Communication Issues in the Mathematical Classroom:
Four Looks on a Taxicab Scenario
Lisser Rye Ejersbo
The Danish School of Education, University of Aarhus
and
Uri Leron
Technion – Israel Institute of Technology
Introduction
This article is concerned with the professional development of mathematics teachers. It starts
with a need, identifies a problem, and suggests some ways of coping with the problem.
The need is for teachers' mathematical communication and reflection skills in their practice.
There is a widespread agreement that these skills are an important component of mathematics
teachers' professional toolbox, as can be seen from curriculum reform documents (e.g., Danish
Ministerial Curriculum, 1993, 2003, 2009). There is an equally widespread agreement on the
problem: that most teachers have not been sufficiently prepared for such skills and, furthermore,
that we do not know much on how to effectively achieve such preparation, whether in preservice or in inservice education. Personally, we have repeatedly ran into this problem during our
intensive involvement in the inservice preparation of Danish teachers to meet the new
requirement of the Danish Ministry of Education.1
In her attempts to cope with this problem in her inservice courses, LRE has developed a method
of adapting research ideas from the mathematics education research literature into experiential
activities in the course (Ejersbo, 2007). In this article we describe how one such idea – the virtual
monologue (Leron & Hazzan, 1997) – has been transposed from a tool for researchers to a tool
for teachers. We do this by means of a case study – the taxicab scenario – which is being
analyzed from four different perspectives, or ‘looks’.
Background
In Denmark, teachers are certified to teach four subjects in grades 1-10. The situation described
here is taken from an inservice course for 20 certified teachers, intended to work on their
mathematics communication skills. This course, and many like it, have been designed and
carried out by the first author. The complete course consists of two separate periods. The first
period extends over one week, six hours per day, for a total of 30 hours. The second period takes
place after a two-month break, and consists of 12 hours over 2 days. During the break, the
1
This paper is based on LRE's many years' work in Danish mathematics inservice teacher education, and on her
doctoral dissertation (Ejersbo, 2007) analyzing this work. Accordingly, the personal pronoun “we” in this paper
sometimes refer to both authors (when we talk about the joint work in writing the paper) and sometimes to LRE
(when we talk about her work as teacher educator).
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participants were asked to collect and send in examples of mathematical communication from
their own classrooms, a task that turned out to be surprisingly hard for many of them. From the
few participants' reports that did live up to the title “mathematical communication”, we now
present one – the taxicab scenario – whose analysis will form the core of this article. The
scenario consists of a dialogue between the teacher – let’s call him Martin – and one of his
pupils, Simon. It has been translated from the original Danish and is quoted verbatim (even
preserving grammatical errors) from Martin's report. The comments in square brackets are ours.2
First Look: Martin's report of the taxicab scenario
The following task is from a textbook for grade 8:
“A taxi costs 25 DKK3 initially, plus 10 DKK for each additional km. Two people traveled in
the taxi. One paid 43 DKK and the other paid 86 DKK.
How many percents [sic] does the second person travel further than the first?”
[Martin:] The result the pupils have to figure out is: 10x + 25 = 43. That is what the pupils
have to find out.
T [Teacher]: What are you going to find out?
P [Pupil]: How far they drive. They drive 1.8 km.
T: How did you find out?
P: 10 DKK for the first km and then 8 DKK up to the 43. It means, 1.8 km.
T: Yes, but you have to put up an equation.
P: But I know what it is!
T: Yes, but you need to put up an equation, so you can practice this. You need to put up
something with an ‘x’.
P: Is it then 25 + 10 + x = 43?
T: No, you shouldn’t plus.
P: Oh, should I then minus?
T: What does x represent?
[Martin:] The pupil has difficulty seeing what x should represent.
T: How did you find out before?
[Martin:] The pupil gives the same explanation as before.
P: Oh, it is multiplying, because x is how many km you drive.
In what follows, we will be taking several more 'looks' on the taxicab scenario and on its follow-up in
the inservice course.
Second Look: What is this task, anyway?
The first look has been Martin's, and would form the raw data for the rest of the analyses. The second
look is our own, but it is still mainly descriptive, keeping interpretation to a minimum.
