spreadsheet, the values in use at step n are used to

2322
derive the value for step n+1. Snapshots of Rita’s
robot as it runs through its first cycle, are shown in
Figure 7.
When this cycle completes, the number in the left-
most box is then passed to the "bird", as a second
element of the sequence, starting the second cycle.
A "bird" is the way that To o n Ta l k uses to
communicate between objects. In this case, it takes
the number of the sequence in turn, and collects
them (on its nest!).
Rita’s robot therefore computes:
An+1 = (An +2) *4
A1 = 2
Nasko’s robot (Figure 8) differs from Rita’s in two
respects. Firstly, its programming style is different;
while Rita uses a bird to carry the sequence
elements out, Nasko displays them in the robot’s
box. The second disparity is more interesting – the
robot appears at first sight to be computing a
completely different sequence!
Nasko’s robot computes:
An= An-1+14) *4n-2
A1 = 2
Are these two robots equivalent? And what, in
any case, does "equivalent" mean (this in itself is
an interesting talking point for the two children
concerned). This may be an interesting exercise in
algebra for the reader.
Copy the value of An to the bird, to
send it out, so the first term is the
number 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 it
by 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 to
explain it. Both groups are scheduled to discuss
the different solutions, compare them and explain
how they appear to generate the same sequence.
The questions they will discuss include:
l Explain how you worked out what the
sequence was, and how you ended up with the
robot you built.
l We think your robots will generate the same
sequence forever, but how can we be sure?
l Discuss the different robots, and explain why
they seem to generate the same sequence.
l 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 for
surprises. We designed the original challenge in
the belief that responders would solve the same
task differently from proposers, and that both sides
would find this interesting enough to have a
mathematical discussion. This time, we’re prepared
to be as surprised as the students. Will they base
their explanations on the ToonTalk representation,
or will they ask for alternative notations? In either
case they will be exploring new mathematical
terrain. The objects of discussion have emerged
from their own activities. This sense of ownership
allows, we believe, children to access structures
and ideas which may normally be considered too
advanced for them. If this is the case, although the
ToonTalk representation cannot help furnish the
proof, it may be a useful tool in motivating an
interesting and non-trivial piece of mathematical
p roblem-posing (the situation is somewhat
analogous to that of Dynamic Geometry in this
respect).
We do not mean to restrict the role of
programming to a matter of motivation. We wrote
"the robot computes the sequence…". In fact, we
should have written "The robot IS the sequence…" in
the same way as we talk of "the sequence
An= a + b * n ". The same sequence can be
represented as "An= An-1+b; A1= a" or by the robot
with a 3-hole box. All three notations are valid,
precise and well-defined. Just as a formal equation
defines a unique sequence, or class of sequences,
so does a ToonTalk robot.
As mathematically-educated people, we may take
for granted the meaning of a statement like
An= a+b*n. In much the same way, participants in
the WebLabs community are coming to share the
meaning of ToonTalk robots as representations of
mathematical entities. Each representation has its
s t rengths. While the algebraic form is more
conducive to formal manipulations and proof, the
To o n Ta l k re p resentation is situated within a
familiar activity, and is easier to tweak in an
exploratory manner. Most important, what you
"write" (i.e. do on the screen) has a real effect in
terms of what the program produces (how do you
know when you've got an algebraic calculation
wrong?).
To illustrate this point, we focus on Rita's response
to Nasko's challenge. Nasko had challenged Rita to
the use the robot he had constructed for her
sequence 2, 16, 72, 296, 1192,... to generate another
sequence: 9.5, 14, 16.25, 17.375, 17.9375
Rita responds to Nasko by comparing each step of
the process that the robot acted out, and imagining
what numbers the robot must have worked on, in
order 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 of
4.5, then in third hole of the box I need to put /2".
Does this count as a mathematical solution? We
leave it to the reader to decide.
Reference
Carraher, D. & Earnest, D. (2003) Guess My Rule Revisited in Proceedings of 27th
International Conference for the Psychology of Mathematics Education,2003:Honolulu
Notes:ToonTalk is a commercial product.Free trial and Beta versions are available
from http://www.ToonTalk.com.We have designed a set of tools to assist in the
programming tasks.These are available on the web reports system under the Tools
section.See weblabs.eu.com To read more about the WebReports system,and to see
it in action,visit:http://www.WebLabs.org.uk/wlplone/Help/about_index_html.The
main "Guess my rule" page can be found on
http://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 of
programming and web-based collaboration in mathematics and science education.Our
focus is on communities of young learners (10-14 years),engaged in collaborative
modelling of mathematical and scientific phenomena,across five EU countries.
Yishay Mor , Celia Hoyles, Ken Kahn,Richard Noss and
Gordon Simpson are involved in The WebLabs Project ,Institute
of Education,University of London