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The mathematical learning environment Bus is based on pupils’ real-life experience and has been elaborated by M. Hejný to develop pupils’ understanding of number as a state, an address, an operator of change and of comparison when it is grasped by the specific language of this environment. The main result of our study is a description of the process in which individual 7-8 years old pupils lose their tendency to use irrelevant aspects of language and effectively select and structure those that are key in a given situation.
P r o c e d i a - S o c i a l a n d B e h a v i o r a l S c i e n c e s 9 3 ( 2 0 1 3 ) 9 7 8 9 8 3
1877-0428 © 2013 The Authors. Published by Elsevier Ltd. Open access under
CC BY-NC-ND license.
Selection and peer review under responsibility of Prof. Dr. Ferhan Odabaşı
doi: 10.1016/j.sbspro.2013.09.314
World Conference on Learning, Teaching and Educational Leadership – WCLTA 2012
Conceptualization of process: Didactic environment Bus
Darina Jirotková*, Jaroslava Kloboučková, Milan Hejný
Faculty of Education, Charles University in Prague, M.D.Rettigové 4, 116 39 Praha 1, Czech Republic
The mathematical learning environment Bus is based on pupils’ real-life experience and has been elaborated by M. Hejný to
develop pupils’ understanding of number as a state, an address, an operator of change and of comparison when it is grasped by
the specific language of this environment. The main result of our study is a description of the process in which individual 7-8
years old pupils lose their tendency to use irrelevant aspects of language and effectively select and structure those that are key in
a given situation.
Keywords: Scheme-oriented education; mathematical environment Bus, number as a state, an address, an operator, record of process;
© 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction
In order to develop quality mathematics thinking and understanding of mathematical objects in pupils in their
early school years, it is important to stress the balance between processual and conceptual development. The neglect
of either leads to future difficulties in solving certain types of mathematical problems. When the processual
understanding of numbers, relationships and operation is underdeveloped at the primary school level, the pupil will
experience hardships in lower secondary school level, e.g. with the solving of dynamic problems. When the
conceptual understanding is underdeveloped, the pupils will encounter difficulties later, for instance when solving
equations and working with algebraic expressions.
Thus, in order to build a good understanding of numbers, it is important that the pupil actively engages with
numbers in a variety of context and roles, i.e. with numbers as quantities, states, operators of change, operators of
comparison and indicators; i.e. numbers as addresses or ordinals and as names (Hejný & Stehlíková, 1999).
Traditional primary school mathematics (ages 6-11) in the Czech Republic, as well as in many other countries,
focuses mainly on pupils’ calculating knowledge and skills. The educational goal to automatize the operation of
addition of one-digit numbers is often achieved by drill of large quantities of “columns”. Theories of cognitive
processes in mathematics (Hejný, 2011a, 2012; Dubinsky & McDonald, 2001) indicate that the understanding of
concepts, relationships, states, and processes is based on activity and real-life experience. Typical school
mathematics reflects this by inserting amounts of word problems. These word problems are very often solvable
using a signal. (By a signal we understand a word or expression that alone gives the problem-solver an indication
* Corresponding Author: Darina Jirotková. Tel.: +420-221-900-226
E-mail address:
Available online at
© 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and peer review under responsibility of Prof. Dr. Ferhan Odabaşı
Darina Jirotková et al. / Procedia - Social and Behavioral Sciences 93 ( 2013 ) 978 – 983
what to do in order to solve the problem. For instance, the expression “five children left“ suggests the operation of
subtraction should be carried out.)
The variety of contexts, their classification and order are the focus of attention for a number of foreign and Czech
researchers (e.g. Vergnaud, 1997; Bell, 1988; Cockburn, 2007; Novotná, 2000). Here we will classify simple word
problems according to their semantic structure, i.e. according to the various roles numbers can have in a problem.