2
It may require a bit of care to distinguish between the different 'voices' involved in the following discussion: The
two voices ('T' and 'P') in the classroom dialogue; the voice of Martin, the participant in the inservice course,
reporting and commenting on this dialog; and the voice of the authors, reporting and reflecting on all the above.
3
Danish Kroner
2
Martin presented his 8th-grade class with the taxicab task, which he had picked up from chapter on
equations in a textbook (Gregersen et al, 2001). One immediate observation is that there is a
considerable confusion about what the pupils are asked to do. The confusion starts already at the
textbook's formulation (“How many percents does the second person travel further than the first?”), and
is further aggravated by Martin's interpretation (“The result the pupils have to figure out is: 10x + 25 =
43.”). The second observation is that there is a clear break in communication between the teacher and
the pupil: Both of them are well-motivated and have clear goals for the activity, but their goals are quite
different – teaching equations for the teacher vs. finding the answer for the pupil – and they fail in this
dialog to bridge the gap. Moreover, it is reasonable to assume that Martin had chosen to submit this
piece of communication for the inservice course because he found it interesting and representing a
successful teaching scenario on his part.
Third Look: Inservice teachers' virtual monologue on the taxicab scenario
As a tool for communication and reflection, we choose the article The world according to Johnny: A
coping perspective in mathematics education (Leron & Hazzan, 1997, henceforth abbreviated L&H), in
particular, the virtual monologue :
[We] wish to take the pupil’s view, adopting what might be called an empathic attitude. […] We try to take
the pupil’s view by ‘looking from within’, by trying to recreate the pupil’s mental state as best we can, and
by trying to communicate our image of this mental state as faithfully as possible. To this end we resort to
writing ‘in the pupil’s own voice’, thus being able to express also non-logical, vague and confused
thinking, such as: “I am confused, I don’t know what to do”; “What does the teacher expect me to do, how
can I come up with something that has a good chance of giving me a passing grade”; “Great! I think I got
it!”. (L&H, 268-9)
In the course, the participants spent about 45 minutes getting acquainted with the VM tool by reading
and re-analyzing the Dina Scenario discussed in L&H (p. 270). ). In this scenario, which L&H picked
up from the research literature, Dina is given a task concerning linear equations with a parameter, and is
being interviewed as she is trying to solve the task. The analysis of the scenario in the original article
had taken a cognitive/mathematical point of view, and had attributed Dina's difficulties to her inability
to think of the solution as a function (instead of number). L&H have used VM to propose an alternative
analysis, based on what they have called a coping perspective, in which Dina in the face of the
unfamiliar task is totally confused and is groping in the dark for an answer – any answer! – that would
meet the expectations of the authority figure involved (in this case, the interviewer). They present their
analysis via “a monologue in Dina’s own voice, incorporating the original data […], augmented by our
own ‘virtual monologue’.” (L&H, 271)
In the workshop activity, for the first 15 minutes, the participants were presented with the task and the
two analyses as they appear in L&H, and were subsequently involved in a lively discussion, taking
alternatively Dina's or the Interviewer's perspectives. For the next 20 minutes, groups of participants
were given new pieces of mathematical communication, with the instruction for some to create a VM
for the pupil, and for others to create a VM for the teacher. However, half of the groups chose to
combine the VM for both the pupil and the teacher in one piece. When the ‘teacher-groups’ and the
‘pupil-groups’ presented their VM at the plenary meeting it was again followed by lively questions,
discussion and laughter, because they all dramatized the presentation by taking up different roles – one
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for the teacher, one for the pupil, and one for the VM interpretation. The discussion focused on the
many different VMs that the groups made, which showed the many choices the teacher have for each
question, answer or comment. Here is one of the results, based on Martin's taxicab scenario, where the
group has chosen to make a combined VM for both the teacher and the pupil. The text consists of the
original dialogue as reported by Martin, with the addition (in italics) of the participants' interpretation of
what was going on in the interlocutors minds.