For example, in the following problem: There are three apples in one bowl and two in another bowl, how many
apples are there in total in the two bowls? both numbers are mentioned in the role of a state. The question is also
asking for a given state. The structure of this problem is therefore State 1 plus State 2 = State 3 (S
+ S
= S
). In the
problem: There were three apples in a bowl and mom added two more apples to the bowl. How many apples were in
the bowl after that? the role of number 2 is that of an operator of change. The structure of the problem is therefore
+ O = S
. These two problem types are most commonly represented in textbooks. Problems of the type O
± O
, where all three numbers are operators (e.g. Three passengers got on and two got off. How many more/fewer
people are now on the bus?) are usually considered as difficult and are neglected. The environment Bus focuses
primarily on these operator-based problems in order to prepare pupils for solving dynamic problems in the future
stages of their mathematics education.
Apart from word problems, the Czech school mathematics rarely offers situations that would encourage
processual or conceptual thinking based on pupils’ real-life experience. The knowledge an individual gains through
real-life experience and through meaningful activities is recorded in mental schemata (Gerrig, 1991). Such
knowledge is operative and enables an individual to make sense of the world around them.
In order to build mathematical schemata, it is necessary to work in numerous mathematical environments
(Wittmann, 2001; Hejný, 2011a, 2011b) and the learning should be taking place in a constructivist perspective, i.e.
the teacher does not explain anything to pupils and, rather, facilitates a group or class discussion. Through solving
problems pupils gradually discover and formulate, argue, clarify and justify, even prove in later stages.
One of the many environments that have been didactically elaborated in textbooks by Hejný et al. (2007-2011)
for ages 6-11 is the Bus environment. We will first describe this environment and later we will present some results
of research conducted with pupils of ages 7 to 9.
2. The Bus environment
The game “Bus” capitalizes on pupils’ experience with travelling in a means of public transport. The process of
people getting on and getting off and its recording in a table offers a rich set of data and numerical relationships
which in turn can be a source of many problems. This environment can be implemented as early as the first grade of
primary school and is also highly effective in lower secondary grades. Let us briefly introduce the environment.
2.1. Description of the activity/performance and the actors
The game simulates travelling by bus on a line connecting several bus stops. The bus is represented for instance
by a cardboard box and the passengers could be represented by plastic bottles or other objects. The bus stops are set
up around the classroom and pupils can give them names, e.g. Red Hill (teacher’s desk), Blue Lake (the washbasin),
Yellow House (the door), Green Meadow (the white board). The bus driver takes the bus from the initial stop to the
terminal stop. At each stop, some passengers may get off (except of the initial stop) and get on (except of the
terminal stop). The getting on and off at each stop is happening at the command of a dispatcher. The number of
passengers moving in and out of the bus at each stop should not exceed number three at first and even later it should
not go into high numbers.
2.2. First stage and didactic commentary
The environment Bus is introduced gradually and as slowly as the age of pupils requires. The first rounds are
performed by the teacher who is also presenting the plot:
980 Darina Jirotková et al. / Procedia - Social and Behavioral Sciences 93 ( 2013 ) 978 – 983
Teacher (T): An empty bus arrives at the bus stop The Blue Lake. One passenger gets here on (the teacher lifts
one bottle above his/her head and places it into the box with an appropriate sound effect).
T: The doors are closing. The teacher gets hold of the box. The bus departs from Red Hill and goes to Blue Lake.
The teacher simulates the path of the bus and moves on to the wash basin.
T: The bus stops at Blue Lake. The teacher puts the box down. No passenger gets off. One passenger gets on the
bus. The teacher repeats the previous action. Then another passenger gets on and finally one more passenger gets
on. The teacher places the passengers into (or takes them out of) the box individually, showing them clearly, but
neither counts them, nor states their number. The bus rides in this way until the final stop.
The role of the bus driver and the role of the dispatchers are soon delegated to pupils. The teacher helps them
follow the plot and use a standardized language commentary. The pupils in the classroom can see the passengers
getting on and off but they cannot see inside the bus (Fig. 1a, 1b). The pupils’ task is to follow the whole process of
the journey and to record it in any way they choose. When the bus reaches the terminal, the teacher asks questions
regarding the activity.