The following task is from a textbook for grade eight:
“A taxi costs 25 DKK initially, plus 10 DKK for each additional km. Two people traveled in
the taxi. One paid 43 DKK and the other paid 86 DKK.
How many percents does the second person travel further than the first?”
[Martin:] The result the pupils have to figure out is: 10x + 25 = 43. That is what the pupils
have to find out.
T: What are you going to find out?
VM/T: It is always a good question to start with
P: How far they drive. They drive 1.8 km.
VM/P: It is so easy, everybody can see it
T: How did you find out?
VM/T: It is correct, but how did he found out? If he can answer, it is fine
P: 10 DKK for the first km and then 8 DKK up to the 43. It means, 1.8 km.
VM/P: I try with this explanation
T: Yes, but you have to put up an equation.
VM/T: I insist that he uses an equation, which is what they have to learn today
P: But I know what it is!
VM/P: Why do I have to do this, why can’t he just go and leave me alone
T: Yes, but you need to put up an equation, so you can practice this. You need to put up
something with an ‘x’.
VM/T: If he can’t do this when it is easy, how will he manage when it is difficult? I can't
sympathize with him, he has to do it.
P: Is it then 25 + 10 + x = 43?
VM/P: I can’t think at all, I try the usual answer.
T: No, you shouldn’t add.
VM/T: That's what I thought; he doesn’t understand what he has to do. I wonder how much he
really understands.
P: Oh, should I then minus?
VM/P: I really don’t know what I have to do, I try this way…
T: What does x represent?
VM/T: I guess I need to ask in another way
[Martin:] The pupil has difficulties seeing, what x should represent.
T: How did you find out before?
[Martin:] The pupil gives the same explanation as before.
VM/T: He doesn’t understand anything, what can I do? I thought that he was good at this.
P: Oh, it is multiplying, because x is how many km you drive.”
VM/P: Oh that was what he wanted me to say. OK, now we are happy…
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Using the VMs produced by the participants, we discussed the different choices they made for the
teacher and for the pupil. We also discussed how they experienced the VM activity itself. Here are some
of teachers’ comments:
-
I don’t have the patience to wait for an answer, I know it. I lose the other pupils if we have
to wait too long.
-
I am afraid that the pupils ask me questions I can’t answer. I really don’t like it. If I am in
charge and can control what is being asked, I feel much better.
-
I know that feeling. I don’t know what to do if I can’t answer, and I don’t know if it is me
or the question that is being stupid.
These comments were typical and part of a longer discussion about what kind of mathematical
communication takes place in classrooms, how they themselves communicated, what kind of habits
they had developed, and why. Next the participants were asked: What did you gain from working with
VM?
Again, here is a sample of the answers.
-
It opened my eyes for a new way to look at my own communication in the classroom, and
perhaps communication in general, I don’t know yet.
-
I am amazed that such little thing can change so much.
-
I think it is fun to think in this new way. We had a great time in the group.
-
I see it, I feel I understand it, but I don’t know yet what to do with it.
We can see from all the above utterances that the participants did pick up on the communication
problem, though they did not quite conceptualize it explicitly. The workshop activity did achieve its
purpose, which was to involve the teachers with communication and reflection activities. Hopefully,
such activities could improve their capacity for empathy and for flexibly shifting perspectives. We
note the dual nature of the teachers' utterances, which is typical of the introduction of new tools and
methods in general. The new tool clearly gives them more power, but it also adds a feeling of
insecurity as it shakes the foundations of their old habits (see especially the last teacher's comment).
Clearly, follow-up meetings for enhancement, consolidation and support are desirable.
Fourth look: Communication issues in the mathematical classroom
The taxicab scenario and its analysis bring up several common themes from the mathematics
classroom. What is immediately observable is the discrepancy between the plan of the lesson and its
actual execution in real time. The teacher, Martin, had it all planned to have a lesson about equations,
and had found in the textbook an activity which was meant to bring up the need for using equations.
Simon, however, has solved the problem correctly and easily without equations, and Martin was
facing an unexpected turn of events.