Figure 1a, 1b. A bus, the bus driver and the dispatcher
In the first stage, the game contributes especially to the development of the following five pupils’ needs and
1) The need to quantify phenomena from everyday life.
2) Focusing their attention during the whole performance.
3) The ability to mathematize a real situation. The game makes the pupil connect life experience with specific
numerical phenomena.
4) Train short-term procedural memory (Bullemer, Nissen, & Willingham, 1989). The pupil is aware that after
each performance the teacher will ask questions. Some questions, e.g. How many passengers rode from stop A to
stop B? Which stop had the most passengers getting on? etc., bring out the need
5) to create a language suitable for grasping the process. This point takes us to the following stage.
2.3. Second stage: From process to concept
The journey from the first pupils’ recordings to an efficient organization of data in a table can be quite long and
requires a considerable amount of patience on the teacher’s part. We will describe this journey in section 3.
The resulting table that provides the base for further problem formulation is shown in Figure 2. Letters A, B, C,
D, and E denote individual stops. The number of passengers is recorded by strokes, or possibly by numbers.
We can read from the table that 3 passengers got on at stop A, they further travelled to stop B. At stop B, two
Got off // / //// ////
Got on /// /// //// /
Travelling /// //// ///// ////
ure 2. Record of a
Darina Jirotková et al. / Procedia - Social and Behavioral Sciences 93 ( 2013 ) 978 – 983
people got off and three got on and so on. Questions such: On which stop did most passengers get off? or How many
passengers were there on the bus on the way from C to D? or In which part of the bus journey were there the most
passengers on the bus?
The second stage of the game enhances the pupils’ intellectual experience especially in the following five
1) Growing aware of the fact that if memory fails to record the whole process one must look for a way to support
it. 2) Looking for support means to find a tool suitable for recording a process. 3) The creation of a suitable tool
usually undergoes gradual improvement. 4) If the process is recorded in the form of a concept, the pupil’s
understanding of the phenomenon is enriched by the amalgam process-concept, i.e. procept (Gray & Tall, 1994).
5) The table is an effective tool for describing a process. The next stages will be described here only briefly, and we
will return to the second stage further in the text.
2.4. Other stages: Work with the table – transition to formalization
The remaining stages consist in solving problem of gradually increasing difficulty. These are formulated through
the use of a table or in words. The level of difficulty is given by the number of boxes that are missing from the table
as well as their specific locations in the table as well as by the number of solutions. Figure 3 illustrates one such
Fill in the 5 missing pieces of information in the table. Now answer these questions: How many passengers went
by this bus in total? How many passengers were travelling by the bus only one stop?
Got off 0 0 3 5
Got on 3 0
Were travelling 4 5 6
Figure 3. Problem
Solving these problems, pupils discover various strategies on their own. At first they choose to act out the
situation, using the trial-and-error strategy the strategy of solving backwards. In the process they unveil relationships
between boxes in the table and gain experience the relation connecting 4 numbers. Each stop (except for the first and
last stops) involves four facts regarding the number of passengers: those who arrived at the stop, those who got off
the bus here, those who got on the bus here, and those who left the bus stop. If we know three of these four numbers,
we can find the fourth one.
The level of difficulty can be further increased by distinguishing between types of passengers, e.g. male and
female. In the lower secondary classes we can use a graph to represent the bus ride, we can use dice to determine the
individual numbers of passengers and bring the game to the realm of probability. We can also add a financial aspect
and solve a problem of optimization.
Let us return to the second stage where pupils discover the way to record the plot in a table, i.e. the transition
from a process to a concept.
3. Research
Our research database includes many video recordings from experimental teaching of 7 primary school teachers
collected in years 2005-2007, from our partial clinical experiments carried out in years 2007-2010, from
experiments of our student included in her diploma theses, series of video recordings collected throughout two
academic years (2009-2011) of experimental teaching of one of the authors, rich photo documentation, notes from
observations and pupils’ written products. These materials are processed using comparative analysis and techniques
of grounded theory.
Pupils looking for a suitable way of making a record of the performance usually proceed in two different
directions: 1. they either try to record everything using letters, numbers, strokes, arrows, drawings (Figures 4, 5 and
982 Darina Jirotková et al. / Procedia - Social and Behavioral Sciences 93 ( 2013 ) 978 – 983
6), or 2. they only try to record the current state of passengers by making strokes and crossing them over or adding
and subtracting numbers (Figures 7 and 8). The first strategy leads to chaos because the pupil fails to record
everything that he/she has intended to. The other strategy leads quite comfortably to the correct answer to the
question how many passengers arrived at the terminal stop. However, it fails to keep a record of the history of the
journey. Figure 9 shows the pupil’s ability to choose important data and structure them in a relevant way.
It is the teacher’s task to guide the pupils to discovery of the table as an efficient tool for description of the bus
ride skit. The teacher has four tools for this: the first is the choice of questions that he/she asks. The second is a
repeated performance of a particular plot. The third tool is monitoring the pupils’ written record and holding a
whole-class discussion. The fourth one is an informal disclosure of the table.
Figure 4. A bus route Figure 5. Stops Figure 6. Stop acronyms
All pictures in Figures 4 through 6 were collected in a class of first-year pupils on three different days (March
, March 3
, March 23
) in 2011. They roughly represent the development of recording skills in individual pupils.
Naturally, the initial stage as well as the pace of the journey towards organizing data into a table was different for
each pupil. The figures illustrate the diversity of approaches within one single classroom. It is evident that the pupils
were able to choose their own pace and direction. That is one of the pillars of constructivist approaches to teaching.
We will now focus on the data that a particular pupil deems relevant.
Figure 4 shows a representation of an entire bus ride. The stops are given names (Camera, Window, Door,
Whiteboard, Terminal). The order of stops made by the bus is sketched by arrows and numbers. All stops (except for
the final stop) are assigned two numbers with the signs – for “got off” and + for “got on”. In Figure 5, the names of
the stops are dominant and, as in the previous case, getting on and off the bus is signaled by the + and – signs.
Figure 6 depicts three different rides. The stop names are not as important, acronyms are employed, e.g. SK means
School). Getting on and off the bus is marked by letters N (on) and V (off).
The way first-graders approach negative numbers, which they will not encounter in the curriculum until three or
four years later, is noteworthy. We believe that a link to different semantic environments is present, namely the
Stepping and Staircase environments. These two environments are discussed in Slezáková et al. (2012).
4. Conclusion
The environment Bus, together with the process of development of new language, is highly motivational for
pupils. It is crucial to allow enough time for the pupils to create and perfect their own recording system. We noted in
a few cases that pupils had difficulties solving problems based on a table. In follow-up interviews with teachers it
transpired that they showed the already designed table to the pupils. They were concerned about the pupils’ ability
to arrive at it and wanted to make things easier for them. The pupils had not made the important link between the
process of the plot and the table, they had not made the journey from a process to a concept. Some pupils learned
procedures for solving certain situations in the table but failed to solve problems at a more difficult level. They were
not able to interpret the table.
In a class where the teacher introduced the Bus environment in the third grade about a half of the pupils were still
using other (non-tabular) representations after three months of working in the environment. We asked the teacher
Darina Jirotková et al. / Procedia - Social and Behavioral Sciences 93 ( 2013 ) 978 – 983
why she wouldn’t show the table to the whole class so that she could solve problems based on tables. We will
conclude this article with her reply: “I am going to wait for all of them to come to the table. They must realize that
they actually need it.”
The study was supported by the research projects No. MSM 0021620862, Teaching profession in the
environment of changing education requirements and No. P407/11/1740 Critical areas of primary school
mathematics – analysis of teachers’ didactic practices.
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