In general, if things do not unfold the way the teacher has planned (which they rarely do!), the teacher
must on the spot recognize that there is a problem, classify the problem, and devise a new plan of
action. It seems unfair to analyze Martin's decisions and behaviour from the comfort of our armchairs,
while he was struggling with so many problems in the messy reality of the classroom. But exactly
because it is so hard to think deeply during the actual classroom activity, it makes sense to reflect on
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the classroom events after the fact, to see if we can do better next time. With this rationalization, then,
let us proceed with the analysis.
Martin seems to have noticed that there was a problem, but he didn't classify it correctly, and then of
course couldn't come up with a good course of action to deal with the problem. He seems to attribute
the problem to Simon's lack of understanding of either the goal of the activity ("Yes, but you have to
put up an equation") or the subject matter itself ("The pupil has difficulties seeing what x should
represent"). In contrast, we tend to classify this as a problem of communication, thus locating it in the
shared space between the teacher and pupil. In fact it is a problem of communication on two levels.
First, the teacher and pupil have different goals for the activity and, second, they do not even
recognize that they are having this problem. The teacher is too intent on the lesson's goal, and the
pupil is too busy defending himself and his (correct) solution, so that none of them is free to really
listen and try to understand the other's point of view.
Had Martin been aware of the communication problem, he could have devised more effective ways to
achieve his goal together with the pupil, not against him. For example, he could have asked the pupil
to solve another task with longer trip or taxi rate of 17 (instead of 10) DKK/km. We can sympathize
with the pupil who has solved the problem correctly but didn't get the teacher's acknowledgement. We
can sympathize with the teacher, too, who has to juggle simultaneously in his working memory (with
its well-known severe limitations) so many things at once. Luckily, this story seems to have had a
(relatively) happy ending, even though Martin's report is not entirely clear on this point. In the end, the
pupil seems to have succeeded in meeting the expectations of the teacher, but he may still not have a
clear idea of why the equation was necessary, what are equations good for anyway, and why his initial
correct solution didn't satisfy the teacher.
Conclusion
This article has been an exercise in interpretation and re-interpretation of a simple classroom story –
the taxicab scenario – as told by Martin, the classroom teacher. The story depicts a typical break in
communication between Martin and one of his pupils, Simon. Simon has solved the taxicab task
correctly but without using equations. Equations, however, were Martin's pre-planned goal for this
lesson, hence a conflict ensued between Martin's and Simon's goals.
The goal of the workshop activity was to work with the teachers on their classroom reflection and
communication skills, and the tool for the activity was the virtual monologue (VM) which has been
transposed for this purpose from a tool for researchers to an inservice workshop activity. The use of
VM has turned the interpretation task for the teachers into an intellectual and emotional experience,
an exercise in empathy, if you will, whereby they were prompted to try to become Martin or Simon,
and describe using first-person narrative what they thought might be going on through their heads.
Our main goal was the process itself – an exercise in empathy and in flexibly shifting perspectives –
which the teachers were involved in. As such, the workshop protocols and transcripts indicate that the
activity has successfully achieved its propose: Beside the fun and excitement it has created, the
teachers have become aware (many for the first time) of the interpretation of familiar classroom
events as communication issues. What the teachers would actually do with these newly acquired
insights and skills in their own classrooms is a topic for further investigation. It is reasonable to hope,
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however, that next time Martin and his colleagues encounter a similar problem, they will at least be
more likely to identify it as a communication problem and perhaps choose a more productive course
of action. A side benefit of this activity has been that while pretending to be Martin or Simon, the
teachers had an opportunity to work in a secure and comfortable way on their own mathematical
knowledge with which they did not feel so secure.
References
Danish Ministry of Education (1993, 2003, 2009): Goals for Curriculum in Mathematics.
Copenhagen.
Ejersbo, L. R. (2007): Design and Redesign of an In-service Course: The interplay of theory and
practice in learning to teach mathematics with open problems. PHD dissertation, DPU, University
of Aarhus.
Gregersen, P. et al (2001): Matematrix 8. Alinea, København.
Leron, U., Hazzan, O. (1997): The world according to Johnny: A coping perspective in mathematics
education. Educational Studies in Mathematics 32, 265-292.
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